Journal of Topology最新文献

We study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension $��2�n�+�1�⩾�7�$ with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension $��d�⩾�2�$: we prove that their total spaces are rationally inessential if $��d�⩾�3�$, and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.

We define the Milnor number of a one-dimensional holomorphic foliation $�F$ as the intersection number of two holomorphic sections with respect to a compact connected component $�C$ of its singular set. Under certain conditions, we prove that the Milnor number of $�F$ on a three-dimensional manifold with respect to $�C$ is invariant by $�C^1$ topological equivalences.

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $�g$ with respect to the Weil–Petersson measure on the moduli space $�M_g$. We show that as $�g$ goes to infinity, a generic surface $��X�\in �M_g�$ satisfies asymptotically:

Given an irreducible, end-periodic homeomorphism $��f�:�S�\to �S�$ of a surface with finitely many ends, all accumulated by genus, the mapping torus, $�M_f$, is the interior of a compact, irreducible, atoroidal 3-manifold $�{\overline{M}}_f$ with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of $�{\overline{M}}_f$ in terms of the translation length of $�f$ on the pants graph of $�S$. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.

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.