Related papers: Homology stratifications and intersection homology

We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally…

We construct GLSM invariants for a general choice of stability in both the narrow and broad sector cases and prove they form a Cohomological Field Theory. This is obtained by forming the analogue of a virtual fundamental class which lives…

Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a…

We show how one can twist the definition of Hochschild homology of an algebra or a DG algebra by inserting a possibly non-additive trace functor. We then prove that many of the usual properties of Hochschild homology survive such a…

This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…

We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is…

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their…

We define a model for the homology of manifolds and use it to describe the intersection product on the homology of compact oriented manifolds and to define homological quantum field theories which generalizes the notions of string topology…

In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham…

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of…

This paper surveys the significant progress over the past couple of decades in the theory of stratified spaces through the application of controlled methods as well as through the application of intersection homology.

We describe a simple homological test for obstructions to graph colorings. The main idea is to combine the framework of Hom-complexes with the following general fact: an arbitrary Z_2-space has nontrivial homology with Z_2-coefficients in…

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…

We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…

In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, i.e. spaces of translation surfaces. In the last decade, several of these have been constructed,…

A subspace arrangement is a finite collection of affine subspaces in $\mathbb{R}^n$. One of the main problems associated to arrangements asks up to what extent the topological invariants of the union of these spaces, and of their complement…

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping…

Interactions in complex systems are widely observed across various fields, drawing increased attention from researchers. In mathematics, efforts are made to develop various theories and methods for studying the interactions between spaces.…

Let G be a finite group acting freely on a compact oriented surface S by homeomorphisms preserving the orientation. Then, there exists a G-invariant Lagrangian subspace in the first homology group of S.