Mathematics

Twisted topological Hochschild homology of C n -equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum M U R . In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson.

Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such properties might extend to graphs arising from nature. We consider the brain network (as reconstructed by Varshney & al.) of the Caenorhabditis elegans nematode, an important model organism in biology. Using an iterative computational procedure based on elementary methods of algebraic topology, namely homology, simplicial collapses and coning operations, we show that its directed flag complex is homotopy equivalent to a wedge of spheres, completely determining, for the first time, the homotopy type of a flag complex corresponding to a brain network. We also consider the corresponding flag tournaplex and show that torsion can be found in the homology of its local directionality filtration. As a toy example, directed flag complexes of tournaments from McKay's collection are classified up to homotopy. Moore spaces other than spheres occur in this classification. As a tool, we prove that the fundamental group of the directed flag complex of any tournament is free by considering its cell structure.

We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its persistent Stiefel-Whitney classes. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2).

We show the homological Serre spectral sequence with coefficients in a field is a spectral sequence of coalgebras. We also identify the comultiplication on the E 2 page of the spectral sequence as being induced by the usual comultiplication in homology. At the end, we provide some example computations highlighting the use the co-Leibniz rule.

For a given bundle ξ:E→M over a manifold, configuration-section spaces on ξ parametrise finite subsets z⊆M equipped with a section of ξ defined on M∖z , with prescribed "charge" in a neighbourhood of the points z . These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the 2 -disc with specified deck transformation group. We prove that configuration-section spaces are homologically stable (with integral coefficients) whenever the underlying manifold M is connected and has non-empty boundary and the charge is "small" in a certain sense. This has a partial intersection with the work on Hurwitz spaces of Ellenberg, Venkatesh and Westerland.

In this paper we show that, besides the usual calculus involving Kähler differentials, it is also possible to define conical calculus on schemes and perfectoid spaces; this can be done via a stratification process. Following some ideas from [1-2], we consider some natural stratifications of these spaces and then we build upon the work of Ayala, Francis, and Tanaka [3] (see also [4-5] and [18]); using their definitions of derivatives, smoothness and vector fields for stratified spaces, and thanks to some particular methods, we are able to transport these concepts to schemes and perfectoid spaces. This also allows us to define conical differential forms and the conical de Rham complex. At the end, we compare this approach with the usual one, noting that it is a useful \textit{addition} to Kähler method.

We show that if the moment-angle complex Z K associated to a simplicial complex K is homotopy equivalent to a connected sum of sphere products with two spheres in each product, then K decomposes as the simplicial join of an n -simplex Δ n and a minimally non-Golod complex. In particular, we prove that K is minimally non-Golod for every moment-angle complex Z K homeomorphic to a connected sum of two-fold products of spheres, answering a question of Grbić, Panov, Theriault and Wu.

The goal of this article is to extend a theorem of Lurie Sh A (X)=Fun( Exit A (X),S) representing constructible sheaves with values in S , the ??-category of spaces, on a stratified space X with poset of strata A , as functors from the exit paths ??-category Exit A (X) to S . Lurie's representation theorem works provided A satisfy the ascending chain condition. This typically rules out infinite dimensional examples of stratified space. Building on it and with the help of a stratified homotopy invariance theorem from Haine, we show that when X is a nice enough A -stratified space and when A is itself stratified A ?? ??A ?? ?�⋯?�A by posets satisfying the ascending chain condition, Hyp A (X)=Fun( Exit A (X),S) the ??-category of A -constructible hypersheaves on X is represented by functors from the exit paths ??-category of X . There are two types of nice stratified spaces on which this extended representation theorem applies: conically stratified spaces and spaces that are sequential colimits of conically stratified spaces. Examples of application include the metric and the topological exponentials of a Fréchet manifold, locally countable simplicial complexes and more generally, locally countable cylindrically normal CW-complexes.

We give a general construction of categorical idempotents which recovers the categorified Jones-Wenzl projectors, categorified Young symmetrizers, and other constructions as special cases. The construction is intimately tied to cell theory in the sense of additive monoidal categories.

We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of LS -category and topological complexity are particular cases of this more general notion.