Recent zbMATH articles in MSC 57Rhttps://zbmath.org/atom/cc/57R2023-02-24T16:48:17.026759ZWerkzeugVirtual classes of representation varieties of upper triangular matrices via topological quantum field theorieshttps://zbmath.org/1502.140332023-02-24T16:48:17.026759Z"Hablicsek, Márton"https://zbmath.org/authors/?q=ai:hablicsek.marton"Vogel, Jesse"https://zbmath.org/authors/?q=ai:vogel.jesseLet \(G\) be a complex affine algebraic group and \(\Gamma\) a finitely generated discrete group. Then the set of group homomorphisms \(\mathrm{Hom}(\Gamma, G)\) has the structure of a complex affine variety. Let \(B_n\subset \mathrm{GL}(n,\mathbb{C})\) be the Borel group of upper-triangular matrices, and \(\Sigma_g\) a closed orientable surface of genus \(g\).
The first main theorem of this interesting paper explicitly computes the Grothendieck class of \(\mathrm{Hom}(\pi_1(\Sigma_g), B_n)\) for \(n=2,3,4\) showing the class to be a polynomial over \(\mathbb{Z}\) in the the class \([\mathbb{A}^1_{\mathbb{C}}]\).
The authors then show that the categorical quotient of \(\mathrm{Hom}(\pi_1(\Sigma_g), B_n)\) by the conjugation action of \(B_n\) exists and that quotient is isomorphic to \((\mathbb{A}^1_{\mathbb{C}}-\{0\})^{2ng}\).
In general, the categorical quotient of \(\mathrm{Hom}(\Gamma, G)\) by \(G\)-conjugation is called the \(G\)-character variety of \(\Gamma\). There is a related moduli space, called the ``variety of characters'' in [\textit{S. Lawton} and \textit{A. S. Sikora}, Algebr. Represent. Theory 20, No. 5, 1133--1141 (2017; Zbl 1400.14123)] when \(G\) admits a faithful representation. Often, but not always, these two moduli spaces are isomorphic.
The second main theorem of the paper under review shows that, with respect to the canonical representation \(B_n\subset \mathrm{GL}(n,\mathbb{C})\), the \(B_n\)-character variety of \(\pi_1(\Sigma_g)\) is \textit{not} isomorphic to the corresponding variety of characters.
Reviewer: Sean Lawton (Fairfax)On tensor fields generated by differentiable functionshttps://zbmath.org/1502.260122023-02-24T16:48:17.026759Z"Salmanova, Gunay H."https://zbmath.org/authors/?q=ai:salmanova.gunay-hSummary: In the paper it is considered the question on differentiable properties of maps generated by real smooth functions. In one dimensional case the set of sequential derivatives well defines the structure of a set of singular points. In this paper we attempt to generalize this theory for multivariate functions.Sharp cohomological bound for uniformly quasiregularly elliptic manifoldshttps://zbmath.org/1502.300732023-02-24T16:48:17.026759Z"Kangasniemi, Ilmari"https://zbmath.org/authors/?q=ai:kangasniemi.ilmariSummary: We show that if a compact, connected, and oriented \(n\)-manifold \(M\) without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring \(H^*(M;\mathbb{R})\) of \(M\) is bounded from above by \(2^n\). This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if \(M\) is not a rational homology sphere, then each such uniformly quasiregular self-map on \(M\) has a Julia set of positive Lebesgue measure.Structure and transitions of line bitangencies in a family of surface pairshttps://zbmath.org/1502.530092023-02-24T16:48:17.026759Z"Dreibelbis, D."https://zbmath.org/authors/?q=ai:dreibelbis.daniel"Olsen, W."https://zbmath.org/authors/?q=ai:olsen.wSummary: We classify the generic structure of line bitangencies in a one-parameter family of surface pairs. Considering the set of line bitangencies as a subset of the Cartesian product of two surfaces, we determine the singularity set when the line bitangencies are projected into one of the two surfaces, along with the generic transitions that can occur in our one-parameter family. All work is connected to the affine invariant geometry of the pair of surfaces.Singular 3-manifolds in \(\mathbb{R}^5\)https://zbmath.org/1502.530102023-02-24T16:48:17.026759Z"Benedini Riul, Pedro"https://zbmath.org/authors/?q=ai:benedini-riul.p"Soares Ruas, Maria Aparecida"https://zbmath.org/authors/?q=ai:ruas.maria-aparecida-soares"Sacramento, Andrea de Jesus"https://zbmath.org/authors/?q=ai:sacramento.andrea-de-jesusIn [\textit{L. F. Martins} and \textit{J. J. Nuño-Ballesteros}, Tôhoku Math. J. (2) 67, No. 1, 105--124 (2015; Zbl 1320.58023)], the authors study singular surfaces in \(\mathbb{R}^n\) with corank-1 singularities for \(n=3\) and the authors in [\textit{P. B. Riul} et al., Q. J. Math. 70, No. 3, 767--795 (2019; Zbl 1460.53007)] study the case when \(n=4\). They define the curvature parabola using the first and second fundamental forms of the surface at the singular point.
For smooth 3-manifolds immersed in \(\mathbb{R}^n\), \(n \geq 5\), the geometry of the second fundamental form at a point is determined by the geometry of a set called the curvature locus.
This paper is inspired by the regular case and by the definition of the curvature parabola for corank-1 surfaces in \(\mathbb{R}^n\), \(n = 3, 4\). The authors define the first and second fundamental forms of a corank-1 3-manifold in \(\mathbb{R}^5\) at \(p\) and study their properties. They also define the curvature locus for a corank-1 3-manifold in \(\mathbb{R}^5\) and study its geometry. The curvature locus contains geometric properties of the second fundamental form.
Reviewer: Daiane Alice Henrique Ament (Lavras)On the structure forms of a projective structurehttps://zbmath.org/1502.530222023-02-24T16:48:17.026759Z"Kuleshov, A. V."https://zbmath.org/authors/?q=ai:kuleshov.a-vSummary: A projective structure on a smooth manifold is a maximal atlas such that all its transition maps are the fractional linear transformations. Our aim is to interpret this notion in terms of the higher order frame bundles and their structure forms. It is shown that the projective structure gener-ates the sequence of differential geometric structures. The construction is following:
Step 1. For a smooth manifold the so-called quotient frame bundle as-sociated to the 2nd order frame bundle on the manifold is constructed.
Step 2. Given projective structure on the manifold, the mappings from the quotient frame bundle to the higher order frame bundles are constructed. These mappings are the differential geometric structures.
Step 3. The pullbacks of the structure forms of the frame bundles via the mappings are considered. These are called structure forms of the projective structure. The expressions of their exterior differentials in terms of the forms themselves are found. These expressions coincide with the structure equations of a projective space.Moduli space of nonnegatively curved metrics on manifolds of dimension \(4k+1\)https://zbmath.org/1502.530492023-02-24T16:48:17.026759Z"Dessai, Anand"https://zbmath.org/authors/?q=ai:dessai.anandIn this paper the author gives examples of closed manifolds of dimension \(4k +1\), for \(k \geq 2\), for which the moduli spaces of metrics of nonnegative sectional curvature have infinitely many path components. The fundamental group of these manifolds is \(\mathbb{Z}_2\). Examples of manifolds with property and finite fundamental group were known only in dimension 5 and for dimensions \(k \in \mathbb{N}\), with \(4k + 3 \geq 7\).
Reviewer: Antonio Masiello (Bari)Thickness of skeletons of arithmetic hyperbolic orbifoldshttps://zbmath.org/1502.530642023-02-24T16:48:17.026759Z"Alpert, Hannah"https://zbmath.org/authors/?q=ai:alpert.hannah"Belolipetsky, Mikhail"https://zbmath.org/authors/?q=ai:belolipetsky.mikhail-vSummary: We show that closed arithmetic hyperbolic 3-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic \(n\)-orbifolds and find nearly optimal geodesic triangulations of arithmetic hyperbolic 3-orbifolds.The positive scalar curvature cobordism categoryhttps://zbmath.org/1502.530832023-02-24T16:48:17.026759Z"Ebert, Johannes"https://zbmath.org/authors/?q=ai:ebert.johannes-felix"Randal-Williams, Oscar"https://zbmath.org/authors/?q=ai:randal-williams.oscarSummary: We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside \(\mathscr{R}^+(S^d)\) if \(d\ge 6\).
To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fiber of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen's Theorem B and instances of the Gromov-Lawson surgery theorem.
We extend some of the surgery arguments by Galatius and the second-named author to the psc setting to pass between different connectivity conditions. Segal's theory of \(\Gamma\)-spaces is then used to construct the claimed infinite loop space structures.
The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen-Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontryagin classes.
A delooped version of the Atiyah-Singer index theorem proved by the first-named author is, moreover, used to show that the secondary index invariant to real \(K\)-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structureshttps://zbmath.org/1502.531092023-02-24T16:48:17.026759Z"Ndawa, Bertuel Tangue"https://zbmath.org/authors/?q=ai:ndawa.bertuel-tangueSummary: We consider a smooth \(2n\)-manifold \(M\) endowed with a bi-Lagrangian structure \((\omega,\mathcal{F}_1,\mathcal{F}_2)\). That is, \(\omega\) is a symplectic form and \((\mathcal{F}_1,\mathcal{F}_2)\) is a pair of transversal Lagrangian foliations on \((M, \omega)\). Such a structure has an important geometric object called the Hess Connection. Among the high importance of Hess connections, they allow to classify affine bi-Lagrangian structures.
In this work, we show that a bi-Lagrangian structure on \(M\) can be lifted as a bi-Lagrangian structure on its trivial bundle \(M\times\mathbb{R}^n\). Moreover, the lifting of an affine bi-Lagrangian structure is also affine. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on \(M\times\mathbb{R}^n\). This lifting can be lifted again on \((M\times\mathbb{R}^{2n})\times\mathbb{R}^{4n}\), and coincides with the initial dynamic (in our sense) to \(M\times\mathbb{R}^n\). By replacing \(M\times\mathbb{R}^{2n}\) with the tangent bundle \(TM\) or cotangent bundle \(T^*M\) of \(M\), results still hold when \(M\) is parallelizable.Braids, fibered knots, and concordance questionshttps://zbmath.org/1502.531122023-02-24T16:48:17.026759Z"Hubbard, Diana"https://zbmath.org/authors/?q=ai:hubbard.diana"Kawamuro, Keiko"https://zbmath.org/authors/?q=ai:kawamuro.keiko"Kose, Feride Ceren"https://zbmath.org/authors/?q=ai:kose.feride-ceren"Martin, Gage"https://zbmath.org/authors/?q=ai:martin.gage-n"Plamenevskaya, Olga"https://zbmath.org/authors/?q=ai:plamenevskaya.olga"Raoux, Katherine"https://zbmath.org/authors/?q=ai:raoux.katherine"Truong, Linh"https://zbmath.org/authors/?q=ai:truong.linh"Turner, Hannah"https://zbmath.org/authors/?q=ai:turner.hannahSummary: Given a knot in \(S^3\), one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in \(S^3\) can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the knot is fibered -- that is, if its complement fibers over \(S^1\) -- one can consider the monodromy of the fibration. One can ask to what extent properties of these surface diffeomorphisms dictate topological properties of the corresponding knot. In this article we collect observations, conjectures, and questions addressing this, from both the braid perspective and the fibered knot perspective. We particularly focus on exploring whether properties of the surface diffeomorphisms relate to four-dimensional topological properties of knots such as the slice genus.
For the entire collection see [Zbl 1478.53006].Stable Hamiltonian structure and basic cohomologyhttps://zbmath.org/1502.531212023-02-24T16:48:17.026759Z"Acakpo, Bill"https://zbmath.org/authors/?q=ai:acakpo.billSummary: Let \((\omega, \lambda)\) be a stable Hamiltonian structure on a closed oriented manifold \(M\) of dimension \(2n-1\), \({\mathcal{F}}\) the stable Hamiltonian foliation, generated by the Reeb vector field \(R\) of \((\omega, \lambda)\), and \(H_B^k(M, {\mathcal{F}})\), the \(k\)th basic cohomology group of \((M, {\mathcal{F}})\); see Section 1 for definitions. In this paper, we give some topological properties of \((\omega, \lambda)\). In particular, we prove the following results:
\begin{itemize}
\item For all \(k\in \{1, \ldots, n-1\}\),
\[0\ne [\omega^k]\in H_B^{2k}(M, {\mathcal{F}}),\]
which allows us to give an example of a manifold with a Hamiltonian structure, which is not stable.
\item If dim \(H^2_B(M, {\mathcal{F}}) = 1\), then \(M\) is a co-symplectic manifold (symplectic mapping torus), or contact manifold.
\end{itemize}Rigid fibers of integrable systems on cotangent bundleshttps://zbmath.org/1502.531232023-02-24T16:48:17.026759Z"Kawasaki, Morimichi"https://zbmath.org/authors/?q=ai:kawasaki.morimichi"Orita, Ryuma"https://zbmath.org/authors/?q=ai:orita.ryumaSummary: (Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.The golden ratio and quantum topologyhttps://zbmath.org/1502.570072023-02-24T16:48:17.026759Z"Marché, Julien"https://zbmath.org/authors/?q=ai:marche.julienSummary: La topologie quantique naît avec les travaux de Jones, Kauffman puis Witten à la fin des années 80: elle associe à des objets de dimension 3 (nœuds, 3-variétés) des ``invariants'' numériques qui sont relativement faciles à définir mais dont il est difficile de percer le sens topologique. \par Bien que ces théories soient relativement récentes, on peut en trouver des racines plus lointaines, et c'est ce que cet article propose d'explorer. En partant des problémes de coloriages de graphes, on entreprend ici une promenade dans le monde de la topologie quantique en se concentrant sur un seul exemple -- lié au nombre d'or -- mais qui en représente toute la complexité et le mystère.Invariants of links and 3-manifolds that count graph configurationshttps://zbmath.org/1502.570082023-02-24T16:48:17.026759Z"Lescop, Christine"https://zbmath.org/authors/?q=ai:lescop.christineSummary: We present ways of counting configurations of uni-trivalent Feynman graphs in 3-manifolds in order to produce invariants of these 3-manifolds and of their links, following Gauss, Witten, Bar-Natan, Kontsevich and others. We first review the construction of the simplest invariants that can be obtained in our setting. These invariants are the linking number and the Casson invariant of integer homology 3-spheres. Next we see how the involved ingredients, which may be explicitly described using gradient flows of Morse functions, allow us to define a functor on the category of framed tangles in rational homology cylinders. Finally, we describe some properties of our functor, which generalizes both a universal Vassiliev invariant for links in the ambient space and a universal finite type invariant of rational homology 3-spheres.On the functoriality of Khovanov-Floer theorieshttps://zbmath.org/1502.570092023-02-24T16:48:17.026759Z"Baldwin, John A."https://zbmath.org/authors/?q=ai:baldwin.john-a"Hedden, Matthew"https://zbmath.org/authors/?q=ai:hedden.matthew"Lobb, Andrew"https://zbmath.org/authors/?q=ai:lobb.andrewSummary: We introduce the notion of a \textit{Khovanov-Floer theory}. We prove that every page (after \(E_1\)) of the spectral sequence accompanying a Khovanov-Floer theory is a link invariant, and that an oriented link cobordism induces a map on each page which is an invariant of the cobordism up to smooth isotopy rel boundary. We then prove that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov-Floer theories and are therefore \textit{functorial} in the manner described above, as had been conjectured for some time.An introduction to Weinstein handlebodies for complements of smoothed toric divisorshttps://zbmath.org/1502.570112023-02-24T16:48:17.026759Z"Acu, Bahar"https://zbmath.org/authors/?q=ai:acu.bahar"Capovilla-Searle, Orsola"https://zbmath.org/authors/?q=ai:capovilla-searle.orsola"Gadbled, Agnès"https://zbmath.org/authors/?q=ai:gadbled.agnes"Marinković, Aleksandra"https://zbmath.org/authors/?q=ai:marinkovic.aleksandra"Murphy, Emmy"https://zbmath.org/authors/?q=ai:murphy.emmy"Starkston, Laura"https://zbmath.org/authors/?q=ai:starkston.laura"Wu, Angela"https://zbmath.org/authors/?q=ai:wu.angela-ySummary: In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we include several complicated examples at the end of the article to showcase the algorithm and the types of Weinstein Kirby diagrams it produces.
For the entire collection see [Zbl 1478.53006].Cellularization for exceptional spherical space forms and the flag manifold of \(\mathsf{SL}_3 (\mathbb{R})\)https://zbmath.org/1502.570162023-02-24T16:48:17.026759Z"Chirivì, Rocco"https://zbmath.org/authors/?q=ai:chirivi.rocco"Garnier, Arthur"https://zbmath.org/authors/?q=ai:garnier.arthur"Spreafico, Mauro"https://zbmath.org/authors/?q=ai:spreafico.mauro\textit{J. W. Milnor} [Am. J. Math. 79, 623--630 (1957; Zbl 0078.16304)] classified the finite groups that act freely on \(S^3\). For many of these actions equivariant cell decompositions are know. The authors provide such a decomposition in the remaining cases, for the free action of the octahedral group \(\mathcal O\) and the binary icosahedral group \(\mathcal I\). Using curved joins these decompositions extend to higher dimensions. The authors' method also provides a decomposition of the action of the tetrahedral group \(\mathcal T\). The main result is:
Theorem. Every sphere \(S^{4n-1}\), endowed with the natural free action of \(\mathcal O\) (resp., of \(\mathcal I\) or \(\mathcal T\)), admits an explicit equivariant cell decomposition. As a consequence, the associated cellular homology chain complex is explicitly given in terms of matrices with entries in the group algebras \(\mathbb Z[\mathcal O]\), \(\mathbb Z[\mathcal I]\), and \(\mathbb Z[\mathcal T]\), respectively.
Reviewer: Karl Heinz Dovermann (Honolulu)On \(C^0\)-continuity of the spectral norm for symplectically non-aspherical manifoldshttps://zbmath.org/1502.570182023-02-24T16:48:17.026759Z"Kawamoto, Yusuke"https://zbmath.org/authors/?q=ai:kawamoto.yusuke|kawamoto.yusuke.1|kawamoto.yusuke.2In the paper under review, it is proven that the spectral norm is \(C^0\)-continuous for complex projective spaces and negative monotone symplectic manifolds. The first result already appeared in work by \textit{E. Shelukhin} [Invent. Math. 230, No. 1, 321--373 (2022; Zbl 07585648)]. Both of the results follow from new bounds on the spectral norm for rational symplectic manifolds.
For a symplectic manifold \((M, \omega)\) we let \(\operatorname{Ham}(M,\omega)\) denote the group of Hamiltonian diffeomorphisms. The \(C^0\)-closure of the group of Hamiltonian diffeomorphisms is called the group of Hamiltonian homeomorphisms, and is denoted by \(\overline{\operatorname{Ham}}(M,\omega)\). The definition of the spectral norm is via the Piunikhin-Salamon-Schwarz (PSS) isomorphism \(\varPhi_{\text{PSS}, H; \mathbb{K}} \colon QH_\ast(M; \mathbb{K}) \longrightarrow HF(H)\) between the quantum cohomology ring and the Floer homology of a non-degenerate time-dependent Hamiltonian \(H \in C^\infty(\mathbb{R}/\mathbb{Z} \times M, \mathbb{R})\). For \(a \in QH_\ast(M; \mathbb{K}) \setminus \left\{0\right\}\) we define the spectral invariant of \(H\) and \(a\) by
\[
c(H,a) := \inf \left\{\tau \mid \varPhi_{\text{PSS},H;\mathbb{K}}(a) \in \operatorname{im}(HF^\tau(H) \longrightarrow HF(H))\right\},
\]
where \(HF^\tau(H)\) is the action filtered Floer homology group. The spectral norm of a Hamiltonian \(H\) is defined by
\[
\gamma(H) := c(H,[M]) + c(\overline H, [M)]),
\]
where \(\overline H(t,x) := -H(t,\phi^t_H(x))\). For a Hamiltonian diffeomorphism \(\phi \in \operatorname{Ham}(M, \omega)\) we define its spectral norm by
\[
\gamma(\phi) := \inf_{\phi = \phi_H} \gamma(H).
\]
For a Hamiltonian homeomorphism \(\phi \in \overline{\operatorname{Ham}}(M,\omega)\), take a sequence \(\phi_k \in \operatorname{Ham}(M,\omega)\) that \(C^0\)-converges to \(\phi\), and define
\[
\gamma(\phi) := \lim_{k\to \infty} \gamma(\phi_k) = \lim_{k\to \infty}\inf_{\phi_k = \phi_{H_k}} \gamma(H_k).
\]
The main result in the paper under review is the following. Let \((M ,\omega)\) be a rational symplectic manifold (i.e. \(\left\langle \omega, \pi_2(M)\right\rangle = \lambda_0 \mathbb{Z}\) for some constant \(\lambda_0 > 0\)). For any \(\varepsilon > 0\), there exists \(\delta > 0\) such that if \(d_{C^0}(\mathrm{id}, \phi_H) < \delta\), then
\[
|\gamma(H) - l \cdot \lambda_0| < \varepsilon
\]
for some integer \(l \in \mathbb{Z}\) depending on \(H\). \(C^0\)-continuity follows from bounding the spectral norm by a number strictly smaller than \(\lambda_0\).
Other than the main result, applications such as \(C^0\)-continuity of barcodes, the \(C^0\)-Arnold conjecture and the displaced disks problem are discussed.
Reviewer: Johan Asplund (New York)On triangulations of orbifolds and formalityhttps://zbmath.org/1502.570192023-02-24T16:48:17.026759Z"Du, Cheng-Yong"https://zbmath.org/authors/?q=ai:du.chengyong|du.chengyong.1"He, Kaimin"https://zbmath.org/authors/?q=ai:he.kaimin"Xue, Han"https://zbmath.org/authors/?q=ai:xue.hanOrbifolds are generalizations of manifolds, locally they are quotients of Euclidean spaces by smooth actions of finite groups. The underlying space of an orbifold can be stratified by the isotropy types, and it admits a triangulation as a stratified space. Orbifolds are naturally associated with two differential graded algebras. One of them is the DGA of orbifold differential forms, i.e., the orbifold de Rham algebra. If the underlying space of the orbifold admits a smooth triangulation, there is also the DGA of piecewise polynomial differential forms of the triangulation. The authors show that these two DGAs are weakly equivalent, which means that they are connected by a chain of quasi-isomorphisms. It follows that one of these DGAs is formal, if and only if the other one is. The authors also prove that global quotient orbifolds and global homogeneous isotropy orbifolds admit smooth triangulations.
Reviewer: Marja Kankaanrinta (Helsinki)A generalization of the Hopf degree theoremhttps://zbmath.org/1502.570202023-02-24T16:48:17.026759Z"Kvalheim, Matthew D."https://zbmath.org/authors/?q=ai:kvalheim.matthew-dSummary: The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth \(n\)-manifold \(M\) to the \(n\)-sphere are classified by their degree. Such a map is equivalent to a section of the trivial \(n\)-sphere bundle over \(M\). A generalization of the Hopf theorem is obtained for sections of nontrivial oriented \(n\)-sphere bundles over \(M\).Pseudo-angle systems and the simplicial Gauss-Bonnet-Chern theoremhttps://zbmath.org/1502.570212023-02-24T16:48:17.026759Z"Klaus, Stephan"https://zbmath.org/authors/?q=ai:klaus.stephanSummary: In a previous article [\textit{S. Klaus}, Front. Math. China 11, No. 5, 1345--1362 (2016; Zbl 1353.51018)], we proved a simplicial version of the Gauss-Bonnet-Chern theorem which expresses the Euler characteristics of a Euclidean simplicial complex \(K\) of any dimension as a sum of its Gauss curvatures over all vertices: \(\sum_{x\in K_0} \kappa(x)=\chi(K)\), where the simplicial Gauss curvature \(\kappa(x)\) is given as a logarithmic sum over the normed dihedral angle defects of all simplices which are adjacent to the vertex \(x\). Here, we generalize this result to systems of pseudo-angles which are certain functions defined on the moduli space of Euclidean simplices. Then we consider combinatorial Riemannian manifolds and introduce simplicial sectional curvature. In the final section, we consider its relation to the simplicial Gauss pseudo-curvature. In particular, the hypothetical existence of a systems of pseudo-angles with certain properties would imply the Hopf conjecture for manifolds of positive sectional curvature.
For the entire collection see [Zbl 07244874].Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotientshttps://zbmath.org/1502.570222023-02-24T16:48:17.026759Z"Wermelinger, Jonathan"https://zbmath.org/authors/?q=ai:wermelinger.jonathanTwo recent results [\textit{M. J. Goodman}, Ann. Global Anal. Geom. 57, No. 2, 305--320 (2020; Zbl 1447.58012); \textit{A. Dessai}, ``On the moduli space of nonnegatively curved metrics on Milnor spheres'', Preprint, \url{arXiv:1712.08821}] concern metrics of non-negative sectional or positive Ricci curvature on total spaces of linear \(S^3 \to E \to S^4\) bundles, respectively: provided these total spaces are rational homology spheres, the corresponding moduli spaces of such metrics have infinitely many path components. Milnor's classical homology \(7\)-spheres are examples of such manifolds.
This paper proves analogues of these results for total spaces of linear \(S^7\)-bundles over \(S^8\), including Shimada's homology \(15\)-spheres [\textit{N. Shimada}, Nagoya Math. J. 12, 59--69 (1957; Zbl 0145.20303)]. More precisely, it is proven that the space of positive Ricci curvature metrics on such a manifold has infinitely many components, provided the manifold is a rational homology sphere. However, it is not known whether these manifolds always support metrics of non-negative sectional curvature.
The antipodal action on the fibers of any sphere bundle gives rise to an involution on the total space. For the examples referred to above, the resulting quotients are called Milnor/Shimada projective spaces. The paper recalls that there are 16 different diffeomorphism types of Milnor projective spaces. Then it is proven that the moduli spaces of metrics of positive sectional or positive Ricci curvature on each of them has infinitely many path components. It is also shown that there are at least 4096 different diffeomorphism types of Shimada projective spaces for which the moduli spaces of positive Ricci curvature metrics has infinitely many path components. The number of different diffeomorphism types of all Shimada projective spaces seems to be unknown, but the paper proves that there are only finitely many.
The tools for the diffeomorphism classification are very classical. In particular, the Browder-Livesay invariant plays an important role, and using it, the Eells-Kuiper invariant [\textit{J. Eells jun.} and \textit{N. H. Kuiper}, Ann. Mat. Pura Appl. (4) 60, 93--110 (1962; Zbl 0119.18704)] is computed for the Shimada projective spaces. For Milnor projective spaces this was already done in [\textit{Z. Tang} and \textit{W. Zhang}, Adv. Math. 254, 41--48 (2014; Zbl 1290.51005)]. The bound of at least 4096 different diffeomorphism types alluded to above is obtained by means of a computer program, as the author did not find a way to solve the resulting arithmetic.
Reviewer: Jens Reinhold (Münster)Deformations in the general position of the optimal functions on oriented surfaces with boundaryhttps://zbmath.org/1502.570232023-02-24T16:48:17.026759Z"Hladysh, B. I."https://zbmath.org/authors/?q=ai:hladysh.bohdana-i"Prishlyak, O. O."https://zbmath.org/authors/?q=ai:pryshlyak.oleksandr-olegovychSummary: We consider simple functions with nondegenerate singularities on smooth compact oriented surfaces with boundary. The relationship between the optimality and polarity of Morse functions, \(m\)-functions and \(mm\)-functions on smooth compact oriented connected surfaces is described. The concept of equipped Kronrod-Reeb graph is used to define deformation in the general position. Moreover, we present the entire list of deformations of simple functions of one of the classes described above on a torus, on a 2-dimensional disk with boundary, and on the connected sum of two tori.Universality in anomaly flowhttps://zbmath.org/1502.810402023-02-24T16:48:17.026759Z"Hosotani, Yutaka"https://zbmath.org/authors/?q=ai:hosotani.yutakaSummary: Universality in anomaly flow by an Aharonov-Bohm phase \(\theta_H\) is shown in the flat \(M^4 \times (S^1/Z_2)\) spacetime and in the Randall-Sundrum (RS) warped space. We analyze the \(SU(2)\) gauge theory with doublet fermions. With orbifold boundary conditions the \(U(1)\) part of the gauge symmetry remains unbroken at \(\theta_H = 0\) and \(\pi\). Chiral anomalies smoothly vary with \(\theta_H\) in the RS space. It is shown that the anomaly coefficients associated with this anomaly flow are expressed in terms of the values of the wave functions of the gauge fields at the UV and IR branes in the RS space. The anomaly coefficients depend on \(\theta_H\), the warp factor of the RS space, and the orbifold boundary conditions for fermions, but not on the bulk mass parameters of fermions.