Recent zbMATH articles in MSC 57https://zbmath.org/atom/cc/572022-09-13T20:28:31.338867ZWerkzeug\(Z\)-oriented triangulations of surfaceshttps://zbmath.org/1491.051352022-09-13T20:28:31.338867Z"Tyc, Adam"https://zbmath.org/authors/?q=ai:tyc.adamSummary: The main objects of the paper are \(z\)-oriented triangulations of connected closed 2-dimensional surfaces. A \(z\)-orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an edge -- zigzags from the \(z\)-orientation pass through this edge in different directions (type I) or in the same direction (type II). Then there are two types of faces in a triangulation: the first type is when two edges of the face are of type I and one edge is of type II and the second type is when all edges of the face are of type II. We investigate \(z\)-oriented triangulations with all faces of the first type (in the general case, any \(z\)-oriented triangulation can be shredded to a \(z\)-oriented triangulation of such type). A zigzag is homogeneous if it contains precisely two edges of type I after any edge of type II. We give a topological characterization of the homogeneity of zigzags; in particular, we describe a one-to-one correspondence between \(z\)-oriented triangulations with homogeneous zigzags and closed 2-cell embeddings of directed Eulerian graphs in surfaces. At the end, we give an application to one type of the \(z\)-monodromy.Excluded checkerboard colourable ribbon graph minorshttps://zbmath.org/1491.051822022-09-13T20:28:31.338867Z"Guo, Xia"https://zbmath.org/authors/?q=ai:guo.xia"Jin, Xian'an"https://zbmath.org/authors/?q=ai:jin.xianan"Yan, Qi"https://zbmath.org/authors/?q=ai:yan.qiSummary: Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.Curves with prescribed symmetry and associated representations of mapping class groupshttps://zbmath.org/1491.140452022-09-13T20:28:31.338867Z"Boggi, Marco"https://zbmath.org/authors/?q=ai:boggi.marco"Looijenga, Eduard"https://zbmath.org/authors/?q=ai:looijenga.eduard-j-nA classical result from Hurwitz asserts that the endomorphism ring of the Jacobian of a very general curve \(C\) of genus at least 2 is the smallest possible, i.e. it is \(\mathbb{Z}\). \textit{S. Lefschetz} [Am. J. Math. 50, 159--166 (1928; JFM 54.0410.02)] proved the same in the case of hyperelliptic curves. \textit{C. Ciliberto} et al. [J. Algebr. Geom. 1, No. 2, 215--229 (1992; Zbl 0806.14020)] studied curves whose Jacobian has endomorphism ring larger than \(\mathbb{Z}\). Finally, \textit{Y. G. Zarhin} [Math. Proc. Camb. Philos. Soc. 136, No. 2, 257--267 (2004; Zbl 1058.14064)] considered curves with automorphisms (of a specific type) and showed that the endomorphism ring of their Jacobians is as smallest as possible.
In the paper under review, the authors show that such a minimality property still holds for curves endowed with an action of a given (but arbitrary) finite group \(G\). Indeed, they show that \(\text{End}_{\mathbb{Q}}(JC)\cong \mathbb{Q}[G]\) for a very general \(G\)-curve, with quotient curve of genus at least 3.
As an application, they also obtain interesting consequences on the natural representation of the centralizer of \(G\) in \(\text{Mod}(C)\), \(\rho_G: \text{Mod}(C)^G\rightarrow \text{Sp}(H^1(C,\mathbb{Q}))^G\), where \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) stands for the centralizer of \(G\) in \(\text{Sp}(H^1(C,\mathbb{Q}))\), regarded as a virtual
linear representation the mapping class group \(\text{Mod}(C/G)\). Indeed, let \(X(\mathbb{Q}[G])\) be the set of rational irreducible characters of \( G \), take the isomorphism \[
\text{Sp}(H^1(C,\mathbb{Q}))^G\cong \prod_{\chi \in X(\mathbb{Q}[G]) }\text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G,
\]
which mirrors the isotypical decomposition of the Jacobian of a \(G\)-curve \(C\), and denote by \(Mon^0(C)\) the identity component of the Zariski closure of the
image of \( \rho_G \) in \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) and by \(\text{Mon}^0(C)_\chi\) the projection of \(\text{Mon}^0(C)\) to the factor \( \text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G \). Then, the authors deduce different interesting properties on \(\text{Mon}^0(C)\) and on its factors \(\text{Mon}^0(C)_\chi\).
Reviewer: Irene Spelta (Pavia)On finiteness theorems of polynomial functionshttps://zbmath.org/1491.140822022-09-13T20:28:31.338867Z"Koike, Satoshi"https://zbmath.org/authors/?q=ai:koike.satoshi"Paunescu, Laurentiu"https://zbmath.org/authors/?q=ai:paunescu.laurentiuSummary: Let \(d\) be a positive integer. We show a finiteness theorem for semialgebraic \(\mathscr{RL}\) triviality of a Nash family of Nash functions defined on a Nash manifold, generalising Benedetti-Shiota's finiteness theorem for semialgebraic \(\mathscr{RL}\) equivalence classes appearing in the space of real polynomial functions of degree not exceeding \(d\). We also prove Fukuda's claim, Theorem 1.3, and its semialgebraic version Theorem 1.4, on the finiteness of the local \({\mathscr{R}}\) types appearing in the space of real polynomial functions of degree not exceeding \(d\).Mirror symmetry on levels of non-abelian Landau-Ginzburg orbifoldshttps://zbmath.org/1491.140852022-09-13T20:28:31.338867Z"Ebeling, Wolfgang"https://zbmath.org/authors/?q=ai:ebeling.wolfgang"Gusein-Zade, Sabir M."https://zbmath.org/authors/?q=ai:gusein-zade.sabir-mSummary: We consider the Berglund-Hübsch-Henningson-Takahashi duality of Landau-Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold monodromy zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants.Flags and tangleshttps://zbmath.org/1491.180172022-09-13T20:28:31.338867Z"Haiden, Fabian"https://zbmath.org/authors/?q=ai:haiden.fabianThe main result of the paper is a proof of equivalence of two braided monoidal categories, exhibited by an explicit monoidal functor \(\Phi:{\mathcal S}|_q\to{\mathcal H}\) between them.
The category \(\mathcal{S}\) is a braided monoidal category of graded Legendrian tangles. More precisely, its objects are finite \(\mathbb{Z}\)-graded subsets of \(\mathbb{R}\). The set of morphisms between two objects is considered as an \(R={\mathbb{Z}}[q^{\pm1},(q-1)^{-1}]\) module generated by isotopy classes of Legendrian curves in \([0,1]\times{\mathbb{R}}^2\) with grading (Maslov potential) whose boundary matches the source and target objects, modulo three local (skein) relations. Specializing to a prime power value \(q\) and considering the evaluation homomorphism \(R\to\mathbb{Q}\) realises a braided monoidal category \(\mathcal{S}|_q\) over \(\mathbb{Q}\).
The category \(\mathcal{H}\) is defined from a choice of fixed finite field \(k=\mathbb{F}_q\), where \(q\) is a prime power. Objects are finite-dimensional \(\mathbb{Z}\)-graded vector spaces over \(k\) together with a complete flag of graded subspaces. The space of morphisms \(Hom_{\mathcal{H}}(V,W)\) between two objects, is the set of rational linear combinations of triples \((d_V,f,d_W)\) where \(d_V\), \(d_W\) are differentials (of degree \(-1\)) on \(V\), \(W\) respectively, and \(f:V\to{}W\) is a quasi-isomorphism.
The result can be considered analogous, in the setting of \(\mathbb{Z}\)-graded chain complexes of vector spaces as opposed to vector spaces, to the result of \textit{V. G. Turaev} [Math. USSR, Izv. 35, No. 2, 411--444 (1990; Zbl 0707.57003); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1073--1107 (1989)] where he defined a Hecke category and identified certain Hom spaces with the classical \(A_n\)-type Iwahori-Hecke algebra. Objects in the Hecke category were finite strings of signs while morphisms were generated by classical oriented tangles (whose boundary matched the source and target orientations) modulo the Jones-Conway skein relation. The \(\mathbb{Z}\)-grading appears in the current setting when replacing classical tangles by Legendrian tangles, from the Maslov potential on Legendrian curves.
Reviewer: Ruth Lawrence (Jerusalem)The equivariant parametrized \(h\)-cobordism theorem, the non-manifold parthttps://zbmath.org/1491.190012022-09-13T20:28:31.338867Z"Malkiewich, Cary"https://zbmath.org/authors/?q=ai:malkiewich.cary"Merling, Mona"https://zbmath.org/authors/?q=ai:merling.monaThis paper is part of the project of developing an equivariant Waldhausen \(K\)-theory and relating it to manifolds with group action.
In [Doc. Math. 24, 815--855 (2019; Zbl 1423.19003)] the authors created a functor \(X\mapsto A_G(X)\) from \(G\)-spaces to \(G\)-spectra, where \(G\) is any finite group. The fixed-point spectrum of \(A_G(X)\) is equivalent to the \(K\)-theory of a suitable category of \(G\)-spaces that contain \(X\) as a retract, and it splits as a product \[ A_G(X)^G\sim \prod_{(H)} A(X^H_{hWH}), \] where \(H\) runs through conjugacy classes of subgroups and \(X^H_{hWH}\) is the homotopy orbit space for the action of the Weyl group on the fixed-point space \(X^H\). Of course this is reminiscent of the familiar tom Dieck splitting in equivariant stable homotopy: \[ \Sigma^\infty_G(X_+)^G\sim \prod_{(H)} \Sigma^\infty(X^H_{hWH+}). \] The main business of the present paper is to construct a natural map of \(G\)-spectra \[ \Sigma^\infty_G(X_+)\to A_G(X), \] that generalizes Waldhausen's map \(\Sigma^\infty (X_+)\to A(X)\) to the equivariant setting. The map is constructed so as to be compatible with the splittings. Thus, denoting the homotopy fiber of the map by \(\mathbf H_G^\infty(X)\), the fixed-point spectrum \(\mathbf H_G^\infty(X)^G\) splits as a product of spectra \(\mathbf H^\infty(X^H_{hWH})\).
The space \(\Omega^\infty\mathbf H^\infty(Y)\) with \(Y=X^H_{hWH}\) has a description in terms of smooth manifolds: it is a homotopy colimit, over compact smooth manifolds \(M\to Y\) of increasing dimension, of the space of smooth \(h\)-cobordisms with base \(M\). This is a consequence of the Waldhausen-Jahren-Rognes work [\textit{F. Waldhausen} et al., Spaces of PL manifolds and categories of simple maps. Princeton, NJ: Princeton University Press (2013; Zbl 1309.57001)] that relates \(A(Y)\) to stabilized spaces of piecewise-linear \(h\)-cobordisms.
This is an important step toward relating \(A_G(X)^G\) to spaces of high-dimensional smooth \(h\)-cobordisms with \(G\)-action. The expected next step is to consider \(\Omega^\infty \mathbf H^\infty_G(M)^G\) when \(M\) is an arbitrary compact smooth \(G\)-manifold, and to interpret it as a space of equivariant or isovariant \(h\)-cobordisms, suitably stabilized with respect to linear representations of \(G\).
A few words about the technical foundations of the paper: \(G\)-spectra are considered as spectral Mackey functors on the category of finite \(G\)-sets, or equivalently as spectrally enriched functors on a certain spectral category \(\mathcal B_G\) introduced by \textit{B. Guillou} and \textit{J. P. May} [``Models of $G$-spectra as presheaves of spectra'', Preprint, \url{arXiv:1110.3571}]. It is necessary to compare \(\mathcal B_G\) with a variant called \(\mathcal B_G^{\text{Wald}}\), because the latter was used in constructing \(A_G(X)\) and obtaining the splitting of \(A_G(X)^G\). The comparison uses ``parameter multicategories''. Except for some detailed explicit constructions in this comparison argument, most of the work has been quite successfully black-boxed for readability.
Reviewer: Thomas Goodwillie (Watertown)On an analogue of \(L^2\)-Betti numbers for finite field coefficients and a question of Atiyahhttps://zbmath.org/1491.200022022-09-13T20:28:31.338867Z"Neumann, Johannes"https://zbmath.org/authors/?q=ai:neumann.johannesSummary: We construct a dimension function for modules over the group ring of an amenable group. This may replace the von Neumann dimension in the definition of \(L^2\)-Betti numbers and thus allows an analogous definition for finite field coefficients. Furthermore we construct examples for characteristic 2 in answer to Atiyah question of irrational \(L^2\)-Betti numbers.On joins and intersections of subgroups in free groupshttps://zbmath.org/1491.200672022-09-13T20:28:31.338867Z"Ivanov, Sergei V."https://zbmath.org/authors/?q=ai:ivanov.sergei-vladimirovich.1Summary: We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs, how to repair the lemma and how to utilize it.Some properties of \(E(G, W, F_TG)\) and an application in the theory of splittings of groupshttps://zbmath.org/1491.200682022-09-13T20:28:31.338867Z"Fanti, E. L. C."https://zbmath.org/authors/?q=ai:fanti.erminia-de-lourdes-campello"Silva, L. S."https://zbmath.org/authors/?q=ai:silva.leticia-s|silva.lillia-s-b|silva.luciano-sSummary: Let us consider \(W\) a \(G\)-set and \(M\) a \(\mathbb{Z}_2G\)-module, where \(G\) is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant \(E(G, W, M)\), defined in [\textit{M. G. C. Andrade} and \textit{E. de L. C. Fanti}, Int. J. Appl. Math. 25, No. 2, 183--190 (2012; Zbl 1282.20057)] and present related results with independence of \(E(G, W, M)\) with respect to the set of \(G\)-orbit representatives in \(W\) and properties of the invariant \(E(G, W, \mathcal{F}_TG)\) establishing a relation with the end of pairs of groups \(\widetilde{e}(G, T)\), defined by \textit{P. H. Kropholler} and \textit{M. A. Roller} [J. Pure Appl. Algebra 61, No. 2, 197--210 (1989; Zbl 0691.20036)]. The main results give necessary conditions for \(G\) to split over a subgroup \(T\), in the cases where \(M=\mathbb{Z}_2(G/T)\) or \(M=\mathcal{F}_TG\).Coloring invariants of knots and links are often intractablehttps://zbmath.org/1491.200802022-09-13T20:28:31.338867Z"Kuperberg, Greg"https://zbmath.org/authors/?q=ai:kuperberg.greg"Samperton, Eric"https://zbmath.org/authors/?q=ai:samperton.ericOne of the main problems in knot theory is the knot recognition problem. The essence of this problem is to construct an effective algorithm that determines by two given knots whether they are equivalent or not. One possible approach to this problem is to construct invariants, i.e. such functions that have the same value on equivalent knots. If we manage to find an invariant \(f\) such that \(f(K_1)\neq f(K_2)\), then we can conclude that \(K_1\) and \(K_2\) are non-equivalent knots.
One of the simplest invariants in knot theory is the \(3\)-coloring invariant of the diagram which is also called the Fox coloring. This invariant is not very strong, i.e. it can be the same for a lot of different knots. However, it allows to distinguish the trefoil knot from the trivial knot and from the figure-eight knot.
More complex invariants are the knot group and the knot quandle. These invariants are very strong (in particular, the knot quandle defines a knot up to simultaneous change of orientation and mirror reflection), however, in order to use these invariants one must be able to solve the isomorphism problem for groups or the isomorphism problem for quandles. In [\textit{A. D. Brooke-Taylor} and \textit{S. K. Miller}, J. Aust. Math. Soc. 108, No. 2, 262--277 (2020; Zbl 1482.20039)] it is shown that the isomorphism problem for quandles is, from the perspective of Borel reducibility, fundamentally difficult (Borel complete).
The invariant \(|Q(K;G,C)|\), where \(K\) is a knot, \(G\) is a nonabelian finite simple group, \(C\) is a generating conjugacy class in \(G\), is a kind of intermediate between the knot group and the knot coloring. The value of this invariant on a knot \(K\) is equal to the number of elements in the quotient of the set of surjective homomorphisms from the group \(G(K)\) of knot \(K\) to the group \(G\) such that the images of the Wirtinger generators of the group \(G(K)\) belong to \(C\), by the group \(\mathrm{Aut}(G,C)\) of automorphisms of \(G\) which fix \(C\).
The authors of the paper under review prove that the problem of computing the invariant \(|Q(K;G,C)|\) for a given group \(G\) and its conjugcay class \(C\) is parsimoniously \(\#P\)-complete.
Reviewer: Timur Nasybullov (Novosibirsk)Proper actions on finite products of quasi-treeshttps://zbmath.org/1491.200892022-09-13T20:28:31.338867Z"Bestvina, Mladen"https://zbmath.org/authors/?q=ai:bestvina.mladen"Bromberg, Ken"https://zbmath.org/authors/?q=ai:bromberg.kenneth-w"Fujiwara, Koji"https://zbmath.org/authors/?q=ai:fujiwara.koji.1Say that a group \(G\) has property (QT) if it acts isometrically on a finite product of quasi-trees such that orbit maps are quasi-isometric embeddings. Here, a \textit{quasi-tree} is a connected graph whose path metric is quasi-isometric to a tree and the product is given the \(\ell^1\)-metric. In this paper, the authors prove that residually finite Gromov hyperbolic groups have (QT), as do mapping class groups of finite-type surfaces.
To build intuition, observe that it is clear that free groups have (QT), and by choosing a filling collection of simple closed curves on a surface and taking the Bass-Serre trees dual to that collection, it is not hard to see that surface groups have (QT); in each of these cases there is in fact a proper action on a finite product of \textit{trees,} a property shared by (subgroups of) Coxeter groups by a result of \textit{A. Dranishnikov} and \textit{T. Januszkiewicz} [Topol. Proc. 24(Spring), 135--141 (1999; Zbl 0973.20029). It follows that \emph{virtually cocompact special} cubulated groups have (QT), a class which includes fundamental groups of closed hyperbolic 3-manifolds. These latter groups act without global fixed point on CAT(0) cube complexes, and thus have the \emph{Haagerup property,} a strong negation of Kazhdan's property (T).
On the other hand, uniform lattices in \(\mathrm{Sp}(n,1)\) have Kazhdan's property (T) and are residually finite and Gromov hyperbolic. Therefore the authors' result shows that property (QT) is not strong enough to imply the Haagerup property. Indeed, it is unknown whether mapping class groups of finite-type surfaces have either the Haagerup property or Kazhdan's property (T).
Since finite products of (quasi-)trees have finite asymptotic dimension, property (QT) is a stronger version of finiteness of the asymptotic dimension, a result due to Gromov for Gromov hyperbolic groups and to the authors [Publ. Math., Inst. Hautes Étud. Sci. 122, 1--64 (2015; Zbl 1372.20029)] for mapping class groups of finite-type surfaces. Indeed, the proof uses the projection complexes techniques developed in the authors' previous paper. The authors remark that lattices in higher-rank Lie groups have finite asymptotic dimension but cannot have (QT). Thus in this regard the mapping class group of a finite-type surface behaves more like a Gromov hyperbolic group than a (higher-rank) lattice.
Reviewer: Rylee Lyman (Newark)The Yagita invariant of symplectic groups of large rankhttps://zbmath.org/1491.200972022-09-13T20:28:31.338867Z"Busch, Cornelia M."https://zbmath.org/authors/?q=ai:busch.cornelia-minette"Leary, Ian J."https://zbmath.org/authors/?q=ai:leary.ian-jamesSummary: Fix a prime \(p\), and let \(\mathcal{O}\) denote a subring of \(\mathbb{C}\) that is either integrally closed or contains a primitive \(p\)th root of 1. We determine the Yagita invariant at the prime \(p\) for the symplectic group \(\mathrm{Sp}(2n,\mathcal{O})\) for all \(n\ge p-1\).On core quandles of groupshttps://zbmath.org/1491.201432022-09-13T20:28:31.338867Z"Bergman, George M."https://zbmath.org/authors/?q=ai:bergman.george-mSummary: We review the definition of a \textit{quandle}, and in particular of the \textit{core quandle} Core\((G)\) of a group \(G\), which consists of the underlying set of \(G\), with the binary operation \(x\lhd y=xy^{-1}x\) This is an \textit{involutory} quandle, i.e., satisfies the identity \(x\lhd(x\lhd y)=y\) in addition to the other identities defining a quandle.
\textit{Trajectories} \((x_i)_{i\in\mathbb{Z}}\) in groups and in involutory quandles (in the former context, sequences of the form \(x_ixz^i(x,z\in G)\) among other characterizations; in the latter, sequences satisfying \(x_{i+1}=x_i\lhd x_{i-1})\) are examined. A family of necessary conditions for an involutory quandle \(Q\) to be embeddable in the core quandle of a group is noted. Some implications are established between identities holding in groups and in their core quandles. Upper and lower bounds are obtained on the number of elements needed to generate the quandle Core\((G)\) for \(G\) a finitely generated group. Several questions are posed.A survey of racks and quandles: some recent developmentshttps://zbmath.org/1491.201452022-09-13T20:28:31.338867Z"Elhamdadi, Mohamed"https://zbmath.org/authors/?q=ai:elhamdadi.mohamedLatin quandles and applications to cryptographyhttps://zbmath.org/1491.201562022-09-13T20:28:31.338867Z"Isere, Abednego Orobosa"https://zbmath.org/authors/?q=ai:isere.abednego-orobosa"Adéníran, John Olúsọlá"https://zbmath.org/authors/?q=ai:adeniran.john-olusola"Jaíyéọlá, Tèmít\'ọp\'ẹ Gb\'ọláhàn"https://zbmath.org/authors/?q=ai:jaiyeola.temitope-gbolahanSummary: This work investigated some properties of Latin quandles that are applicable in cryptography. Four distinct cores of an Osborn loop (non-diassociative and non-power associative) were introduced and investigated. The necessary and sufficient conditions for these cores to be (i) (left) quandles (ii) involutory quandles (iii) quasi-Latin quandles and (iv) involutory quasi-Latin quandles were established. These conditions were judiciously used to build cipher algorithms for cryptography in some peculiar circumstances.Extension of the Švarc-Milnor lemma to gyrogroupshttps://zbmath.org/1491.201632022-09-13T20:28:31.338867Z"Wattanapan, Jaturon"https://zbmath.org/authors/?q=ai:wattanapan.jaturon"Atiponrat, Watchareepan"https://zbmath.org/authors/?q=ai:atiponrat.watchareepan"Suksumran, Teerapong"https://zbmath.org/authors/?q=ai:suksumran.teerapongSummary: A strongly generated gyrogroup is a gyrogroup that comes with a specific generating set invariant under its gyroautomorphisms, which may be viewed as a suitable generalization of a group. The achievement of constructing a word metric on a strongly generated gyrogroup, together with the recent notion of a gyrogroup action, prompts us to extend the Švarc-Milnor lemma (also known as the fundamental lemma of geometric group theory) to the case of gyrogroups.Configuration Lie groupoids and orbifold braid groupshttps://zbmath.org/1491.220012022-09-13T20:28:31.338867Z"Roushon, S. K."https://zbmath.org/authors/?q=ai:roushon.sayed-kSummary: We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the class of Lie groupoids corresponding to global quotient orbifolds with nonempty singular set, the fibration theorems do not hold. Secondly, we prove a short exact sequence of fundamental groups (called \textit{pure orbifold braid groups}) of one of the configuration Lie groupoids of the Lie groupoid corresponding to the punctured complex plane with cone points. This shows the possibility of a quasifibration type Fadell-Neuwirth theorem for Lie groupoids.
As consequences, first we see that the pure orbifold braid groups have poly-virtually free structure, which generalizes the classical braid group case. We also provide an explicit set of generators of the pure orbifold braid groups. Secondly, we prove that a class of affine and finite complex Artin groups are virtually poly-free, which partially answers the question if all Artin groups are virtually poly-free [\textit{M. Bestvina}, Geom. Topol. 3, 269--302 (1999; Zbl 0998.20034), Question 2]. Finally, combining this poly-virtually free structure and a recent result [\textit{M. Bestvina} et al., The Farrell-Jones conjecture for hyperbolic-by-cyclic groups. (2021; \url{arXiv:2105.13291}], we deduce the Farrell-Jones isomorphism conjecture for the above class of orbifold braid groups. This also implies the conjecture for the case of the affine Artin group of type \(\widetilde{D}_n\), which was left open in [the author's paper, Can. J. Math. 73, No. 4, 1153--1170 (2021; Zbl 1485.19003); erratum ibid. 74, No. 2, 602 (2022), Problem].On a higher integral invariant for closed magnetic lines, revisitedhttps://zbmath.org/1491.353292022-09-13T20:28:31.338867Z"Akhmet'ev, Peter M."https://zbmath.org/authors/?q=ai:akhmetev.petr-mSummary: We recall a definition of an asymptotic invariant of classical link, which is called \(M\)-invariant. \(M\)-invariant is a special Massey integral, this integral has an ergodic form and is generalized for magnetic fields with open magnetic lines in a bounded \(3D\)-domain. We present a proof that this integral is well defined. A combinatorial formula for \(M\)-invariant using the Conway polynomial is presented. The \(M\)-invariant is a higher invariant, it is not a function of pairwise linking numbers of closed magnetic lines. We discuss applications of \(M\)-invariant for MHD.Combinatorial vs. classical dynamics: recurrencehttps://zbmath.org/1491.370412022-09-13T20:28:31.338867Z"Mrozek, Marian"https://zbmath.org/authors/?q=ai:mrozek.marian"Srzednicki, Roman"https://zbmath.org/authors/?q=ai:srzednicki.roman"Thorpe, Justin"https://zbmath.org/authors/?q=ai:thorpe.justin"Wanner, Thomas"https://zbmath.org/authors/?q=ai:wanner.thomasSummary: Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in discrete contexts, such as graph theory or in the recently developed field of combinatorial dynamics, is straightforward and computationally feasible. In this paper, we present an approach to study classical dynamical systems as given by semiflows or flows using techniques from combinatorial topological dynamics. More precisely, we present a general existence theorem for periodic orbits of semiflows which is based on suitable phase space decompositions, and indicate how combinatorial techniques can be used to satisfy the necessary assumptions. In this way, one can obtain computer-assisted proofs for the existence of periodic orbits and even certain chaotic behavior.An iterated graph construction and periodic orbits of Hamiltonian delay equationshttps://zbmath.org/1491.370522022-09-13T20:28:31.338867Z"Albers, Peter"https://zbmath.org/authors/?q=ai:albers.peter"Frauenfelder, Urs"https://zbmath.org/authors/?q=ai:frauenfelder.urs-adrian"Schlenk, Felix"https://zbmath.org/authors/?q=ai:schlenk.felixSummary: According to the Arnold conjectures and \textit{A. Floer}'s proofs [J. Differ. Geom. 28, No. 3, 513--547 (1988; Zbl 0674.57027); Commun. Pure Appl. Math. 42, No. 4, 335--356 (1989; Zbl 0683.58017)], there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.New families of highly neighborly centrally symmetric sphereshttps://zbmath.org/1491.520132022-09-13T20:28:31.338867Z"Novik, Isabella"https://zbmath.org/authors/?q=ai:novik.isabella"Zheng, Hailun"https://zbmath.org/authors/?q=ai:zheng.hailunSummary: \textit{W. Jockusch} [J. Comb. Theory, Ser. A 72, No. 2, 318--321 (1995; Zbl 0844.52006)] constructed an infinite family of centrally symmetric (cs, for short) triangulations of 3-spheres that are cs-2-neighborly. Recently, \textit{I. Novik} and \textit{H. Zheng} [Adv. Math. 370, Article ID 107238, 15 p. (2020; Zbl 1454.57017)] extended Jockusch's construction: for all \(d\) and \(n>d\), they constructed a cs triangulation of a \(d\)-sphere with \(2n\) vertices, \( \Delta^d_n\), that is cs-\(\lceil d/2\rceil \)-neighborly. Here, several new cs constructions, related to \(\Delta^d_n\), are provided. It is shown that for all \(k>2\) and a sufficiently large \(n\), there is another cs triangulation of a \((2k-1)\)-sphere with \(2n\) vertices that is cs-\(k\)-neighborly, while for \(k=2\) there are \(\Omega (2^n)\) such pairwise non-isomorphic triangulations. It is also shown that for all \(k>2\) and a sufficiently large \(n\), there are \(\Omega (2^n)\) pairwise non-isomorphic cs triangulations of a \((2k-1)\)-sphere with \(2n\) vertices that are cs-\((k-1)\)-neighborly. The constructions are based on studying facets of \(\Delta^d_n\), and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale's evenness condition. Along the way, it is proved that Jockusch's spheres \(\Delta^3_n\) are shellable and an affirmative answer to Murai-Nevo's question about 2-stacked shellable balls is given.The holonomy of a singular leafhttps://zbmath.org/1491.530312022-09-13T20:28:31.338867Z"Laurent-Gengoux, Camille"https://zbmath.org/authors/?q=ai:laurent-gengoux.camille"Ryvkin, Leonid"https://zbmath.org/authors/?q=ai:ryvkin.leonid\textit{I. Androulidakis} and \textit{G. Skandalis} [J. Reine Angew. Math. 626, 1--37 (2009; Zbl 1161.53020)] gave a construction for the holonomy groupoid of any singular foliation. Androulidakis and Zambon formulated a holonomy map for singular foliations, which is defined on the holonomy groupoid, rather than the fundamental group of a leaf, as it happens with regular foliations. On the other hand, Laurent-Gengoux, Lavau and Strobl established a universal Lie-\(\infty\) algebroid to every singular foliation.
In the paper under review, the authors construct higher holonomy maps, defined on \(\pi_n(L)\), where \(L\) is a singular leaf \(L\). They take values in the \((n-1)\)-th homotopy group of the universal Lie-\(\infty\) algebroid associated with the transversal foliation to \(L\). Moreover, they show that these holonomy maps form a long exact sequence.
Reviewer: Iakovos Androulidakis (Athína)On irregular Sasaki-Einstein metrics in dimension 5https://zbmath.org/1491.530622022-09-13T20:28:31.338867Z"Süß, H."https://zbmath.org/authors/?q=ai:suss.helmut|suss.hendrikIn the present paper the author studies Fano cone singularities \(X\) polarised by a Reeb field \(\xi\) in the sense of Collins and Sźekelyhidi. The choice of such a polarisation induces a Sasakian metric structure on the link of \(X\). This Sasakian structure is called quasi-regular if the Reeb field generates a one-dimensional torus action; it is called irregular if the corresponding torus is at least two-dimensional. Motivated by questions posed by J. Sparks: ``Are there continuous families of irregular Sasaki-Einstein structures in dimension 5?'' and ``Are there any nontoric irregular Sasaki-Einstein structures in dimension 5?'' and by T. C. Collins, G. Sźekelyhidi such as ``Are there any irregular Sasaki-Einstein structures on \(S^5\)?'', the author proves that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. Also, using K-stability he proves the existence of continuous families of nontoric irregular Sasaki-Einstein structures on odd connected sums of \(S^2 \times S^3\).
Reviewer: Andreas Arvanitoyeorgos (Patras)The stable converse soul question for positively curved homogeneous spaceshttps://zbmath.org/1491.530672022-09-13T20:28:31.338867Z"González-Álvaro, David"https://zbmath.org/authors/?q=ai:gonzalez-alvaro.david"Zibrowius, Marcus"https://zbmath.org/authors/?q=ai:zibrowius.marcusAuthors' abstract: The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E \times \mathbb{R}^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is ``stable under tangential homotopy equivalence'': if it has an affirmative answer for all vector bundles over a certain manifold \(M\), then the same is true for any manifold tangentially homotopy equivalent to \(M\). Our main tool is topological K-theory. Over \(B^{13}\), there is essentially one stable class of real vector bundles for which our method fails.
Reviewer: V. V. Gorbatsevich (Moskva)A note on Griffiths' conjecture about the positivity of Chern-Weil formshttps://zbmath.org/1491.530782022-09-13T20:28:31.338867Z"Fagioli, Filippo"https://zbmath.org/authors/?q=ai:fagioli.filippoLet \((E, h)\) be a Griffiths semipositive Hermitian holomorphic vector bundle of rank 3 over a complex manifold. In this paper, the author proves the positivity of the characteristic differential form
\[
c_1(E, h)\wedge c_2(E, h) -c_3(E, h).
\]
Here, for \(k=1,2,3\), \(c_k(E,h)\) denotes the Chern form of bidegree \((k,k)\), which represents the Chern class \(c_k(E)\) of the vector bundle \(E\). This provides a new evidence towards Griffiths' conjecture about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, the author establishes the following new chain of inequalities between Chern forms
\[
c_1(E, h)^3 \ge c_1(E, h) \wedge c_2(E, h) \ge c_3(E, h).
\]
The author also shows how to obtain the positivity of the second Chern form \(c_2(E, h)\) in any rank, if \((E, h)\) is Griffiths positive. This is obtained by adapting the Griffiths' result on the positivity of \(c_2(E, h)\) in rank 2.
The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.
Reviewer: Riccardo Piovani (Parma)Liouville hypersurfaces and connect sum cobordismshttps://zbmath.org/1491.530812022-09-13T20:28:31.338867Z"Avdek, Russell"https://zbmath.org/authors/?q=ai:avdek.russellIn symplectic/contact geometry, the Moser/Gray trick to prove existence of the unique local model (up to symplectomorphism), constructions of isotopies, the flow of a natural vector field provide powerful tools to find nicer local shapes (or the model structure), to decompose a symplectic manifold into building blocks of the model structure, to glue pieces and to assemble topological/homological/categorical data.
In [\textit{M. Abouzaid}, Ann. Math. (2) 175, No. 1, 71--185 (2012; Zbl 1244.53089)], the author defined a gluing operation for contact manifolds along nice neighborhoods of Liouville hypersurfaces, called the Liouville connected sum, which generalizes the Weinstein handle attachment for Weinstein (symplectic) manifolds. To do it, he introduced the concept of Liouville hypersurfaces (more generally, Liouville submanifolds) in contact manifolds which generalize ribbons of Legendrian graphs and pages of open book decomposition of contact manifolds. A fundamental construction is that any Liouville hypersurface \(\Sigma\) of a contact manifold can have a standard tubular neighborhood \(\mathcal{N}(\Sigma)\) with smooth, convex boundary, which can be considered as a flattening of \(\Sigma\) by the Reeb vector fields. A local model of Liouville connected sum is convex gluing of unit-cotangent bundles of two smooth compact manifolds with non-empty boundaries with orientation-reversing diffeomorphism on the boundary of the base manifolds (Example 2.14).
A Liouville domain is a smooth compact symplectic manifold with boundary with exact symplectic form with Liouville vector field trasversely outward along the boundary. A Liouville hypersurface of a contact manifold \((M^{2n+1}, \xi\subset \text{Ker}(\alpha))\) is the image of an embedding \(i:\Sigma \to M\) of a Liouville domain \((\Sigma^{2n}, d\beta)\) of codimension \(1\) with \(i^{*}\alpha=\beta\). Since a Liouville hypersurface is transverse to the Reeb vector field \(R\) for \(\alpha\) (\(\alpha(R)=1\) and \(d\alpha(R,\cdot)=0\)), a map \(\Phi:[-\epsilon,\epsilon]\times \Sigma\to M\), defined by a time \((z \in [-\epsilon,\epsilon])\) flow of \(R\) starting from \(x\in \Sigma\), gives a small tubular neighborhood \(N(\Sigma)\) of \(\Sigma\) inside \(M\) satisfying \(\alpha|_{N(\Sigma)}=dz+\beta\). After edge-rounding on the neighborhood \(N(\Sigma)\), one get a standard neighborhood \(\mathcal{N}(\Sigma)\) of the Liouville hypersurface (Figure 3, Definition 3.6). The author also proves a neighborhood theorem for Liouville submanifolds of high codimension.
The second half of the paper consists of applications of the gluing construction, for example, the construction of a Weinstein handle attachment from a standard neighborhood of a Liouville hypersurface, the symplectic handle attachment to the positive boundary of a weak symplectic cobordism, fillabilities preserved under contact connected sum, the monoid structure on non-vanishing contact homology, extension of contact \((1/k)\)-surgeries to arbitrary dimension, and certain generalized Dehn twists that create exotic structures.
The main idea of [\textit{R. Avdek}, J. Symplectic Geom. 19, No. 4, 865--957 (2021; Zbl 07455583)], that was released in the earlier arXiv preprint version, was adapted to glue of Weinstein pairs in [\textit{Y. Eliashberg}, Proc. Symp. Pure Math. 99, 59--82 (2018; Zbl 1448.53083), Section 3.1], where the skeleta of Weinstein pairs are glued along the skeleta of glued hypersurfaces. It would also be interesting to consider the concept of sutured Liouville manifolds and Liouville sectors in [\textit{S. Ganatra} et al., Publ. Math., Inst. Hautes Étud. Sci. 131, 73--200 (2020; Zbl 07209675); ``Sectorial descent for wrapped Fukaya categories'', Preprint, \url{arXiv:1809.03427}; ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}].
Reviewer: Dahye Cho (Stony Brook)Reeb chords of Lagrangian sliceshttps://zbmath.org/1491.530832022-09-13T20:28:31.338867Z"Chantraine, Baptiste"https://zbmath.org/authors/?q=ai:chantraine.baptisteSummary: In this short note, we observe that the boundary of a properly embedded compact exact Lagrangian sub-manifold in a subcritical Weinstein domain \(X\) necessarily admits Reeb chords. The existence of a Reeb chord either follows from an obstruction to the deformation of the boundary to a cylinder over a Legendrian sub-manifold or from the fact that the wrapped Floer homology of the Lagrangian vanishes once this boundary has been ``collared''.Formality of Floer complex of the ideal boundary of hyperbolic knot complementhttps://zbmath.org/1491.530902022-09-13T20:28:31.338867Z"Bae, Youngjin"https://zbmath.org/authors/?q=ai:bae.youngjin"Kim, Seonhwa"https://zbmath.org/authors/?q=ai:kim.seonhwa"Oh, Yong-Geun"https://zbmath.org/authors/?q=ai:oh.yong-geunA knot \(K\) in the three-sphere \(S^{3}\) is an embedding of the circle \(S^{1}\) into \(S^{3}\). Since topology of the knot complement \(S^{3}\setminus K\) determines that of the knot \(K\) (or mirror of \(K\)), understanding topology of the knot complement is important in knot theory. Given a knot \(K\subset S^{3}\), the boundary \(T:=\partial (U(K))\) of a tubular neighborhood \(U(K)\) of \(K\) in \(S^{3}\) is a torus. The authors take the conormal \(\nu^{*}T\) of the torus \(T\) in the cotangent bundle \(T^{*}(S^{3}\setminus K)\) of the knot complement, which is a Lagrangian submanifold of \(T^{*}(S^{3}\setminus K)\) and defined the wrapped Floer complex which is generated by the set of Hamiltonian chords of kinetic energy Hamiltonian associated to a Riemannian metric on the complement with boundaries on the Lagrangian [\textit{Y. Bae} et al., ``A wrapped Fukaya category of knot complement'', Preprint, \url{arXiv:1901.02239}]. The \(A_{\infty}\)-structure on the complex is defined by counting pseudo-holomorphic disks bounding the Hamiltonain chords and the Langrangian submanifolds (under the Hamiltonian diffeomorphism).
A knot is called hyperbolic if the knot complement admits a hyperbolic metric. For example, the figure-eight knot is hyperbolic. Using special properties of hyperbolic geometry, the authors prove that the \(A_{\infty}\) structure maps of the wrapped Floer complex of hyperbolic-knot complement vanishes except degree \(2\) and the resulting reduced wrapped Floer cohomology(equivalently, knot Floer cohomology) is concentrated to degree \(0\) [\textit{Y. Bae} et al., Asian J. Math. 25, No. 1, 117--176 (2021; Zbl 07413486)]. They also showed that, for torus knots, the reduced wrapped Floer cohomology is non-zero for degree \(0,1\) and that the wrapped Floer cohomology distinguishes hyperbolic knots and torus knots.
A very rough outline of the proof of the main theorem is following: The hyperbolic-knot complement has the hyperbolic \(3\)-space \(\mathbb{H}^{3}:=\{(x,y,z)\in \mathbb{R}^{2}\times \mathbb{R}^{+}\}\) with isometric \(\mathrm{PSL}(2,\mathbb{C})\)-action as its universal cover. By the Margulis' thick-thin decomposition, one can take the torus \(T:=\partial (U(K))\) as a level set of the Busemann function which is lifted to \(log z\), under which \(T\) is lifted to a horosphere centered at \(\{z=+\infty\}\). Using property of horo-sphere and \(\mathrm{PSL}(2,\mathbb{C})\)-action, the set of geodesic cord in \(S^{3}\setminus K\) is bijective to the set of images of all infinite tame geodesics in \(S^{3}\setminus K\) (Proposition 3.6). Moreover, the set of geodesic cord in \(S^{3}\setminus K\) is bijective to the set of Hamiltonian chord of \(\nu^{*}T\subset S^{3}\setminus K\) (Lemma 5.2). From the non-negativity of the second variation of the energy functional, both nullity and Morse index of any geodesic cord vanish (Theorem 1.1). Therefore, all the degrees of generators are \(0\), which implies, by the dimension formula, that the structure maps vanish except degree 2 (Theorem 1.3). By the maximum principle, for any Floer disk bounding triple of Hamiltonian chords, there exists a totally geodesic immersed ideal triangle \(\delta\subset S^{3}\setminus K\) such that the ideal edges contain geodesic triples corresponding to the chords and that it contains the projected image of the Floer disk (Theorem 1.5). Technical results include vertical \(C^{0}\) estimates, uniform horizontal \(C^{0}\) estimates and the maximum principle which lead to the well-definedness of the wrapped Floer cohomology.
There are also other contact/symplectic approaches to knot invariants, for example, [\textit{T. Ekholm} et al., Geom. Topol. 17, No. 2, 975--1112 (2013; Zbl 1267.53095); \textit{T. Ekholm} et al., Invent. Math. 211, No. 3, 1149--1200 (2018; Zbl 1385.57015); \textit{V. Shende}, Forum Math. Pi 7, Paper No. e6, 16 p. (2019; Zbl 1426.57021)].
Reviewer: Dahye Cho (Stony Brook)A degree-zero, monotone, surjective self-map of the Pontryagin surfacehttps://zbmath.org/1491.540152022-09-13T20:28:31.338867Z"Daverman, Robert J."https://zbmath.org/authors/?q=ai:daverman.robert-j"Thickstun, Thomas L."https://zbmath.org/authors/?q=ai:thickstun.thomas-lIn this paper a space is called \textit{nice} if it is locally compact, locally path-connected, separable and metrizable. A compact, connected, nice space \(P\) is a closed Pontryagin surface if there exists a countable pairwise disjoint family \(\mathcal{E}\) of figure-eights in \(P\) such that \(\mathcal{E}\) is null in \(P\) and for any cofinite subfamily \(\mathcal{D}\) of \(\mathcal{E}\), the image of \(P\) under the decomposition map \(P\to P/\mathcal{D}\) is a closed, orientable surface. Such a family \(\mathcal{E}\) is called a sufficient family for \(P\). A closed Pontryagin surface \(P\) is said to be rich if it has a sufficient family \(\mathcal{E}\) whose union is dense in \(P\). The symbol \(\mathbb{P}\) refers to a compact, connected, rich Pontryagin surface (any two such spaces are homeomorphic).
Let us now quote from the Introduction. ``The main result of \textit{R. J. Daverman} and \textit{T. L. Thickstun} [Pac. J. Math. 303, No. 1, 93--131 (2019; Zbl 1436.57018)] promises that a map of the standard Pontryagin surface to itself is a near-homeomorphism if and only if it is monotone and has absolute degree one. Here we supplement that effort by producing an example showing that the degree one hypothesis cannot be discarded.'' The second sentence is confusing, so we sought clarification from the authors. The point they were trying to make is that if they could have proved that any monotone self map of the Pontryagin surface must have degree one, then \textit{degree one} could have been expunged from the statement of the result just mentioned.
As to the degree of a map \(F:M\to M'\) between closed, connected, orientable \(2\)-manifolds or closed Pontryagin surfaces, this is an integer \(k\geq0\) such that the \(F\)-induced homomorphism of the second Čech homology with \(\mathbb{Z}\) coefficients of these spaces (these groups being isomorphic to \(\mathbb{Z}\)), amounts to multiplication by \(k\), up to sign. With these notions in mind, we now state the main result of the paper which consists of a constructed example.
\textbf{Theorem 3.1.} There exists a surjective monotone map of \(\mathbb{P}\) to \(\mathbb{P}\) of degree \(0\).
The example itself is given on page 307, but Sections 4--7 are needed to prove the claim.
Reviewer: Leonard R. Rubin (Norman)Resolving compacta by free \(p\)-adic actionshttps://zbmath.org/1491.550012022-09-13T20:28:31.338867Z"Levin, Michael"https://zbmath.org/authors/?q=ai:levin.michael-w|levin.michael-e|levin.michael-y|levin.michael-jIn this paper \(p\) is a prime number, \(A_p\) denotes the group of \(p\)-adic integers, \(\mathbb{Z}_p\) the \(p\)-cyclic group, and \(\mathbb{Z}_{p^\infty}=\mathbb{Z}[1/p]/\mathbb{Z}\), the \(p\)-adic circle. Motivation for the research in this paper is described by the author in the following way.
``The Hilbert-Smith conjecture asserts that a compact group effectively (and continuously) acting on a manifold must be a Lie group. This assertion is equivalent to the following one: there is no effective action of \(A_p\) on a manifold. The Hilbert-Smith conjecture is proved for manifolds of dimension \(\leq3\) [\textit{J. Pardon}, Proc. Symp. Pure Math. 102, 187--193 (2019; Zbl 1453.57026) and J. Am. Math. Soc. 26, No. 3, 879--899 (2013; Zbl 1273.57024)] and open even for free actions of \(A_p\) in dimension \(>3\). \textit{C.-T. Yang} [Mich. Math. J. 7, 201--218 (1960; Zbl 0094.17502)] showed that if \(A_p\) effectively acts on an \(n\)-manifold \(M\) then either \(\mathrm{dim}\, M/A_p=\infty\) or \(\mathrm{dim}\, M/A_p=n+2\). This naturally suggests examining if the latter dimensional relations may occur in a more general setting, namely when \(M\) is just a finite-dimensional compactum (\(=\) compact metric space). One of these relations was confirmed by \textit{F. Raymond} and \textit{R. F. Williams} [Ann. Math. (2) 78, 92--106 (1963; Zbl 0178.26003)] who constructed an effective action of \(A_p\) on an \(n\)-dimensional compactum \(X\), \(n\geq2\), with \(\mathrm{dim}\,X/A_p=n+2\). However, it remains open for more than \(50\) years now whether there exists a free action of \(A_p\) on a finite-dimensional compactum \(X\) such that \(\mathrm{dim}\,X\in\{\infty,n+2\}\).''
The following definition plays a prominent role in this work.
\textbf{Definition.} A compactum \(Y\) is said to be \(p\)-resolvable by a compactum \(X\) if there is a free action of the \(p\)-adic integers \(A_p\) on \(X\) such that \(Y=X/A_p\).
It is stated by the author that ``in this paper we study compacta \(Y\) that are \(p\)-resolvable by compacta of a lower dimension and mainly focus on compacta with \(\mathrm{dim}_{\mathbb{Z}[1/p]}\,Y=1\), and in particular, on compacta which are \(p\)-resolvable by \(1\)-dimensional compacta, a case that turns out to be highly non-trivial.''
The main theorems of the paper are stated in Section 1:
\textbf{Theorem 1.2.} If \(A_p\) acts on a finite-dimensional compactum \(X\) so that \(Y=X/A_p\) is infinite-dimensional then there exists an invariant compactum \(X'\subset X\) on which the action of \(A_p\) is free and whose orbit space \(Y'=X'/A_p\) is infinite-dimensional with \(\mathrm{dim}_{[1/p]}\,Y'=1\).
\textbf{Theorem 1.3.} Let \(Y\) be a finite-dimensional compactum with \(\mathrm{dim}_{\mathbb{Z}[1/p]}\,Y=1\). Then:
\((i)\) \(Y\) is \(p\)-resolvable by a compactum of dimension \(\leq\mathrm{dim}\,Y-1\) if \(\mathrm{dim}\,Y\geq2\);
\((ii)\) \(Y\) is not \(p\)-resolvable by a compactum of dimension \(\leq\mathrm{dim}\,Y-2\) if \(\mathrm{dim}\,Y\geq4\).
\textbf{Theorem 1.4.} There is a \(3\)-dimensional compactum \(Y\) with \(\mathrm{dim}_{\mathbb{Z}[1/p]}\,Y=1\) and \(\mathrm{dim}_{\mathbb{Z}_p}\,Y=2\) that is not \(p\)-resolvable by a \(1\)-dimensional compactum.
\textbf{Theorem 1.5.} There is an infinite-dimensional compactum \(Y\) such that \(\mathrm{dim}_{\mathbb{Z}[1/p]}\,Y=1\) and \(\mathrm{dim}_{\mathbb{Z}}\,Y=2\) that is not \(p\)-resolvable by a finite-dimensional compactum.
Theorem 1.6, which we shall not state, is used in proving Theorems 1.2, 1.4, and 1.5. However, let us quote:
\textbf{Conjecture 1.7.} No compactum \(Y\) with \(\mathrm{dim}_{\mathbb{Z}[1/p]}\,Y=1\) and \(\mathrm{dim}\,Y\geq n+2\) is \(p\)-resolvable by an \(n\)-dimensional compactum.
Reviewer: Leonard R. Rubin (Norman)The degrees of maps between \((n-1)\)-connected \((2n+1)\)-dimensional manifolds or Poincaré complexes and their applicationshttps://zbmath.org/1491.550032022-09-13T20:28:31.338867Z"Grbić, J."https://zbmath.org/authors/?q=ai:grbic.jelena"Vučić, A."https://zbmath.org/authors/?q=ai:vucic.aleksandarThe main result of this paper provides necessary and sufficient algebraic conditions for the existence of maps of a given degree between \((n-1)\)-connected \((2n+1)\)-dimensional Poincaré duality complexes without 2-torison in homology. These conditions allow for the explicit construction of all homotopy classes of maps of a given degree. The authors consider the degree of a map between CW-complexes with one top cell and use homotopy theoretic methods, in particular, the Hilton-Milnor Theorem and Hilton-Hopf invariants, to describe the homotopy invariants of the attaching maps of the top cells. The main result is used to state a sufficient condition for a map of degree \(\pm 1\) between \((n-1)\)-connected \((2n+1)\)-dimensional Poincaré duality complexes to be a homotopy equivalence. For low \(n\), the authors ``classify, up to homotopy, torsion free \((n-1)\)-connected \((2n+1)\)-dimensional Poincaré complexes.''
Reviewer: Beatrice Bleile (Armidale)On the topological complexity of manifolds with abelian fundamental grouphttps://zbmath.org/1491.550042022-09-13T20:28:31.338867Z"Cohen, Daniel C."https://zbmath.org/authors/?q=ai:cohen.daniel-c"Vandembroucq, Lucile"https://zbmath.org/authors/?q=ai:vandembroucq.lucileA standard upper bound for the topological complexity of a CW complex \(X\) is given by \(\mathrm{TC}(X) \leq 2 \dim X\). (Here we adapt the article's convention that \(\mathrm{TC}(\mathrm{pt})=0\).) There are plenty of examples of manifolds whose topological complexity coincides with this upper bound. In this case we shall say that the topological complexity is \emph{maximal}.
\textit{A. Costa} and \textit{M. Farber} [Commun. Contemp. Math. 12, No. 1, 107--119 (2010; Zbl 1215.55001)] have studied the maximality of topological complexity for cell complexes. They introduced a cohomology class \(v \in H^1(X \times X;I)\), where \(I\) is a certain local coefficient system, and show that \(\mathrm{TC}(X)\) of a \(2n\)-dimensional finite cell complex \(X\) is maximal if and only if \(v^{\cup 2n} \neq 0\). However, this condition is hard to check explicitly.
In the present article, \(v\) is called the Costa-Farber class. The authors consider arbitrary closed manifolds with \emph{abelian} fundamental groups and establish a more tangible cohomological criterion that is equivalent to the maximality of topological complexity in this case. This criterion only depends on the fundamental group of the manifold, but no longer on the manifold itself. Methodically, the authors rely on the fact that for an abelian group \(\pi\) the map \(\pi \times \pi \to \pi\), \((g,h) \mapsto gh^{-1}\), is a group homomorphism to relate the Costa-Farber class to the better-explored Berstein-Schwarz class, see for example [\textit{A. N. Dranishnikov} and \textit{Y. B. Rudyak}, Math. Proc. Camb. Philos. Soc. 146, No. 2, 407--413 (2009; Zbl 1171.55002)]. In addition, they establish similar results for the Lusternik-Schnirelmann category of the cofiber \(C_\Delta\) of the diagonal inclusion \(\Delta: X \to X \times X\). There are several classes of spaces for which \(\mathrm{TC}(X)\) and \(\mathrm{cat}(C_\Delta)\) are known to coincide, see for example [\textit{J. M. García Calcines} and \textit{L. Vandembroucq}, Math. Z. 274, No. 1--2, 145--165 (2013; Zbl 1275.55003)]. The authors of the present article show among other things that in the abelian case, \(\mathrm{TC}(X)\) is maximal if and only if \(\mathrm{cat}(\Delta_X)\) is maximal and the two numbers coincide. These results are carried out in Section 2 of the article.
In Section 3 the authors derive explicit classes of fundamental groups for which the topological complexity of closed manifolds is non-maximal. They discuss chain-level Pontryagin algebra structures in group homology and use them to carry out explicit calculations. In their main result, the authors provide a list of conditions on abelian groups for which the desired non-maximality holds. All of them are abelian groups whose torsion parts are products of primary cyclic groups for the same.
Finally, in Section 4 the authors show by providing examples that their conditions are sharp. More precisely, for various slight relaxations of the conditions, they provide examples of manifolds with such fundamental groups whose topological complexity is maximal.
Reviewer: Stephan Mescher (Halle)Integral foliated simplicial volume and \(S^1\)-actionshttps://zbmath.org/1491.550052022-09-13T20:28:31.338867Z"Fauser, Daniel"https://zbmath.org/authors/?q=ai:fauser.danielIn this paper, the author proves that the integral foliated simplicial volume of a smooth manifold admitting certain \(S^1\)-actions vanishes.
The integral foliated simplicial volume is a generalization of the classical simplicial volume of oriented compact manifolds. For an oriented compact connected \(m\)-manifold \(M\) (possibly with boundary) with \(\Gamma=\pi_1(M)\) and a measure preserving action \(\alpha: \Gamma \curvearrowright (Z,\mu)\) on a Borel probability space, the parametrized \(l^1\)-norm \(|M,\partial M|^{\alpha}\) is the infimum of \(l^1\)-norms of cycles in the twisted chain group \[C_m(M, \partial M;\alpha)=L^{\infty}(Z,\mu;\mathbb{Z})\otimes_{\mathbb{Z}\Gamma}C_n(\tilde{M},\partial \tilde{M};\mathbb{Z})\] that represents the fundamental class \([M,\partial M]^{\alpha}\in H_*(M,\partial M; \alpha)\). The (relative) integral foliated simplicial volume \(|M,\partial M|\) of \(M\) is the infimum of \(|M,\partial M|^{\alpha}\) when \(\alpha\) runs over all measure preserving \(\Gamma\)-actions on Borel probability spaces.
The main result of this paper (Theorem 1.1) states that: for any oriented compact smooth manifold \(M\) that admits a smooth \(S^1\)-action without fixed points such that every orbit is \(\pi_1\)-injective, \(|M,\partial M|=0\) holds. Moreover, for any essentially free measure preserving action \(\alpha:\pi_1(M)\curvearrowright (Z,\mu)\) on a Borel probability space, \(|M,\partial M|^{\alpha}=0\) holds.
The proof of Theorem 1.1 relies on the so-called \emph{hollowing construction} of \textit{K. Yano}, given in [J. Fac. Sci., Univ. Tokyo, Sect. I A 29, 493--501 (1982; Zbl 0518.57017)], which is used to prove the vanishing of simplicial volume of smooth manifolds with nontrivial smooth \(S^1\)-action. Roughly speaking, the hollowing construction gives a non-proper map \(p:M_{n-2}\to M\) between \(m\)-dim manifolds such that \(M_{n-2}\) splits as \(N\times S^1\) and the map \(p\) is nice. The \(S^1\)-product structure of \(M_{n-2}\) gives a representative of \([M_{n-2},\partial M_{n-2}]^{\alpha\circ p_*}\) with small \(l^1\)-norm. Although \(p\) does not push-forward \([M_{n-2},\partial M_{n-2}]^{\alpha \circ p_*}\) to the fundamental class \([M,\partial M]^{\alpha}\), it can be adjusted by filling to a fundamental cycle of \(M\) without changing the norm too much.
Reviewer: Hongbin Sun (Piscataway)Homotopy groups of highly connected Poincaré duality complexeshttps://zbmath.org/1491.550102022-09-13T20:28:31.338867Z"Beben, Piotr"https://zbmath.org/authors/?q=ai:beben.piotr"Theriault, Stephen"https://zbmath.org/authors/?q=ai:theriault.stephen-dThe authors consider the following case related to the cell attachment problem and Ganea-type results. Suppose that there is a cofibration $A\xrightarrow{f}Y\to Y\cup CA$; there is a map $Y\to Z$ which induces a principal fibration $\Omega Z\to E \to Y$; and the map $Y \to Z$ extends to a map $Y \cup CA \to Z$, inducing a principal fibration $\Omega Z \to E' \xrightarrow{p'} Y \cup CA$. They develop new techniques that relate the action of a principal fibration to relative Whitehead products in order to identify the homotopy type of $E'$ and the homotopy class of $p'$ in terms of the homotopy type of $E$ and the homotopy classes of $p$ and $f$. This requires certain hypotheses on the spaces and maps involved, but these are fulfilled in a wide variety of contexts. The new methods are powerful and should have numerous applications.
Reviewer: Jean-Claude Thomas (Angers)Archipelago groups are locally free. Corrigendum to: ``Cotorsion and wild homology''https://zbmath.org/1491.550132022-09-13T20:28:31.338867Z"Herfort, Wolfgang"https://zbmath.org/authors/?q=ai:herfort.wolfgang-n"Hojka, Wolfram"https://zbmath.org/authors/?q=ai:hojka.wolframSummary: An Archipelago group is the quotient of the topologist's product \(G = \circledast_{i \ge 1}G_i\) of a sequence \((G_i)_{i \ge 1}\) of groups modulo the normal closure of the subset \(\bigcup_{i \ge 1}G_i\) in \(G\). In this note we provide a simple proof of a result from [the authors, ibid. 221, No. 1, 275--290 (2017; Zbl 1421.55012)], namely that Archipelago groups are locally free.Torsion in thin regions of Khovanov homologyhttps://zbmath.org/1491.570012022-09-13T20:28:31.338867Z"Chandler, Alex"https://zbmath.org/authors/?q=ai:chandler.alex"Lowrance, Adam M."https://zbmath.org/authors/?q=ai:lowrance.adam-m"Sazdanović, Radmila"https://zbmath.org/authors/?q=ai:sazdanovic.radmila"Summers, Victor"https://zbmath.org/authors/?q=ai:summers.victorKhovanov homology categorifies the Jones polynomial in the sense that the Jones polynomial of a link can be recovered as the graded Euler characteristic of its Khovanov homology. Torsion in Khovanov homology is a new phenomenon in knot theory which does not appear in the theory of the Jones polynomial.
Homologically thin links, that is, links whose Khovanov homology is supported on two adjacent diagonals,
are known to contain only \(\mathbb{Z}_2\) torsion.
In this paper, the authors ``prove a local version of this result. If the Khovanov homology of a link is supported on two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only \(\mathbb{Z}_2\) torsion over that range of homological gradings.''
The result of this paper complements and contrasts with the result of \textit{L. Helme-Guizon} et al. [Fundam. Math. 190, 139--177 (2006; Zbl 1105.57012)], extending work by \textit{A. M. Lowrance} and \textit{R. Sazdanović} [Topology Appl. 222, 77--99 (2017; Zbl 1371.57011)], saying in a range of homological gradings, Khovanov homology contains only \(\mathbb{Z}_2\) torsion.
Reviewer: Meili Zhang (Dalian)Conway's potential function via the Gassner representationhttps://zbmath.org/1491.570022022-09-13T20:28:31.338867Z"Conway, Anthony"https://zbmath.org/authors/?q=ai:conway.anthony"Estier, Solenn"https://zbmath.org/authors/?q=ai:estier.solennSummary: We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the Alexander-Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which have appeared in the literature.Higher arity self-distributive operations in cascades and their cohomologyhttps://zbmath.org/1491.570032022-09-13T20:28:31.338867Z"Elhamdadi, Mohamed"https://zbmath.org/authors/?q=ai:elhamdadi.mohamed"Saito, Masahico"https://zbmath.org/authors/?q=ai:saito.masahico"Zappala, Emanuele"https://zbmath.org/authors/?q=ai:zappala.emanueleEven though distributive structures have been studied for a long time, self distributivity of binary operations and their cohomology theories have been studied extensively in the recent years in the context of knot theory. In this paper, the authors generalize the notion of mutual distributive operations that were introduced in [\textit{J. H. Przytycki}, Demonstr. Math. 44, No. 4, 823--869 (2011; Zbl 1286.55004)] to \(n\)-ary operations. This provides an algebraic flavour to the definitions of \(2\)-cocycle framed link invariants. Moreover, the authors define the notion of \(n\)-ary self distributivity in a symmetric monoidal category which provides a higher arity version of work done by \textit{J. S. Carter} et al. [J. Homotopy Relat. Struct. 3, No. 1, 13--63 (2008; Zbl 1193.18009)].
The authors call this process \textit {internalization of higher order self-distributivity} and produce interesting examples of self-distributive objects among coalgebras. Many examples are provided throughout the paper including revisits of those examples again as they move along to show the connection in different algebras. They provide detailed computations related to Lie algebras and Hopf algebras at the end in the appendices which may be useful in their own right for interested readers.
Reviewer: Indu R. U. Churchill (Oswego)A note on the four-dimensional clasp number of knotshttps://zbmath.org/1491.570042022-09-13T20:28:31.338867Z"Feller, Peter"https://zbmath.org/authors/?q=ai:feller.peter"Park, Junghwan"https://zbmath.org/authors/?q=ai:park.junghwanFor knots that are the connected sum of two torus knots with cobordism distance 1 this paper characterizes those that have 4-dimensional clasp number at least 2 and shows that their \(n\)-fold connected self-sum has 4-dimensional clasp number at least \(2n\). It also builds a family of topologically slice knots for which the \(n\)-fold connected self-sum has 4-ball genus \(n\) and 4-dimensional clasp number at least \(2n\).
Reviewer: Kenneth A. Perko Jr. (Scarsdale)Slope of orderable Dehn filling of two-bridge knotshttps://zbmath.org/1491.570052022-09-13T20:28:31.338867Z"Gao, Xinghua"https://zbmath.org/authors/?q=ai:gao.xinghua|gao.xinghua.1|gao.xinghua.2For genus one, double twist knots, the interval consisting of slopes which yield closed \(3\)-manifolds with left-orderable fundamental groups is known from [\textit{R. Hakamata} and \textit{M. Teragaito}, Algebr. Geom. Topol. 14, No. 4, 2125--2148 (2014; Zbl 1311.57011)] and [\textit{A. T. Tran}, J. Math. Soc. Japan 67, No. 1, 319--338 (2015; Zbl 1419.57028)].
The purpose of the present paper is to give such an interval for the class of double twist knots with higher genus. (Parts of this result have also been proved by A. Tran.)
The main argument goes as follows. Examine real roots of the Riley polynomial of the knot, and construct a non-trivial representation of the knot group into the universal covering group of \(\mathrm{SL}_2(\mathbb{R})\), which is known to be left-orderable. Consider the translation extension locus introduced by \textit{M. Culler} and \textit{N. Dunfield} [Geom. Topol. 22, No. 3, 1405--1457 (2018; Zbl 1392.57012)] and the holonomy extension locus introduced by the author of the present paper [``Orderability of homology spheres obtained by Dehn filling'', Preprint, \url{arXiv:1810.11202}]. They are locally finite unions of analytic arcs and isolated points in \(H^1(\partial E(K);\mathbb{R})\cong \mathbb{R}^2\), where \(E(K)\) is the knot exterior. For a slope \(r\), if the line \(L_r\) through the origin with slope \(-r\) intersects the extension locus at a nonzero point which is neither parabolic nor ideal, then \(r\)-surgery yields a \(3\)-manifold whose fundamental group is left-orderable. A similar result holds for the holonomy extension locus.
The last part of the paper contains various examples of the translation extension locus and holonomy extension locus of some double twist knots and other \(2\)-bridge knots, made by a computer program. Based on these, a conjecture for general \(2\)-bridge knots is proposed.
Reviewer: Masakazu Teragaito (Hiroshima)Distance and the Goeritz groups of bridge decompositionshttps://zbmath.org/1491.570062022-09-13T20:28:31.338867Z"Iguchi, Daiki"https://zbmath.org/authors/?q=ai:iguchi.daiki"Koda, Yuya"https://zbmath.org/authors/?q=ai:koda.yuyaA bridge decomposition of a link \(L\) in a closed orientable 3-manifold \(M\) is a Heegaard splitting of \(M\) such that \(L\) intersects each handlebody in properly embedded trivial arcs. In this article, the authors discuss the Goeritz Group (Mapping class group) of a bridge decomposition. They prove that if the distance of a bridge decomposition of a link with respect to a Heegaard splitting of a 3-manifold is at least 6, then the Goeritz group is a finite group, which improves a result from [\textit{S. Hirose} et al., Int. Math. Res. Not. 2022, No. 12, 9308--9356 (2022; Zbl 07542591)]. The key tool for the proof is the double sweep-out technique involving the theory of graphics introduced by \textit{H. Rubinstein} and \textit{M. Scharlemann} [Topology 35, No. 4, 1005--1026 (1996; Zbl 0858.57020)].
Reviewer: Qiang E (Dalian)Tunnel-number-one knot exteriors in \(S^3\) disjoint from proper power curveshttps://zbmath.org/1491.570072022-09-13T20:28:31.338867Z"Kang, Sungmo"https://zbmath.org/authors/?q=ai:kang.sungmoConsider a genus \(2\) Heegaard splitting of \(\mathbb{S}^3\), and a simple closed curve lying on the surface. The curve is called primitive/primitive if it is primitive in each of the handlebodies (that is, adding a \(2\)-handle to each handlebody along the curve yields a solid torus). In this case the curve is a Berge knot, and has a lens space surgery. Such knots have tunnel number one, and it is known that these are the only tunnel-number-one knots in \(\mathbb{S}^3\) that admit lens space surgeries.
A generalisation of such knots are those that are primitive/Seifert. That is, rather than the Heegaard splitting and the knot giving rise to two solid tori, they instead give rise to one solid torus and one Seifert-fibred space. The knot in \(\mathbb{S}^3\) therefore has a surgery that produces a Seifert-fibred space. The knots of this form again have tunnel number one, but it is not known whether all tunnel-number-one knots with Seifert-fibred surgeries are of this form.
The author states that this paper contributes to a program of Berge to classify all hyperbolic primitive/Seifert knots. Specifically, it studies pairs of disjoint, non-separating simple closed curves \(R\) and \(\beta\) on the boundary of a genus \(2\) handlebody. The curve \(R\) is required to be such that adding a \(2\)-handle along it gives a manifold that embeds into \(\mathbb{S}^3\) (and so is the exterior of a tunnel-number-one knot \(k\)). Meanwhile, the curve \(\beta\) is required to represent a proper power of a non-trivial element of the fundamental group of the handlebody. Along with more technical results, the paper shows that, given \(R\), there exists such a curve \(\beta\) if and only if \(k\) is the unknot, a torus knot, or a tunnel-number-one cable of a torus knot.
The paper uses three types of diagrams to record curves on a genus \(2\) surface: Heegaard diagrams, R-R diagrams, and hybrid diagrams (which combine features of the previous two types). Possible meridian curves are identified by locating `waves' in an R-R diagram, and the `culling lemma' is used to reduce the number of candidate meridians.
Reviewer: Jessica Banks (Liverpool)Extending fibrations of knot complements to ribbon disk complementshttps://zbmath.org/1491.570082022-09-13T20:28:31.338867Z"Miller, Maggie"https://zbmath.org/authors/?q=ai:miller.maggieThe famous slice-ribbon conjecture of \textit{R. H. Fox} [in: Topology of 3-manifolds and related topics. Proceedings of the University of Georgia Institute 1961. Englewood Cliffs, N.J.: Prentice-Hall, Inc.. 168--176 (1962; Zbl 1246.57011)] asks whether every knot that is slice (i.e. bounds a smoothly embedded disc in the 4-ball) is also ribbon (bounds a ribbon-immersed disc in the 3-sphere or, equivalently, bounds a slice disc without local maxima with respect to the radial Morse function on the 4-ball). An intermediate notion is that of being homotopy-ribbon, or bounding a slice disc such that the map induced by the inclusion from the fundamental group of the knot exterior to the fundamental group of the disc exterior is surjective. Despite a great deal of work on the matter, it remains open whether either of the inclusions \(\{\)ribbon\(\} \subseteq \{\)homotopy-ribbon\(\} \subseteq\{\)slice\(\}\) are proper. Recall that a knot \(K\) in \(S^3\) is fibered if \(S^3 \smallsetminus \nu(K)\) is homeomorphic to \(\Sigma \times_{\phi} S^1\) for \(\Sigma\) a Seifert surface for \(K\) and \(\phi\) a self-homeomorphism of \(\Sigma\) fixing \(\partial \Sigma\) pointwise. \textit{A. J. Casson} and \textit{C. McA. Gordon} [Invent. Math. 74, 119--137 (1983; Zbl 0538.57003)] gave a homotopy ribbon obstruction for fibered knots, proving that a fibered knot \(K\) is homotopy-ribbon if and only if its closed monodromy extends over some handlebody if and only if there exists a homotopy 4-ball \(V\) and a slice disc \(D \subset V\) such \((S^3, K)= \partial (V, D)\) and such that \(V \smallsetminus \nu(D)\) is itself fibered over \(S^1\) by handlebodies.
This paper addresses the question of whether ribbon discs with fibered knot boundaries necessarily have fibered exteriors. The main result, Theorem 1.7, is as follows: Let \(K\) be a fibered knot bounding a ribbon disc \(D \subset B^4\), and view the index one critical points of \(D\) as fission bands attached to \(K\). If these bands can be isotoped to be transverse to the fibration on \(S^3 \smallsetminus \nu(K)\), then this fibration extends to a fibration by handlebodies on \(B^4 \smallsetminus \nu(D)\). As a consequence, the author obtains the result (Corollary 1.13) that any ribbon disc with exactly two minima and boundary a fibered knot has its exterior fibered by handlebodies. Section 6 of the paper also contains explicit fibered ribbon discs for many small fibered ribbon knots. The proof of the main result is constructive, and relies on using smoothly varying singular fibrations on the point pre-images of a Morse function on a 4-manifold to build a smooth fibration on that 4-manifold. This requires a careful analysis of different singularity types and painstakingly explicit construction of no fewer than twenty local movies representing birth/death of singularities, handle additions, and more. In addition to the impressive mathematical content, this paper is a graphical tour de force -- this reviewer particularly enjoyed contemplating Figures 29, 30, and 36.
Reviewer: Allison N. Miller (Swarthmore)Knots, quandles and homologyhttps://zbmath.org/1491.570092022-09-13T20:28:31.338867Z"Vendramin, Leandro"https://zbmath.org/authors/?q=ai:vendramin.leandroSummary (translated from Spanish): We present an informal introduction to the combinatorial theory of knots. We discuss the fundamental group of a knot, coloring invariants and invariants given by quandles and quandle homology.Concordances to prime hyperbolic virtual knotshttps://zbmath.org/1491.570102022-09-13T20:28:31.338867Z"Chrisman, Micah"https://zbmath.org/authors/?q=ai:chrisman.micah-whitneyEvery concordance class of classical knots contains a prime satellite knot and a hyperbolic knot (necessarily prime). The paper under review proves that the same is true in the case of knots in thickened surfaces. Specifically, Theorems A and B state that every virtual knot is concordant to a prime satellite virtual knot, and a prime hyperbolic virtual knot, respectively. (Note that hyperbolic virtual knots are not necessarily prime.) Theorem C presents stronger versions of these results for almost classical virtual knots (those that possess a homologically trivial representative).
Reviewer: William Rushworth (Kingston Upon Hull)Parity, virtual closure and minimality of knotoidshttps://zbmath.org/1491.570112022-09-13T20:28:31.338867Z"Gügümcü, N."https://zbmath.org/authors/?q=ai:gugumcu.neslihan"Kauffman, L. H."https://zbmath.org/authors/?q=ai:kauffman.louis-hirschIn this paper, the authors explore the topic of virtual knotoids. A virtual knotoid is an extension of the concept of a knot diagram in two ways: first, through the introduction of virtual crossings, and second, through allowing the knot diagram to have ``endpoints'' on its arcs. In the version studied by Gügügmcü and Kauffman, the endpoints are not allowed to move over or under any arcs. These knotoids were introduced by \textit{V. Turaev} in [Osaka J. Math. 49, No. 1, 195--223 (2012; Zbl 1271.57030)], although previously, a similar idea was used by Satoh to represent ribbon knottings of spheres in four dimensions [\textit{S. Satoh}, J. Knot Theory Ramifications 9, No. 4, 531--542 (2000; Zbl 0997.57037)]. It should be noted, however, that Satoh allowed endpoints to slide under other arcs (but not over them).
The paper uses the concept of the \textit{odd writhe} of a knotoid diagram. Form a Gauss code for the knotoid, which consists of a sequence of labels showing the order in which we encounter each crossing, and whether we do so by crossing over or under it, as we traverse from one end of the knotoid to the other. A crossing is called odd, and assigned a parity of \(1\), if there are an odd number of labels in between the two appearances of the crossing, and otherwise it is called even and assigned a parity of \(0\). The odd writhe of the knotoid is the sum of the signs of the odd crossings. That this is an invariant is checked using Reidemeister moves. In a previous paper [Eur. J. Comb. 65, 186--229 (2017; Zbl 1373.57012)], the same authors have defined a polynomial invariant for knotoids called the odd writhe polynomial, which is defined using a state sum similar to the Kauffman bracket polynomial construction of the Jones polynomial, based on work of \textit{V. O. Manturov} [Sb. Math. 201, No. 5, 693--733 (2010; Zbl 1210.57010)].
Every virtual knotoid gives rise to a virtual knot by connecting the two endpoints of the knotoid using an arc that has only virtual crossings. This is known as the \textit{virtual closure} of the knotoid. For a classical knotoid (a knotoid on \(S^2\) without virtual crossings), the resulting virtual knot will have a genus of either \(0\) or \(1\). A knotoid whose virtual closure is classical is said to be of \textit{knot-type}. The height of a (classical) knotoid diagram is the minimum number of virtual crossings created when forming the diagram of the virtual closure; clearly, being of knot-type is equivalent to having a diagram with height \(0\).
The main result of this paper is to prove, using the invariants the authors have constructed, a conjecture of Turaev [loc. cit.], namely:
Theorem. If \(K\) is a classical knotoid whose virtual closure is classical, then its minimal diagram has height \(0\).
This of course makes it much easier to check whether a given knotoid is of knot-type.
Reviewer: Blake Winter (Buffalo)A spinning construction for virtual \(1\)-knots and \(2\)-knots, and the fiberwise and welded equivalence of virtual \(1\)-knotshttps://zbmath.org/1491.570122022-09-13T20:28:31.338867Z"Kauffman, Louis H."https://zbmath.org/authors/?q=ai:kauffman.louis-hirsch"Ogasa, Eiji"https://zbmath.org/authors/?q=ai:ogasa.eiji"Schneider, Jonathan"https://zbmath.org/authors/?q=ai:schneider.jonathanVirtual knot theory, which was introduced by L. Kauffman in the 1990s, is a generalized knot theory which concerns the embeddings of knots in thickened surfaces up to stabilizations and destabilizations. A connection between virtual knots and 4-dimensional topology was found by \textit{S. Satoh} in [J. Knot Theory Ramifications 9, No. 4, 531--542 (2000; Zbl 0997.57037)], who constructed a ribbon torus knot in \(\mathbb{R}^4\) for a given virtual knot such that the fundamental group of the complement of this torus is isomorphic to that of the virtual knot. A similar interpretation for welded knots was later given by \textit{C. Rourke} [Ser. Knots Everything 40, 263--270 (2007; Zbl 1202.57011)].
For a given virtual knot diagram \(\alpha\) on \(\mathbb{R}^2_b\), there is a submanifold \(\mathcal{S}(\alpha)\subset\mathbb{R}^2_b\times\mathbb{R}^2_f=\mathbb{R}^4\). Denote the projection from \(\mathbb{R}^2_b\times\mathbb{R}^2_f\) to \(\mathbb{R}^2_b\) by \(p\), then for a normal point \(x\) of \(\alpha\), \(p^{-1}(x)\) is a circle. If \(x\) is a classical crossing, the corresponding fiber are two concentric circles, where the small one corresponds to the lower point and the big one corresponds to the upper point. For a virtual crossing, the corresponding fiber are two adjacent circles. Two virtual knot diagrams \(\alpha\) and \(\beta\) are called \emph{fiberwise equivalent} if they are fiberwise isotopic. The main result of this paper is: two virtual knot diagrams are smooth fiberwise equivalent if and only if they are smooth rotational welded equivalent, i.e. they are related by generalized Reidemeister moves and the first forbidden moves, however the virtual version of the first Reidemeister move is forbidden. To this end, the authors introduce the concepts of virtual 2-knots and welded 2-knots.
This well-written long research paper greatly extends virtual knot theory from dimension one to higher dimensions. It is worth pointing out that a detailed discussion on the motivation of studying virtual knots is provided in Appendix A of this paper.
Reviewer: Zhiyun Cheng (Beijing)Circle patterns on surfaces of finite topological typehttps://zbmath.org/1491.570132022-09-13T20:28:31.338867Z"Ge, Huabin"https://zbmath.org/authors/?q=ai:ge.huabin"Hua, Bobo"https://zbmath.org/authors/?q=ai:hua.bobo"Zhou, Ze"https://zbmath.org/authors/?q=ai:zhou.zeLet \(S\) be a compact oriented surface with constant curvature metric \(\mu\), and suppose that \({\mathcal T}\) is a triangulation of \(S\) with geodesic edges. A circle pattern is a collection of circles centered at the vertices of \({\mathcal T}\) such that two circles intersect whenever their centers are the endpoints of an edge. Consider the function \(\Theta:E\rightarrow[0,\pi)\) on the edges \(E\) of \({\mathcal T}\), where \(\Theta(e)\) is the exterior intersection angle between the two circles centered at the endpoints of \(e\). In the case when the Euler characteristic of \(S\) is non-positive, the authors enumerate sufficient conditions on \(\Theta\) for which there exists a hyperbolic (or Euclidean) metric \(\mu\), unique up to isometry (or similarity), for which there exists a circle pattern. This extends a theorem of Thurston to include surfaces with boundary, as well as exterior intersection angles that are not necessarily acute. The results are obtained by examining a Ricci flow type system of differential equations involving the circle radii and the angular defects for each vertex of \({\mathcal T}\).
Reviewer: Jason Hanson (Redmond)Unicity for representations of the Kauffman bracket skein algebrahttps://zbmath.org/1491.570142022-09-13T20:28:31.338867Z"Frohman, Charles"https://zbmath.org/authors/?q=ai:frohman.charles-d"Kania-Bartoszynska, Joanna"https://zbmath.org/authors/?q=ai:kania-bartoszynska.joanna"Lê, Thang"https://zbmath.org/authors/?q=ai:le-tu-quoc-thang.Summary: This paper resolves the unicity conjecture of \textit{F. Bonahon} and \textit{H. Wong} [Quantum Topol. 10, No. 2, 325--398 (2019; Zbl 1447.57017)] for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine \(k\)-algebra over an algebraically closed field \(k\), that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebra is finitely generated as a module over its center. It is shown that for any orientable surface the center of the skein algebra at any root of unity is the coordinate ring of an affine algebraic variety.Lagrangian cobordisms and Legendrian invariants in knot Floer homologyhttps://zbmath.org/1491.570152022-09-13T20:28:31.338867Z"Baldwin, John A."https://zbmath.org/authors/?q=ai:baldwin.john-a"Lidman, Tye"https://zbmath.org/authors/?q=ai:lidman.tye"Wong, C.-M. Michael"https://zbmath.org/authors/?q=ai:wong.c-m-michaelA fundamental question in contact and symplectic geometry is whether given two Legendrian links in \((\mathbb{R}^3,\xi_{std})\) there is an exact Lagrangian cobordism between them. There are many obstructions known towards this end, both in terms of classical invariants [\textit{B. Chantraine}, Algebr. Geom. Topol. 10, No. 1, 63--85 (2010; Zbl 1203.57010)] and more subtle invariants such as the Chekanov-Eliashberg DGA [\textit{T. Ekholm} et al., J. Eur. Math. Soc. (JEMS) 18, No. 11, 2627--2689 (2016; Zbl 1357.57044)] and knot Floer homology.
In this paper, the authors show that the GRID invariants \(\widehat{\lambda}^{\pm}\) of \textit{P. Ozsváth} et al. [Geom. Topol. 12, No. 2, 941--980 (2008; Zbl 1144.57012)] can be used to obstruct the existence of \textit{decomposable} cobordisms, i.e. a special class of exact Lagrangian cobordisms that corresponds to combinatorial moves on front projections. This is very interesting in light of the following open question: is every exact Lagrangian cobordism between non-empty links decomposable? The authors exhibit concrete examples to show that \(\widehat{\lambda}^{\pm}\) are effective (i.e. they give obstructions in examples for which classical invariants do not). They also provide a very useful discussion on the strengths and weaknesses of their invariant in comparison to other ones in the literature. In particular, their invariant is the first one in knot Floer homology that allows to obstruct positive genus cobordisms (with the assumption that they are decomposable).
Reviewer: Francesco Lin (New York)Lagrangian torus invariants using \(ECH=SWF\)https://zbmath.org/1491.570162022-09-13T20:28:31.338867Z"Gerig, Chris"https://zbmath.org/authors/?q=ai:gerig.chrisEmbedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology, cf. [\textit{C. H. Taubes}, Geom. Topol. 14, No. 5, 2497--2581 (2010; Zbl 1275.57037)]. The author of the paper under review describes distinguished elements in ECH of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. These invariants are not new, ``they repackage the Gromov (and Seiberg-Witten) invariants of various torus surgeries''. Finally, the autor ``recovers a result of Morgan-Mrowka-Szabo on product formulas for the Seiberg-Witten invariants along 3-tori [\textit{J. W. Morgan} et al., Math. Res. Lett. 4, No. 6, 915--929 (1997; Zbl 0892.57021)].''
Reviewer: Roman Golovko (Praha)On \(\mathbb{Z}_2\)-Thurston norms and pseudo-horizontal surfaces in orientable Seifert 3-manifoldshttps://zbmath.org/1491.570172022-09-13T20:28:31.338867Z"Du, Xiaoming"https://zbmath.org/authors/?q=ai:du.xiaomingIn this paper, the author gives a method to compute the \(\mathbb{Z}_2\)-Thurston norm for every \(\mathbb{Z}_2\)-homology class in an orientable Seifert manifold \(M\) with orientable base orbifold.
For a possibly disconnected and possibly non-orientable closed surface \(S\), written as the union of its components \(S=S_1\cup \cdots\cup S_n\), the complexity of \(S\) is defined to be \(\sum_{i=1}^n\max{\{0,-\chi(S_i)\}}\). For any \(\mathbb{Z}_2\)-coefficient homology class \(\alpha \in H_2(M;\mathbb{Z}_2)\), its \(\mathbb{Z}_2\)-Thurston norm \(\|\alpha\|_{\mathbb{Z}_2-Th}\) is defined to be the minimum of the complexity of embedded subsurfaces of \(M\) that represent \(\alpha\). An embedded subsurface of \(M\) containing no \(S^2\) or \(\mathbb RP^2\) component that represents \(\alpha\) and realizes \(\|\alpha\|_{\mathbb{Z}_2-Th}\) is called \(\mathbb{Z}_2\)-taut, and a \(\mathbb{Z}_2\)-taut surface must be geometrically incompressible.
A pseudo-vertical (pseudo-horizontal) surface in a Seifert manifold is a generalization of the classical vertical (horizontal) surface, and it can be non-orientable. More precisely, a pseudo-vertical (pseudo-horizontal) surface is vertical (horizontal) in the complement of a neighborhood of singular fibers, and its intersection with a neighborhood of each singular fiber is either a family of meridian disks or a geometrically incompressible non-orientable surface (classified by \textit{G. E. Bredon} and \textit{J. W. Wood} [Invent. Math. 7, 83--110 (1969; Zbl 0175.20504)]). By \textit{C. Frohman}'s work [Topology Appl. 23, 103--116 (1986; Zbl 0606.57007)], up to isotopy, every geometrically incompressible surface in a Seifert manifold is vertical, or horizontal, or pseudo-vertical, or pseudo-horizontal.
Since vertical and horizontal surfaces are classical objects, and there are only finitely many types of pseudo-vertical surfaces, the bulk of this paper is devoted to studying pseudo-horizontal surfaces. Each pseudo-horizontal surface is determined by pairs of integers \((\lambda_1,\mu_1),\dots, (\lambda_n,\mu_n)\) (one pair for each singular fiber) satisfying certain conditions, and the author gives a formula for the genus of this pseudo-horizontal surface (Theorem 3.5).
To compute the \(\mathbb{Z}_2\)-Thurston norm in Seifert manifolds, the author first computes a uniform upper bound for \(\mathbb{Z}_2\)-Thurston norms of all elements in \(H_2(M;\mathbb{Z}_2)\), and uses this uniform upper bound to give an upper bound of all \(|\lambda_i|\) and \(|\mu_i|\). So for each \(\alpha\in H_2(M;\mathbb{Z}_2)\), to compute \(\|\alpha\|_{\mathbb{Z}_2-Th}\), it remains to compute the complexity of finitely many surfaces.
The author also gives examples for which a pseudo-vertical surface is not \(\mathbb{Z}_2\)-taut, a pseudo-horizontal surface is not \(\mathbb{Z}_2\)-taut, and a pseudo-vertical surface is isotopic to a pseudo-horizontal surface and they are both \(\mathbb{Z}_2\)-taut.
Reviewer: Hongbin Sun (Piscataway)Topological 4-manifolds with 4-dimensional fundamental grouphttps://zbmath.org/1491.570182022-09-13T20:28:31.338867Z"Kasprowski, Daniel"https://zbmath.org/authors/?q=ai:kasprowski.daniel"Land, Markus"https://zbmath.org/authors/?q=ai:land.markusLet \(\pi\) be a group that satisfies the Farrell-Jones Conjecture such that \(B{\pi}\) is a 4-dimensional Poincaré complex (i.e. \(\pi\) is a 4-dimensional Poinaré group). Let \(w\) be the orientation character of \(B{\pi}\). The main result of the paper is a type of topological rigidity for the group \(\pi\). Let \(M^4\) and \(N^4\) be topological, closed and connected manifolds with orientation character \(w\) that admit \({\pm}1\)-degree maps to \(B{\pi}\). Then \(M\) and \(N\) are \(s\)-cobordant if and only if the equivariant intersection forms are isometric and they have the same Kirby-Siebenmann invariant. If the manifolds are almost spin, then they are \(s\)-cobordant if and only if the equivariant intersection forms are isometric. Notice that, under the conditions of the theorem, the manifolds are homotopy equivalent. As a Corollary, the authors obtain that if, furthermore, \(\pi\) is a good group in the sense of Freedman, under the same conditions, the manifolds are homeomorphic.
As a first step, the authors prove that if a 4-dimensional Poinaré group satisfies the Farrell-Jones Conjecture, then it satisfies the conditions;
\begin{itemize}
\item[(W)] \(Wh({\pi}) = 0\).
\item[(A1)] The assembly map \(L{\langle}1{\rangle}_4^w(B{\pi}) \to L_4^q(\mathbb{Z}{\pi}, w)\) is injective.
\item[(A2)] The assembly map \(L{\langle}1{\rangle}_5^w(B{\pi}) \to L_5^q(\mathbb{Z}{\pi}, w)\) is surjective.
\end{itemize}
Using these conditions, the authors prove that, under the assumptions of the theorem, if there is a homotopy equivalence between \(M\) and \(N\) with trivial normal invariant, then the manifolds are \(s\)-cobordant. This is the first step towards proving the main theorem.
As a more direct application to the Borel Conjecture, if \(X\) is an aspherical \(4\)-manifold with good Farrell-Jones group and \(L\) is a simply connected \(4\)-manifold and if \(M\) is a \(4\)-manifold with \(w_1(M) = w_1(X)\) which is homotopy equivalent to \(X{\#}L\), then it is homeomorphic to \(X{\#}L\) or to \(X{\#}{\star}L\). Also, if the equivariant intersection form on \(M\) is induced from an integral form \(\lambda\), then \(M \cong X{\#}K\), where \(K\) is a simply connected \(4\)-manifold whose intersection form is \(\lambda\).
Reviewer: Stratos Prassidis (Karlovasi)Connected components of Isom\((\mathbb{H}^3)\)-representations of non-orientable surfaceshttps://zbmath.org/1491.570192022-09-13T20:28:31.338867Z"Durán Batalla, Juan Luis"https://zbmath.org/authors/?q=ai:duran-batalla.juan-luisLet \(M\) be a closed surface and \(G\) a Lie group. Let \(\pi_{1}(M)\) denote the fundamental group of the surface. The set of homomorphisms \(\phi:\pi_{1}(M)\to G\) is called the variety of representations of the fundamental group of \(M\) in \(G\) and denoted hom(\(\pi_{1}(M),G\)). The goal of the paper under review is to find topological invariants which classify the connected components of hom(\(\pi_{1}(M),G\)), for \(M\) non-orientable and \(G = \mathrm{Isom}(\mathbb H^3)\cong\mathrm{PSL}(2,\mathbb C)\rtimes\mathbb Z_2.\) The author proves the following result: Let \(N_k\) denote the closed non-orientable surface of genus \(k.\) The representation variety hom(\(\pi_{1}(N_k),G\)) has \(2^{k+1}\) connected components.
In the proof he uses the so-called `square map' \([A]\in \mathrm{PSL}(2,\mathbb C)\mapsto A^2\in\mathrm{SL}(2,\mathbb C\)). Moreover, it is proved that the above connected components are distinguished by the Stiefel-Whitney classes of the associated flat \(G\)-bundle over \(M.\) In this sense, these cohomological classes are constant on connected components.
Reviewer: Andrzej Szczepański (Gdańsk)Invariants of \(\mathrm{PSL}_n\mathbb{R}\)-Fuchsian representations and a slice of Hitchin componentshttps://zbmath.org/1491.570202022-09-13T20:28:31.338867Z"Inagaki, Yusuke"https://zbmath.org/authors/?q=ai:inagaki.yusukeLet \(S\) be a compact orientable surface of genus \(g > 1\), \(\tilde{S}\) its universal cover and \(\pi\) its fundamental group. The space \(\mathrm{Hom}(\pi, \mathrm{PSL}_2(\mathbb{R}))\) has \(4g-3\) components and one of these, \(R_2(S)\), consists of discrete representations [\textit{W. M. Goldman}, Invent. Math. 93, No. 3, 557--607 (1988; Zbl 0655.57019)]. The group \(\mathrm{PSL}_2(\mathbb{R})\) acts on \(R_2(S)\) by equivalence of representations. One definition of Teichmüller space of \(S\) is the quotient \( \mathcal{T}(S) := R_2(S)/\mathrm{PSL}_2(\mathbb{R}), \) which is a ball of \(\mathbb{R}\)-dimension \(6(g-1)\).
Identify \(\tilde{S}\) with the upper half-plane \(\mathbb{H}\) which is the symmetric space of \(\mathrm{PSL}_2(\mathbb{R})/\mathrm{U}(1)\), having both the natural complex structure and the hyperbolic (Poincaré) metric. \(\mathrm{PSL}_2(\mathbb{R})\) acts on \(\mathbb{H}\), preserving both structures. Any \(\rho \in R_2(S)\) induces a free and faithful \(\pi\)-action on \(\mathbb{H}\). This identifies \(S\) with \(\mathbb{H}/\rho(\pi)\). The \(\mathrm{PSL}_2(\mathbb{R})\)-action extends to \(\partial \tilde{S}\). Identify the ideal boundary via a \(\pi\)-equivariant map \(\xi_\rho : \partial \tilde{S} \longrightarrow \mathbb{P}_1(\mathbb{R})\).
Fix a hyperbolic structure on \(S\), corresponding to a point in \([\rho] \in \mathcal{T}(S)\). A lamination \(\lambda\) on \(S\) is a set of non-intersecting complete geodesics on \(S\) [\textit{W. P. Thurston}, Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy. Princeton, NJ: Princeton University Press (1997; Zbl 0873.57001)]. We assume \(\lambda\) to be finite and maximal. Then \(\lambda\) has \(6(g-1)\) bi-infinite leaves and \(s\) closed geodesics with \(1 \le s \le 3(g-1)\) and \(S \setminus \lambda\) contains \(4(g-1)\) ideal triangles.
Each bi-infinite leaf \(B \in \lambda\) (with vertices \(x, y\)) is the diagonal of an ideal quadrilateral. Choose a lifting of the quadrilateral in \(\tilde{S}\) with the lifted vertices in \(x,z^L,y,z^R \in \partial \tilde{S}\), ordered counter-clock-wise. The shearing parameter of \(B\) is \[ \sigma(B,[\rho]) = \log(-cr(x,y,z^L,z^R)), \] where \(cr\) is the cross-ratio. For each closed curve \(C\) there are two bi-infinite leaves \(B_1\) and \(B_2\) spiraling toward \(C\). Each is a side of an ideal triangle. Lifting \(C\) and the two triangles to \(\tilde{S}\), we obtain the two ends \(x, y \in \partial \tilde{S}\) of \(\tilde{C}\) and the two vertices \(z^L, z^R\) opposite of \(\tilde{B}_1\) and \(\tilde{B}_2\), respectively. Again, assume that \(x,z^L,y,z^R\) are ordered counter-clock-wise. The twist parameter of \(C\) is \[ \theta(C,[\rho]) = \log(-cr(x,y,z^L,z^R)). \] Since \(cr\) is \(\mathrm{PSL}_2(\mathbb{R})\)-invariant, \(\sigma\) and \(\theta\) are independent of the lifting; hence, both are invariants of \([\rho]\). Together the shearing and the twist parameters form a Thurston coordinate system of \(\mathcal{T}(S)\) [loc. cit.].
For \(n \ge 2\), the space \(\mathrm{Hom}(\pi_1, \mathrm{PSL}_n(\mathbb{R}))\) also has a component \(R_n(S)\) and \( H_n(S) = R_n(S)/\mathrm{PSL}_n(\mathbb{R}) \) is a ball of \(\mathbb{R}\)-dimension \(d = 2(n^2-1)(g-1)\) [\textit{N. J. Hitchin}, Topology 31, No. 3, 449--473 (1992; Zbl 0769.32008)]. There is a unique \(n\)-dimensional irreducible representation of \(\mathrm{PSL}_2(\mathbb{R})\) giving rise to an embedding \[\iota_n : \mathrm{PSL}_2(\mathbb{R}) \longrightarrow \mathrm{PSL}_n(\mathbb{R}), \qquad \iota_n : \mathcal{T}(S) \longrightarrow H_n(S).\] The component \(H_n(S)\) was discovered by algebraic-geometric methods. The effort is to understand the representations in \(H_n(S)\) in terms of hyperbolic geometry.
Bonahon-Dreyer extended the Thurston coordinates to \(H_n(S)\) [\textit{F. Bonahon}, Ann. Fac. Sci. Toulouse, Math. (6) 5, No. 2, 233--297 (1996; Zbl 0880.57005); \textit{F. Bonahon} and \textit{G. Dreyer}, Duke Math. J. 163, No. 15, 2935--2975 (2014; Zbl 1326.32023); Acta Math. 218, No. 2, 201--295 (2017; Zbl 1388.32006)]. Let \(\mathrm{B} \subset \mathrm{PSL}_n(\mathbb{R})\) be a Borel subgroup. Then the (complete) flag manifold of \(\mathbb{R}^n\) is \(\mathrm{PSL}_n(\mathbb{R})/\mathrm{B}\). Labourie showed that each \([\rho] \in H_n(S)\) induces a map [\textit{F. Labourie}, Invent. Math. 165, No. 1, 51--114 (2006; Zbl 1103.32007)] \[\xi_\rho : \partial \tilde{S} \longrightarrow \mathrm{PSL}_n(\mathbb{R})/\mathrm{B}.\] Let \(\{F_i\}_{i = 0}^n\) be a complete flag of \(\mathbb{R}^n\) and let \(f^d\) be a generator of \(\wedge^d F_d\). For flags \(E, F, G, G'\) and \(b + c + 1 = n\), the \(b\)-th double ratio is \[ D_b(E, F, G, G') = -\frac{e^b \wedge f^c \wedge g^1 \cdot e^{b-1} \wedge f^{c+1} \wedge g'^1}{e^b \wedge f^c \wedge g'^1 \cdot e^{b-1} \wedge f^{c+1} \wedge g^1} \] For flags \(E, F, G\) and \(p+q+r=1\), the triple ratio is \[ T_{pqr}(E, F, G) = \frac{e^{p+1} \wedge f^q \wedge g^{r-1} \cdot e^p \wedge f^{q-1} \wedge g^{r+1} \cdot e^{p-1} \wedge f^{q+1} \wedge g^r}{e^{p-1} \wedge f^q \wedge g^{r+1} \cdot e^p \wedge f^{q+1} \wedge g^{r-1} \cdot e^{p+1} \wedge f^{q-1} \wedge g^r}. \]
The quadrilateral containing a bi-infinite geodesic \(B\) gives rise to four flags \(\xi_\rho(x), \xi_\rho(y), \xi_\rho(z^L), \xi_\rho(z^R) \in \mathrm{PSL}_n(\mathbb{R})/\mathrm{B}\). The generalized shearing invariant is \[\sigma_b(B, [\rho]) = \log D_b(\xi_\rho(x), \xi_\rho(y), \xi_\rho(z^L), \xi_\rho(z^R)), \qquad 0 < b < n.\] Similarly, the twist invariant for a closed leaf \(C \in \lambda\) is \[\sigma_c(C, [\rho]) = \log D_c(\xi_\rho(x), \xi_\rho(y), \xi_\rho(z^L), \xi_\rho(z^R)), \qquad 0 < c < n.\] Finally, each ideal triangle \(t\) in \(\lambda\) with bi-infinite leaves lifts to an ideal triangle in \(\tilde{S}\) with vertices \(v, v', v'' \in \partial \tilde{S}\). For each \(p+q+r = n\), the triangle invariant is \[\tau_{pqr}(t, [\rho]) = \log T_{pqr}(\xi_\rho(v), \xi_\rho(v'), \xi_\rho(v'')).\] Again, the generalized shearing, twist and the triangle invariants are independent of the lifting (notice the symmetry in the expressions of \(D_b\) and \(T_{pqr}\)).
Suppose \([\rho] \in H_n(S)\) and \(\rho = \iota_n \circ \eta\) with \(\eta \in R_2(S)\). The main results of the paper are
\begin{enumerate}
\item \(\tau_{pqr}(t_i, [\rho]) = 0\) for all \(p, q, r\).
\item \(\sigma_b(B_j, [\rho]) = \sigma(B_i, [\eta])\) for all \(b\).
\item \(\theta_c(C_k, [\rho])\) is independent of \(c\),
\end{enumerate}
where \(i, j, k\) range over all triangles of bi-infinite sides, bi-infinite leaves and closed leaves in \(\lambda\), respectively. Notice that if \(n=2\), then \(\iota_2\) is identity and \(b=c=1\). Moreover, \[\xi_\rho : \partial \tilde{S} \longrightarrow \mathrm{PSL}_2(\mathbb{R})/\mathrm{B} \cong \mathbb{P}_1(\mathbb{R})\] and we are back in the situation of Thurston coordinates. In other words, the Bonahon-Dreyer coordinates are natural extensions of the Thurston coordinates.
The first four chapters of the paper give a nice introduction on geodesic laminations, shearing and twist invariants and the triangle invariants, the Hitchin component and the Bonahon-Dreyer generalization of the Thurston coordinates. Chapter 5 and 6 prove the results above. The proof is technical and computational. Notice that the number of invariants is \(d + s(n-1)\), greater than the dimension of \(H_n(S)\). This means that there are relations among the invariants. The last Chapter 7 is a brief discussion of the case of surfaces with geodesic boundary.
Reviewer: Eugene Xia (Tainan)Rational Pontryagin classes of Euclidean fiber bundleshttps://zbmath.org/1491.570212022-09-13T20:28:31.338867Z"Weiss, Michael S."https://zbmath.org/authors/?q=ai:weiss.michael-sLet \(\text{TOP}(n)\) denote the (topological or simplicial) group of homeomorphisms of \(\mathbb R^n\). Classical yet deep results by Sullivan and Kirby-Siebenmann imply that the canonical map \(BO \to B\text{TOP} = \text{colim}_{n \to \infty} B \text{TOP}(n)\) is a rational equivalence, hence the Pontryagin classes \(p_i \in \text{H}^{\ast}(B\text{TOP}; \mathbb Q)\) are well-defined. The main result of this paper states that for sufficiently big \(n\) and all \(k\) in some non-trivial range that grows linearly with \(n\), \(p_{n+k} \in \text{H}^{4(n+k)}(B\text{TOP}; \mathbb Q)\) remains non-zero when pulled back to \(\text{H}^{4(n+k)}(B\text{TOP}(2n); \mathbb Q)\). This is in stark contrast to the case of classifying spaces of linear \(\mathbb R^n\)-bundles where it is known that \(0 = p_{n+k} \in \text{H}^{4(n+k)}(BO(2n); \mathbb Q)\) for all \(k > 0\).
The high-level strategy to prove this result is to combine work by \textit{S. Galatius} and \textit{O. Randal-Williams} on the cohomology of diffeomorphism groups (usually referenced by the keyword \emph{moduli spaces of manifolds}, cf. [Acta Math. 212, No. 2, 257--377 (2014; Zbl 1377.55012)]), with manifold calculus invented by \textit{T. G. Goodwillie} and \textit{M. Weiss} (also known as embedding calculus, cf. [Geom. Topol. 3, 103--118 (1999; Zbl 0927.57028)]), to construct a certain fiber bundle. This bundle \(E \to M\) is such that \(M\) is a \((2n+4k)\)-dimensional smooth closed stably parallelizable manifold, the fibers are homeomorphic to \(W_g = \#^g S^n \times S^n\), and the total space \(E\) has vanishing decomposable Pontryagin classes, but non-zero signature. Deducing the main result from the existence of such a bundle is uncomplicated. In the final chapter, it is proven that the `surreal' Pontryagin numbers appearing in the main theorem evaluate non-trivially against the image of the Hurewicz.
On the more technical side, the paper starts with a computational, homotopically flavored analysis of certain \(\kappa\)-classes whose non-vanishing for \(W_g\) follows from the work by Galatius and Randal-Williams alluded to above. Then manifold/embedding calculus is used to study configuration categories (cf. [\textit{P. B. De Brito} and \textit{M. Weiss}, J. Topol. 11, No. 1, 65--143 (2018; Zbl 1390.57018)]) which in turn are helpful to understand spaces of self-embeddings. The essential geometric property that the manifolds \(W_{g,1} = W_g \backslash \mathring{D^{2n}}\) have and that is needed for this part of the paper is that their homotopical dimension is \(n\), whereas their geometric dimension is \(2n\). Thus for \(2n \geq 6\) the difference of the two is at least \(3\) and hence embedding calculus is applicable. By combining both approaches using rational homotopy theory, the existence of the aforementioned fiber bundle becomes evident.
As a side-product, a fiber sequence \[B\text{Diff}_{\partial}(D^{2n}) \to B\text{Diff}_{\partial}(W_{g,1}) \to B\text{Diff}_{\partial}(W_{g,1+\varepsilon})\] is produced, where \(W_{g,1+\varepsilon}\) is obtained from \(W_{g,1}\) by removing a point in the boundary. The third term in the sequence is equivalent to the space of self-embeddings of \(W_{g,1}\) fixing half of the boundary; so this space is amenable to the methods of manifold calculus. The aforementioned fiber sequence was used with great success by Kupers (cf. [\textit{A. Kupers}, Geom. Topol. 23, No. 5, 2277--2333 (2019; Zbl 1437.57035)]) and others to study \(B\text{Diff}_{\partial}(D^{2n})\) and is now known by the name \emph{Weiss fiber sequence} in the literature.
Hirzebruch's signature theorem is used multiple times; at one point, the non-vanishing of certain particular coefficients of Hirzebruch's \(\mathcal L\)-polynomials is needed. This was shown directly in a preliminary version of the paper, but in the meantime a much more general result was proven by \textit{A. Berglund} and \textit{J. Bergström} [Math. Ann. 372, No. 1--2, 125--137 (2018; Zbl 1406.55008)].
Without doubt, this paper is one of the biggest breakthroughs in homotopical geometric topology in recent years. The main result is rather surprising -- the author states he tried to prove the opposite statement for a long time -- and probably even more important are the new methods he developed to prove it. There are several preliminary versions available on the arXiv [\url{https://arxiv.org/abs/1507.00153}] also worth looking at. In the peculiar section \emph{Acknowledgements and Apologies} at the end of the introduction, it is explained how the paper grew out of some notes that were written to clarify a story told by the author at a satellite conference of the ICM 2014 in Dalian (China).
Reviewer: Jens Reinhold (Münster)Fundamental class, Poincaré duality and finite oriented \(\mathrm{FC}_4\)-complexeshttps://zbmath.org/1491.570222022-09-13T20:28:31.338867Z"Cavicchioli, Alberto"https://zbmath.org/authors/?q=ai:cavicchioli.alberto"Hegenbarth, Friedrich"https://zbmath.org/authors/?q=ai:hegenbarth.friedrich"Spaggiari, Fulvia"https://zbmath.org/authors/?q=ai:spaggiari.fulviaAn FC\(_{4}\) complex is obtained by taking a finite \(3\)-complex \(K\) and considering all \(4\)-complexes \(X \ = K \cup_{\phi} D^{4}\) such that \(H_{4}(X, {\mathbb Z}) \ \cong \ {\mathbb Z}\) with a fixed generator \([X] \in H_{4}(X, {\mathbb Z})\). The paper is concerned with two questions. First for which \(3\)-complexes \(K\) does an element \([\phi] \in \pi_{3}(K)\) exist for which the complex \(K \cup_{\phi} D^{4}\) is a Poincaré complex? Second if there is one \(\phi\) how many others can be constructed from \(K\)? A necessary and sufficient condition for the existence of such a \(\phi\) is given and a result that shows how under certain conditions additional \(FC_{4}\)-complexes can be constructed.
Reviewer: Jonathan Hodgson (Swarthmore)Existence of pseudoheavy fibers of moment mapshttps://zbmath.org/1491.570232022-09-13T20:28:31.338867Z"Kawasaki, Morimichi"https://zbmath.org/authors/?q=ai:kawasaki.morimichi"Orita, Ryuma"https://zbmath.org/authors/?q=ai:orita.ryumaA partial symplectic quasi-state on a closed symplectic manifold \(M\) is a functional on the space of continuous functions on \(M\) which satisfies certain axioms. Among these axioms, the functional should take a constant function \(c\) to the corresponding value \(c\), it should vanish on functions supported on a set which is displaceable from itself by a Hamiltonian isotopy, it should be invariant by precomposition of functions by a Hamiltonian diffeomorphism (this list is not exhaustive). Once a partial symplectic quasi-state \(\zeta\) is given on \(M\) one can define \(\zeta\)-heavy and \(\zeta\)-superheavy subsets of \(M\). All these definitions are due to \textit{M. Entov} and \textit{L. Polterovich} [Comment. Math. Helv. 81, No. 1, 75--99 (2006; Zbl 1096.53052); Compos. Math. 145, No. 3, 773--826 (2009; Zbl 1230.53080)]. The main interest in these definitions is that \(\zeta\)-heavy subsets are not displaceable (i.e. cannot be made disjoint from themselves by a Hamiltonian isotopy). This allows to build examples of singular non-displaceable subsets of closed symplectic manifolds (historically, the typical example of a non-displaceable subset was a Lagrangian submanifold).
In this article the authors prove that given a partial symplectic quasi-state \(\zeta\) on a closed symplectic manifold \(M\), and given a finite dimensional space \(A\) of smooth Poisson-commuting functions of \(M\), there exists a fiber of the corresponding moment map which is \(\zeta\)-pseudoheavy. The notion of being \(\zeta\)-pseudoheavy is introduced in the article (and is a weakening of the notion of being \(\zeta\)-heavy). This is the main theorem, which answers a question raised by Entov and Polterovich [op. cit.] (with the notion of pseudoheavy set instead of heavy set). The article also contains other results, among them are some rigidity results for certain singular Lagrangian submanifolds of \(\mathbf{S}^{2}\times\mathbf{S}^{2}\).
Reviewer: Pierre Py (Strasbourg)Symplectic fillings of asymptotically dynamically convex manifolds Ihttps://zbmath.org/1491.570242022-09-13T20:28:31.338867Z"Zhou, Zhengyi"https://zbmath.org/authors/?q=ai:zhou.zhengyiThis detailed paper presents numerous results on symplectic fillings of contact manifolds.
Given a contact manifold, it is natural to try to find and classify its symplectic fillings. A great deal of previous work, by Eliashberg, Gromov, McDuff, Floer and many others, has considered such questions. The contact and symplectic manifolds involved may come in various flavours, and this paper considers several different possibilities.
In the present paper, the author's main focus is on \emph{asymptotically dynamically convex} (ADC) contact manifolds. The ADC notion was introduced by \textit{O. Lazarev} [Geom. Funct. Anal. 30, No. 1, 188--254 (2020; Zbl 1436.53055)], generalising a notion of dynamical convexity given by positivity of index of Reeb orbits. A contact structure on \(Y^{2n-1}\) is \emph{index-positive} if it has a nondegenerate contact form such that every Reeb orbit has positive degree, i.e. Cohnley-Zehnder index greater than \(3-n\). The index-positivity condition, under certain circumstances, avoids holomorphic curves extending into a symplectic filling. The ADC condition on a contact \(2n-1\) manifold \((Y, \xi)\) is a more subtle condition, requiring a decreasing sequence of contact forms \(\alpha_1 > \alpha_2 > \cdots\), and an increasing sequence of real numbers \(D_1 < D_2 < \cdots\) tending to infinity, such that all contractible Reeb orbits of each \(\alpha_i\) of action \(<D_i\) are nondegenerate and have degree bounded below. A \emph{strong} version of the ADC condition additionally requires all Reeb orbits of \(\alpha_i\) of action \(<D_i\) to be contractible.
ADC manifolds are common. Lazarev showed that subcritical and flexible surgeries preserve the ADC property, and that all \(\pi_1\)-injective Liouville fillings of an ADC manifold have isomorphic positive symplectic cohomology. In the present paper, the author expands upon these and other results. Essentially, the present paper shows that it is not just symplectic cohomology which is independent of the choice of filling. Structure maps are constructed and shown to be invariant. These maps are used in various applications to derive (and re-derive) numerous results for symplectic fillings. Thirteen theorems are selected by the author as the main Theorems A--M, grouped into five sets.
Given a contact \(Y^{2n-1}\) and an filling \(W^{2n}\), a first set of results (Theorems A--F) considers to what extent the filling and related data are unique. These results derive from considering a structure map \(\delta_\partial \colon SH_+^* (W) \rightarrow H^{*+1} (Y)\). This map from positive symplectic cohomology of \(W\) to singular cohomology of \(Y\) is the composition of a connecting map in symplectic homology \(SH_+^* (W) \rightarrow H^{*+1}(W)\) and the restriction map \(H^{*+1}(W) \rightarrow H^{*+1}(Y)\). These results consider fillings which are \emph{topologically simple} in the sense that \(c_1 (W) = 0\) and the inclusion \(Y \hookrightarrow W\) is \(\pi_1\)-injective (a strong version of the ADC condition makes \(\pi_1\)-injectivity redundant).
The paper's first main theorem (Theorem A) is that when \(Y\) is strongly ADC and the filling is exact and topologically simple, the structure map \(\delta_\partial \colon SH_+^* (W) \rightarrow H^{*+1}(Y)\) is independent of the choice of filling \(W\). As a corollary (Corollary B), in the case where \(SH^*(W) = 0\), the restriction map \(H^*(W) \rightarrow H^*(Y)\) is shown to be independent of the choice of such filling \(W\).
The author also proves numerous more specific and detailed results along these lines. Exact fillings \(W\) with vanishing \(c_1\) of certain contact manifolds \(Y\) arising from Liouville domains and flexible cotangent bundles are shown to be unique up to diffeomorphism (Theorem C). Detailed invariance results are obtained for fillings of unit cotangent bundles \(ST^* Q\) when the Hurewicz map \(\pi_2(Q) \rightarrow H_2 (Q)\) is nonzero (Theorem D). And when a topologically simple exact filling \(W\) of an ADC \(Y\) with \(SH^*(W)=0\) has the property that \(\pi_1(Y) \rightarrow \pi_1 (W)\) is an isomorphism, it is shown that \(\pi_1(Y) \rightarrow \pi_1 (W')\) is an isomorphism for any other topologically simple exact filling \(W'\) (Theorem E).
In a slightly different direction, Theorem F shows that if \(Y\) satisfies a \emph{tamed} version of the ADC condition (weaker than index-positive but stronger than ADC), and has a topologically simple exact filling \(W\) with \(SH^*(W;\mathbb{Q})=0\) such that the restriction map \(H^2 (W; \mathbb{Q}) \rightarrow H^2 (Y; \mathbb{Q})\) is injective and \(H^1 (W; \mathbb{Q}) \rightarrow H^1 (Y; \mathbb{Q})\) is surjective, then any topologically simple strong filling of \(Y\) is exact.
A second set of results (Theorems G--H) derives from considering ideas related to the \emph{symplectic dilation} introduced by \textit{P. Seidel} and \textit{J. P. Solomon} [Geom. Funct. Anal. 22, No. 2, 443--477 (2012; Zbl 1250.53078)]. Symplectic cohomology comes equipped with a BV operator \(\Delta\), and a symplectic dilation is an \(x \in SH^1 (W)\) such that \(\Delta(x) = 1\). Restricting to positive symplectic cohomology yields a map \(\Delta_+ \colon SH_+^* (W) \rightarrow SH_+^{*-1} (W)\). The author also defines a structure map \(\Delta_\partial \colon \ker \Delta_+ \rightarrow \text{coker} \, \delta_\partial\). The author shows (Theorem G) that if \(Y\) is ADC and \(W_1, W_2\) are topologically simple exact fillings, then there is an isomorphism \(\Gamma \colon SH_+^* (W_1) \rightarrow SH_+^* (W_2)\) such that \(\Gamma\) commutes with \(\Delta_+\), \(\delta_\partial\), and \(\Delta_\partial\), in appropriate senses. Using this, it is shown (Corollary H) that if \(Y\) is ADC of dimension at least \(5\), then the existence of a symplectic dilation is independent of a choice of Weinstein filling.
A third set of results (Theorems I--J) focuses on Weinstein fillings and variants thereof. The structure maps \(\delta_\partial\) and \(\Delta_\partial\) are used to obtain obstructions to the existence of Weinstein fillings and symplectic cobordisms. The author is able to prove (Theorem I) that for \( k \geq 1\) there are infinitely many non-contactomorphic \((4k+3)\)-dimensional contact manifolds which are exactly fillable and almost Weinstein fillable but not Weinstein fillable. Further (Corollary J), the author shows that if \(Y^{2n-1}\) (\(n \geq 3\)) is ADC with Weinstein filling \(W\) satisfying \(c_1 (W) = 0\), and \(V\) is another Weinstein domain, then various conditions involving \(\delta_\partial\), \(\Delta_\partial\) and \(\Delta\) imply that there is no Weinstein cobordism from \(\partial V\) to \(Y\).
Fourth, Theorem K gives constructions of ADC manifolds, adding to results of Lazarev. If \(V\) is an exact domain with \(c_1 (V) = 0\) and \(\dim V > 0\) then \(\partial(V \times \mathbb{C})\) is shown to be ADC. Moreover, if \(V,W\) are exact ADC domains of dimension at least 4 with vanishing \(c_1\), then \(\partial(V \times W)\) is ADC.
Finally, Theorems L--M concern uniruledness, which has been studied in the symplectic context by \textit{M. McLean} [Duke Math. J. 163, No. 10, 1929--1964 (2014; Zbl 1312.53107)] An exact domain \(W\) is \emph{\((k, \Lambda)\)-uniruled} if for each \(p\) in the interior of \(W\), there is a proper holomorphic curve passing through \(p\) with area at most \(\Lambda\) and domain Riemann surface \(S\) having \(H_1 (S; \mathbb{Q})\) of rank \(\leq k-1\). Theorem L shows that if \(W\) is an exact domain with a symplectic dilation, then \(W\) is \((1, \Lambda)\)-uniruled for some positive \(\Lambda\). Using the fact that log Kodaira dimension provides an obstruction to algebraic uniruledness, the author shows (Corollary M) that if \(V\) is an affine variety of non-negative log Kodaira dimension, then \(SH^* (V) \neq 0\) and has no symplectic dilation. In particular, a complex exotic \(\mathbb{C}^n\) with non-negative log Kodaira dimension is a symplectic exotic \(\mathbb{C}^n\).
Aside from further results throughout the paper, the author discusses numerous connections to other results in the literature.
Reviewer: Daniel Mathews (Monash)A Boothby-Wang theorem for Besse contact manifoldshttps://zbmath.org/1491.570252022-09-13T20:28:31.338867Z"Kegel, Marc"https://zbmath.org/authors/?q=ai:kegel.marc"Lange, Christian"https://zbmath.org/authors/?q=ai:lange.christianThe authors consider closed connected contact manifolds with a preferred choice of a contact form, whose Reeb orbits are all periodic. By the work of \textit{D. Sullivan} [J. Pure Appl. Algebra 13, 101--104 (1978; Zbl 0402.57015)] and \textit{A. W. Wadsley} [J. Differ. Geom. 10, 541--549 (1975; Zbl 0336.57019)] this is to say that the Reeb flow is periodic. Analogously to the theory of closed geodesics those contact manifolds are called Besse contact manifolds. In the same way, if the minimal periods of the periodic Reeb orbits are all the same, equal to the circumference of the unit circle, say, the Besse contact manifold is called Zoll.
Zoll contact manifolds are precisely the total spaces of Boothby-Wang bundles, i.e. principal circle bundles over closed connected symplectic manifolds, cf. [\textit{H. Geiges}, An introduction to contact topology. Cambridge: Cambridge University Press (2008; Zbl 1153.53002)], such that: (1) The symplectic form normalized by the circumference of the unit circle is integral and the negative of its cohomology class is the real Euler class of the circle bundle. (2) The symplectic form appears as the curvature form of a connection one-form that turns out to be the contact form of the Zoll contact manifold. (3) The Reeb flow generates the circle action. Observe that the real Euler class may have several integral lifts, the integral Euler classes, that topologically classify principal circle bundles.
The authors prove a similar characterization for a Reeb flow with period equal to the circumference of the unit circle, the so-called Besse case. This time the orbit space has the structure of an orbifold and the Besse contact manifold is the total space of a principal circle orbibundle. This is because the circle action induced by the Reeb flow allows finite isotropy. The curvature form is induced by the exterior differential of the contact form, which is odd-symplectic, invariant under the Reeb flow and vanishes in Reeb direction. This form induces an integral symplectic orbifold structure on the orbit space. The real Euler class of the orbibundle is an element in the second orbifold cohomology, the equivariant cohomology of the total space. The fact that the total space is a manifold results in a sequence of isomorphisms in the orbifold cohomology ring induced by the degree two action of the integral Euler class in all degrees equal to the dimension of the Besse contact manifold or higher.
In dimension three this characterization allows a complete classification of Besse contact manifolds (up to conjugation of Reeb flows) in terms of a suitable subclass of Seifert fibrations.
Reviewer: Kai Zehmisch (Bochum)The Poincaré-Lefschetz pairing viewed on Morse complexeshttps://zbmath.org/1491.570262022-09-13T20:28:31.338867Z"Laudenbach, François"https://zbmath.org/authors/?q=ai:laudenbach.francoisFor a compact manifold with non-empty boundary and equipped with a generic Morse function, the author constructs a short exact sequence from the two Morse complexes (one yielding the absolute homology and the other the relative homology) and the Morse complex of the boundary (Theorem 1.1). Moreover, a pairing is defined of the relative Morse complex with the absolute Morse complex which induces the intersection product in homology, in the form due to S. Lefschetz (Theorem 1.2). The following two papers by the author are directly connected to the above mentioned results: [\textit{J.-M. Bismut} and \textit{W. Zhang}, An extension of a theorem by Cheeger and Müller. With an appendix by François Laudenbach. Paris: Société Mathématique de France (1992; Zbl 0781.58039) and \textit{F. Laudenbach}, Geom. Dedicata 153, 47--57 (2011; Zbl 1223.57020)].
Reviewer: Dorin Andrica (Riyadh)What does a vector field know about volume?https://zbmath.org/1491.570272022-09-13T20:28:31.338867Z"Geiges, Hansjörg"https://zbmath.org/authors/?q=ai:geiges.hansjorgA vector field \(X\) without zeros on a manifold \(M\) is called geodesible if there is a Riemannian metric on \(M\) such that the flow lines of \(X\) are unit speed geodesics -- equivalently, if there exists a one-form \(\alpha\) invariant under the flow of \(X\) such that \(\alpha(X) = 1\). It was observed by \textit{C. B. Croke} and \textit{B. Kleiner} [J. Differ. Geom. 39, No. 3, 659--680 (1994; Zbl 0807.53035)] that if \(M^{2n+1}\) is closed and orientable, the integral \(\int_M \alpha\wedge (d\alpha)^n\) is independent of the form \(\alpha\); in the paper at hand this number is hence called the volume of \(X\). In particular, the volume of a closed contact manifold is determined by the Reeb vector field alone. This raised the question, posed to the author by Claude Viterbo, whether there exist nondiffeomorphic contact forms on the same manifold with the same Reeb vector field. This paper contains a variety of results motivated by this question (which is answered in the affirmative by giving explicit examples as Boothby-Wang bundles, using a construction of \textit{D. McDuff} [Invent. Math. 89, 13--36 (1987; Zbl 0625.53040)] of cohomologous, non-diffeomorphic symplectic forms) as well as the above notion of volume. Amongst others, the question whether one can compute the volume of a geodesible vector field explicitly from the vector field alone is answered positively in some special cases, such as Seifert fibered \(3\)-manifolds. Moreover, theorems of Gauß-Bonnet and Poincaré-Hopf are proved for \(2\)-dimensional orbifolds.
Reviewer: Oliver Goertsches (Marburg)Rank conditions for finite group actions on 4-manifoldshttps://zbmath.org/1491.570282022-09-13T20:28:31.338867Z"Hambleton, Ian"https://zbmath.org/authors/?q=ai:hambleton.ian"Pamuk, Semra"https://zbmath.org/authors/?q=ai:pamuk.semra``Let \(M\) be a closed, connected, orientable topological 4-manifold, and \(G\) a finite group acting topologically and locally linearly on \(M\). In this paper, we investigate the spectral sequence for the Borel cohomology \(H_G^*(M)\) and establish new bounds on the rank of \(G\) for homologically trivial actions with discrete singular set.'' Here the rank of \(G\) is the maximum of the \(p\)-ranks of \(G\) over all primes \(p\), the \(p\)-rank is the maximum rank \(r\) of an elementary abelian \(p\)-group \((\mathbb Z_p)^r \le G\). As the authors note, a basic case considered in various papers (but still not completely understood) is that of free actions on simply connected 4-manifolds (or equivalently, of closed topological 4-manifolds with finite fundamental group). ``We will extend the scope of previous work by including \textit{nonsimply connected} manifolds and concentrate on nonfree actions.'' ``Beyond the rank restrictions, we would like to know which finite groups \(G\) can act.'' (See also a paper by \textit{M. Mecchia} and \textit{B. Zimmermann} [Pac. J. Math. 243, No. 2, 357--374 (2009; Zbl 1198.57012)] for a discussion of the finite nonsolvable and simple groups which may act on a 4-manifold with trivial first homology).
Reviewer: Bruno Zimmermann (Trieste)Topologically nontrivial counterexamples to Sard's theoremhttps://zbmath.org/1491.580052022-09-13T20:28:31.338867Z"Goldstein, Paweł"https://zbmath.org/authors/?q=ai:goldstein.pawel"Hajłasz, Piotr"https://zbmath.org/authors/?q=ai:hajlasz.piotr"Pankka, Pekka"https://zbmath.org/authors/?q=ai:pankka.pekkaSummary: We prove the following dichotomy: if \(n=2,3\) and \(f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)\) is not homotopic to a constant map, then there is an open set \(\Omega \subset \mathbb{S}^{n+1}\) such that \(\mathrm{rank} df=n\) on \(\Omega\) and \(f(\Omega )\) is dense in \(\mathbb{S}^n\), while for any \(n\geq 4\), there is a map \(f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)\) that is not homotopic to a constant map and such that \(\mathrm{rank} df<n\) everywhere. The result in the case \(n\geq 4\) answers a question of Larry Guth.On approximation theorems for the Euler characteristic with applications to the bootstraphttps://zbmath.org/1491.600492022-09-13T20:28:31.338867Z"Krebs, Johannes"https://zbmath.org/authors/?q=ai:krebs.johannes-t-n"Roycraft, Benjamin"https://zbmath.org/authors/?q=ai:roycraft.benjamin"Polonik, Wolfgang"https://zbmath.org/authors/?q=ai:polonik.wolfgangThe authors study approximation theorems for the Euler characteristic of the Vietrois-Rips and Čech filtrations obtained from a Poisson or binomial sampling scheme in the critical regime. The main results concern to functional central limits theorems for the Euler characteristic curve associated to such filtrations, the results are applied to the smooth bootstrap of the Euler characteristic where the rate of convergence is determined relative to the Kantorovich-Wasserstein and Kolmogorov metrics. Computer simulations are also provided.
The paper is well written and recommended to those researchers interested in topological data analysis from its statistical point of view.
Let \(X_n\) be a random process that generates point clouds of \(n\) points in \([0,T]^d\). Over these point clouds we can consider the Vietoris-Rips or the Čech filtrations, denoted by \(\mathcal{K}_t(X_n)\), \(t\in[0,T]\). For each \(t\) we can consider the mean Euler characteristic of these filtrations, that define the so called mean Euler characteristic curve. This curve depends only in the underlying distribution of \(X_n\) and the main problem to solve is to provide good estimations for this curve depending on \(n\).
The functional central limit theorem presented in this work is the first to consider the Čech complex, in this sense it is an extension of [\textit{A. M. Thomas} and \textit{T. Owada}, Adv. Appl. Probab. 53, No. 1, 57--80 (2021; Zbl 07458579)]. Moreover, the application to the binomial case is also new and it is achieved by means of an approximation by Possion schemes. Remark that central limit theorems for persistent Betti numbers obtained from a stationary Poisson process can be found in [\textit{Y. Hiraoka} et al., Ann. Appl. Probab. 28, No. 5, 2740--2780 (2018; Zbl 1402.60059)].
The authors use a bootstrap technique to estimate the Euler characteristic curve: Given an iid generated cloud point of size \(n\) relative to an unknown density, the authors use a density estimate to replicate the cloud point and compute a bootstrapped Euler characteristic curve from these replicates. The authors show precise estimations for the Kantorovich-Wasserstein and Kolmogorov distances between the bootstrapped and the true curves depending on \(n\) and the supremum distance of the densities. The use of bootstrap for estimation of persistent invariants is not new, see for instance [\textit{F. Chazal} et al., J. Mach. Learn. Res. 18, Paper No. 159, 40 p. (2018; Zbl 1435.62452)].
Reviewer: Carlos Meniño (Vigo)Maintaining Reeb graphs of triangulated 2-manifoldshttps://zbmath.org/1491.680532022-09-13T20:28:31.338867Z"Agarwal, Pankaj K."https://zbmath.org/authors/?q=ai:agarwal.pankaj-kumar"Fox, Kyle"https://zbmath.org/authors/?q=ai:fox.kyle"Nath, Abhinandan"https://zbmath.org/authors/?q=ai:nath.abhinandanSummary: Let \(M\) be a triangulated, orientable 2-manifold of genus \(g\) without boundary, and let \(h\) be a height function over \(M\) that is linear within each triangle. We present a kinetic data structure (KDS) for maintaining the Reeb graph \(\mathcal{R}\) of \(h\) as the heights of \(M\)'s vertices vary continuously with time. Assuming the heights of two vertices of \(M\) become equal only \(O(1)\) times, the KDS processes \(O((\kappa+g)n\operatorname{polylog}n)\) events; \(n\) is the number of vertices in \(M\), and \(\kappa\) is the number of external events which change the combinatorial structure of \(\mathcal{R}\). Each event is processed in \(O(\log^2n)\) time, and the total size of our KDS is \(O(gn)\). The KDS can be extended to maintain an augmented Reeb graph as well.
For the entire collection see [Zbl 1388.68010].Embeddability in \(R^3\) is NP-hardhttps://zbmath.org/1491.680792022-09-13T20:28:31.338867Z"de Mesmay, Arnaud"https://zbmath.org/authors/?q=ai:de-mesmay.arnaud"Rieck, Yo'av"https://zbmath.org/authors/?q=ai:rieck.yoav"Sedgwick, Eric"https://zbmath.org/authors/?q=ai:sedgwick.eric"Tancer, Martin"https://zbmath.org/authors/?q=ai:tancer.martinComparing parametric and non-parametric velocity-dependent one-scale models for domain wall evolutionhttps://zbmath.org/1491.830292022-09-13T20:28:31.338867Z"Avelino, P. P."https://zbmath.org/authors/?q=ai:avelino.pedro-pina(no abstract)Spiky CMB distortions from primordial bubbleshttps://zbmath.org/1491.830322022-09-13T20:28:31.338867Z"Deng, Heling"https://zbmath.org/authors/?q=ai:deng.heling(no abstract)