Journal of Topology最新文献

We develop a theory of stated SL($�n$)-skein modules, $��S_n��(�M�,�N�)��$, of 3-manifolds $�M$ marked with intervals $�N$ in their boundaries. These skein modules, generalizing stated SL(2)-modules of the first author, stated SL(3)-modules of Higgins', and SU(n)-skein modules of the second author, consist of linear combinations of framed, oriented graphs, called $�n$-webs, with ends in $�N$, considered up to skein relations of the $��U_q��\left(�s�l_n�\right)��$-Reshetikhin–Turaev functor on tangles, involving coupons representing the anti-symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3-manifold $�M$ along a disk resulting in a 3-manifold $�M^{\prime }$ yields a homomorphism $��S_n$

We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

In this paper, we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a $�C^{\alpha }$-submanifold of the real projective space for some $��\alpha �\in �(�1�,�2�)�$. We also calculate the optimal value of $�\alpha$ in terms of the eigenvalue data of the Anosov representation.

We show that for many right-angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: They induce the same coarse median structure on the group. These are the first examples of non-hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank $��⩾�3�$, we show that all automorphisms preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo–Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation.

We introduce $�k$-positive representations, a large class of $��\{�1�,�\text{…}�,�k�\}�$-Anosov surface group representations into $��\mathrm{PGL}�\left(�E�\right)�$ that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least $��(�k�-�3�)�$-positive and irreducible limits are $��(�k�-�1�)�$-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations.

We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2-manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2-manifolds.

The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.

For a germ of a variety $��V�,�0�\subset �C^N�,�0�$, a singularity $�V_0$ of “type $�V$”  is given by a germ $��f_0�:�C^n�,�0�\to �C^N�,�0�$, which is transverse to $��V�\setminus �\left\{�0�\right\}�$ in an appropriate sense, such that $��V_0�=�f_0^{�-�1�}��\left(�V�\right)��$. In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for $�$

We develop a version of Seiberg–Witten Floer cohomology/homotopy type for a $�{\mathrm{spin}}^c$ 4-manifold with boundary and with an involution that reverses the $�{\mathrm{spin}}^c$ structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3-manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov-type inequalities that relate topological quantities of 4-manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4-manifolds, and nonsmoothable unoriented surfaces in 4-manifolds.

Let $��(�\Gamma �,�P�)�$ be a relatively hyperbolic group pair that is relatively one ended. Then, the Bowditch boundary of $��(�\Gamma �,�P�)�$ is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of $�\Gamma$ and no restriction on the peripheral subgroups $��P�\in �P�$.