Journal of Topology and Analysis

In this paper, we propose two algorithms for determining the Nielsen–Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in 〈X〉. We show that if ψ represents a reducible mapping class in Mod(S), then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use this fact to find the canonical reduction system of ψ. We also prove an effective conjugacy separability result for π₁(S) which allows us to lift the action of ψ to a finite cover of S whose degree depends computably on the word length of ψ, and to use the homology action of ψ on to determine the Nielsen–Thurston classification of ψ.

Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on M whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

Aramayona and Leininger have provided a "finite rigid subset" 𝔛(Σ) of the curve complex of a surface , characterized by the fact that any simplicial injection is induced by a unique element of the mapping class group Mod(Σ). In this paper we prove that, in the case of the sphere with n ≥ 5 marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a Mod(Σ)-module generator for the reduced homology of the curve complex , answering in the affirmative a question posed in [1]. For the surface with g ≥ 3 and n ∈ {0, 1} we find that the finite rigid set 𝔛(Σ) of Aramayona and Leininger contains a proper subcomplex X(Σ) whose reduced homology class is a Mod(Σ)-module generator for the reduced homology of but which is not itself rigid.

The waist inequality states that for a continuous map from Sⁿ to ℝ^q, not all fibers can have small (n - q)-dimensional volume. We construct maps for which most fibers have small (n - q)-dimensional volume and all fibers have bounded (n - q)-dimensional volume.

In this paper we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As applications we obtain deformation theorems for fixed point properties on Banach spaces. In particular, our results yield fixed point theorems for affine actions in which the linear part is not uniformly bounded. Our proofs are effective and allow for quantitative estimates.

We investigate a generalization to non-Witt stratified spaces of the intersection homology theory of Goresky–MacPherson. The second-named author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L² de Rham cohomology. In this paper we first extend both of these cohomology theories by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. Our main result is that this refined intersection cohomology theory coincides with the analytic de Rham theory on Thom–Mather stratified spaces. The word "refined" is motivated by the fact that the definition of this cohomology theory depends on the choice of an additional structure (mezzo-perversity) which is automatically zero in the case of a Witt space.

The noncommutative stable homotopy category NSH is a triangulated category that is the universal receptacle for triangulated homology theories on separable C*-algebras. We show that the triangulated category NSH is topological as defined by Schwede using the formalism of (stable) infinity categories. More precisely, we construct a stable presentable infinity category of noncommutative spectra and show that NSH^op sits inside its homotopy category as a full triangulated subcategory, from which the above result can be deduced. We also introduce a presentable infinity category of noncommutative pointed spaces that subsumes C*-algebras and define the noncommutative stable (co)homotopy groups of such noncommutative spaces generalizing earlier definitions for separable C*-algebras. The triangulated homotopy category of noncommutative spectra admits (co)products and satisfies Brown representability. These properties enable us to analyze neatly the behavior of the noncommutative stable (co)homotopy groups with respect to certain (co)limits. Along the way we obtain infinity categorical models for some well-known bivariant homology theories like KK-theory, E-theory, and connective E-theory via suitable (co)localizations. The stable infinity category of noncommutative spectra can also be used to produce new examples of generalized (co)homology theories for noncommutative spaces.