Journal of Topology最新文献

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $�Z$ whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type $�F_2$ but not $�{FP}_3$, and examples where the RACG is hyperbolic and the kernel is finitely generated and non-hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz–Norin–Wise involving Bestvina–Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

We define on any affine invariant orbifold $�M$ a foliation $�F^M$ that generalizes the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in the rank 1 case. We establish a criterion that ensures the density of the leaves and provide two applications of this criterion. The first one is a classification of the dynamical behavior of the leaves of $�F^M$ when $�M$ is a connected component of a Prym eigenform locus in genus 2 or 3 and the second provides the first examples of dense isoperiodic leaves in the stratum $��H�(�2�,�1�,�1�)�$.

The earring tangle consists of four strands $��4�\text{pt}�\times �I�\subset �S^2�\times �I�$ and one meridian around one of the strands. Equipping this tangle with a nontrivial $��S�O�\left(�3�\right)�$ bundle, we show that its traceless $��S�U�\left(�2�\right)�$ flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties.

We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.

We prove that if a fibered knot $�K$ with genus greater than 1 in a three-manifold $�M$ has a sufficiently complicated monodromy, then $�K$ induces a minimal genus Heegaard splitting $�P$ that is unique up to isotopy, and small genus Heegaard splittings of $�M$ are stabilizations of $�P$. We provide a complexity bound in terms of the Heegaard genus of $�M$. We also provide global complexity bounds for fibered knots in the three-sphere and lens spaces.

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy separability theorem that may be of independent interest. The infinite sequence of geometric triangulations is supported in a geometric submanifold associated to one cusp, and can be organized into an infinite trivalent tree of Pachner moves.

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex of the graph. This extends previously established results for the pants graph and the separating curve graph to a broad family of graphs associated to surfaces.

We show that the stable module $�\infty$-category of a finite group $�G$ decomposes in three different ways as a limit of the stable module $�\infty$-categories of certain subgroups of $�G$. Analogously to Dwyer's terminology for homology decompositions, we call these the centraliser, normaliser, and subgroup decompositions. We construct centraliser and normaliser decompositions and extend the subgroup decomposition (constructed by Mathew) to more collections of subgroups. The key step in the proof is extending the stable module $�\infty$-category to be defined for any $�G$-space, then showing that this extension only depends on the $�S$-equivariant homotopy type of a $�G$-space. The methods used are not specific to the stable module $�\infty$-category, so may also be applicable in other settings where an $�\infty$-category depends functorially on $�G$.

We use Galois group actions on étale cohomology to prove results of formality for dg-operads and dg-algebras with torsion coefficients. Our theory applies, among other related objects, to the dg-operad of singular chains of the operad of little disks and to the dg-algebra of singular cochains of the configuration space of points in the complex space. The formality that we obtain is only up to a certain degree, which depends on the cardinality of the field of coefficients.