Recent zbMATH articles in MSC 57Rhttps://zbmath.org/atom/cc/57R2024-04-02T17:33:48.828767ZWerkzeugLeafwise quasigeodesic foliations in dimension three and the funnel propertyhttps://zbmath.org/1529.370202024-04-02T17:33:48.828767Z"Chanda, Anindya"https://zbmath.org/authors/?q=ai:chanda.anindya"Fenley, Sérgio R."https://zbmath.org/authors/?q=ai:fenley.sergio-rA foliation \(\mathcal{G}\) is said to subfoliate a foliation \(\mathcal{F}\) if each leaf of \(\mathcal{F}\) has a foliation made up of leaves of \(\mathcal{G}\). In this context, \(\mathcal{G}\) is called the subfoliation and \(\mathcal{F}\) the superfoliation.
The authors make a study of one-dimensional foliations which are subfoliations of two-dimensional foliations in 3-manifolds. They analyze whether certain geometric conditions imply that a one-dimensional foliation in a 3-manifold is the foliation by flow lines of a topological Anosov flow. Such a one-dimensional subfoliation made of quasigeodesics in the leaves of a superfoliation is called a leafwise quasigeodesic foliation. When all leaves of a leafwise quasigeodesic subfoliation in a leaf of the superfoliation have a common ideal point (limit point), we call that leaf a funnel leaf. If all leaves of the superfoliation are funnel leaves, then the leafwise quasigeodesic foliation is said to have the funnel property.
In their main results, the authors provide interesting examples showing that the funnel property is not a consequence of leafwise quasigeodesic behavior.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Hyperbolicity from contact surgeryhttps://zbmath.org/1529.370242024-04-02T17:33:48.828767Z"Hasselblatt, Boris"https://zbmath.org/authors/?q=ai:hasselblatt.boris"Heberle, Curtis"https://zbmath.org/authors/?q=ai:heberle.curtis-rSummary: A Dehn surgery on the \textit{periodic} fiber flow of the unit tangent bundle of a surface produces a uniformly \textit{hyperbolic} Cantor set for the resulting contact flow.Topological and geometric rigidity for spaces with curvature bounded belowhttps://zbmath.org/1529.530012024-04-02T17:33:48.828767Z"Núñez-Zimbrón, Jesús"https://zbmath.org/authors/?q=ai:nunez-zimbron.jesusThis paper is an extended version of a survey talk given by the author on topological and geometric rigidities. The author mainly focuses on rigidity results from his works for two types of spaces.
He starts with the following three well-known conjectures on rigidity:
(1) Poincaré conjecture: Every closed \(n\)-manifold which is homotopy equivalent to the \(n\)-sphere is homeomorphic to it;
(2) Borel conjecture: If two aspherical closed \(n\)-manifolds are homotopy equivalent then they are homeomorphic;
(3) Mostow rigidity: Let \(M\) and \(N\) be closed hyperbolic \(n\)-manifolds with \(n\geq 3\). If \(M\) is homotopy equivalent to \(N\), then \(M\) and \(N\) are isometric.
Milnor's exotic manifolds imply that there exist examples of Riemannian manifolds which are homotopy equivalent or even homeomorphic, and which are not diffeomorphic. Therefore, it is natural to require certain geometric conditions for rigidity results such as the curvature condition in Mostow rigidity or the volume entropy (see Theorem 1.4 and Theorem 1.5, respectively).
In Section 2 the author gives a brief review of two types of geometric spaces: Alexandrov spaces and \(\mathrm{RCD}/\mathrm{RCD}^*\) spaces.
An Alexandrov space is defined by comparing geodesic triangles with the corresponding geodesic triangles in model spaces, where model spaces are essentially the standard 2-sphere, the standard Euclidean plane and the standard hyperbolic plane. The author gives a list of examples of Alexandrov spaces.
In order to define \(\mathrm{RCD}/\mathrm{RCD}^*\) spaces, one needs the \(\mathrm{CD}\)-spaces (Definition 2.5) and infinitesimally Hilbertian metric measure spaces (Definition 2.6). Then a metric measure space \((X,d,m)\) is a \(\mathrm{RCD}(K,N)\) space if it is an infinitesimally Hilbertian \(\mathrm{CD}(K,N)\) space.
As pointed out by the author, one of the central problems in the theory of \(\mathrm{CD}(K,N)\) spaces is ``to determine whether, as in the case of Alexandrov spaces, \(\mathrm{CD}\) spaces exhibit a local-to-global property, that is, whether satisfying \(\mathrm{CD}(K,N)\) for all subsets of a covering implies the condition on the full space.'' However, \textit{T. Rajala}'s examples [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 15, 45--68 (2016; Zbl 1339.53045)]
answer this negatively. To address this issue, a modified condition was introduced: the \(\mathrm{CD}^*(K,N)\)-condition, or reduced curvature-dimension condition. Then \(\mathrm{RCD}^*(K,N)\) spaces are defined as infinitesimally Hilbertian \(\mathrm{CD}^*(K,N)\) spaces in Definition 2.8. Several constructions of \(\mathrm{RCD}^*\) spaces are discussed.
In Section 3 the author presents several of his results on topological rigidity of closed Alexandrov 3-spaces. At the end, the author proposes the following conjecture.
Conjecture. Every closed aspherical Alexandrov 3-space is homeomorphic to a 3-manifold.
In Section 4 the author discusses the volume entropy rigidity of \(\mathrm{RCD}^*\) spaces. The volume entropy of a compact Riemannian manifold \((M^n,g)\) is defined as
\[
h(M,g) = \lim_{R\to\infty}\frac{\log(\mathrm{vol}(B(\tilde{x},R)))}{R},
\]
where \(B(\tilde{x},R)\) is a ball in the universal cover \(\tilde{M}\) of \(M\). It can be shown that the limit exists and does not depend on the choice of \(\tilde{x}\). The author lists several results on volume entropy rigidity. Theorem 4.1 is a result of \textit{L. Chen} et al. [J. Differ. Geom. 113, No. 2, 227--272 (2019; Zbl 1430.53054)] connecting Ricci curvature, volume entropy and diffeomorphisms to a hyperbolic manifold. The author introduces the related concept of volume growth entropy and proves an analogous result in Theorem 4.2 for \(\mathrm{RCD}^*\)-spaces. This theorem implies Theorem 4.3 on stability under measured Gromov-Hausdorff convergence. After the definition of the Sobolev-to-Lipschitz property (Definition 4.5), isomorphisms via duality with Sobolev norms (Theorem 4.6), and a weak Bochner inequality (Theorem 4.7), the author concludes with Theorem 4.8 on the rigidity of \(\mathrm{RCD}\)-spaces in terms of the comparison of \(N\)-dimensional Hausdorff measure and the volume of a closed \(N\)-manifold of constant curvature \(-1\).
For the entire collection see [Zbl 1506.53001].
Reviewer: Lin Shan (San Juan)Involute-evolute and pedal-contrapedal curve pairs on \(S^2\)https://zbmath.org/1529.530042024-04-02T17:33:48.828767Z"Li, Enze"https://zbmath.org/authors/?q=ai:li.enze"Pei, Donghe"https://zbmath.org/authors/?q=ai:pei.donghe(no abstract)Evolutes and focal surfaces of \((1, k)\)-type curves with respect to Bishop frame in Euclidean 3-spacehttps://zbmath.org/1529.530052024-04-02T17:33:48.828767Z"Li, Pengcheng"https://zbmath.org/authors/?q=ai:li.pengcheng"Pei, Donghe"https://zbmath.org/authors/?q=ai:pei.donghe(no abstract)A foliated Hitchin-Kobayashi correspondencehttps://zbmath.org/1529.530202024-04-02T17:33:48.828767Z"Baraglia, David"https://zbmath.org/authors/?q=ai:baraglia.david"Hekmati, Pedram"https://zbmath.org/authors/?q=ai:hekmati.pedramIn this paper the authors prove an analogue of the Hitchin-Kobayashi correspondence for compact, taut Riemannian foliated manifolds with transverse Hermitian structure.
Let \(X\) be a compact oriented manifold with a taut Riemannian foliation which has a transverse Hermitian structure. On the one side of the correspondence is a transverse Hermitian-Einstein metric on a foliated holomorphic vector bundle \(E\) on \(X\), i.e., a transverse Hermitian metric such that the curvature form \(F_{A}\) of its Chern connection \(A\) is of type \((1,1)\) and satisfies the equation \(i\Lambda F_{A}=\gamma_{A}\mathrm{id}_{E}\), where \(\Lambda\) is the adjoint of the Lefschetz operator and \(\gamma_{A}\) is a constant.
On the other side are the foliated analogues of polystable holomorphic vector bundles. Let \(E\) be a transverse holomorphic vector bundle which admits a transverse Hermitian metric. We say that \(E\) is stable (resp. semistable) if for each transverse coherent subsheaf \(\mathcal{F}\) of \(E\) with \(0<\mathrm{rk}(\mathcal{F}) < \mathrm{rk}(E)\) and such that the quotient \(\mathcal{O}(E)/\mathcal{F}\) is torsion-free, we have:
\begin{align*}
\mathrm{deg}(\mathcal{F})/\mathrm{rk}(\mathcal{F}) < \mathrm{deg}(E)/\mathrm{rk}(E) \\
(\text{resp.}, \mathrm{deg}(\mathcal{F})/\mathrm{rk}(\mathcal{F}) \leq \mathrm{deg}(E)/\mathrm{rk}(E)).
\end{align*}
Moreover, \(E\) is polystable if \(E\) is the direct sum of stable bundles of the same slope.
Theorem. Let \(E\) be a transverse holomorphic vector bundle which admits transverse Hermitian metrics. Then \(E\) admits a transverse Hermitian metric \(h\) which is Hermitian-Einstein if and only if \(E\) is polystable. Moreover, if \(E\) is simple, then \(h\) is unique up to constant rescaling.
The proof is based on the Uhlenbeck-Yau method of continuity proof of the Hitchin-Kobayashi correspondence from [\textit{K. Uhlenbeck} and \textit{S. T. Yau}, Commun. Pure Appl. Math. 39, S257--S293 (1986; Zbl 0615.58045)]. As a consequence, the transverse Hitchin-Kobayashi correspondence holds on any compact Sasakian manifold.
Reviewer: Andrea Tamburelli (Houston)Contact structures with singularities: from local to globalhttps://zbmath.org/1529.530722024-04-02T17:33:48.828767Z"Miranda, Eva"https://zbmath.org/authors/?q=ai:miranda.eva"Oms, Cédric"https://zbmath.org/authors/?q=ai:oms.cedricContact manifolds are well studied, being the odd dimensional counterpart of symplectic manifolds. In this connection between symplectic and contact manifolds, it seems that singular forms have not been sufficiently studied. In recent years, a class of singular symplectic forms called \(b^m\)-symplectic forms has been investigated. In this paper, the authors take a first step towards studying the geometry and local topology of the odd-sized counterpart of these manifolds, called \(b^m\)-contact manifolds. These singular contact structures are determined by the kernel of non-smooth differential forms, called \(b^m\)-contact forms, which have an associated critical hypersurface \(Z\). The topology of the manifolds equipped with such singular contact forms is related to the smooth contact structures by desingularization. It is proved that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a \(b^m\)-contact structure. Also, the topology of the \(b^m\)-contact manifolds, depending on the parity of \(m\), is related to that of contact manifolds or so-called folded contact manifolds. The paper ends with some open problems.
Reviewer: Liviu Popescu (Craiova)Kirwan surjectivity and Lefschetz-Sommese theorems for a generalized hyperkähler reductionhttps://zbmath.org/1529.530752024-04-02T17:33:48.828767Z"Fisher, Jonathan"https://zbmath.org/authors/?q=ai:fisher.jonathan-m"Jeffrey, Lisa"https://zbmath.org/authors/?q=ai:jeffrey.lisa-c"Malusà, Alessandro"https://zbmath.org/authors/?q=ai:malusa.alessandro"Rayan, Steven"https://zbmath.org/authors/?q=ai:rayan.stevenThere is a highly successful method developed by \textit{F. C. Kirwan} [Cohomology of quotients in symplectic and algebraic geometry. Princeton University Press, Princeton, NJ (1984; Zbl 0553.14020)], \textit{M. F. Atiyah} and \textit{R. Bott} [Philos. Trans. R. Soc. Lond., Ser. A 308, 523--615 (1983; Zbl 0509.14014)]
and others to compute the (rational) cohomology of symplectic quotients \(X// G = \mu^{-1}(0)/G\), where \(X\) is compact symplectic and \(G\) is a compact Lie group, which is assumed to act in a Hamiltonian fashion with moment map \(\mu\colon X\to \mathfrak g^*\). A key step is to show that the Kirwan map, i.e., the natural restriction map \(H_G^*(X)\to H_G(\mu^{-1}(0))\cong H^*(X// G)\) induced by \(\mu^{-1}(0)\hookrightarrow X\), is surjective, by showing that \(|\mu|^2\) is a minimally degenerate Morse-Bott function. In the hyper-Kähler category, the non-compactness of \(M\) prevents a straightforward generalization of Kirwan surjectivity. \textit{L. Jeffrey} et al. [Transform. Groups 14, No. 4, 801--823 (2009; Zbl 1186.53061)] have provided a Morse-theoretic criterion, which is however difficult to check in practice.
This paper studies the so-called hyper-reductions of decorated \(G\)-manifolds. Here a decorated \(G\)-manifold is a (non-compact) Kähler manifold \(M\) with a Hamiltonian \(G\times S^1\)-action and a \(G\)-invariant holomorphic function \(s\) which is homogeneous of some positive degree with respect to the \(S^1\)-action. The hyper-reduction is then \(M/// G = (\mu_G^{-1}(0)\cap s^{-1}(0))/G\). The notion is motivated in particular by hyper-Kähler quotients by tori in the presence of an additional rotating \(S^1\)-action, where \(s = \mu_{\mathbb C}\), the complex moment map. The authors then prove that under some assumptions the hyper-Kirwan map \[\kappa_{G,s}\colon H_G^*(M)\to H^*(M/// G )\] is surjective.
The strategy of the proof is to write \(\kappa_{G,s} = \iota^* \circ \kappa\), where \(\kappa\colon H_G(M)\to H^*(M//G)\) is the usual Kirwan map, and \(\iota\colon (\mu_G^{-1}(0)\cap s^{-1}(0)) \to \mu_G^{-1}(0)\) is the inclusion, and to examine \(\kappa\) and \(\iota^*\) separately. The authors show in \S 2 that \(\kappa\) is surjective if \(M\) and the symplectic reduction \(M// G\) are circle compact with smooth compactification \(\overline{M//G}\). In Theorem 3.9 they show that \(\iota^*\) is in fact an isomorphism under some regularity assumptions. In the final section the authors apply their theory to the situation of a toric hyper-Kähler quotient mentioned above.
Reviewer: Markus Röser (Hamburg)The symplectic cohomology of magnetic cotangent bundleshttps://zbmath.org/1529.530792024-04-02T17:33:48.828767Z"Groman, Yoel"https://zbmath.org/authors/?q=ai:groman.yoel"Merry, Will J."https://zbmath.org/authors/?q=ai:merry.will-jThe notion of a magnetic geodesic flow, which models the motion of a charged particle in a magnetic field, is a natural generalization of the notion of geodesic flow on a Riemannian manifold. The symplectic framework is as follows. Take the cotangent bundle \(T^*N\) of a manifold \(N\) equipped with its canonical symplectic form, \(\omega\); assume that \(\sigma\) is a closed 2-form on \(N\) and construct the symplectic form
\[ \omega_\sigma = \omega + \pi^* \sigma, \]
where \(\pi : T^*N \longrightarrow N\) is the canonical projection. We call \(\omega_\sigma\) a magnetic symplectic form, and we refer to the symplectic manifold \((T^*N, \omega_\sigma)\) as a magnetic cotangent bundle.
The aim of the current paper is to study how the symplectic topology of \((T^*N, \omega_\sigma)\) is influenced by the choice of \(\sigma\). The authors construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, using the dissipative method for compactness introduced by \textit{Y. Groman} in [Geom. Topol. 27, No. 4, 1273--1390 (2023; Zbl 07699412)]. So they obtain a lot of new results as well as interesting applications. For instance, if \(N\) is a closed orientable manifold and \(\sigma\) is a magnetic form that is not weakly exact, then the \(\pi_1\)-sensitive Hofer-Zehnder capacity of any compact set in the magnetic cotangent bundle determined by \(\sigma\) is finite.
Reviewer: Manuel de León (Madrid)Lectures on factorization homology, \(\infty\)-categories, and topological field theories. With contributions by Araminta Amabel, Artem Kalmykov and Lukas Müllerhttps://zbmath.org/1529.550012024-04-02T17:33:48.828767Z"Tanaka, Hiro Lee"https://zbmath.org/authors/?q=ai:tanaka.hiro-leeThese lecture notes provide an introduction to factorization homology, via ideas in category theory and in homotopy theory, culminating in a discussion of the Cobordism Hypothesis. These are very much lecture notes -- the chapters were clearly written to accompany lectures.
There is very little discussion in these notes -- they don't advertise the topic or the results, and these notes are not light reading. There are also cases where the text refers to things that would be known only to participants of the school, such as the content of another lecture in the summer school not discussed in the book (e.g. Example 2.10).
For me, the highlight of these notes are the numerous exercises of various levels of difficulty, designed to give the reader the tools to work in the field. These, together with a good technical discussion of factorization homology in dimensions 1 and higher, make these notes a valuable tool for the serious reader seeking to learn factorization homology and related topics.
Reviewer: Daniel Moskovich (Beer-Sheva)Geometric Hodge filtered complex cobordismhttps://zbmath.org/1529.550052024-04-02T17:33:48.828767Z"Haus, Knut Bjarte"https://zbmath.org/authors/?q=ai:haus.knut-bjarte"Quick, Gereon"https://zbmath.org/authors/?q=ai:quick.gereonGiven a topological rationally even spectrum \(E\) and a prime \(p\), \textit{M. J. Hopkins} and \textit{G. Quick} [J. Topol. 8, No. 1, 147--183 (2015; Zbl 1349.32009)] have defined Hodge filtered \(E\)-cohomology groups with twist \(p\) for complex manifolds \(X\). For each \(n\), these are \(E^n_{\mathcal D}(p) (X) = \mathrm{Hom}_{hoSp(\mathbf{ sPre_*})}(\Sigma^\infty(X_+), \Sigma^nE_{\mathcal D}(p))\), where \(\mathbf{ sPre_*}\) is the homotopy category of presheaves of spectra on \(\mathbf{Man}_{\mathbb{C}}\), the category of complex manifolds and holomorphic maps. \(E_{\mathcal D}(p)\) is the homotopy pullback of \(H(F^p\mathcal{A}^*(\mathcal{V}_*)) \rightarrow H(\mathcal{A}^*(\mathcal{V}_*)) \leftarrow \mathrm{sing}(E)\) as maps of presheaves of spectra on \(\mathbf{Man}_{\mathbb{C}}\). Here, \(\mathcal{A}^*(\mathcal{V}_*)\) denotes the space of smooth forms on \(\mathcal{V}_* = E_* \otimes_{\mathbb{Z}} \mathbb{C}\), the \((F^p\mathcal{A}^*(\mathcal{V}_*))\) represent the \textit{Hodge filtration on forms} (see Definition 2.8), and \(\mathrm{sing}(E)\) is the spectrum of singular simplicial sets for \(E\). The \(E^n_{\mathcal D}(p) (X)\) are an example of a Hodge filtered extension of a rationally even cohomology theory, in line e.g. with complex analytic Deligne cohomology or Karoubi's multiplicative \(K\)-theory.
The purpose of this paper is to obtain a more concrete and geometric description of the \(E^n_{\mathcal D}(p) (X)\) whenever \(E=MU\), the complex cobordism spectrum. The authors define thus their \textit{geometric Hodge filtered complex cobordism groups} \(MU^n(p)(X)\), but do it not simply for complex manifolds but for smooth manifolds (together with a descending filtration \(F^*\) of \(\mathcal{A}^*\) on the manifold). In their words, this generality makes it easier to work with products of a complex manifold with a smooth one, an advantage for future applications of the results. On the other hand, it is reasonable to expect that Hopkins-Quick's definition of the \(E^n_{\mathcal D}(p) (X)\) might be extended to the category of smooth manifolds.
The \(MU^n(p)(X)\) are obtained in Definition 2.18 as a quotient between \(ZMU^n(p)(X)\), the group of \textit{Hodge filtered cycles} of Definition 2.13, and \(BMU^n(p)(X)\), the group of \textit{Hodge filtered cobordism relations} stemming from Proposition 2.17 (and using the so-called \textit{nullbordant} cycles). Some of the properties of the \(MU^n(p)(X)\) are then presented, and special mention should be given to the long exact sequence in Theorem 2.21, and to the attempt at an axiomatic interpretation of Hodge filtered cohomology theories in Section 2.9.
For the main result of the paper, Theorem 1.2, the authors consider again just complex manifolds. This result states that, given one such manifold \(X\) with a Hodge filtration, the two theories agree, that is, there is an isomorphism between the cohomology groups \(MU^n_{\mathcal D}(p) (X)\) and \(MU^n(p)(X)\) (for each \(n\) and \(p\)). The proof of this theorem uses a form of Pontryagin-Thom construction that takes into account the geometric data given by the Hodge filtration. The isomorphism is in fact constructed as a composition of two, and is achieved by means of an intermediate theory \(MU_{hs}(p)\) which shares the homotopy-theoretic flavor of \(MU^n_{\mathcal D}(p) (X)\) but is more accessible. This new theory (for smooth manifolds with the Hodge filtration) appears as Definition 3.15 and uses Mathai-Quillen forms. Section 4.2 shows that the new model is represented by a presheaf of spectra which fits into a homotopy pullback similar to the one defining \(MU^n_{\mathcal D}(p)\), and which permits the direct comparison of the two models in Theorem 4.7, leading to the isomorphism \(MU^n_{\mathcal D}(p) (X) \cong MU^n_{hs}(p) (X)\) of Theorem 4.9. Finally, Section 5 relates the \(MU^n_{hs}(p) (X)\) to the \(MU^n(p)(X)\) by a geometric Pontryagin-Thom map that induces the isomorphism of Definition 5.7, later allowing for Theorem 5.9, which gives \(MU^n_{hs}(p) (X) \cong MU^n(p)(X)\) for each \(X\), \(n\) and \(p\). This result, combined with Theorem 4.7, then gives the main result of Theorem 1.2.
Reviewer: Rui Miguel Saramago (Porto Salvo)Geometric triangulations and highly twisted linkshttps://zbmath.org/1529.570022024-04-02T17:33:48.828767Z"Ham, Sophie L."https://zbmath.org/authors/?q=ai:ham.sophie-l"Purcell, Jessica S."https://zbmath.org/authors/?q=ai:purcell.jessica-shepherdIt is conjectured that every cusped hyperbolic 3-manifold admits a geometric triangulation into positive-volume ideal hyperbolic tetrahedra. This is known, for example, for 2-bridge knot complements and for punctured torus bundles [\textit{F. Guéritaud}, Geom. Topol. 10, 1239--1284 (2006; Zbl 1130.57024)]; also, every cusped hyperbolic 3-manifold admits a finite cover with a geometric triangulation [\textit{F. Luo} et al., Proc. Am. Math. Soc. 136, No. 7, 2625--2630 (2008; Zbl 1144.57016)] and has a decomposition into into convex ideal polyhedra [\textit{D. B. A. Epstein} and \textit{R. C. Penner}, J. Differ. Geom. 27, No. 1, 67--80 (1988; Zbl 0611.53036)]. In the present paper, the authors show that sufficiently highly twisted knots admit a geometric triangulation. Specifically, the first main result states that, for every \(n \ge 2\) there exists a constant \(A_n\) such that, if \(K\) is a link in \(S^3\) with a prime, twist-reduced diagram with \(n\) twist regions and at least \(A_n\) crossings in each twist region, then the complement of \(K\) admits a geometric triangulation (such highly twisted links are known to be hyperbolic when there are at least six crossings in each twist region, cf. [\textit{D. Futer} and \textit{J. S. Purcell}, Comment. Math. Helv. 82, No. 3, 629--664 (2007; Zbl 1134.57003)]).
The authors prove also a more general result for \textit{fully augmented links} obtained by adding to each twist region an unknot (``crossing circle'') encircling the twist region, and then removing all crossings in each twist region (the original link is obtained by Dehn filling of the crossing circles); the more general result allows arbitrary Dehn fillings of the crossing circles (or leaving some crossing circles unfilled). Finally, the authors give quantified versions with explicit constants of their results for infinite families of examples (extending work of \textit{F. Guéritaud} and \textit{S. Schleimer} [Geom. Topol. 14, No. 1, 193--242 (2010; Zbl 1183.57013)] on Dehn fillings on one cusp of the Whitehead link complement).
Reviewer: Bruno Zimmermann (Trieste)A survey of the homology cobordism grouphttps://zbmath.org/1529.570122024-04-02T17:33:48.828767Z"Şavk, Oğuz"https://zbmath.org/authors/?q=ai:savk.oguzSummary: In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology \(3\)-spheres and the structure of \(\Theta_{\mathbb{Z}}^3\). Finally, we briefly discuss the knot concordance group \(\mathcal{C}\) and the rational homology cobordism group \(\Theta_{\mathbb{Q}}^3\), focusing on their algebraic structures, relating them to \(\Theta_{\mathbb{Z}}^3\), and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology \(3\)-spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.Stable diffeomorphism classification of some unorientable 4-manifoldshttps://zbmath.org/1529.570172024-04-02T17:33:48.828767Z"Debray, Arun"https://zbmath.org/authors/?q=ai:debray.arunAs it is well-known, two closed 4-manifolds \(M\) and \(N\) are said to be \textit{stably diffeomorphic} if two non-negative integers \(m,n\) exist, such that \(M \# m (\mathbb S^2 \times \mathbb S^2)\) is diffeomorphic to \(N \# n (\mathbb S^2 \times \mathbb S^2).\) Thanks to Kreck's modified surgery theory (see [\textit{M. Kreck}, Ann. Math. (2) 149, No. 3, 707--754 (1999; Zbl 0935.57039)]), classification of 4-manifolds with fixed fundamental group up to stable diffeomorphism turns out to be tractable, since it is reduced to a collection of bordism computations.
The present paper applies this approach to unorientable 4-manifolds.
In particular, the author proves that, if \(\pi_1(M)\) is a finite group of order 2 mod 4 and a suitable hypothesis on cohomology holds, then there exist exactly fourteen classes of closed connected unorientable 4-manifolds \(M\) up to stable diffeomorphism: nine for which \(M\) is \(pin^+\), one for which \(M\) is \(pin^-\), and four for which \(M\) is neither. Furthermore, the corresponding stable homeomorphism classes are determined, too.
Reviewer: Maria Rita Casali (Modena)Corks, involutions, and Heegaard Floer homologyhttps://zbmath.org/1529.570202024-04-02T17:33:48.828767Z"Dai, Irving"https://zbmath.org/authors/?q=ai:dai.irving"Hedden, Matthew"https://zbmath.org/authors/?q=ai:hedden.matthew"Mallick, Abhishek"https://zbmath.org/authors/?q=ai:mallick.abhishekIn this paper, Dai, Hedden, and Abhishek use Heegard Floer homology to study corks. A cork is a contractible \(4\)-manifold with an involution on the boundary that does not extend as any diffeomorphism to the entire \(4\)-manifold. A cork is called strong if the involution on the boundary does not extend as a diffeomorphism to any homology ball. The first proofs that certain diffeomorphisms did not extend across used embeddings of these manifolds into closed \(4\)-manifolds and demonstrated that using the diffeomorphism to twist the manifold would result in a different smooth structure. Since Floer theory is the natural setting for gluing results for invariants related to gauge theory, it is natural to expect that cork twists would act in interesting ways on the Floer theory of the boundary. This is indeed the case, but it has been difficult to compute the action of a diffeomorphism on the relevant Floer theory. The authors use involutive Heegaard Floer theory combined with a monotonicity result for the theory under negative definite cobordisms to make conclusions about the action of the diffeomorphism group on the Floer groups. In particular they use the charge conjugation to define a pair of invariants taking values in the group of iota-complexes modulo local equivalence. The authors emphasize the fact that their invariants are \(3\)-dimensional in nature. This paper greatly extends our knowledge of the interaction between cork twists and Floer theory, and uses this new knowledge to exhibit many new examples of strong corks.
Reviewer: David Auckly (New York)A cohomological Seiberg-Witten invariant emerging from the adjunction inequalityhttps://zbmath.org/1529.570212024-04-02T17:33:48.828767Z"Konno, Hokuto"https://zbmath.org/authors/?q=ai:konno.hokutoAuthor's summary: ``We construct an invariant of closed spin\(^c\) 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a 4-manifold. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This nonvanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.''
Reviewer: Ximin Liu (Dalian)On the order of Dehn twistshttps://zbmath.org/1529.570302024-04-02T17:33:48.828767Z"Keating, Ailsa"https://zbmath.org/authors/?q=ai:keating.ailsa-m"Randal-Williams, Oscar"https://zbmath.org/authors/?q=ai:randal-williams.oscarIn this note, the authors compute the order of the (higher-dimensional) Dehn twist in several mapping class groups of automorphisms of the cotangent bundle \(T^{\ast} S^n\) of the \(n\)-sphere. More concretely, the relevant mapping class groups are those of compactly-supported symplectomorphisms, diffeomorphisms, homeomorphisms and homotopy autmorphisms of \(T^{\ast} S^n\), respectively. These mapping class groups are connected by neglectful homomorphisms.
The answer is that for \(n\) odd, the order of the Dehn twist is infinite in all these groups. For \(n\) even, the order is infinite in \(\pi_0 \text{Symp}_{c}(T^{\ast} S^n)\). For \(n = 2, 6\), the order in \(\pi_0 \text{Diff}_{c}(T^{\ast} S^n)\), \(\pi_0 \text{Homeo}_{c}(T^{\ast} S^n)\) and \(\pi_0 \text{hAut}_{c}T(^{\ast} S^n)\) is \(2\). For all other even \(n\), the order in the latter two groups is \(4\), whereas the order in \(\pi_0 \text{Diff}_{c}(T^{\ast} S^n)\) is \(4\) or \(8\), depending on whether the \((2n+1)\)-dimensional Kervaire sphere is trivial or not. Thus, the only ambiguity arises for \(n = 62\).
The authors explain that the majority of their results were known before. In fact, their biggest contribution comprises assembling and popularising known results; in particular, they rely on work by \textit{P. Seidel} [J. Differ. Geom. 52, No. 1, 145--171 (1999; Zbl 1032.53068); Bull. Soc. Math. Fr. 128, No. 1, 103--149 (2000; Zbl 0992.53059)] and \textit{L. H. Kauffman} and \textit{N. A. Krylov} [Topology Appl. 148, No. 1--3, 183--200 (2005; Zbl 1066.57025)]. To derive the orders in the mapping class group of homotopy automorphisms, the authors employ methods from algebraic topology.
In the final chapter, the almost-complex mapping class group of \(T^{\ast} S^n\) is introduced. It is shown that for \(n\) even, the order of the Dehn twist in this group is finite and of the form \(n! \cdot 2^j\) with \(-2 \leq j \leq 4\).
Reviewer: Jens Reinhold (Münster)Round fold maps of \(n\)-dimensional manifolds into \((n-1)\)-dimensional Euclidean spacehttps://zbmath.org/1529.570312024-04-02T17:33:48.828767Z"Kitazawa, Naoki"https://zbmath.org/authors/?q=ai:kitazawa.naoki"Saeki, Osamu"https://zbmath.org/authors/?q=ai:saeki.osamuGiven a smooth closed manifold \(M\), a smooth map \(f:M\to\mathbb R^p\) with \(n\geq p\geq 1\) is called a fold map it its only singular points are fold points. Fold singularities are the simplest type of stable singularities which these maps may have, and, therefore, fold maps can be seen as a generalization of Morse functions. For fold maps, the singular set is a regular closed submanifold of \(M\) of dimension \(p-1\) and \(f\) restricted to its singular set is an immersion.
In this paper the authors study round fold maps, which are fold maps where the map restricted to the singular set is an embedding onto a disjoint union of concentric spheres in \(\mathbb R^p\). These maps have nice properties such as being simple, i.e. each component of the preimage of a point in \(\mathbb R^p\) contains at most one singular point.
In particular, they succeed in characterizing which smooth closed \(n\)-manifolds admit round fold maps into \(\mathbb R^{n-1}\) for \(n\geq 4\). They also classify round fold maps up to \(\mathscr A\)-equivalence (smooth changes of coordinates in source and target).
Reviewer: Raúl Oset Sinha (València)On necessary and sufficient conditions for the real Jacobian conjecturehttps://zbmath.org/1529.580042024-04-02T17:33:48.828767Z"Tian, Yuzhou"https://zbmath.org/authors/?q=ai:tian.yuzhou"Zhao, Yulin"https://zbmath.org/authors/?q=ai:zhao.yulinSummary: This paper is an expository one on necessary and sufficient conditions for the real Jacobian conjecture, which states that if \(F = \left(f^1, \dots, f^n\right): \mathbb{R}^n\rightarrow\mathbb{R}^n\) is a polynomial map such that \(\det DF\neq 0\), then \(F\) is a global injective. In Euclidean space \(\mathbb{R}^n\), the Hadamard's theorem asserts that the polynomial map \(F\) with \(\det DF\neq 0\) is a global injective if and only if \(\|F(\mathbf{x})\|\) approaches to infinite as \(\|\mathbf{x}\|\rightarrow\infty\). The first part of this paper is to study the two-dimensional real Jacobian conjecture via Bendixson compactification, by which we provide a new version of \textit{M. Sabatini}'s result [Nonlinear Anal., Theory Methods Appl. 34, No. 6, 829--838 (1998; Zbl 0949.34018)]. This version characterizes the global injectivity of polynomial map \(F\) by the local analysis of a singular point (at infinity) of a suitable vector field coming from the polynomial map \(F\). Moreover, applying the above results we present a dynamical proof of the two-dimensional Hadamard's theorem. In the second part, we give an alternative proof of the \textit{A. Cima} et al.'s result [Nonlinear Anal., Theory Methods Appl. 26, No. 4, 877--885 (1996; Zbl 0845.58014)] on the \(n\)-dimensional real Jacobian conjecture by the \(n\)-dimensional Hadamard's theorem.Generalized osculating-type ruled surfaces of singular curveshttps://zbmath.org/1529.580152024-04-02T17:33:48.828767Z"Yazıcı, Bahar Doğan"https://zbmath.org/authors/?q=ai:yazici.bahar-dogan"İşbilir, Zehra"https://zbmath.org/authors/?q=ai:isbilir.zehra"Tosun, Murat"https://zbmath.org/authors/?q=ai:tosun.muratSummary: In this study, we introduce generalized osculating-type ruled surfaces of special singular curves. We give some theories and results about the geometric structure of the surface. In addition, the singular point classes of the surface are examined, and the conditions for being a cross-cap surface are expressed. Generalized osculating-type ruled surface is considered as a framed surface and its basic invariants are found and some results are given. Finally, we give some examples and figures to support the theories.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Sculpting the standard model from low-scale gauge-Higgs-matter \(\mathrm{E}_8\) grand unification in ten dimensionshttps://zbmath.org/1529.810742024-04-02T17:33:48.828767Z"Aranda, Alfredo"https://zbmath.org/authors/?q=ai:aranda.alfredo"de Anda, Francisco J."https://zbmath.org/authors/?q=ai:de-anda.francisco-j"Morais, António P."https://zbmath.org/authors/?q=ai:morais.antonio-p"Pasechnik, Roman"https://zbmath.org/authors/?q=ai:pasechnik.romanSummary: The construction and general implications of a model with complete supersymmetric unification of the Standard Model matter content, interactions and families' replication into a single \(\mathrm{E}_8\) gauge superfield in ten dimensions is presented. The gauge and extended Poincaré symmetries are broken through compactification of the \(\mathbb{T}^6 /(\mathbb{Z}_3\times\mathbb{Z}_3)\) orbifold with Wilson lines, which reduces the original symmetry and matter content into those of the Standard Model plus additional heavier states. Proton decay can be suppressed automatically while the compactification scale may be as low as \(10^9\) GeV, so that the corresponding GUT-scale physics may be potentially accessible and testable by future collider measurements.Suspended fixed pointshttps://zbmath.org/1529.810982024-04-02T17:33:48.828767Z"Antinucci, Andrea"https://zbmath.org/authors/?q=ai:antinucci.andrea"Bianchi, Massimo"https://zbmath.org/authors/?q=ai:bianchi.massimo"Mancani, Salvo"https://zbmath.org/authors/?q=ai:mancani.salvo"Riccioni, Fabio"https://zbmath.org/authors/?q=ai:riccioni.fabioSummary: We study the orientifold of the \(\mathcal{N} = 1\) superconformal field theories describing D3-branes probing the Suspended Pinch Point singularity, as well as the orientifolds of non-chiral theories obtained by a specific orbifold \(\mathbb{Z}_n\) of SPP. We find that these models realize a mechanism analogous to the one recently found for the orientifold of the complex Calabi-Yau cone over the Pseudo del Pezzo surface \(\mathrm{PdP}_{3 c}\): they all flow to a new IR fixed point such that the value of the \(a\)-charge is less than half the one of the oriented theory. We also find that the value of \(a\) coincides with the charge of specific orientifolds of the toric singularities \(L^{(\bar{n}, \bar{n}, \bar{n})}\) with \(\bar{n} = 3 n / 2\) for \(n\) even or \(L^{(\bar{n}, \bar{n} + 1, \bar{n})}\) with \(\bar{n} = (3n - 1)/2\) for \(n\) odd, suggesting the existence of an IR duality.Librations with large periods in tunneling: efficient calculation and applications to trigonal dimershttps://zbmath.org/1529.811022024-04-02T17:33:48.828767Z"Anikin, A. Yu."https://zbmath.org/authors/?q=ai:anikin.a-yu"Dobrokhotov, S. Yu."https://zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Nosikov, I. A."https://zbmath.org/authors/?q=ai:nosikov.i-aSummary: In studying tunnel asymptotics for lower energy levels of the Schrödinger operator (such as the energy splitting in a symmetric double well or the width of a spectral band in a periodic problem), there naturally arise librations, i.e., periodic solutions of a classical system with inverted potential, which reach the boundary of the domain of possible motions twice during the period. In the limit, they give double-asymptotic solutions with two symmetric unstable equilibria (instantons). The tunnel asymptotics can be written in two ways: either in terms of the action on the instanton and the linearized dynamics in its neighborhood or in terms of a certain libration, called a tunnel libration. The second way is more constructive, since when used in numerical calculations, it reduces to two operations: finding a libration with a given energy and calculating the Floquet coefficients for a given libration. To apply this approach in practice, we propose to find librations with a given energy by using a numerical variational method that generalizes the ideas of the nudged elastic band method. As an application, we find the asymptotics for the widths of the lower spectral bands and gaps, expressed in terms of tunnel libration in a four-dimensional system describing the dimer in a trigonal-symmetric field, which was proposed by \textit{M. I. Katsnelson} [Graphene. Carbon in Two Dimensions. Cambridge: Cambridge University Press (2012; \url{doi:10.1017/CBO9781139031080})].Quantum statistical and squeezing properties in a semiconductor cavity QEDhttps://zbmath.org/1529.811092024-04-02T17:33:48.828767Z"Ayehu, Desalegn"https://zbmath.org/authors/?q=ai:ayehu.desalegn"Hirpo, Deribe"https://zbmath.org/authors/?q=ai:hirpo.deribeSummary: We consider a quantum well with one exciton in a microcavity that initially contains photons and is coupled to a one-mode squeezed vacuum reservoir. The solution of the quantum Langevin equation is used to investigate the intensity, second-order correlation function, intensity spectrum, and quadrature squeezing of the exciton mode in the low-excitation regime. Although the initial cavity photons increase the intensity of the exciton mode, they have an adverse effect on the quadrature squeezing. We also demonstrate that the exciton mode exhibits quadrature squeezing, with the amount of squeezing increased by the input squeezed light.\(Y(su(2))\) symmetry of Landau levels on the spherehttps://zbmath.org/1529.811112024-04-02T17:33:48.828767Z"Sun, En-Xin"https://zbmath.org/authors/?q=ai:sun.en-xin"Ma, Liu-Biao"https://zbmath.org/authors/?q=ai:ma.liu-biao"Yuan, Zi-Gang"https://zbmath.org/authors/?q=ai:yuan.zi-gang"Jing, Jian"https://zbmath.org/authors/?q=ai:jing.jianSummary: We find that there is a Yangian symmetry, i.e., \(Y(su(2))\), in the model of a charged particle on the surface of a sphere with a magnetic monopole situating at the center. We construct generators of \(Y(su(2))\) algebra explicitly and derive energy spectrum by employing its representation theory. We also show that this model is integrable from RTT relation.Exact solutions for a spin-orbit coupled ultracold atom held in a driven double wellhttps://zbmath.org/1529.811172024-04-02T17:33:48.828767Z"Luo, Yunrong"https://zbmath.org/authors/?q=ai:luo.yunrong"Wang, Xuemei"https://zbmath.org/authors/?q=ai:wang.xuemei"Yi, Jia"https://zbmath.org/authors/?q=ai:yi.jia"Li, Wenjuan"https://zbmath.org/authors/?q=ai:li.wenjuan"Xie, Xin"https://zbmath.org/authors/?q=ai:xie.xin"Luo, Zhida"https://zbmath.org/authors/?q=ai:luo.zhida"Hai, Wenhua"https://zbmath.org/authors/?q=ai:hai.wenhuaSummary: Exact solutions for spin-orbit (SO)-coupled cold atomic systems are very important and rare in physics. In this paper, we propose a simple method of combined modulations to generate the exactly analytic solutions for a single SO-coupled ultracold atom held in a driven double well. For the cases of synchronous combined modulations and the spin-conserving tunneling, we obtain the generally accurate solutions of this system respectively. For the case of spin-flipping tunneling under asynchronous combined modulations, we get the specially exact solutions in simple form when the driving parameters are appropriately chosen. Based on these obtained exact solutions, we reveal some intriguing quantum spin dynamical phenomena, for instance, the arbitrary coherent population transfer with and/or without spin-flipping, the controlled coherent population conservation, and the controlled coherent population inversion. The results may provide a possibility for generating the accurate quantum entangled states and the exact control of spin dynamics for a SO-coupled ultracold atomic system.Initial conditions problem in cosmological inflation revisitedhttps://zbmath.org/1529.830832024-04-02T17:33:48.828767Z"Garfinkle, David"https://zbmath.org/authors/?q=ai:garfinkle.david"Ijjas, Anna"https://zbmath.org/authors/?q=ai:ijjas.anna"Steinhardt, Paul J."https://zbmath.org/authors/?q=ai:steinhardt.paul-jSummary: We present first results from a novel numerical relativity code based on a tetrad formulation of the Einstein-scalar field equations combined with recently introduced gauge/frame invariant diagnostics. The results provide support for the argument that inflation does not solve the homogeneity and isotropy problem beginning from generic initial conditions following a big bang.