Ramón Flores, Juan González-Meneses, Porfirio L. León-Álvarez
Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the classifying space $E_{\mathcal{F}_n}G$ for the pure braid group of a surface of non-negative Euler characteristic with at least one boundary component or one puncture is equal to the virtual cohomological dimension of $G$ plus $n$. We prove an analogous result for the full braid group of the sphere. As an application, we compute the minimal dimension of a model for the classifying space associated to the family of amenable subgroups of pure surface braid groups.
Rubén A. Hidalgo
In this paper, we introduce a collection of purely loxodromic free Kleinian groups, called infinite Schottky group, which are defined by a suitable collection of simple loops in a similar way as in the case for Schottky groups of finite rank. An infinite Schottky group $Γ$ admits a $Γ$-invariant connected component $Ω$ of its region of discontinuity, such that every other component is a topological disc and has trivial $Γ$-stabilizer, and $Ω/Γ$ is an infinite type Riemann surface without planar ends. Every infinite type Riemann surface $Σ_{F}$ without planar ends can be so obtained (retrosection theorem). If $G < {\rm Aut}(Σ_{F})$ acts freely and $Σ_{F}/G$ is of finite type, then we observe that it lifts to a group of automorphisms of $Ω$, for a suitable infinite Schottky uniformization of it by a infinite Schottky group $Γ$, if and only if there is a $G$-invariant collection ${\mathcal F}$ of pairwise disjoint essential simple loops on $Σ_{F}$ such that each connected component of $Σ_{F} \setminus {\mathcal F}$ is a finite planar surface, generalizing the situation for the case of Schottky groups of finite rank.
Hokuto Konno
This article provides a survey of gauge theory for families, with a particular focus on its applications to diffeomorphism groups of $4$-manifolds that were developed during the period 2021--2025.
Hokuto Konno
This article surveys gauge theory for families and its applications to the comparison between the diffeomorphism group and the homeomorphism group of $4$-manifolds, up to 2021.
Blake K Winter, Amanda Taylor Lipnicki
We give a method for constructing an interactive art piece which illustrates two different definitions of the braid groups, along with their faithful action on the free group. The box also demonstrates how all motions of points in the plane can be realized by motions in a single T-shaped subspace of the plane. This helps students and those who are not specialists in algebraic topology to understand these important topological objects.
Marián Poppr
Given a thin torus $T_K$ around a knot $K\subset \mathbb{R}^3$, we construct Morse models of cord algebra $Cord(T_K)$ with $\mathbb{Z}$ and loop space coefficients. Using the Multiple time scale dynamics we identify $Cord(T_K; \mathbb{Z})$ with $Cord(K; \mathbb{Z})$. In combination with the works of Cieliebak-Ekholm-Latschev-Ng and Petrak this indirectly relates $Cord(T_K)$ to $0$-th degree Legendrian contact homology $LCH_0(\mathcal{L}^\ast_+ T_K)$ of one component of the unit conormal bundle over $T_K$. Our definition of $Cord(T_K)$ is motivated by $J$-holomorphic curves with boundary on the Lagrangian submanifold $L^\ast_+ T_K\cup\mathbb{R}^3$ with an arboreal singularity along the torus $T_K$.
Michael Dougherty, Jon McCammond
This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra.
Panagiotis Dimakis, Duong Dinh, Shengjing Xu
On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast C$ (respectively, the twisted cotangent bundles of $C$). Central to these constructions are Higgs bundles (respectively, holomorphic connections) which are transversal to line subbundles of the underlying bundles: these naturally induce divisors on $C$ together with auxiliary parameters, namely lifts to divisors on spectral curves for Higgs bundles and residue parameters of apparent singularities for holomorphic connections. We discuss the evidence showing that the Dolbeault geometric Langlands correspondence is generically realized by these Lagrangian correspondences; we expect that the de Rham geometric Langlands correspondence can be realized by their quantization, following Drinfeld's construction of Hecke eigensheaves. We also discuss the relations of our constructions to various topics, including reductions of Kapustin-Witten equations, the conformal limit, separation of variables, and degenerate fields in conformal field theories.
Maya Kayali
The triple-cup product form $μ$ is a classical invariant of $3$-manifolds, determining the cohomology ring up to torsion. Given a closed, connected, oriented $3$-manifold $M$, we describe an explicit formula for computing $μ$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $μ$ can be recovered as a reduction of Turaev's homotopy intersection form $η$ of the Heegaard surface.
Daniel Groves, Emily Stark, Genevieve S. Walsh, Kevin Whyte
We prove that if a proper metric space is quasi-isometric to a finitely generated group and to a space with a horoball over a finitely generated group, then that space is quasi-isometric to a rank-one symmetric space or the real line.
Raphael Appenzeller, José Pedro Quintanilha
We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a trick by depicting the target knot as a special type of knot diagram, which we call a "knotholder diagram". By proving that all knots admit knotholder diagrams, we obtain variants of the trick for producing every knot.
Khushbu Gulati, Parameswaran Sankaran
A perforated surface is the complement $\mathringΣ:=Σ\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $Σ$. It is known that the topological type of $Σ\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring Σ$ arises from a covering of a surface $Σ'$ such that $\mathringΣ\cong \mathringΣ'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring Σ$, the Sierpiński curve and the Menger curve are not Hopfian.
Ayodeji Lindblad
For $X$ any complete intersection of even complex dimension or any connected sum thereof (or, more generally, any space among certain broad classes of smooth manifolds), we concretely construct orientation-preserving diffeomorphisms $a,c$ of punctured $X$ rel boundary whose commutator $[a,c]$ represents the smooth mapping class rel boundary of the boundary Dehn twist. This shows that boundary Dehn twists on 4-manifolds known to be nontrivial in the smooth mapping class group rel boundary by work of Baraglia-Konno, Kronheimer-Mrowka, J. Lin, and Tilton become trivial after abelianization, generalizing work of Y. Lin which applied an argument based on the global Torelli theorem and an obstruction of Baraglia-Konno to prove that the abelianized boundary Dehn twist on the punctured $K3$ surface is trivial.
Yongsheng Jia, Yusen Long
Drawing inspiration from [BBF15], we construct a family of unbounded quasi-trees for a connected closed oriented surface \(S_g\) of genus \(g\geq 2\), upon which the group \(\Homeo_0(S_g)\) acts coboundedly by isometries. As an application, we show that some surface homeomorphisms preserving a non-sporadic essential subsurface or an essential subsurface homeomorphic to a once-bordered torus can have positive stable commutator length in \(\Homeo_0(S_g)\). Moreover, we provide a version of projection complex that does not require the finiteness conditions.
Georg Frenck
The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the boundary satisfies a certain split-condition on fundamental groups. Our proof is based on surgery-techniques for positive scalar and mean curvature. If the boundary is non-connected, we use existence of area-minimizing hypersurfaces and the monotonicity-formula. Furthermore, we investigate if a psc-metric on a closed manifold can be adjusted so that a given embedded hypersurface is minimal, stable minimal or totally geodesic. While not true in general, such an adjustment is possible in many cases.
Román Aranda, Sarah Blackwell, Geunyoung Kim, Patrick Naylor, Puttipong Pongtanapaisan
We introduce and study bridge decompositions for 3-manifolds embedded in the 5-sphere. These generalize both the classical notion of bridge position for knots in the 3-sphere and the bridge trisections of surfaces in the 4-sphere due to Meier and Zupan. Our main technical tool is the multisections of 5-manifolds introduced by Aribi, Courte, Golla, and Moussard. We prove that every embedded 3-manifold admits such a decomposition; in particular, any such embedding is encoded by four trivial tangle diagrams. We also present a range of explicit examples, including $S^2$-spun knots and ribbon 3-knots.
Tolga Talha Yıldız, Uğur Önal, Ergün Akleman, Vinayak Krishnamurthy
We present Twisted Edges, a unified framework for designing Linked Knot (LK) structures using labeled non-manifold surface meshes. While the concept of edge twists, originating in topological graph theory, is foundational to these designs, prior approaches have been strictly limited to binary states. We identify this restriction as a critical barrier; binary twisting fails to capture the full spectrum of topological possibilities, rendering a vast class of structural and dynamic behaviors inaccessible. To overcome this limitation, we generalize the twist formulation to support arbitrary integer twist labels. This expansion reveals that while zero twists may introduce disconnections, applying even twists to 2-manifold meshes robustly preserves connectivity, transforming surfaces into fully connected, chainmail-like structures where faces form consistently linked cycles. Furthermore, we extend this framework to non-manifold meshes, where specific integer assignments prevent cycle merging. This capability, unattainable with binary methods, enables the design of partial connectivity and functional hinges, supporting dynamic folding and articulation. Theoretically, we show that these integer-twisted meshes correspond to knotted surfaces in four dimensions, with LK structures arising as their immersions into $\mathbb{R}^3$. By breaking the binary constraint, this work establishes a coherent paradigm for the systematic exploration of previously unstudied woven and articulated structures.
Charles Bordenave, Xinlong Dong, Dragomir D. Šarić
Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. The vertices of the corresponding dual graph G are pairs of pants and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on X is ergodic if and only if the random walk on G is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows. In addition, we show that rough isometry of surfaces does not preserve the ergodicity of the geodesic flow while rough isometry of their dual graphs does. The above equivalence result uses a characterization of the measured geodesic laminations on X that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on G.
Jonathan A. Hillman, Riccardo Pedrotti
We address the question of existence of sections of fibrations in two settings. First, we show that a bundle with base a finite 2-complex admits a section if and only if the inclusion of the fiber is $π_1$-injective and the associated short exact sequence of fundamental groups splits. Second, for Lefschetz fibrations over the disk we provide a complete algebraic criterion characterizing which loops in the boundary mapping torus extend to continuous or smooth sections over the disk. Finally, we apply our results to achiral Lefschetz fibrations over the sphere obtained by doubling along the vertical boundary, and give a criterion ensuring the existence of at least two homologically distinct sections.
Shinichiro Kakuta
We study the volume conjecture of the colored Jones invariants with sequences of colors corresponding to the deformation of the hyperbolic structure of a link complement. In particular, we investigate certain limits of the colored Jones invariants of the figure-eight knot and the Borromean rings and show that the limits are related to the volumes of hyperbolic cone manifolds whose singular sets are the links.