Recent zbMATH articles in MSC 57https://zbmath.org/atom/cc/572024-02-28T19:32:02.718555ZWerkzeugExotic Lagrangian tori in Grassmannianshttps://zbmath.org/1527.140982024-02-28T19:32:02.718555Z"Castronovo, Marco"https://zbmath.org/authors/?q=ai:castronovo.marcoLagrangian tori of symplectic manifolds have relevance in various areas, e.g. Hamiltonian mechanics, mirror symmetry, low-dimensional topology, Seiberg-Witten invariants. In the paper under review, the author gives a construction of a Lagrangian torus \(L_{\mathfrak{s}}\) of the Grassmannian \(Gr(k, n)\) of \(k\)-dimensional subspaces of \(\mathbb{C}^n\). The torus \(L_{\mathfrak{s}}\) is attached to a Plucker sequence \(\mathfrak{s}\) of type \((k, n)\) and it is obtained by a technique of degeneration to a toric variety. In this way, the author gives examples of Lagrangian tori that are not displaceable and are not Hamiltonian isotopic one to the other. Moreover, the derived Fukaya category of \(Gr(2, n)\), in various cases is shown to be split generated by objects supported on a finite number of these tori.
Reviewer: Francesco Esposito (Padova)A geometric model of Brauer graph algebrashttps://zbmath.org/1527.160162024-02-28T19:32:02.718555Z"Adachi, Takahide"https://zbmath.org/authors/?q=ai:adachi.takahide"Chan, Aaron"https://zbmath.org/authors/?q=ai:chan.aaron-c-sSummary: In this note, we give a construction of pretilting complexes for a Brauer graph algebra from certain curves on a surface.
For the entire collection see [Zbl 1415.16002].Unrestricted quantum moduli algebras. I: The case of punctured sphereshttps://zbmath.org/1527.170062024-02-28T19:32:02.718555Z"Baseilhac, Stéphane"https://zbmath.org/authors/?q=ai:baseilhac.stephane"Roche, Philippe"https://zbmath.org/authors/?q=ai:roche.philippe|roche.philippe-eSummary: Let \(\Sigma\) be a finite type surface, and \(G\) a complex algebraic simple Lie group with Lie algebra \(\mathfrak{g}\). The quantum moduli algebra of \((\Sigma,G)\) is a quantization of the ring of functions of \(X_G(\Sigma)\), the variety of \(G\)-characters of \(\pi_1(\Sigma)\), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid 1990s. It can be realized as the invariant subalgebra of so-called graph algebras, which are \(U_q(\mathfrak{g})\)-module-algebras associated to graphs on \(\Sigma\), where \(U_q(\mathfrak{g})\) is the quantum group corresponding to \(G\). We study the structure of the quantum moduli algebra in the case where \(\Sigma\) is a sphere with \(n+1\) open disks removed, \(n\geq 1\), using the graph algebra of the ``daisy'' graph on \(\Sigma\) to make computations easier. We provide new results that hold for arbitrary \(G\) and generic \(q\), and develop the theory in the case where \(q=\epsilon\), a primitive root of unity of odd order, and \(G=\mathrm{SL}(2,\mathbb{C})\). In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring \(\mathcal{O}(G^n)\). We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on \(\mathcal{O}(G^n)\). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of \(\mathbb{C}[X_G(\Sigma)]\) endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra \(K_{\zeta}(\Sigma)\) at \(\zeta:=\mathrm{i}\epsilon^{1/2}\) with this quantum moduli algebra specialized at \(q=\epsilon\). This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on \(K_{\zeta}(\Sigma)\).Sewn sphere cohomologies for vertex algebras.https://zbmath.org/1527.170142024-02-28T19:32:02.718555Z"Zuevsky, Alexander"https://zbmath.org/authors/?q=ai:zuevsky.alexanderSummary: We define sewn elliptic cohomologies for vertex algebras by sewing procedure for coboundary operators.From braid groups to mapping class groupshttps://zbmath.org/1527.200542024-02-28T19:32:02.718555Z"Chen, Lei"https://zbmath.org/authors/?q=ai:chen.lei.3"Mukherjea, Aru"https://zbmath.org/authors/?q=ai:mukherjea.aru\textit{F. Castel} [Geometric representations of the braid groups. Paris: Société Mathématique de France (SMF) (2016; Zbl 1441.20001)] gave a complete classification of homomorphisms from the Artin braid group \(B_n\) to the pure mapping class group PMod\((S_{g,p})\) of the surface \(S_{g,p}\) of genus \(g\) with \(p\) punctures, for \(n \ge 6\) and \(g \le \frac{n}{2}\).
Using totally symmetric sets and totally symmetric multicurves as the main technical tools, the authors of this paper give a new proof of the result by Castel [loc. cit.] and also extend the classification to the case of \(g < n-2\) but in a reduced range of \(n \ge 23\). The main theorem in this paper states the following:
(Main theorem) For \(k \ge 13\), \(n=2k\) or \(n=2k+1\) and \(g \le 2k-3\), any homomorphism \(\rho : B_n \to \mathrm{PMod}(S_{g,p})\) is either trivial, standard or negative-standard homomorphism, up to transvection.
As a corollary to the above classification theorem, the authors partially recover a result by \textit{J. Aramayona} and \textit{J. Souto} [Geom. Topol. 16, No. 4, 2285--2341 (2012; Zbl 1262.57003)] as follows:
(Corollary) For \(g \ge 23\) and \(h < 2g\), any homomorphism \(\rho : \mathrm{PMod}(S_{g,p}) \to \mathrm{PMod}(S_{h,q})\) is either trivial or induced by an embedding.
Reviewer: Soumya Dey (Chennai)Surface-like boundaries of hyperbolic groupshttps://zbmath.org/1527.200622024-02-28T19:32:02.718555Z"Beeker, Benjamin"https://zbmath.org/authors/?q=ai:beeker.benjamin"Lazarovich, Nir"https://zbmath.org/authors/?q=ai:lazarovich.nirThe authors prove that if a hyperbolic group \(G\) acts properly and cocompactly on a \(\mathrm{CAT}(0)\) cube complex such that
\begin{itemize}
\item each hyperplane stabilizer is virtually isomorphic to the fundamental group of a closed surface of genus at least \(2\), and
\item any two distinct hyperplane stabilizers have virtually cyclic intersection
\end{itemize}
then the Gromov boundary \(\partial G\) of \(G\) is homeomorphic to the \(2\)-sphere \(S^2\), to the Pontryagin sphere \(P\) or to the Pontryagin surface \(\Pi(2)\).
The spaces \(P\) and \(\Pi(2)\) are inverse limits of closed surfaces and are particular examples of \textit{trees of manifolds} (also known as \textit{Jakobsche spaces}) which have seen much study in relation to boundaries of hyperbolic groups [\textit{W. Jakobsche}, Fundam. Math. 137, No. 2, 81--95 (1991; Zbl 0727.57018); \textit{J. Świątkowski}, Geom. Topol. 24, No. 2, 593--622 (2020; Zbl 07256596)].
Reviewer: Nima Hoda (Ithaca)Klein-Maskit combination theorem for Anosov subgroups: free productshttps://zbmath.org/1527.200632024-02-28T19:32:02.718555Z"Dey, Subhadip"https://zbmath.org/authors/?q=ai:dey.subhadip"Kapovich, Michael"https://zbmath.org/authors/?q=ai:kapovich.michaelThe classical Klein-Maskit combination theorem for Kleinian groups (discrete isometry groups of the hyperbolic space \(\mathbb{H}^{n}\)) establishes sufficient conditions for a subgroup \(\Gamma < G = \mathrm{Isom}(\mathbb{H}^{n})\) generated by two discrete subgroups \(\Gamma_{A}\), \(\Gamma_{B}\) of \(G\) to be discrete and isomorphic to the free product \(\Gamma_{A} \ast \Gamma_{B}\).
In the paper under review, the authors prove a generalization of the Klein-Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if \(\Gamma_{A}\) and \(\Gamma_{B}\) are Anosov subgroups of a semisimple Lie group \(G\) of noncompact type, then under suitable topological assumptions, the subgroup \(\big \langle \Gamma_{A}, \Gamma_{B} \big \rangle \leq G\) is again Anosov and is naturally isomorphic to the free product \(\Gamma_{A} \ast \Gamma_{B}\).
Reviewer: Enrico Jabara (Venezia)Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groupshttps://zbmath.org/1527.200682024-02-28T19:32:02.718555Z"Schesler, Eduard"https://zbmath.org/authors/?q=ai:schesler.eduard"Zaremsky, Matthew C. B."https://zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderLet \(\mathcal{P}\) be a group theoretic property. A group \(G\) is said to algebraically \(\mathcal{P}\)-fiber if it admits an epimorphism \(G \rightarrow \mathbb{Z}\) whose kernel has property \(\mathcal{P}\). The question raised by this paper is to find which right-angled Coxeter groups \(\mathrm{F}_{n}\)-fibers or \(\mathrm{FP}_{n}\)-fibers, and which do not.
Let \(G\) be a right-angled Coxeter group with underlying graph \(\mathcal{G}=(V,E)\). In the paper under review, the authors determine a sufficient condition on \(\mathcal{G}\) under which \(G\) algebraically \(\mathrm{F}_{n}\)-fibers. Then they prove that this condition holds with high probability when \(\mathcal{G}\) is the 1-skeleton of a finite building. Furthermore, they use their techniques to present examples where the kernel is of type \(\mathrm{F}_{2}\) but not \(\mathrm{FP}_{3}\) and examples where the right-angled Coxeter group is hyperbolic and the kernel is finitely generated and non-hyperbolic.
Reviewer: Egle Bettio (Venezia)Reflection trees of graphs as boundaries of Coxeter groupshttps://zbmath.org/1527.200692024-02-28T19:32:02.718555Z"Świątkowski, Jacek"https://zbmath.org/authors/?q=ai:swiatkowski.jacekSummary: To any finite graph \(X\) (viewed as a topological space) we associate an explicit compact metric space \(\mathcal{X}^r(X)\), which we call \textit{the reflection tree of graphs \(X\)}. This space is of topological dimension \(\leq 1\) and its connected components are locally connected. We show that if \(X\) is appropriately triangulated (as a simplicial graph \(\Gamma\) for which \(X\) is the geometric realization) then the visual boundary \(\partial_\infty(W,S)\) of the right-angled Coxeter system \((W,S)\) with the nerve isomorphic to \(\Gamma\) is homeomorphic to \(\mathcal{X}^r(X)\). For each \(X\), this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space \(\mathcal{X}^r(X)\).Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaceshttps://zbmath.org/1527.200702024-02-28T19:32:02.718555Z"Choi, Inhyeok"https://zbmath.org/authors/?q=ai:choi.inhyeokLet \((X,d)\) either be a Gromov hyperbolic space or Teichmüller space of a finite-type hyperbolic surface. Fix a reference point \(o \in X\) and denote by \(G\) the isometry group of \(X\). For \(g \in G\) we define the translation length \(\tau(g) := \lim_{n \to \infty} \frac{1}{n} d(o,g^n o)\).
Let \(\mu\) be a non-elementary discrete probability measure on \(G\) and denote by \(\omega = (\omega_n)_{n=1}^\infty\) the random walk generated by it. Theorem A of the paper states: If \(\mu\) has finite first moment then there exists a constant \(K < \infty\) such that
\[
\limsup_{n \to \infty} \frac{1}{\log n} |\tau(\omega_n) - d(o,\omega_n o)| < K
\]
for almost every \(\omega\).
The paper goes on to prove several more theorems related to the central limit theorem and about geodesic tracking. To illustrate the latter, let me state Theorem D.(1): Assume that \(X\) is geodesic. If \(\mu\) has finite \(p\)-th moment for some \(p > 0\), then for almost every \(\omega\) there exists a quasi-geodesic \(\gamma\) such that
\[
\lim_{n \to \infty} \frac{1}{n^{1/2p}} d(\omega_n o,\gamma) = 0\,.
\]
Reviewer: Alexander Engel (Greifswald)Cone-equivalent nilpotent groups with different Dehn functionshttps://zbmath.org/1527.200712024-02-28T19:32:02.718555Z"Llosa Isenrich, Claudio"https://zbmath.org/authors/?q=ai:llosa-isenrich.claudio"Pallier, Gabriel"https://zbmath.org/authors/?q=ai:pallier.gabriel"Tessera, Romain"https://zbmath.org/authors/?q=ai:tessera.romainAuthor's abstract: For every \(k\geq 3\), we exhibit a simply connected \(k\)-nilpotent Lie group \(N_k\) whose Dehn function behaves like \(n^k\), while the Dehn function of its associated Carnot graded group \(\mathrm{gr}(N_k)\) behaves like \(n^{k+1}\). This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bi-Lipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer \(k\geq 4\), the centralised Dehn function of \(N_k\) behaves like \(n^{k-1}\) and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bi-Lipschitz equivalences (SBEs). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasi-isometries where the additive error is replaced by a sublinearly growing function \(v\). We show that a \(v\)-SBE between \(N_k\) and \(\mathrm{gr}(N_k)\) must satisfy \(v(n)\geq n^{1/(2k+2)}\), strengthening the fact that those two groups are not quasi-isometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.
Reviewer: Alexander Ivanovich Budkin (Barnaul)Topological groups with a compact open subgroup, relative hyperbolicity and coherencehttps://zbmath.org/1527.220082024-02-28T19:32:02.718555Z"Arora, Shivam"https://zbmath.org/authors/?q=ai:arora.shivam"Martínez-Pedroza, Eduardo"https://zbmath.org/authors/?q=ai:martinez-pedroza.eduardoIn this article, the pairs \((\mathcal G, H)\) are studied where \(\mathcal G\) is a topological group with a compact open subgroup, and \(\mathcal H\) is a finite collection of open subgroups.
Let \(G\) be a topological group. \(G\) is compactly generated if it admits a compact generating set. It is compactly presented if it admits a standard presentation \(\langle S\mid R\rangle\) where \(S\) is a compact subset of \(G\) and \(R\) is a set of words in \(S\) of uniformly bounded length. A topological group is said to be coherent if every closed compactly generated subgroup is compactly presented. There is also a notion of \(\mathcal G\) being compactly generated and compactly presented relative to \(\mathcal H\). In this case the authors show if \(\mathcal G\) is compactly generated, then each subgroup \(H\in\mathcal H\) is compactly generated and if each subgroup \(H\in\mathcal H\) is compactly presented, then \(G\) is compactly presented.
An approach is presented to relative hyperbolicity for pairs \((\mathcal G, H)\) based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, it is shown that if \(\mathcal G\) is hyperbolic relative to \(\mathcal H\) then \(\mathcal G\) is compactly presented relative to \(\mathcal H\).
As an application combination results for coherent topological groups with a compact open subgroup are shown, and the McCammond-Wise perimeter method is applied to this general framework.
Reviewer: Stephan Rosebrock (Karlsruhe)Concentration of measure for classical Lie groupshttps://zbmath.org/1527.220142024-02-28T19:32:02.718555Z"Cacciatori, Sergio L."https://zbmath.org/authors/?q=ai:cacciatori.sergio-luigi"Ursino, Pietro"https://zbmath.org/authors/?q=ai:ursino.pietroSummary: We study the concentration of measure in metric-measurable (mm)-spaces. We define the notion of concentration locus of a flag sequence of metric-measurable (mm)-spaces. Some examples of infinite group action on an infinite dimensional compact and non-compact manifold show the role played by the trajectory of concentration locus. We also provide some applications in physics, which emphasize the role of concentration of measure in gravitational effects.On the limit set of a spherical CR uniformizationhttps://zbmath.org/1527.220162024-02-28T19:32:02.718555Z"Acosta, Miguel"https://zbmath.org/authors/?q=ai:acosta.miguelSummary: We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of \(\mathbb{R} \)-circles and contains a Hopf link with three components. We also show that the fundamental group of its complement in \(S^3\) is not finitely generated. Additionally, we prove that rank-one spherical CR cusps are quotients of horotubes.Multiply transitive Lie group of transformations as a physical structurehttps://zbmath.org/1527.220292024-02-28T19:32:02.718555Z"Kyrov, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:kyrov.vladimir-aleksandrovichSummary: We establish a connection between physical structures and Lie groups and prove that the physical structure of rank \((n+1,2)\), \(n\in\mathbb{N} \), on a smooth manifold is isotopic to an almost \(n\)-transitive Lie group of transformations. Afterwards, we prove that an almost \(n\)-transitive Lie group of transformations is isotopic to a physical structure of rank \((n+1,2)\).Transversal families of nonlinear projections and generalizations of Favard lengthhttps://zbmath.org/1527.280022024-02-28T19:32:02.718555Z"Bongers, Rosemarie"https://zbmath.org/authors/?q=ai:bongers.rosemarie"Taylor, Krystal"https://zbmath.org/authors/?q=ai:taylor.krystalThe Favard length of a set \(E\subset\mathbb{R}^2\) is defined by \(\mathrm{Fav}(E)=\frac{1}{\pi}\int_0^{\pi}|P_{\theta}(E)|d\theta\), where \(P_{\theta}\) is the orthogonal projection into a line \(L_{\theta}\) through the origin at angle \(\theta\) from the positive \(x\)-axis and \(|\cdot|\) denotes the 1-dimensional Hausdorff measure. The aim of the paper is to prove a generalization of a theorem due to \textit{P. Mattila} [Indiana Univ. Math. J. 39, No. 1, 185--198 (1990; Zbl 0682.28003)] concerning the Favard length for neighbourhoods in the case when families of projections are not orthogonal projections. It is assumed that the families of projections satisfy a so-called s-transversality condition (def. 1.2 and 1.3), which means some compatibility condition for different parameters. The most important case considered in the paper is when \(s=m\), where \(m\) is the dimension of the target space. The natural example of a transversal family is the collection of orthogonal projections from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) for some \(m<n\). Another example is the family of radial maps defined in the following way: for a point \(a\in\mathbb{R}^n\) the radial projection with a center \(a\) maps \(\mathbb{R}^n\diagdown\{a\}\) into \(S^{n-1}\) via \(P_a(x)=\frac{x-a}{|x-a|}\).
The generalization of Mattila's theorem is the following result
Theorem 1.5. [average nonlinear projection length for neighborhoods] Assume that \(\{ \widetilde{\pi_{\alpha}} : \alpha \in A \}\) is an \(m\)-transversal family of projections into an \(m\)-dimensional space. Fix a positive Borel probability measure \(\mu\) supported on a compact set \(F\subseteq\Omega\), so that
\[
\mu(B(x,r))\lesssim r^t
\]
for all \(x\in\Omega\) and \(0<r<\infty\).
\begin{itemize}
\item If \(t<m\), then
\[
\int_A h(\widetilde{\pi_{\alpha}}F(r))d\psi(\alpha)\gtrsim r^{m-1}.
\]
\item If \(t=m\), then
\[
\int_A h(\widetilde{\pi_{\alpha}}F(r))d\psi(\alpha)\gtrsim (\log r^{-1})^{-1}.
\]
\end{itemize}
The authors have introduced also the notion of the Favard surface length for sets is higher dimensions and proved the following theorem (for dimension 3 with the observation that the change for greater dimension offers no special difficulty):
Theorem 1.10. [Favard surface length of neighborhoods.] Let \(E\) be a compact set in the plane and \(\Gamma\) denote a surface in \(\mathbb{R}^3\) defined by \(\Gamma=\{ (t,\gamma(t)) : t\in I \}\), where \(\gamma : \mathbb{R}^2 \to \mathbb{R}\), \(\gamma(s) = f(|s|)\), and \(f : \mathbb{R} \to \mathbb{R}\) is a \(C^2\) function on a nonempty compact interval \(I\) satisfying \(f(x) = f(-x)\), with \(f''>0\) on \(I\). Assume further that \(E\) supports a Borel probability measure \(\mu\) with the \(t\)-dimensional growth condition \(\mu (B(x,r))\lesssim r^t\) for all \(x \in E\), \(0 < r < \infty\). The following statements hold:
\begin{itemize}
\item The family of curve projections \(\Phi_{\alpha}\) is 1-transversal.
\item If \(t < 1\), then for all sufficiently small \(r\) we have
\[
\mathrm{Fav}_{\Gamma}(E(r))\gtrsim r^{1-t}.
\]
\item If \(t = 1\), then for all sufficiently small \(r\) we have
\[
\mathrm{Fav}_{\Gamma}(E(r))\gtrsim (\log r^{-1})^{-1}.
\]
\end{itemize}
The notation \(A\lesssim B\) means always that there exists a constant \(C\) so that \(A \leq C \cdot B\) and \(A \sim B\) means that \(A \lesssim B\) and \(B \lesssim A\).
Reviewer: Władysław Wilczyński (Łódź)There are no exotic ladder surfaceshttps://zbmath.org/1527.300302024-02-28T19:32:02.718555Z"Basmajian, Ara"https://zbmath.org/authors/?q=ai:basmajian.ara-s"Vlamis, Nicholas G."https://zbmath.org/authors/?q=ai:vlamis.nicholas-gA Riemann surface \(X\) is \textit{quasiconformally homogeneous} (QCH) if there exists \(K\geq 1\) such that for any two points \(x,y\in X\) there exists a \(K\)-quasiconformal map \(f\colon X\to X\) such that \(f(x)=y\). Although we have a complete characterization of \textit{conformally homogeneous} surfaces (i.e., with \(K=1\)), it remains an open problem to characterize QCH surfaces that are hyperbolic.
It is known that there are QCH exotic surfaces (i.e., that are not quasiconformally equivalent to regular covers of closed orbifolds), a phenomenon that does not occur in higher dimensions, as proved by \textit{P. Bonfert-Taylor} et al. [Math. Ann. 331, No. 2, 281--295 (2005; Zbl 1063.30020)]. In the present paper the authors prove that every QCH surface topologically covers a closed surface. As a consequence, the only non-compact QCH surfaces, up to homeomorphism, are
\begin{enumerate}
\item the plane,
\item the annulus,
\item the Cantor tree surface (i.e., the planar surface whose space of ends is a Cantor space),
\item the blooming Cantor tree surface (i.e., the infinite-genus surface with no planar ends and whose space of ends is a Cantor space),
\item the Loch Ness monster surface (i.e., the one-ended infinite genus surface), and
\item the ladder surface (i.e., the two-ended infinite genus surface).
\end{enumerate}
Out of those six cases, note that all simply connected surfaces are already QCH, whereas an annulus is QCH if and only if its universal cover is \(\mathbb C\); in fact, in these two cases QCH surfaces are actually conformally homogeneous. Therefore, there are only four cases remaining to consider.
The main result of the paper classifies completely the 6th case, i.e., QCH ladder surfaces: a hyperbolic ladder surface is QCH if and only if it is quasiconformally equivalent to a regular cover of a closed hyperbolic surface. In other words, there are no QCH exotic ladder surfaces.
Reviewer: Dimitrios Ntalampekos (Stony Brook)Riemann surfaces of second kind and effective finiteness theoremshttps://zbmath.org/1527.320092024-02-28T19:32:02.718555Z"Jöricke, Burglind"https://zbmath.org/authors/?q=ai:joricke.burglindOne of the oldest finiteness theorems in the theory of Riemann surfaces says that for a closed connected Riemann surface \(X\) there are (up to isomorphism) only finitely many non-constant holomorphic mappings \(f: X \rightarrow Y\), where \(Y\) ranges over all closed Riemann surfaces of genus at least 2.
In the paper under review, the author generalizes this result to some classes of open Riemann surfaces. More specifically, the domain \(X\) is a connected Riemann surface of genus \(0\) and \(m+1 \geq 1\) holes and the target \(Y\) is \(\mathbb{P}^{1}\setminus \{1, -1, \infty\}\). The author gives an explicit upper bound on the number of irreducible holomorphic maps \(f:X\rightarrow Y\) up to homotopy, in terms of a conformal invariant of the thrice punctured sphere defined in terms of extreman length.
Reviewer: Andrea Tamburelli (Houston)Singular holomorphic foliations by curves. II: Negative Lyapunov exponenthttps://zbmath.org/1527.320192024-02-28T19:32:02.718555Z"Viêt-Anh Nguyên"https://zbmath.org/authors/?q=ai:nguyen-viet-anh.Summary: Let \(\mathscr{F}\) be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface \(X\) satisfying the following two conditions: the singular points of \(\mathscr{F}\) are all hyperbolic; \(\mathscr{F}\) is Brody hyperbolic. Then we establish cohomological formulas for the Lyapunov exponent and the Poincaré mass of an extremal positive \({dd^c}\)-closed current tangent to \(\mathscr{F}\). If, moreover, there is no nonzero positive closed current tangent to \(\mathscr{F}\), then we show that the Lyapunov exponent \(\chi (\mathscr{F})\) of \(\mathscr{F}\), which is, by definition, the Lyapunov exponent of the unique normalized positive \({dd^c}\)-closed current tangent to \(\mathscr{F}\), is a strictly negative real number. As an application, we compute the Lyapunov exponent of a generic foliation with a given degree in \(\mathbb{P}^2\).
For Part I see [the author, Invent. Math. 212, No. 2, 531--618 (2018; Zbl 1484.32037)].Correlation of the renormalized Hilbert length for convex projective surfaceshttps://zbmath.org/1527.370382024-02-28T19:32:02.718555Z"Dai, Xian"https://zbmath.org/authors/?q=ai:dai.xian"Martone, Giuseppe"https://zbmath.org/authors/?q=ai:martone.giuseppeSummary: In this paper, we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The \textit{correlation number} in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of \textit{R. Schwartz} and \textit{R. Sharp} [Commun. Math. Phys. 153, No. 2, 423--430 (1993; Zbl 0772.58045). In contrast, we provide examples of diverging sequences, defined via \textit{cubic rays}, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.Rotated odometershttps://zbmath.org/1527.370402024-02-28T19:32:02.718555Z"Bruin, Henk"https://zbmath.org/authors/?q=ai:bruin.henk"Lukina, Olga"https://zbmath.org/authors/?q=ai:lukina.olgaThe authors study a family of infinite interval exchange transformations called rotated odometers which can be viewed as perturbations of the von Neumann-Kakutani map. They prove that any rotated odometer \(F_\pi\) can be obtained as the first return map of a flow of rational slope on a translation surface with infinite genus. Another topic of this paper is to consider which properties of the von Neumann-Kakutani map are preserved after perturbation. It is shown that the minimality of the von Neumann-Kakutani map may be lost, but the minimal subset of the aperiodic subsystem persists.
The authors also show that the Lebesgue measure is ergodic for \(F_\pi\) if and only if there are no periodic points. They also obtain a bound on the number of ergodic invariant measures on the aperiodic subsystem and show that the unique minimal subsystem is uniquely ergodic. This is done by showing that each aperiodic subsystem is measurably isomorphic to a Bratteli-Vershik system on a Cantor set. Furthermore, they show that \(F_\pi\) has zero topological entropy.
Since the von Neumann-Kakutani map is measurably isomorphic to the dyadic odometer and some properties of the dyadic odometer such as minimality may be lost under perturbation, the authors investigate if the dyadic odometer is preserved as a factor for both the aperiodic system and its unique minimal set.
Reviewer: Steve Pederson (Atlanta)The Schwarz-Milnor lemma for braids and area-preserving diffeomorphismshttps://zbmath.org/1527.370432024-02-28T19:32:02.718555Z"Brandenbursky, Michael"https://zbmath.org/authors/?q=ai:brandenbursky.michael"Marcinkowski, Michał"https://zbmath.org/authors/?q=ai:marcinkowski.michal"Shelukhin, Egor"https://zbmath.org/authors/?q=ai:shelukhin.egorThis paper studies the identity component of the group of area-preserving diffeomorphisms of orientable surfaces (denoted by \(\mathcal{G}\), and more precisely defined in the paper). Specifically, the authors show how to obtain a number of results using a nice strategy that relies on the Schwarz-Milnor lemma. Given a diffeomorphism \(\phi\) isotopic to the identity, we can define a norm using the ``shortest'' isotopy from \(1\) to \(\phi\). We can also compute the word length (in the fundamental group of the surface, after suitably closing the arc) of the trajectory of a point under an isotopy, and we can average this word length norm over all trajectories in the configuration space of \(n\) points in the surface. The main technical result bounds this average word length norm by \(l_1(\phi)\).
This has a number of corollaries. First, it controls the image of a natural class of quasimorphisms (Gambaudo-Ghys quasimorphisms) on the space of area-preserving diffeomorphisms and shows they are continuous in the \(L^p\) metric. It shows that the \(L^1\) diameter of \(\mathcal{G}\) is infinite. It shows that \(\mathcal{G}\) admits a quasi-isometric embedding of any right-angled Artin group, and also admits a bi-Lipschitz embedding of \(\mathbb{R}^k\) for any \(k\).
Although the proofs are unavoidably technical, the paper does an excellent job of explaining the background and context and leading the reader through the story of the motivations and arguments.
Reviewer: Alden Walker (Chicago)Combining rational maps and Kleinian groups via orbit equivalencehttps://zbmath.org/1527.370472024-02-28T19:32:02.718555Z"Mj, Mahan"https://zbmath.org/authors/?q=ai:mj.mahan"Mukherjee, Sabyasachi"https://zbmath.org/authors/?q=ai:mukherjee.sabyasachiSummary: We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are orbit equivalent to Fuchsian punctured sphere groups. We call these \textit{higher Bowen-Series} maps. The existence of this class ensures that the Teichmüller space of matings has one component corresponding to Bowen-Series maps and one corresponding to higher Bowen-Series maps. We also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers boundary groups that are mateable in our framework.
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}On foliations of bounded mean curvature on closed three-dimensional Riemannian manifoldshttps://zbmath.org/1527.530202024-02-28T19:32:02.718555Z"Bolotov, Dmitry"https://zbmath.org/authors/?q=ai:bolotov.dmitrii-valerevichGiven a closed oriented three-dimensional Riemannian manifold \(M\), the author provides constants \(H_0\) and \(C_0\) (depending on the geometry of \(M\)) such that any codimension-one transversely oriented foliation \(\mathcal F\) of \(M\), whose leaves have mean curvature bounded from above by \(H_0\) contains at most \(C_0\) Reeb components.
Reviewer: Paweł Walczak (Łódź)Foliations on closed three-dimensional Riemannian manifolds with small modulus of mean curvature of the leaveshttps://zbmath.org/1527.530212024-02-28T19:32:02.718555Z"Bolotov, Dmitry V."https://zbmath.org/authors/?q=ai:bolotov.dmitrii-valerevichGiven a closed and oriented \(3\)-manifold \(M\), the author provides a constant \(H_0\) such that every smooth transversely orientable codimension-one foliation \(\mathcal{F}\) of \(M\) such that the mean curvature \(H\) of the leaves satisfies the inequality \(|H| < H_0\) everywhere on \(M\) is \textit{taut}, that is all the leaves of \(\mathcal{F}\) are minimal for some Riemannian metric on \(M\).
Reviewer: Paweł Walczak (Łódź)Cohomogeneity one manifolds and homogeneous spaces of positive scalar curvaturehttps://zbmath.org/1527.530282024-02-28T19:32:02.718555Z"Frenck, Georg"https://zbmath.org/authors/?q=ai:frenck.georg"Galaz-García, Fernando"https://zbmath.org/authors/?q=ai:galaz-garcia.fernando"Reiser, Philipp"https://zbmath.org/authors/?q=ai:reiser.philippThe question ``Which smooth manifolds do admit complete Riemannian metrics of positive scalar curvature?'' is a long-standing problem in Riemannian geometry. In the paper this question is studied for cohomogeneity-one manifolds and homogeneous spaces with actions of compact Lie groups \(G\). The main results of this paper characterise completely the case when such a manifold admits a not necessary \(G\)-invariant complete metric of positive scalar curvature. The proofs are based on structure results for cohomogeneity-one manifolds and homogeneous spaces, existence results for metrics of non-negative Ricci curvature on such manifolds and obstructions to the existence of positive scalar curvature metrics.
Reviewer: Michael Wiemeler (Münster)Noncollapsed degeneration of Einstein 4-manifolds. Ihttps://zbmath.org/1527.530452024-02-28T19:32:02.718555Z"Ozuch, Tristan"https://zbmath.org/authors/?q=ai:ozuch.tristanIn this paper, the author studies the long-standing open question whether Einstein orbifolds can be resolved to smooth Einstein metrics in the Gromov-Hausorff topology. The basic setting is as follows. Let \((M^4 ,g)\) be an Einstein \(4\)-manifold with \(\mathrm{Vol}_g(M^4) \geq v > 0\), \(\mathrm{diam}_g(M^4) \leq D\), and \(|\mathrm{Ric}_g| \leq 3\). The main result of the paper states that, if \(M^4\) sufficiently degenerate in the sense that there exists some Einstein orbifold \((X, d)\) with \[d_{GH}((M^4, g), (X,d)) < \delta,\] for some small number \(\delta\), then \((M^4,g)\) is the result of a certain gluing-perturbation procedure.
The proof of the theorem requires a very delicate analysis of the bubbling and neck regions at different scales. The main goal is to set up some weighted estimates to approximate the original Einstein metrics by some glued metrics. Then one needs to identify the neck regions in Einstein manifolds which are GH-close to an orbifold.
This result is a major progress in understanding the degenerations of Einstein metrics in the non-collapse and orbifold case.
Reviewer: Ruobing Zhang (Stony Brook)Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank twohttps://zbmath.org/1527.530542024-02-28T19:32:02.718555Z"Solonenko, Ivan"https://zbmath.org/authors/?q=ai:solonenko.ivanA proper isometric action of a connected Lie group \(G\) on a complete connected Riemannian manifold \(M\) is said to be of cohomogeneity one if it has a codimension-one orbit, and two such actions are called orbit equivalent if there is an isometry of \(M\) mapping the orbits of one action onto the orbits of the other. In the paper under review the author obtains the explicit classification of cohomogeneity-one actions and classifies, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces \(\mathrm{SL}(3,{\mathbb H})/ \mathrm{Sp}(3)\), \(\mathrm{SO}(5, {\mathbb C})/ \mathrm{SO}(5)\) and \(\mathrm{Gr}^*(2, {\mathbb C}^{n+4})= \mathrm{SU}(n+2,2)/(\mathrm{U}(n+2) \mathrm{U}(2))\), \(n\ge 1\).
Reviewer: Sergei S. Platonov (Petrozavodsk)On the topology of real Lagrangians in toric symplectic manifoldshttps://zbmath.org/1527.530652024-02-28T19:32:02.718555Z"Brendel, Joé"https://zbmath.org/authors/?q=ai:brendel.joe"Kim, Joontae"https://zbmath.org/authors/?q=ai:kim.joontae"Moon, Jiyeon"https://zbmath.org/authors/?q=ai:moon.jiyeonSymplectic toric manifolds are classified by their Delzant polytopes [\textit{T. Delzant}, Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)]. Many properties of these symplectic manifolds may be characterized or expressed by the combinatorial data of the corresponding Delzant (moment) polytopes.
A Lagrangian submanifold in a symplectic toric manifold \((M, \omega)\) is called \textit{real} if it is the fixed point set of an antisymplectic involution \(R:(M, \omega)\to (M, \omega)\), i.e., a diffeomorphism \(R:M\to M\) satisfying \(R^2=-\mathrm{id}_M\) and \(R^\ast\omega=\omega\).
Let \((M, \omega)\) be a toric symplectic manifold with moment map \(\mu\) and moment polytope \(\Delta\subset (\mathfrak{t}^n)^\ast\), and let \(\mathcal{S}_\Delta=\{\sigma\in\mathrm{Aut}_{\mathbb{Z}}(\mathfrak{t}^n)^\ast\,|\,\sigma(\Delta)=\Delta\}\) be the group of the symmetries of \(\Delta\).
For each involution \(\sigma\in\mathcal{S}_\Delta\) of \(\Delta\), the authors show that:
(i) \(\sigma\) lifts to an antisymplectic involution \(R^\sigma\) of \(M\);
(ii) The diffeomorphism type of \(L =\mathrm{Fix}(R^\sigma)\) is completely determined by \(\Delta\) and \(\sigma\);
(iii) \(\mu(L)=\mathrm{Fix}(\sigma)\) is convex and \(\dim H_\ast(L;\mathbb{Z}_2)=\dim H_\ast(\mathrm{Crit}({H_\xi}|_L);\mathbb{Z}_2)\) for any \(\xi\in\mathfrak{t}\), where \(H_\xi\) is the smooth function \(\langle\mu, \xi\rangle\) on \(M\) and \(\mathrm{Crit}({H_\xi}|_L)\) denotes the set of critical points of \({H_\xi}|_L\).
Claims (ii) and (iii) imply that \(L =\mathrm{Fix}(R^\sigma)\) is not empty. As an application, it is proved that each connected real Lagrangian in a toric symplectic del Pezzo surface (i.e., the product of two \(2\)-spheres or the \(k\)-fold monotone symplectic blow-up of \(\mathbb{C}P^2\) for \(0\le k\le 3\)) is diffeomorphic to one of the following five manifolds: \(S^2\), \(T^2\), \(\mathbb{R}P^2\), \(\mathbb{R}P^2\sharp\mathbb{R}P^2\), \(\sharp_3\mathbb{R}P^2\) or \(\sharp_4\mathbb{R}P^2\).
Reviewer: Guang-Cun Lu (Beijing)Notes on a category-theoretic definition of Maslov classes of a Lagrangian manifoldhttps://zbmath.org/1527.530672024-02-28T19:32:02.718555Z"Mishchenko, A. S."https://zbmath.org/authors/?q=ai:mishchenko.andrey-s|mishchenko.alexander-s(no abstract)A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectorieshttps://zbmath.org/1527.530722024-02-28T19:32:02.718555Z"Jin, Rongrong"https://zbmath.org/authors/?q=ai:jin.rongrong"Lu, Guangcun"https://zbmath.org/authors/?q=ai:lu.guangcunSummary: In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by \textit{S. Artstein-Avidan} and \textit{Y. Ostrover} [Int. Math. res. Not. 2008, Article ID mn044, 31 p. (2008; Zbl 1149.52006)] to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by \textit{S. Artstein-Avidan} and \textit{Y. Ostrover} [Int. Math. res. Not. 2014, No. 1, 165--193 (2014; Zbl 1348.37059)].Diffeomorphism type via aperiodicity in Reeb dynamicshttps://zbmath.org/1527.530732024-02-28T19:32:02.718555Z"Kwon, Myeonggi"https://zbmath.org/authors/?q=ai:kwon.myeonggi"Wiegand, Kevin"https://zbmath.org/authors/?q=ai:wiegand.kevin"Zehmisch, Kai"https://zbmath.org/authors/?q=ai:zehmisch.kaiThe article investigates compact strict contact manifolds with boundary and discusses boundary-shaped disc-like neighborhoods of certain isotropic submanifolds in terms of aperiodicity of Reeb flows. The aim of the authors is to generalise earlier results from [\textit{K. Barth} et al., Münster J. Math. 12, No. 1, 31--48 (2019; Zbl 1422.53067)] using the so called degree method (see [\textit{H. Geiges} and \textit{K. Zehmisch}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 15, 663--681 (2016; Zbl 1356.53080)]). The main result of the work is to establish uniqueness of homotopy and diffeomorphism type of strict contact manifolds (satisfying certain conditions) assuming non-existence of short periodic Reeb orbits.
Reviewer: Alexander Schmeding (Trondheim)Basic Kirwan injectivity and its applicationshttps://zbmath.org/1527.530742024-02-28T19:32:02.718555Z"Lin, Yi"https://zbmath.org/authors/?q=ai:lin.yi|lin.yi.1|forrest.jeffrey-yi-lin|lin.yi.3"Yang, Xiangdong"https://zbmath.org/authors/?q=ai:yang.xiangdong.1|yang.xiangdong|yang.xiangdong.2Summary: Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell-Liberman theorem. As an application, this confirms a conjecture raised by \textit{F. Battaglia} and \textit{D. Zaffran} [Int. Math. Res. Not. 2015, No. 22, 11785--11815 (2015; Zbl 1351.14032)] on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifoldshttps://zbmath.org/1527.530762024-02-28T19:32:02.718555Z"Ganor, Yaniv"https://zbmath.org/authors/?q=ai:ganor.yaniv"Tanny, Shira"https://zbmath.org/authors/?q=ai:tanny.shiraAuthors' abstract: We study the relation between spectral invariants of disjointly supported Hamiltonians and of their sum. On aspherical manifolds, such a relation was established by \textit{V. Humilière} et al. [Geom. Topol. 20, No. 4, 2253--2334 (2016; Zbl 1356.53082)]. We show that a weaker statement holds in a wider setting, and derive applications to Polterovich's Poisson bracket invariant and to \textit{M. Entov} and \textit{L. Polterovich}'s notion [Compos. Math. 145, No. 3, 773--826 (2009; Zbl 1230.53080)] of superheavy sets.
Reviewer: Alexander Felshtyn (Szczecin)Fukaya-Seidel categories of Hilbert schemes and parabolic category \(\mathcal{O}\)https://zbmath.org/1527.530782024-02-28T19:32:02.718555Z"Mak, Cheuk Yu"https://zbmath.org/authors/?q=ai:mak.cheuk-yu"Smith, Ivan"https://zbmath.org/authors/?q=ai:smith.ivanThe authors realize Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type \(A\) nilpotent/Slodowy slice with two Jordan blocks \((n,m-n)\) arising from its inclusion in the \(n\)-th Hilbert scheme of the Milnor fiber of the \(A_{m-1}-\)surface singularity, and hence obtain a symplectic interpretation of certain parabolic two-block versions of the Bernstein-Gelfan'd-Gelfan'd category \(\mathcal{O}\), i.e., a Morita equivalence to the principal block \(\mathcal{O}^{m,n}\) of \(BGG\) parabolic category \(\mathcal{O}\) associated to the partition \(m=n+(m-n)\). The authors also give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. That is, they describe a new semi-orthogonal decomposition of the dg-category of perfect modules \(\text{perf-}\mathcal{O}^{n,2n}\). The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces.
Reviewer: Mee Seong Im (Annapolis)Braid loops with Infinite monodromy on the Legendrian contact DGAhttps://zbmath.org/1527.530802024-02-28T19:32:02.718555Z"Casals, Roger"https://zbmath.org/authors/?q=ai:casals.roger"Ng, Lenhard"https://zbmath.org/authors/?q=ai:ng.lenhard-lThe authors consider certain families of Legendrian links in \(\mathbb S^3\) and construct for each member of the family a loop of Legendrian links. The main result is that the action of the constructed loop on the Legendrian contact DGA is of infinite order. As a consequence, it allows the construction of infinitely many non-Hamiltonian exact Lagrangian fillings.
This is not the first instance of Legendrian links admitting infinitely many fillings. Such links were first found by \textit{R. Casals} and \textit{H. Gao} [Ann. Math. (2) 195, No. 1, 1--43 (2022; Zbl 1494.53090)] using the theory of microlocal sheaves. Among the new family of Legendrian links admitting infinitely many fillings are Legendrian links that are not rainbow closures of positive braids, so that the current techniques using microlocal sheaves do not apply.
The proof of the main result uses Floer theoretic techniques. Given a Lagrangian filling one can construct a new Lagrangian filling by concatenating it with the exact Lagrangian concordance induced by the loop of Legendrians. Each such filling gives an augmentation of the Legendrian contact DGA, and the strategy for the proof is to show that they are all different from each other.
Reviewer: Johan Asplund (Stony Brook)Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomialshttps://zbmath.org/1527.530812024-02-28T19:32:02.718555Z"Ekholm, Tobias"https://zbmath.org/authors/?q=ai:ekholm.tobias"Ng, Lenhard"https://zbmath.org/authors/?q=ai:ng.lenhard-lSummary: We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large \(N\) duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov-Witten disk potentials established in [ibid. 18, No. 4, 827--956 (2014; Zbl 1315.81076)] by \textit{M. Aganagic} et al., and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.A note on the weighted Yamabe flowhttps://zbmath.org/1527.530902024-02-28T19:32:02.718555Z"Popelensky, Theodore Yu."https://zbmath.org/authors/?q=ai:popelensky.theodore-yuThe author introduces a natural generalization of the classical combinatorial Ricci flow introduced by \textit{B. Chow} and \textit{F. Luo} [J. Differ. Geom. 63, No. 1, 97--129 (2003; Zbl 1070.53004)], which the author calls the weighted Ricci flow. To study its properties, the author uses the discrete conformal transformation introduced by \textit{F. Luo} [Commun. Contemp. Math. 6, No. 5, 765--780 (2004; Zbl 1075.53063)], which is usually called vertex scaling now. In the discrete conformal class on a triangulated surface, the author studies the invariants, monotonic quantities, singularities along the combinatorial curvature flow. Following Luo's original work, the author further investigates the combinatorial and topological obstacles for the existence of discrete metrics with constant weighted curvature. He does surgery by edge flipping on the triangulations of the surface under the Delaunay condition along the weighted Ricci flow, and uses Gu-Luo-Sun-Wu's discrete conformal theory to study the longtime behavior of the weighted Ricci flow with surgery.
Reviewer: Xu Xu (Wuhan)The Reidemeister trace of an \(n\)-valued maphttps://zbmath.org/1527.550022024-02-28T19:32:02.718555Z"Staecker, P. Christopher"https://zbmath.org/authors/?q=ai:staecker.p-christopherRecently, several works have treated Nielsen theory of \(n\)-valued maps, where well-known concepts from Nielsen fixed-point theory for single-valued maps were introduced/extended to \(n\)-valued maps (multivalued maps where the image of a point \(x\in X\) is a subset with exactly \(n\) points of a given space \(Y\)). Among those concepts for single-valued maps, we have the so-called ``Reidemeister trace'', which now in the present work, is appropriately extended to \(n\)-valued maps. A suitable definition of Reidemeister classes for \(n\)-valued maps, denoted by \(\mathcal{R}(\tilde f_{\#})\) where \(\tilde f\) is a lifting of the given map \(f\), has been given by \textit{R. F. Brown} et al. [Topology Appl. 274, Article ID 107125, 26 p. (2020; Zbl 1437.55003)], which satisfies properties similar to the ones the Reidemeister classes of single-valued maps satisfy. Using this concept, in addition to other ingredients, the article under review introduces the concept of Reidemeister trace for \(n\)-valued maps. The definition of the Reidemeister trace for \(n\)-valued maps is given as follows:
Definition 3. Let \(f : X \to D_n(X)\) be an \(n\)-valued map, and let \(f : \tilde X \to F_n(\tilde X, \pi)\)) be a lifting with \(\tilde f= (\tilde f_1,\dots,\tilde f_n )\). Then we define the Reidemeister trace of \(f\) and \(\tilde f\) as: \[ RT(f,\tilde f) =\qquad \sum_{[(\alpha, k)]\in \mathcal{R}(\tilde f_{\#})}ind(f, U_{[(\alpha,k)]})[(\alpha,k)]_{\tilde {f}_{\#}} \in \mathbb{Z}\mathcal{R}(\tilde f_{\#}),\] \noindent where each \(U_{[(\alpha,k)]}\subset X\) is an open set containing \(pFix(\alpha^{-1}\tilde f_{k})\) and no other fixed points of \(f\).
Then many properties, which parallel those satisfied by the Reidemeister trace for single valued maps, are proved. Notably, in the context of simplicial complexes it is proved in detail that such object can be obtained as the trace of certain matrices. Some examples with the calculation of the Reidemeister trace for the circle are provided. In Section 4 the author shows the relation between the trace for two different lifts \(\tilde f,\tilde f'\) of \(f\). In Section 5, the last section, the local version of the Reidemeister trace is defined and it is shown that it satisfies the \textit{Homotopy property}, the \textit{Excision property} and the \textit{Additivity property}. Then it is shown:
Corollary 25: Any function satisfying the three properties above is the local Reidemeister trace.
Finally with respect to a regular finite covering \(q:\bar X \to X\) which corresponds to a normal subgroup \(G\) of \(\pi\), an average formula is shown which relates the trace of the map on \(X\) with the trace of the liftings of \(f\) to the finite covering and the cardinality \([\pi, G]\), the index of the subgroup \(G\) on \(\pi\). The paper provides enough details and the presentation is very well organized.
Reviewer: Daciberg Lima Gonçalves (São Paulo)Iterated integrals with values in Azumaya algebrashttps://zbmath.org/1527.550122024-02-28T19:32:02.718555Z"Glass, Cheyne"https://zbmath.org/authors/?q=ai:glass.cheyne-j"Tradler, Thomas"https://zbmath.org/authors/?q=ai:tradler.thomas"Zeinalian, Mahmoud"https://zbmath.org/authors/?q=ai:zeinalian.mahmoudIn general, Chen's iterated integrals with values in a bundle of algebras fail to have a well defined product. By making use of parallel transport, the authors generalize Chen's iterated integrals to the setting of forms with values in an Azumaya algebra and prove that if \(M\) is simply connected, then there exist generalizations of the two sided bar construction, and the Hochschild complex along with iterated integrals to forms on the path and loop space respectively, which form a commutative diagram induced by the inclusion \(LM\longrightarrow PM\). In the last section they note that in the case of a finite dimensional vector bundle, endomorphism-bundle valued forms can be related to real-valued forms by the usual trace map.
Reviewer: Jean-Claude Thomas (Angers)Good covers for vortex nerve cell complexes. Free group presentation of intersecting nested cycles in planar CW spaceshttps://zbmath.org/1527.550132024-02-28T19:32:02.718555Z"Peters, J. F."https://zbmath.org/authors/?q=ai:peters.john-f|peters.james-f-iiiSummary: This paper introduces good covers for cell complexes in the form of what are known as path nerve complexes in a planar Whitehead CW space together with their Rotman free group presentations. A \textbf{path} is a mapping \(h:[0,1]\to K\) over a space \(K\). Paths provide the backbone of homotopy theory introduced by J.H.C. Whitehead during the 1930s. A \textbf{path nerve complex} is a collection of sets of path-connected points (path images) that have nonempty intersection.
A space \(X\) has a good cover, provided \(X=\bigcup E\) for subsets \(E\) in \(X\) and the space is contractible. This form of good cover was introduced by K. Tanaka in 2021. The focus here is on intersecting path cycles (sequences of paths attached to each other in vortexes with nonempty interiors) that form a path nerve. A path nerve results from the nonvoid intersection of a collection of path cycles. The geometric realization of a path cycle is a 1-cycle. A 1-cycle is a finite sequence of path-connected 0-cells (vertexes) with no end vertex and with a nonvoid interior. A 1-cycle has the structure of a path cycle in which sequences of paths are replaced by edges.
A group \(G(V,+)\) containing a basis \(\mathcal{B}\) is \textit{free}, provided every member of \(V\) can be written as a linear combination of elements (generators) of the basis \(\mathcal{B}\subset V\). Let \(\bigtriangleup\) be the members \(v\) of \(V\), each written as a linear combination of the basis elements of \(\mathcal{B}\). A presentation of \(G(V,+)\) is a mapping \(\mathcal{B}\times\bigtriangleup\to G(\bigg\{v\in V:=\sum_{\substack{k\in\mathbb{Z} g\in\mathcal{B}}}{kg}\bigg\},+)\). This form of presentation of structures was introduced by J. J. Rotman during the 1960s as part of a study of groups. The main results in this paper are (1) Every path triangle cluster has a free group presentation, (2) Every path triangle cluster has a free group presentation, (3) Every path vortex has a free group presentation, (4) Every path vortex nerve has a free group presentation, (5) A vortex nerve and the union of the sets in the nerve have the same homotopy type and (6) Every path triangulaton of a cell complex has a good cover.
For the entire collection see [Zbl 1522.34005].Subgroup collections controlling the homotopy type of a \(p\)-local compact grouphttps://zbmath.org/1527.550162024-02-28T19:32:02.718555Z"Belmont, Eva"https://zbmath.org/authors/?q=ai:belmont.eva"Castellana, Natàlia"https://zbmath.org/authors/?q=ai:castellana.natalia"Lesh, Kathryn"https://zbmath.org/authors/?q=ai:lesh.kathrynA \(p\)-local compact group is a triple \((S,\mathcal{F},\mathcal{L})\), where \(S\) is a discrete \(p\)-toral group and \(\mathcal{F}\) and \(\mathcal{L}\) are categories which must satisfy certain properties. The classifying space of this triple is the \(p\)-completion of the geometric realization of \(\mathcal{L}\). These structures are used to model \(p\)-completed classifying spaces of compact Lie groups, \(p\)-compact groups and other spaces.
Since the morphisms of \(\mathcal{F}\) are generated by inclusions and automorphisms of \(\mathcal{F}\)-centric, \(\mathcal{F}\)-radical subgroups, it is natural and convenient to express properties and invariants of \(p\)-local compact groups in terms of these subgroups. The authors consider families of \(\mathcal{F}\)-centric subgroups which are closed under \(\mathcal{F}\)-conjugacy and the bullet construction \((\mbox{ })^{\bullet}\), and which contain all subgroups of \(S\) which are both \(\mathcal{F}\)-centric and \(\mathcal{F}\)-radical. If one considers the full subcategory \(\mathcal{L}^{\mathcal{H}}\) of \(\mathcal{L}\) whose object set is one such family \(\mathcal{H}\), the main result states that the natural map \(|\mathcal{L}^{\mathcal{H}}| \to |\mathcal{L}|\) is a homotopy equivalence. Remarkably, it is not necessary to \(p\)-complete these spaces for this homotopy equivalence to hold.
The proof uses a filtration of \(\mathcal{L}^{\bullet}\) by full subcategories that begins at \(\mathcal{L}^{\mathcal{H}}\) and adds a conjugacy class of objects at a time. This filtration is finite because \(\mathcal{L}^{\bullet}\) has a finite number of conjugacy classes of objects, and the undercategories of each of the inclusions in this filtration have contractible geometric realizations.
Reviewer: José Maria Cantarero Lopez (Mérida)Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospanshttps://zbmath.org/1527.550172024-02-28T19:32:02.718555Z"Steinebrunner, Jan"https://zbmath.org/authors/?q=ai:steinebrunner.janThe classifying space of an \(\infty\)-category is the homotopy type obtained by universally inverting all of its morphisms. Many interesting homotopy types arise in this way, for example, the classifying space of the \(\infty\)-category of \(k\)-dimensional bordisms embedded in \(\mathbb{R}^n \times [0,1]\) yields the \(n\)-fold loop space of the space of \(k\)-planes in \(\mathbb{R}^{n+1}\) (including the ``empty plane'').
Similarly to the situation for topological spaces, it is useful for computations to know which commutative squares of \(\infty\)-categories are homotopy pullbacks, i.e., produce pullback squares of classifying spaces. A functor of \(\infty\)-categories is called a \textbf{realisation fibration} if any pullback along it is a homotopy pullback in this sense. The main theorem of the paper under review is:
\textbf{Theorem:} Any functor which is both a locally Cartesian fibration and a locally coCartesian fibration is a realisation fibration.
Thus, the main theorem generalises a result of \textit{W. Steimle} [Algebr. Geom. Topol. 21, No. 2, 601--646 (2021; Zbl 1475.57046)] stating that functors which are both Cartesian fibrations and coCartesian fibrations are realisation fibrations.
With a view towards applications, the author characterises such functors in three different models: quasi-categories, weakly unital Segal spaces, and weakly unital topological categories. In one such application, functors between several variants of weakly unital topological categories of cospans of finite categories are shown to be realisation fibrations, allowing for explicit computations of their fibres.
The paper is clearly structured, recalls all relevant definitions (thus making it largely self-contained), and carefully carries out all arguments in all three models.
Reviewer: Adrian Clough (Abu Dhabi)Collected works of William P. Thurston with commentary: I. Foliations, surfaces and differential geometry. Edited by Benson Farb, David Gabai and Steven P. Kerckhoffhttps://zbmath.org/1527.570012024-02-28T19:32:02.718555Z"Thurston, William P."https://zbmath.org/authors/?q=ai:thurston.william-pThis is the first of the four volume set of William Thurston's \textit{Collected Works}. It is divided into three parts: 1. \textit{Foliations}; 2. \textit{Surfaces and mapping class groups}; 3. \textit{Differential geometry}. Each part starts with an introductory essay by the editors. Part 1 contains, besides Thurston's published papers on foliations, his PhD thesis, titled \textit{Foliations of three-manifolds which are circle bundles}, and two other unpublished papers: \textit{Three-manifolds, foliations and circles, I and II}. Part 2 contains, besides the published papers, one unpublished paper \textit{Minimal stretch maps between hyperbolic surfaces}.
The list of papers contained in each volume is the following:
\textit{Part 1. Foliations: } Foliations of three-manifolds which are circle bundles (This is Thurston's PhD thesis, 1972); (with J. Plante) Anosov flows and the fundamental group (1972, [Zbl 0246.58014]); Non cobordant foliations of \(S^3\) (1972, [Zbl 0266.57004]); (with H. Rosenberg) Some remarks on foliations (1973, [Zbl 0286.57014]); Foliations and groups of diffeomorphisms (1974, [Zbl 0295.57014]); A generalization of the Reeb stability theorem (1974, [Zbl 0305.57025]); The theory of foliations of codimension greater than one (1974, [Zbl 0295.57013]); (With M. Hirsch) Foliated bundles, invariant measures and flat manifolds, [Zbl 0321.57015]; The theory of foliations of dimension greater than one (1975, [Zbl 0316.57013]); (With H. E. Wilkenkemperer) On the existence of contact forms (1975, [Zbl 0312.53028]); A local construction of foliations for three-manifolds (1975, [Zbl 0323.57014]); On the construction and classification of foliations (1975, [Zbl 0332.57014]); Existence of codimension-one foliations (1976, [Zbl 0347.57014]); (With J. Plante) Polynomial growth in holonomy groups of foliations (1976, [Zbl 0348.57009]); (with M. Handel) Anosov flows on three-manifolds, [Zbl 0435.58019]; A norm for the homology of 3-manifolds (1986, [Zbl 0585.57006]); (with Y. Eliashberg) Contact structures and foliations on 3-manifolds (1996, [Zbl 0879.57021]); (with Y. Eliashberg) Confoliations (1996, [Zbl 0893.53001]); Three-manifolds, foliations and circles, I (1997, [arXiv:math/9712268]); (unfinished) Three-manifolds, foliations and circles, II: The transverse asymptotic geometry of foliations (1998).
\textit{Part 2. Surfaces and mapping class groups:} (with A. Hatcher) A presentation for the mapping class group of a closed orientable surface (1980, [Zbl 0447.57005]); (with M. Handel) New proofs of some results of Nielsen (1985, [Zbl 0584.57007]); On the geometry and dynamics of diffeomorphisms of surfaces (1988, [Zbl 0674.57008]); Earthquakes in 2-dimensional hyperbolic geometry (1986, [Zbl 0628.57009]); Minimal stretch maps between hyperbolic surfaces (1986, [arXiv:math/9801039]); (with S. P. Kerckhoff) Non-continuity of the action of the modular group at Bers's boundary of Teichmüller space (1990, [Zbl 0698.32014]).
\textit{Part 3. Differential geometry: } Some simple examples of symplectic manifolds (1976, [Zbl 0324.53031]); (with J. Milnor) Characteristic numbers of 3-manifolds (1977, [Zbl 0393.57002]); (with D. B. A. Epstein) Transformation groups and natural bundles (1979, [Zbl 0409.58001]); (with D. Sullivan) Manifolds with canonical coordinate charts: some examples (1983, [Zbl 0529.53025]) (with M. Gromov and H. B. Lawson, Jr.) Hyperbolic 4-manifolds and conformally flat 3-manifolds (1998, [Zbl 0692.57012]); Shapes of polyhedra and triangulations of the sphere (1998, [Zbl 0931.57010]); (with J. H. Conway, O. Delgado-Friedrichs and D. H. Huson) On three-dimensional space groups (2001, [Zbl 0991.20036]).
I will make a few comments on some of these papers, especially on the unpublished ones, since they are not reviewed in Zentralblatt. I start with Thurston's PhD thesis, \textit{Foliations of 3-manifolds which are circle bundles} (1972).
The main result in this thesis says the following: On any 3-manifold that is an oriented circle bundle over a surface which is not a torus, any \(C^2\) transversely oriented foliation with no torus compact leaf is isotopic to a foliation that is transverse to the fibers. Thurston deduces several corollaries, including the first examples of 3-manifolds that do not carry any analytic foliation, and of manifolds with contractible universal coverings on which every foliation has a compact leaf. The thesis was never published: It was submitted to \textit{Inventiones}, the referee asked for more detailed proofs, and Thurston withdrew it. This is confirmed by André Haefliger, who writes in Thurston's memorial article which appeared in the Notices of the AMS [\textit{A. Haefliger} et al., Notices Am. Math. Soc. 62, No. 11, 1318--1332 (2015; Zbl 1338.55002)]: ``Thurston [after receiving the referee's report], who was busy proving more theorems, decided not to publish it.'' Thurston had several times shown the same reaction to referees' reports.
The same year, Thurston published a three-page note, which was his first published paper, \textit{Noncobordant foliations of \(S^3\)}. He proves there that there exists a family of foliations whose Godbillon-Vey invariant takes all possible real values. This answered in a spectacular way an open problem, asking whether there exist foliations with non-zero Godbillon-Vey invariant. It is interesting to see that Thurston's proof of this result uses a construction in hyperbolic geometry, a subject which was still dormant (before Thurston revived it, 3 or 4 years later): Thurston constructed foliations of the 3-sphere that depend on convex polygons in the hyperbolic plane whose area is equal to the Godbillon-Vey invariant of the foliation. As a consequence of the main result, Thurston proved that there exists an uncountable family of non-cobordant foliations on \(S^3\). The precise results are stated in terms of Haefliger's classifying spaces of geometric structures. As a matter of fact, Thurston uses in this paper a notion of geometric structure, called Haefliger structure, introduced by Haefliger in his thesis (1958), which generalizes the notion of foliation. The definition is close to the more general notion of geometric structure which plays a paramount role in the subsequent work of Thurston. The formalization of the idea of geometric structure originates in the work of Charles Ehresmann (who was Haefliger's advisor). In conclusion, in this paper of Thurston one finds two essential ingredients for Thurston's later works: hyperbolic geometry and geometric structures. Two years after his thesis, Thurston was appointed full professor at Princeton (at age 27).
Let me mention another remarkable result on foliations, contained in Thurston's paper \textit{Existence of codimension-one foliations} (1976): There exists a \(C^\infty\) codimension-one foliation on any closed manifold with zero Euler characteristic. The result solved one of the main conjectures in the field.
Going over the rest of the first part of the volume under review will give an idea of how Thurston, in a period of five years, made a transformation of the topic of foliations. He proved the main conjectures, including all the existence theorems that were hoped for.
The other unpublished work included in Part 1 of the book is titled \textit{Three-manifolds, foliations and circles, II: The transverse asymptotic geometry of foliations} (1998). Thurston writes in the introduction: ``In this paper, we will develop some theory that describes how leaves of a foliation fit together in 3-manifolds.'' He notes that 3-dimensional topology has developed several impressive strands, including theories like (1) incompressible surfaces, (2) geometrization, (3) foliations and laminations, and that these three strands ``have separately produced major insights which have been partially woven together; however, they have not been connected well enough to give a sense of unitary whole.'' Thurston, in this paper, makes connections. He discusses Teichmüller spaces, pseudo-Anosov homeomorphisms and their mapping tori, discrete group actions, Brownian motion, harmonic measures for group actions on the circle and for foliated bundles, the Milnor-Wood inequality for circle bundles over a surface, 3-manifolds fibering over the circle, Finsler geometry of Teichmüller space, Brownian motion on this space and many other topics. An emphasis is given to the subject of linear and circular order, group actions on the line and on the circle, and foliated line and circle bundles, with a special mention of work of Étienne Ghys. Some work in this paper is closely related to the subject of Part 2 of the present volume (surfaces and mapping class groups), and to the subject of Volume II of the Collected Works (Three-dimensional manifolds).
Among Thurston's unpublished works on geometry that are not contained in the \textit{Collected Works} and which could have been included in the present volume, let me mention the set of notes written by J. H. Conway, P. G. Doyle, J. Gilman, W. P. Thurston, \textit{Geometry and the Imagination in Minneapolis}, which consists of lecture notes of a course given at Princeton and the Geometry Center in Minneapolis in 1991. The notes are elementary, but this is not a reason for not including them in the \textit{Collected Works}. They contain interesting material. There are sections on maps (decompositions of the sphere or other surfaces), orbifolds, polyhedra, curvature, etc. besides course projects and ``exercises in imagining''. According to the authors, ``this course aims to convey the richness, diversity, connectedness, depth and pleasure of mathematics.''
\par Let me now pass to Part 2 of the volume, titled \textit{Surfaces and mapping class groups}. The most important results of Thurston on this topic are included in the Orsay seminar notes [Travaux de Thurston sur les surfaces. Séminaire Orsay. Société Mathématique de France (SMF), Paris (1979; Zbl 0406.00016)], edited by Albert Fathi, François Laudenbach and Valentin Poénaru and published in 1979. The notes contain complete proofs of Thurston's result on the classification of surface mapping classes, with a detailed exposition of the background material on hyperbolic geometry, Teichmüller spaces and foliations, together with additional chapters on the dynamics of pseudo-Anosov homeomorphisms and on the presentation of the mapping class group using Cerf's theory, a result which was published in the later paper \textit{A presentation for the mapping class group of a closed orientable surface} by Thurston and Hatcher (1980).
Thurston's ``announcement'', titled \textit{On the geometry and dynamics of diffeomorphisms of surfaces} (1988), on which the work \textit{Travaux de Thurston sur les surfaces} is based (actually, it was based on an earlier manuscript version), is still a wonderful summary, resubmitted and published 12 years after it was first circulated, after a few attempts to publish it. Thurston writes there, as a postscript, that at the time he worked out the classification of surfaces, he was unaware, on the one hand, of Riemann surfaces, quasiconformal maps and Teichmüller theory, and on the other hand, of Nielsen's work on the classification of surface mapping classes by looking at the dynamical behavior on the universal cover, and (Thurston writes) of Nielsen's ``near understanding of geodesic laminations''. This brings us to another paper published in this volume, titled \textit{New proofs of some results of Nielsen} (1985) by Thurston and Mike Handel. This is part of an effort by Thurston, Handel, Jane Gilman, Joan Birman, R. T. Miller, Heiner Zieschang, Robert Brown and probably others to present the work of Jacob Nielsen (1890-1959) on the classification of mapping classes, and how far he got in his project. Nielsen developed a beautiful theory of surface mapping classes, but not comparable in depth to Thurston's theory, in terms of depth and scope. It is good to know that in 1986, a two volume set of \emph{Collected Mathematical Papers} of Jacob Nielsen was published, with several articles (including his his pioneering work on surface mapping classes) translated into English [Zbl 0609.01050].
Part 2 of the present volume also contains Thurston's unpublished paper \textit{Minimal stretch maps between hyperbolic surfaces}, and I would like to say a few words on it. In this paper, Thurston introduces a new metric on the Teichmüller space of complete finite area hyperbolic surfaces. The distance between two such surfaces is taken to be the logarithm of the smallest Lipschitz constant of a homeomorphism from the first surface to the second one. This defines an asymmetric metric, known today as the \textit{Thurston metric} on Teichmüller space. The theory that Thurston developed is meant to be a counterpart to the quasiconformal theory, whose bases (especially worked out by Ahlfors and Bers) involve a significant amount of complex analysis, whereas the present theory is based on elementary 2-dimensional hyperbolic geometry.
In the paper \textit{Minimal stretch maps between hyperbolic surfaces}, Thurston proves several important results, including an equivalent definition of the metric based on a comparison of the lengths of simple closed curves, the fact that this metric is Finsler (and Thurston gives a formula for the Finsler infinitesimal norm as well as a description of the Finsler unit co-ball at each point, obtained as an embedding of the space of projective measured laminations in that space). Thurston also describes a distinguished class of geodesics, called stretch lines, showing that any two points can be joined by a geodesic which is a concatenation of stretch lines.
The paper, when it was circulated, was considered (rightly) very difficult to read, although it is self-contained and uses only elementary hyperbolic geometry; in fact, the difficulty of the paper probably lies in the fact that the theory developed makes use of elementary but profound ideas, and requires some amount of imagination from the reader. The paper was submitted to the journal Topology, and the referee sent a report asking for clarifications at every step, but Thurston decided not to make any change, and the paper remained a preprint. The referee's report is long and was circulated among a certain number of mathematicians. A posteriori one can see that the remarks there do not add much to the paper. The work done and the literature on this metric (called now Thurston's metric) has been increasing gradually, especially since the years 2000, with generalizations to various settings: surfaces with boundary (works by Liu, Papadopoulos, Su, Théret, Alessadrini, Disarlo, Huang and Yamada), higher dimensions (work by Kassel and Guéritaud); analogues in various Euclidean settings (works by Belkhirat, Papadopoulos, Troyanov, Miyachi, Ohshika and Saglam); rigidity and local rigidity (works by Walsh, Huang, Pan, Ohshika and Papadopoulos); higher Teichmüller theory point of view (works by Huang, Zhou, Carvajales, Dai, Pozzetti and Wienhard); the limiting behavior of geodesics and the ray structure (works by Pan, Papadopoulos, Théret, Wolf); a geometric analysis approach to the metric (work by Daskalopoulos and Uhlenbeck), and there are many other works on the subject. Today the Thurston metric is an object of intense research.
Let me pass now to Part 3.
There are no unpublished papers in this part, but I would like to highlight two published papers, on different topics, which, like all of Thurston's papers, opened up the way to a large amount of developments. The first one is the paper \emph{Some simple examples of symplectic manifolds}, in which Thurston gives the first examples of (closed) symplectic manifolds which are not Kähler. This result contradicted a result published by H. Guggenheimer and it gave rise to the question of whether there exist simply connected non-Kähler symplectic manifolds, which had many consequences. The second paper is \emph{Shapes of polyhedra and triangulations of the sphere}, in which Thurston studies the geometry of the space of Euclidean metrics on the 2-sphere with cone singularities of positive curvature. He proves that the space of such metrics of fixed area has a natural Kähler metric and is locally isometric to a complex hyperbolic space. The metric completion of this space is a hyperbolic cone manifold. The work turns out to be closely related to works of Picard, Deligne and Mostow on discrete subgroups of \(\mathrm{PU}(n,1)\). (Again, Thurston was not aware of this when he started lecturing on this topic.) Much has been done on this topic after Thurston, and many natural questions that arise from his ideas may still be explored.
The present volume, as the other volumes of the collection, is now an invaluable element of our mathematical literature, and I do not have enough words to recommend that every geometer acquire this collection, insofar as possible. The editors have done a great job in collecting these papers. My reservation is the same as for the other volumes, regarding the meticulousness of the editing work: typos in the few introductory pages, including many missing accents in foreign words (Poincare instead of Poincaré, Kahler instead of Kähler, etc.). The reviewer is surprised that the publisher's editor did not fix these problems.
Reviewer: Athanase Papadopoulos (Strasbourg)On the cosmetic crossing conjecture for special alternating linkshttps://zbmath.org/1527.570022024-02-28T19:32:02.718555Z"Boninger, Joe"https://zbmath.org/authors/?q=ai:boninger.joeThe cosmetic crossing conjecture, attributed to Xiao-Song Lin [\textit{R. Kirby}, in: Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 2, 273--312 (1978; Zbl 0394.57002)], posits that changing a nontrivial crossing in a link diagram must change the isotopy type of the link. This conjecture has been verified for many classes of links. In this paper, the author verifies this conjecture for a class of links, which includes all special alternating knots.
Reviewer: Khaled Qazaqzeh (Irbid)Maps to virtual braids and braid representationshttps://zbmath.org/1527.570032024-02-28T19:32:02.718555Z"Manturov, Vassily O."https://zbmath.org/authors/?q=ai:manturov.vassily-olegovich"Nikonov, Igor M."https://zbmath.org/authors/?q=ai:nikonov.igor-mikhailovichFrom the introduction: The aim of our note is to construct new representations of the (coloured) braid group using methods of parity theory and picture-valued invariants [\textit{V. O. Manturov}, Sb. Math. 201, No. 5, 693--733 (2010; Zbl 1210.57010) and \textit{V. O. Manturov} et al., Invariants and pictures. Low-dimensional topology and combinatorial group theory. Hackensack, NJ: World Scientific (2020; Zbl 1447.57001)].Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance grouphttps://zbmath.org/1527.570042024-02-28T19:32:02.718555Z"Miller, Allison N."https://zbmath.org/authors/?q=ai:miller.allison-n"Powell, Mark"https://zbmath.org/authors/?q=ai:powell.markA \emph{strongly invertible knot} \((K,\tau)\) is a pair given by a knot \(K\subset S^3\) and an orientation preserving involution \(\tau\) of \(S^3\) such that \(\tau(K)=K\) and which reverses the orientation of \(K\). ``The study of strongly invertible knots up to equivariant concordance was instigated by \textit{M. Sakuma} [in: Algebraic and topological theories. Papers from the symposium dedicated to the memory of Dr. Takehiko Miyata held in Kinosaki, October 30- November 9, 1984. Tokyo: Kinokuniya Company Ltd.. 176--196 (1986; Zbl 0800.57001)], who defined an equivariant knot concordance group of directed strongly invertible knots.''
\textit{J. Levine} [Comment. Math. Helv. 44, 229--244 (1969; Zbl 0176.22101)] used the slice obstruction coming from the Seifert form to define the algebraic concordance group as a Witt group of Seifert forms. As shown by \textit{C. Kearton} [J. Lond. Math. Soc., II. Ser. 10, 406--408 (1975; Zbl 0305.57015)], the Blanchfield pairing provides the same obstruction to sliceness and hence an equivalent and more intrisic approach to algebraic concordance.
In this paper the authors obtain an equivariant sliceness obstruction from the \textit{equivariant Blanchfield pairing} of a strongly invertible knot \((K,\tau)\), i.e. the data given by the Blanchfield pairing on the Alexander module of \(K\), together with the action induced by the strong inversion \(\tau\).
In particular they define an \textit{equivariant algebraic concordance group} \(\mathcal{AC}^{SI}\) as the Witt group of abstract equivariant Blanchfield pairings and a homomorphism from the equivariant concordance group of strongly invertible knot to \(\mathcal{AC}^{SI}\).
Moreover, using the equivariant Blanchfield pairing they derive a new lower bound on the equivariant slice genus of a strongly invertible knot. As an application of this result, the authors provide new examples of slice strongly invertible knots with arbitrarily large equivariant slice genus, giving an alternative proof of an analogous result proven by \textit{I. Dai} et al. [J. Topol. 16, No. 3, 1167--1236 (2023; Zbl 07738258)] using knot Floer homology.
Reviewer: Alessio Di Prisa (Pisa)Combinatorial approach to Milnor invariants of welded linkshttps://zbmath.org/1527.570052024-02-28T19:32:02.718555Z"Miyazawa, Haruko A."https://zbmath.org/authors/?q=ai:miyazawa.haruko-aida"Wada, Kodai"https://zbmath.org/authors/?q=ai:wada.kodai"Yasuhara, Akira"https://zbmath.org/authors/?q=ai:yasuhara.akiraIn this paper, the authors seek to give a combinatorial proof of results first demonstrated by \textit{M. Chrisman} [Algebr. Geom. Topol. 22, No. 5, 2293--2353 (2022; Zbl 1511.57003)] using topological methods based on Satoh's correspondence [\textit{S. Satoh}, ``Virtual knot presentation of ribbon torus-knots'', WSPC Proc. (2018), \url{http://msp.warwick.ac.uk/~cpr/ftp/zrourke.pdf}]. between welded link diagrams and knotted tori in four dimensions. In particular, the authors are able to give a combinatorial extension of Milnor's concordance invariants to welded links, and show that these numbers are invariant modulo a factor determined by the links themselves.
It is interesting to see this done in an explicitly combinatorial way, but this approach particularly helps when considering the results in Section 6. In this section, the authors show the following theorem:
Theorem. Given two welded link diagrams \(D, D'\) which are related by a finite sequence of self-crossing virtualizations and welded isotopies, the Milnor invariants of the two diagrams will be equal.
A self-crossing virtualization consists of replacing a classical crossing involving over and under arcs from the same link component with a virtual crossing. Such a change is rather simple combinatorially, but more complicated when considered geometrically.
It would, however, be very interesting to see if the corresponding geometric interpretation of this result would yield any interesting information about knotted tori in four dimensions.
Finally, the authors apply their approach to the study of welded string links, links with endpoints fixed on a boundary, and show that for these, the Milnor numbers are an invariant.
Reviewer: Blake Winter (Buffalo)On the Jones polynomial modulo primeshttps://zbmath.org/1527.570062024-02-28T19:32:02.718555Z"Aiello, Valeriano"https://zbmath.org/authors/?q=ai:aiello.valeriano"Baader, Sebastian"https://zbmath.org/authors/?q=ai:baader.sebastian"Ferretti, Livio"https://zbmath.org/authors/?q=ai:ferretti.livioSummary: We derive an upper bound on the density of Jones polynomials of knots modulo a prime number \(p\), within a sufficiently large degree range: \(4/p^7\). As an application, we classify knot Jones polynomials modulo two of span up to eight.Categorical lifting of the Jones polynomial: a surveyhttps://zbmath.org/1527.570072024-02-28T19:32:02.718555Z"Khovanov, Mikhail"https://zbmath.org/authors/?q=ai:khovanov.mikhail-g"Lipshitz, Robert"https://zbmath.org/authors/?q=ai:lipshitz.robertSummary: This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.Sutured instanton homology and Heegaard diagramshttps://zbmath.org/1527.570082024-02-28T19:32:02.718555Z"Baldwin, John A."https://zbmath.org/authors/?q=ai:baldwin.john-a"Li, Zhenkun"https://zbmath.org/authors/?q=ai:li.zhenkun"Ye, Fan"https://zbmath.org/authors/?q=ai:ye.fanSummary: Suppose \(\mathcal{H}\) is an admissible Heegaard diagram for a balanced sutured manifold \((M, \gamma)\). We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology \(\mathit{SHI}(M, \gamma)\). It follows, in particular, that strong L-spaces are instanton L-spaces.Generating the extended mapping class group by three involutionshttps://zbmath.org/1527.570092024-02-28T19:32:02.718555Z"Altunöz, Tülin"https://zbmath.org/authors/?q=ai:altunoz.tulin"Pamuk, Mehmetcik"https://zbmath.org/authors/?q=ai:pamuk.mehmetcik"Yildiz, Oğuz"https://zbmath.org/authors/?q=ai:yildiz.oguzSummary: We prove that the extended mapping class group, \(\mathrm{Mod}^*(\Sigma_g)\), of a connected orientable surface of genus \(g\), can be generated by three involutions for \(g\geq 5\). In the presence of punctures, we prove that \(\mathrm{Mod}^*(\Sigma_{g,p})\) can be generated by three involutions for \(g\geq 10\) and \(p\geq 6\) (with the exception that for \(g\geq 11\), \(p\) should be at least \(15)\).On the involution generators of the mapping class group of a punctured surfacehttps://zbmath.org/1527.570102024-02-28T19:32:02.718555Z"Altunöz, Tülin"https://zbmath.org/authors/?q=ai:altunoz.tulin"Pamuk, Mehmetcik"https://zbmath.org/authors/?q=ai:pamuk.mehmetcik"Yildiz, Oğuz"https://zbmath.org/authors/?q=ai:yildiz.oguzLet \(\Sigma_{g,p}\) denote the closed orientable surface of genus \(g\) with \(p\) punctures (with the understanding that \(\Sigma_{g,0} := \Sigma_g\)), and let \(\mathrm{Mod}(\Sigma_{g,p})\) be the mapping class group of \(\Sigma_{g,p}\). The problem of generating \(\mathrm{Mod}(\Sigma_{g,p})\) with involutions was first examined by \textit{J. McCarthy} and \textit{A. Papadopoulos} [Enseign. Math. (2) 33, 275--290 (1987; Zbl 0655.57005)]. It was subsequently shown by \textit{T. E. Brendle} and \textit{B. Farb} [J. Algebra 278, No. 1, 187--198 (2004; Zbl 1051.57019)] that for \(g \geq 3\), \(\mathrm{Mod}(\Sigma_g)\) is generated by six involutions. This result was further improved by the works of \textit{M. Korkmaz} [Math. Res. Lett. 27, No. 4, 1095--1108 (2020; Zbl 1459.57020)] and \textit{T. Altunöz} et al. [Osaka J. Math. 60, No. 1, 61--75 (2023; Zbl 1527.57009)], who showed that \(\mathrm{Mod}(\Sigma_g)\) is generated by four involutions for \(g \geq 3\), and three involutions for \(g \geq 6\). Though \(\mathrm{Mod}(\Sigma_{g,p})\) is known [\textit{N. Monden}, ``On minimal generating sets for the mapping class group of a punctured surface'', Preprint, \url{arXiv:2103.01525}; \textit{B. Wajnryb}, Topology 35, No. 2, 377--383 (1996; Zbl 0860.57040)] to be two-generated for \(g \geq 3\), these generators cannot be involutions since \(\mathrm{Mod}(\Sigma_{g,p})\) has non-abelian free subgroups.
In this paper, it is shown that when \(p\) is even, for \(p \geq 10\) and \(g \geq 14\), three involutions generate \(\mathrm{Mod}(\Sigma_{g,p})\), and for \(p \geq 4\) and \(g \in \{3,4,5,6\}\), four involutions generate \(\mathrm{Mod}(\Sigma_{g,p})\). Furthermore, this result also holds when \(p = 2 \text{ or } 3\). Moreover, when \(p\) is odd, it is established that for \(p \geq 9\) and \(g \geq 13\), four involutions generate \(\mathrm{Mod}(\Sigma_{g,p})\), and for \(p \geq 5\) and \(3 \leq g \leq 6\), five involutions generate \(\mathrm{Mod}(\Sigma_{g,p})\).
Reviewer: Kashyap Rajeevsarathy (Bhopal)Flip graphs for infinite type surfaceshttps://zbmath.org/1527.570112024-02-28T19:32:02.718555Z"Fossas, Ariadna"https://zbmath.org/authors/?q=ai:fossas.ariadna"Parlier, Hugo"https://zbmath.org/authors/?q=ai:parlier.hugoThe paper discusses infinite type surfaces and the associated flip graphs \(\mathcal{F}(\Sigma)\). The use of flip graphs to study surfaces, their geometric structures, and moduli spaces has been studied in the literature. Flip graphs provide a way to measure distances between triangulations and are adapted to both finite and infinite type surfaces.
The main result of the present paper is Theorem 1.1, which establishes a criterion for two triangulations to be in the same connected component of the flip graph. The flip graph \(\mathcal{F}(\Sigma)\) has multiple connected components, as shown by Corollary 1.2, and it has uncountably many connected components for surfaces of infinite type. The paper mentions the construction of triangulations that are not related by flip transformations, and also provides upper and lower bounds on the flip distance between the triangulations.
The concept of flip graphs associated with infinite type surfaces, the properties of arcs, intersection numbers and triangulations are discussed in detail. Different approaches to define flip graphs and triangulations for infinite type surfaces are compared, and examples of surfaces with ideal vertices are given.
Reviewer: Keerti Vardhan (Chandigarh)Some properties of Pin\({}^\pm\)-structures on compact surfaceshttps://zbmath.org/1527.570122024-02-28T19:32:02.718555Z"Klug, Michael R."https://zbmath.org/authors/?q=ai:klug.michael-r"Stehouwer, Luuk"https://zbmath.org/authors/?q=ai:stehouwer.luukSummary: We show that two Pin-structures on a surface differ by a diffeomorphism of the surface if and only if they are cobordant (for comparison, the analogous fact has already been shown for Spin-structures). We give a construction that shows that this does not extend to dimensions greater than two. In addition, we count the number of Pin-structures on a surface in a given cobordism class.Heegaard genus and complexity of fibered knotshttps://zbmath.org/1527.570132024-02-28T19:32:02.718555Z"Cengiz, Mustafa"https://zbmath.org/authors/?q=ai:cengiz.mustafaIn this paper, the author considers the uniqueness of Heegaard splittings induced by a fibered knot with genus more than 1 in a 3-manifold. He proves that if the monodromy is sufficiently complicated, then the fibered knot induces a minimal genus Heegaard splitting which is unique up to isotopy. He uses the distance in the arc-curve complex of the page surface to describe the complexity of the monodromy and gives a complexity bound in terms of the Heegaard genus of the 3-manifold. The result are proved based on standard thin position and double sweepout arguments.
Reviewer: Junhua Wang (Shanghai)The finiteness conjecture for skein moduleshttps://zbmath.org/1527.570142024-02-28T19:32:02.718555Z"Gunningham, Sam"https://zbmath.org/authors/?q=ai:gunningham.sam"Jordan, David"https://zbmath.org/authors/?q=ai:jordan.david-andrew"Safronov, Pavel"https://zbmath.org/authors/?q=ai:safronov.pavelPrzytycki and Turaev introduced skein modules for arbitrary \(3D\) manifolds, spanned over certain rings, e.g. \(\mathcal{Q}(A)\) for \(A\) a distinguished indeterminate by all framed links in the manifold, up to isotopies (obvious) and skein relations understood as local linear relations for knots and links that are identical (up to isotopies) outside a \(3D\) ball but differ inside the ball. The motivation seemed to be to embed the theory of knot invariants in algebraic geometry. An important example of so defined skein modules is obtained for Kauffman relations and is in this case called the Kauffman skein module. In the case of a \(3D\) sphere it produces an invariant essentially equivalent to the celebrated Jones polynomial. In the case of general \(3D\) manifolds the modules were calculated only in special cases, but the known examples lead to formulation of conjectures about the general case. One of the conjectures was formulated by Witten and it states that for any closed oriented \(3D\) manifold the skein module is finite dimensional (over \(\mathcal{Q}(A)\)).
The authors prove the theorem but they do much more than that. Namely they reformulate skein modules to be even more algebraic, so that they can use tools from representation theory of quantum groups and from deformation quantization modules. Another aspect of the theory developed in the paper is formulation of the theory of so called internal skein modules (the authors suggested this notion is equivalent to the notion of stated skein modules, introduced in parallel to their definition), which have nice behavior under Heegard splitting, so that the skein module of a \(3D\) manifold split into two submanifolds meeting at a certain \(2D\) surface consists of invariant elements of the tensor product of the internal skein modules of the component submanifolds. This approach works particularly well in certain interesting special cases and is the essence of the proof of the general case. Additionally, the authors advertise that internal skein modules are easier to deal with than the standard skein modules since they are defined in terms of generators and relations, making computations more straightforward. The paper deals with many technicalities that this approach introduces, in quite some detail.
Reviewer: Robert Owczarek (Los Alamos)3-manifolds and Vafa-Witten theoryhttps://zbmath.org/1527.570152024-02-28T19:32:02.718555Z"Gukov, Sergei"https://zbmath.org/authors/?q=ai:gukov.sergei"Sheshmani, Artan"https://zbmath.org/authors/?q=ai:sheshmani.artan"Yau, Shing-Tung"https://zbmath.org/authors/?q=ai:yau.shing-tungSummary: We initiate explicit computations of Vafa-Witten invariants of 3-manifolds, analogous to Floer groups in the context of Donaldson theory. In particular, we explicitly compute the Vafa-Witten invariants of 3-manifolds in a family of concrete examples relevant to various surgery operations (the Gluck twist, knot surgeries, log-transforms). We also describe the structural properties that are expected to hold for general 3-manifolds, including the modular group action, relation to Floer homology, infinite-dimensionality for an arbitrary 3-manifold, and the absence of instantons.Rectangular knot diagrams classification with deep learninghttps://zbmath.org/1527.570162024-02-28T19:32:02.718555Z"Kauffman, L. H."https://zbmath.org/authors/?q=ai:kauffman.louis-hirsch"Russkikh, N. E."https://zbmath.org/authors/?q=ai:russkikh.n-e"Taimanov, I. A."https://zbmath.org/authors/?q=ai:taimanov.iskander-aIn their paper, Kauffman, Russkikh, and Taimanov present a novel approach to the classification of rectangular knot diagrams using deep learning techniques. Their aim is to utilize neural networks to distinguish between different classes of knots, focusing primarily on the unknotting problem. The authors use rectangular Dynnikov diagrams and apply neural networks to distinguish different classes from a given finite family of topological types.
The authors provide a thorough and detailed explanation of their methodology, including a comprehensive discussion of the theory behind rectangular diagrams and the elementary moves involved. They also present a clear and detailed explanation of their application of deep learning to the unknotting problem, providing a valuable contribution to the field of knot theory.
The paper's strength lies in its innovative approach and its successful combination of mathematical theory and machine learning techniques. The authors effectively demonstrate how deep learning can contribute to the classification of knot diagrams, providing valuable insights into the problem's complexity.
However, the paper would benefit from further discussion and analysis of the results. While the authors provide some analysis of their results, a more detailed discussion would add depth to their findings and provide readers with a better understanding of the implications of their work.
Reviewer: Qingyun Zeng (Philadelphia)Legendrian ribbons and strongly quasipositive links in an open bookhttps://zbmath.org/1527.570172024-02-28T19:32:02.718555Z"Hayden, Kyle"https://zbmath.org/authors/?q=ai:hayden.kyleThe main theorem (Theorem 1) of this paper states that a link in an open book is strongly quasipositive if and only if it bounds a Legendrian ribbon in the associated contact manifold. This is a generalization the work of \textit{S. Baader} and \textit{M. Ishikawa} [Ann. Fac. Sci. Toulouse, Math. (6) 18, No. 2, 285--305 (2009; Zbl 1206.57005)] for the standard tight contact structure on the 3-sphere. The same result has been proven in a different manner in the AIM SQuaREs project by Baykur, Etnyre, Hedden, Kawamuro and Van Horn-Morris.
Strongly quasipositive links were introduced by Rudolph and have been a very important class of links in contact geometry. A strongly quasipositive link is defined to be a braid in an open book that bounds a Bennequin surface with only positively twisted bands. It has been conjectured by a number of people that a link is strongly quasipositive if and only if it achieves equality in the Bennequin-Eliashberg inequality: \(\mathrm{sl}(L, [\Sigma]) \leq -\chi(\Sigma)\). The only-if part of the conjecture is known to be true by the definition of a strongly quasipositive link. Theorem 1 gives a new approach to the conjecture.
The only if part of Theorem 1 has been proven in the author's earlier paper [Geom. Topol. 25, No. 3, 1441--1477 (2021; Zbl 1490.57004)]. The outline of the if-part of Theorem 1 is as follows. By definition, a Legendrian ribbon is a certain collar neighborhood of a Legendrian graph. Starting with Gay-Licata's front projection diagram of a Legendrian graph, Hayden arranges it by isotopy into a cusped arc position, which is similar to an arc presentation of the graph. From the cusped arc diagram, a recipe to form the front projection of the Legendrian ribbon and then to turn it into a Bennequin surface is given with helpful illustrations.
Plenty of applications of Theorem 1 are shown. The author shows that if \(J \subset S^1\times D^2\) is a strongly quasipositive braid and \(K\) is a strongly quasipositive link, then the satellite \(J(K)\) is also a strongly quasipositive link. This result generalizes \textit{M. Hedden}'s work [J. Knot Theory Ramifications 19, No. 5, 617--629 (2010; Zbl 1195.57029)] on iterated torus knots in \(S^3\). It also generalizes Rudolph's constructions of quasipositive annuli and Whitehead doubles. The author also studies genus bounds, the relationship between strongly quasipositivity and fiberedness for knots in a tight contact structure, and Markov braid destabilization operation.
Reviewer: Keiko Kawamuro (Iowa City)New developments on the topology at infinity of discrete groupshttps://zbmath.org/1527.570182024-02-28T19:32:02.718555Z"Otera, Daniele Ettore"https://zbmath.org/authors/?q=ai:otera.daniele-ettoreSummary: We report our recent results on the geometry and the topology at infinity of finitely generated groups. In particular, we describe few generalizations of the (simple)-connectivity at infinity and of the QSF property (we actually define a sort of growth functions for them), and we present new classes of QSF groups.Some `converses' to intrinsic linking theoremshttps://zbmath.org/1527.570192024-02-28T19:32:02.718555Z"Karasev, Roman"https://zbmath.org/authors/?q=ai:karasev.roman-n"Skopenkov, Arkadiy"https://zbmath.org/authors/?q=ai:skopenkov.arkadij-bSummary: A low-dimensional version of our main result is the following `converse' of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph \(K_6\) in 3-space: \textit{For any integer \(z\) there are six points \(1, 2, 3, 4, 5, 6\) in \(3\)-space, of which every two \(i, j\) are joined by a polygonal line \(ij\), the interior of one polygonal line is disjoint with any other polygonal line, the linking number of any pair of disjoint \(3\)-cycles except for \(\{123,456\}\) is zero, and for the exceptional pair \(\{123,456\}\) is \(2z+1\).} We prove a higher-dimensional analogue, which is a `converse' of a lemma by Segal-Spież.Agol cycles of pseudo-Anosov 3-braidshttps://zbmath.org/1527.570202024-02-28T19:32:02.718555Z"Aceves, Elaina"https://zbmath.org/authors/?q=ai:aceves.elaina"Kawamuro, Keiko"https://zbmath.org/authors/?q=ai:kawamuro.keikoAn \textit{Agol cycle} is a combinatorial object (a sequence of measured train tracks encoding the transverse measured foliation of a pseudo-Anosov map) which is a complete invariant of the conjugacy class of a pseudo-Anosov mapping class of a surface [\textit{I. Agol}, Contemp. Math. 560, 1--17 (2011; Zbl 1335.57026)]; cf. a paper by \textit{C. D. Hodgson} et al. [Exp. Math. 25, No. 1, 17--45 (2016; Zbl 1337.57043)]. In the present paper, the authors consider the case of the 3-punctured disk and of pseudo-Anosov 3-braids, studying necessary and sufficient conditions for Agol cycles of pseudo-Anosov 3-braids to be equivalent. They present infinitely many pseudo-Anosov 3-braids belonging to distinct conjugacy classes but with the same dilatation. Considering then pseudo-Anosov 3-braids with the same dilatation, they give three equivalent conditions (a topological, a number theoretical and a numerical one) for two such 3-braids to have combinatorially isomorphic Agol cycles.
Reviewer: Bruno Zimmermann (Trieste)Global fixed points of mapping class group actions and a theorem of Markovichttps://zbmath.org/1527.570212024-02-28T19:32:02.718555Z"Chen, Lei"https://zbmath.org/authors/?q=ai:chen.lei.3"Salter, Nick"https://zbmath.org/authors/?q=ai:salter.nick\textit{V. Markovic} proved [Invent. Math. 168, No. 3, 523--566 (2007; Zbl 1131.57021)] that the mapping class group \(\mathrm{Mod}(\Sigma_g)\) of a closed surface \(\Sigma_g\) of genus \(g\) cannot be realized by homeomorphisms (in the sense of the Nielsen realization problem, originally formulated by Nielsen for finite groups of mapping classes and resolved by Kerckhoff); Markovic' proof is quite long and involved, using many analytic tools. In the present paper, the authors give a ``short and elementary proof'' of the non-realizability of the mapping class group by homeomorphisms. ``With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.'' Specifically, any nontrivial continuous action of the ``pointed mapping class group'' \(\mathrm{Mod}(\Sigma_{g,1})\) (of a surface \(\Sigma_{g,1}\) with a marked point) on Euclidean space \(\mathbb{R}^2\) (respectively \(\mathbb{R}^3\)) has a unique global fixed point (respectively an invariant line); in particular, for \(g \ge 4\), there is no nontrivial action of \(\mathrm{Mod}(\Sigma_{g,1})\) on \(\mathbb{R}^2\) and \(\mathbb{R}^3\) by \(C^1\)-diffeomorphisms. All these results are consequences of a basic structural result that there exists an order 6 element \(\alpha_g \in\mathrm{Mod}(\Sigma_{g,1})\) such that the centralizers of \(\alpha_g^2\) and \(\alpha_g^3\) together generate the full mapping class group \(\mathrm{Mod}(\Sigma_{g,1})\).
Reviewer: Bruno Zimmermann (Trieste)Cusps, Kleinian groups, and Eisenstein serieshttps://zbmath.org/1527.570222024-02-28T19:32:02.718555Z"Liu, Beibei"https://zbmath.org/authors/?q=ai:liu.beibei.1|liu.beibei"Wang, Shi"https://zbmath.org/authors/?q=ai:wang.shi|wang.shi.1Let \(\Gamma < \mathrm{Isom}^{+}(\mathbb{H}^{n})\) be a Kleinian group with Poincaré series \(f_{s}=\sum_{\gamma \in \Gamma} e^{-sd(\gamma(o),o)}\) where \(o \in \mathbb{H}^{n}\) is a base-point and \(d\) is the hyperbolic metric on \(\mathbb{H}^{n}\). The critical exponent of \(\Gamma\) is \(\delta(\Gamma)=\inf \{ s \mid f_{s} < \infty \}\) (the critical exponent has several alternative descriptions).
The purpose of the paper under review is to study the number of cusps in the associated quotient manifold \(\Gamma \backslash \mathbb{H}^{n}\). The authors consider Eisenstein's series associated to the full rank cusps in a complete hyperbolic manifold and they show that, for a Kleinian group \(\Gamma\), each full rank cusp corresponds to a cohomology class in \(H^{n}(\Gamma,V)\), where \(V\) is either the trivial coefficient or the adjoint representation. In the case of trivial coefficient, they prove Theorem 1.1: Let \(\Gamma < \mathrm{Isom}^{+}(\mathbb{H}^{n+1})\) be a torsion free discrete group. If the critical exponent \(\delta(\Gamma)\) is less than \(n\) or \(\Gamma\) is of convergence type, then for any parabolic subgroup \(\Gamma_{i} < \Gamma\) of rank \(n\), and any generating cohomology class \(\alpha_{i} \in H^{n}(\Gamma_{i}, \mathbb{R})\simeq \mathbb{R}\), there is a harmonic form \(E(\alpha_{i})\) on \(\Gamma \backslash \mathbb{H}^{n+1}\) constructed via the Eisenstein series, such that the restriction homomorphism \(H^{n}(\Gamma, \mathbb{R}) \rightarrow H^{n}(\Gamma_{i}, \mathbb{R})\) sends \([E(\alpha_{i})]\) to \(\alpha_{i}\) and, if \(\Gamma_{i}\), \(\Gamma_{j}\) are not \(\Gamma\)-conjugate, then the restriction homomorphism sends \([E(\alpha_{j})]\) to 0.
Reviewer: Egle Bettio (Venezia)On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifoldshttps://zbmath.org/1527.570232024-02-28T19:32:02.718555Z"Baraglia, David"https://zbmath.org/authors/?q=ai:baraglia.david"Konno, Hokuto"https://zbmath.org/authors/?q=ai:konno.hokutoThe article establishes a cohomological formula to show how the Seiberg-Witten invariants (SW-invariants) of a family of smooth 4-manifolds can be recovered from the Bauer-Furuta invariants (BF-invariants) of the same family. A family of smooth 4-manifolds means a smooth locally trivial fibre bundle over a smooth base manifold whose fibres are a fixed compact smooth 4-manifold \(X\). From a family of smooth 4-manifolds results a family of smooth invariants.
Several properties of families SW-invariants are shown to yield from this cohomological formula, e.g., it is shown that the Steenrod squares of the families SW-invariants lead to a series of relations between the SW-invariants and the Chern classes of the \(\mathrm{spin}^{c}\) index bundle of the family. As a by-product, the article shows there is an obstruction to the existence of certain families of 4-manifolds with fibres diffeomorphic to a 4-manifold \(X\). A non-smoothable family of \(K3\) surfaces is explicitly shown. The article simplifies the wall crossing formula for families. It also introduces \(K\)-theoretic SW-invariants. A formula expressing the Chern character of the \(K\)-theoretic SW-invariants in terms of the cohomological SW-invariants is obtained, and so, new divisibility properties of the families SW-invariants are achieved.
In references [\textit{T.-J. Li} and \textit{A.-K. Liu}, Commun. Anal. Geom. 9, No. 4, 777--823 (2001; Zbl 1034.53090); \textit{A.-K. Liu}, J. Differ. Geom. 56, No. 3, 381--579 (2000; Zbl 1036.14014); \textit{N. Nakamura}, Asian J. Math. 7, No. 1, 133--138 (2003; Zbl 1074.57016); \textit{D. Ruberman}, Math. Res. Lett. 5, No. 6, 743--758 (1998; Zbl 0946.57025); \textit{D. Ruberman}, Geom. Topol. 5, 895--924 (2001; Zbl 1002.57064)], the SW-invariants of smooth 4-manifolds have been extended to invariants of families of smooth manifolds. The family is assumed to be equipped with a \(\mathrm{spin^{c}}\)-structure on the vertical tangent bundle. The BF-invariants can also be extended to a family of smooth 4-manifolds, as implicitly described in [\textit{S. Bauer} and \textit{M. Furuta}, Invent. Math. 155, No. 1, 1--19 (2004; Zbl 1050.57024), Theorem 2.6] and further developed by \textit{M. Szymik} [Forum Math. 22, No. 3, 509--523 (2010; Zbl 1200.57019)]. However, neither of these works establishes how one can recover the families SW-invariants from the families BF-invariant.
The article's main result (Theorem 3.6) shows the families SW-invariants can be recovered from the families BF-invariants through an explicit cohomological formula relating the two. The formula is used to extract a number of results concerning the families SW-invariants:
(1) The Steenrod powers of the families Seiberg-Witten invariants are computed (Section 4).
(2) A simple new proof of the wall crossing formula, of Li and Liu [loc. cit.], is achieved in the context of families SW-invariants (Section 5).
(3) \(K\)-theoretic families SW-invariants are introduced and it is shown how they are related to the usual cohomological families of invariants via the Chern character, so leading to certain divisibility properties of the families Seiberg-Witten invariants (Section 6).
Reviewer: Celso M. Doria (Florianópolis)Relative trisection embeddings of 4-manifoldshttps://zbmath.org/1527.570242024-02-28T19:32:02.718555Z"Pandit, Suhas"https://zbmath.org/authors/?q=ai:pandit.suhas"Selvakumar, A."https://zbmath.org/authors/?q=ai:selvakumar.aSummary: In this article, we discuss relative trisection embeddings of compact orientable \(4\)-manifolds with nonempty boundary into a trivial relative trisection of the \(8\)-disc \(D^8 \subset \mathbb{R}^8\).A complete characterization of reversibility in \(PL (S)\)https://zbmath.org/1527.570252024-02-28T19:32:02.718555Z"Ben Rejeb, Khadija"https://zbmath.org/authors/?q=ai:ben-rejeb.khadijaAn element of a group \(G\) is reversible if it is conjugate to its inverse. It is strongly reversible if the conjugacy is given by an involution. Hence, strongly reversible elements can be written as the product of two involutions. See the monograph [\textit{A. G. O'Farrell} and \textit{I. Short}, Reversibility in dynamics and group theory. Cambridge: Cambridge University Press (2015; Zbl 1331.37003)] for a complete introduction to this property. Here the author characterizes reversibility in the group \(PL(S)\) of piecewise-linear homeomorphisms of the circle. The principle is that for reversible elements, conjugacy-invariant properties must coincide with those of their inverse, and the goal is to find a finite list of conjugacy-invariant properties that are enough to detect reversibility. More precisely, the author considers the following very natural invariants.
\begin{itemize}
\item[1.] The rotation number \(\rho(f)\) of an orientation-preserving homeomorphism \(f\in PL^+(S)\).
\item[2.] When \(f\in PL^+(S)\) has a periodic orbit of order \(n\), the power \(f^n\) has fixed points, therefore one can consider the minimal decomposition of the circle into finitely many intervals such that the sign of \(f(x)-x\) is constant on any such interval.
\item[3.] The collection of derivatives at fixed points.
\item[4.] The so-called Mather invariant (see [\textit{F. Matucci}, in: Combinatorial and geometric group theory. Dortmund and Ottawa-Montreal conferences. Selected papers of the conferences on ``Combinatorial and geometric group theory with applications'' (GAGTA), Dortmund, Germany, August 27--31, 2007, ``Fields workshop in asymptotic group theory and cryptography'', Ottawa, Canada, December 14--16, 2007, and the workshop on ``Action on trees, non-Archimedian words, and asymptotic cones'', Montreal, Canada, December 17--21, 2007. Basel: Birkhäuser. 251--260 (2010; Zbl 1200.37037)]) on any such interval where the sign is not zero.
\end{itemize}
The author denotes by \(\Sigma(f^n)\) the collection of the last three invariants, and defines it in such a way that it makes sense to say that the \(\Sigma\)-invariant of one element is the opposite of another. After [\textit{M. G. Brin} and \textit{C. C. Squier}, Commun. Algebra 29, No. 10, 4557--4596 (2001; Zbl 0986.57025)], these four invariants characterize conjugacy classes in \(PL^+(S)\).
The first main result (Theorem 1.1) characterizes reversibility in \(PL^+(S)\). Note that in this situation (orientation-preserving homeomorphisms), a reversible element necessarily satisfies that its rotation number is either 0 or 1/2. When \(\rho(f)=0\), the element \(f\) is reversible in \(PL^+(S)\) if and only if there exists an element \(h\in PL^+(S)\) such that \(\Sigma(f)\) and \(\Sigma(hfh^{-1})\) are opposite. Moreover, \(f\) is strongly reversible if and only if there exits an element \(h\in PL^+(S)\) with \(\rho(h)=1/2\) and such that \(\Sigma(f)\) and \(\Sigma(hfh^{-1})\) are opposite. When \(\rho(f)=1/2\), the same conditions must be satisfied for \(f^2\), respectively.
As a corollary (Theorem 1.3), the author can give an example of an element which is reversible in \(PL^+(S)\) but not strongly reversible.
The second main result (Theorem 1.4) characterizes reversibility in \(PL(S)\) of elements of \(PL^+(S)\). When the rotation number is rational, the author gets a characterization similar to the one in Theorem 1.1. For irrational rotation number, reversibility is characterized by \((D)\) -- the property, which means that the product of jumps of derivatives at points in a single orbit must be 1, and this for any orbit, and the fact that a slightly different product of jumps is also 1.
Finally, Theorem 1.5 says that any element in \(PL(S)\) can be written as the product of involutions in \(PL^-(S)\).
Reviewer: Michele Triestino (Dijon)Free circle actions on Dold and Wall manifoldshttps://zbmath.org/1527.570262024-02-28T19:32:02.718555Z"Paiva, Thales Fernando Vilamaior"https://zbmath.org/authors/?q=ai:paiva.thales-fernando-vilamaior"dos Santos, Edivaldo Lopes"https://zbmath.org/authors/?q=ai:dos-santos.edivaldo-lopesThe \textit{ Dold manifold} \(P(m,n)\) is the quotient \(\mathbb{S}^m\times \mathbb{CP}^n/\sim\) where \((x,[z]) \sim (-x,[z]) \). The \textit{ Wall manifold} is the mapping torus of the involutive homeomorphism on the corresponding Dold manifold given by the reflection \((x_0,\dots, x_{m-1},x_m)\, \mapsto\, (x_0,\dots, x_{m-1}, -x_m)\) on the first factor and identity on the second factor. Such manifolds are important in the bordism theory.
In this paper, the authors construct examples of free actions of \(\mathbb{S}^1\) on the Dold manifold and compute the cohomology algebra of the orbit space \(X/\mathbb{S}^1\) where \(X\) is any finitistic space with mod 2 cohomology of a Dold manifold \(P(m,\mathrm{odd} )\). For any integers \(m, n\), the Wall manifold of type \(Q(m,n)\) can not admit any free circle action. The proofs involve cohomology computations.
Reviewer: Chandan Maity (S.A.S. Nagar)The cohomology rings of homogeneous spaceshttps://zbmath.org/1527.570272024-02-28T19:32:02.718555Z"Franz, Matthias"https://zbmath.org/authors/?q=ai:franz.matthias-o|franz.matthiasLet \(G\) be a compact connected Lie group and \(K\) a closed connected subgroup. Real cohomology of a homogeneous space \(G/K\) was first calculated by A. Cartan. This result was followed by a series of clarifications and generalizations. It is significant here that the real cohomology of the classifying space of a connected Lie group is a polynomial algebra on even-degree generators. An obvious question here is whether a result analogous to Cartan's holds for other principal ideal domains \(k\), for which cohomologies of classifying spaces \(H^\ast (BG)\) and \(H^\ast (BK)\) are polynomial algebras with even-degree generators. An equivalent condition here is that the orders of the torsion subgroups of \(H^\ast (G; \mathbb Z)\) and \(H^\ast(K;\mathbb Z)\) are invertible in \(k\). It holds for example for \(U (n),SU (n)\) and \(Sp(n)\) over any \(k\), and for \(SO(n)\) and Spin\((n)\) if 2 is invertible in \(k\). The main result of this paper it the following. Assume that 2 is invertible in \(k\). If \(H^\ast (BG)\) and \( H^\ast (BK)\) are polynomial algebras, then there is an isomorphism of graded \(k\)-algebras \(H^\ast (G/K ) \cong \mathrm{Tor}^\ast_ {H ^\ast (BG)} (k, H^\ast(BK))\), which is natural with respect to maps of pairs \( (G,K) \to (G^\prime, K^\prime)\). It is important to note, that here we have an isomorphism of \(k\)-algebras, but not only of \(k\)-modules, as was usually the case until now. Some examples of calculations of \(H^\ast (G/K)\) are given. It is interesting to note that in this article symbolic calculation with aid of packages Maple and Sage was used to derive some of the formulas.
Reviewer: V. V. Gorbatsevich (Moskva)On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networkshttps://zbmath.org/1527.682052024-02-28T19:32:02.718555Z"Grigsby, J. Elisenda"https://zbmath.org/authors/?q=ai:grigsby.julia-elisenda"Lindsey, Kathryn"https://zbmath.org/authors/?q=ai:lindsey.kathryn-aSummary: Let \(F: \mathbb{R}^n \rightarrow \mathbb{R}\) be a feedforward ReLU neural network. It is well-known that for any choice of parameters, \(F\) is continuous and piecewise affine-linear. We lay some foundations for a systematic investigation of how the \textit{architecture} of \(F\) impacts the geometry and topology of its possible decision regions, \(F^{-1}((-\infty, t))\) and \(F^{-1}((t, \infty))\), for binary classification tasks. Following the classical progression for smooth functions in differential topology, we first define the notion of a \textit{generic, transversal} ReLU neural network and show that almost all ReLU networks are generic and transversal. We then define a partially oriented linear 1-complex in the domain of \(F\) and identify properties of this complex that yield an obstruction to the existence of bounded connected components of a decision region. We use this obstruction to prove that a decision region of a generic, transversal ReLU network \(F: \mathbb{R}^n \rightarrow \mathbb{R}\) with a single hidden layer of dimension \(n+1\) can have no more than one bounded connected component.Classroom-suitable matrix method for one-dimensional stepped quantum potentialshttps://zbmath.org/1527.810512024-02-28T19:32:02.718555Z"Reed, B. Cameron"https://zbmath.org/authors/?q=ai:reed.bruce-cameron(no abstract)Considerations about the incompleteness of the Ehrenfest's theorem in quantum mechanicshttps://zbmath.org/1527.810602024-02-28T19:32:02.718555Z"Giordano, Domenico"https://zbmath.org/authors/?q=ai:giordano.domenico"Amodio, Pierluigi"https://zbmath.org/authors/?q=ai:amodio.pierluigi(no abstract)Bound state eigenvalues from transmission coefficientshttps://zbmath.org/1527.811122024-02-28T19:32:02.718555Z"Ahmed, Zafar"https://zbmath.org/authors/?q=ai:ahmed.zafar"Bhattacharya, Koushik"https://zbmath.org/authors/?q=ai:bhattacharya.koushik(no abstract)\(f(R,\Sigma,T)\) gravityhttps://zbmath.org/1527.830322024-02-28T19:32:02.718555Z"Bakry, M. A."https://zbmath.org/authors/?q=ai:bakry.m-a"Ibraheem, Shymaa K."https://zbmath.org/authors/?q=ai:ibraheem.shymaa-kSummary: We used the absolute parallelism geometry to obtain a new formula for the Ricci scalar. We consider \(f(R,\Sigma,T)\) modified theories of gravity, where the gravitational Lagrangian is given by three arbitrary functions of the Ricci scalar \(R\), Ricci torsion scalar \(\Sigma\), and the trace of the stress-energy tensor \(T\). We obtain the gravitational field equations in the metric formalism. The evolution of the function \(f(R)\) withr time is studied, and we discuss the parameters that make up the function and impose constraints on these parameters. The solution of the \(f(R,\Sigma,T)\) gravity equations are obtained under a varying polynomial deceleration parameter. The effect of torsion on cosmological models is also discussed. Physical aspects of the energy density, pressure, and energy conditions of the cosmological models proposed in this article are studied, and the evolution of the physical parameters is shown in figures. Evolution of the fluid pressure and energy density parameter as a function of redshift has been obtained. The \(f(R)\) gravity and \(f(R,T)\) gravity theories as special cases could be inferred from \(f(R,\Sigma,T)\) gravity. Several special cases have been studied, with illustrations for each case.Exact geometries from boundary gravityhttps://zbmath.org/1527.830352024-02-28T19:32:02.718555Z"Gupta, Rohit K."https://zbmath.org/authors/?q=ai:gupta.rohit-k"Kar, Supriya"https://zbmath.org/authors/?q=ai:kar.supriya-k"Nitish, R."https://zbmath.org/authors/?q=ai:nitish.r"Verma, Monika"https://zbmath.org/authors/?q=ai:verma.monikaSummary: We show that the extremal Reissner-Nordström type multi-black holes in an emergent scenario are exact in General Relativity. It is shown that an axion in the bulk together with a geometric torsion ensures the required energy momentum to source the \((3 + 1)\) geometry in the Einstein tensor. Analysis reveals a significant role of dark energy in curved space-time.