Mathematics

We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.

We correct some oversights in the paper "A spectral sequence for stratified spaces and configuration spaces of points" by the second named author. In particular we explain that an additional hypothesis should be added to Theorem 4.15 in said paper.

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction -- vertices for the complex are either the rows or the columns of the matrix representing the relation -- the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes. Finally, we explore a different cosheaf that detects the failure of the Dowker complex itself to be a faithful functor.

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK-equivalences.

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K -theory, and applying these tools to studying the Grothendieck ring of varieties. In this paper we give a new application of their framework: we construct a spectrum that recovers the classical SK ("schneiden und kleben," German for "cut and paste") groups for manifolds on π 0 , and we construct a derived version of the Euler characteristic.

Let A be a topological Azumaya algebra of degree mn over a CW complex X . We give conditions for the positive integers m and n , and the space X so that A can be decomposed as the tensor product of topological Azumaya algebras of degrees m and n . Then we prove that if m<n and the dimension of X is higher than 2m+1 , A may not have such decomposition.

We consider moduli spaces of d -dimensional manifolds with embedded particles and discs. In this moduli space, the location of the particles and discs is constrained by the d -dimensional manifold. We will compare this moduli space with the moduli space of d -dimensional manifolds in which the location of such decorations is no longer constrained, i.e. the decorations are decoupled. We generalise work by Bödigheimer--Tillmann for oriented surfaces and obtain new results for surfaces with different tangential structures as well as to higher dimensional manifolds. We also provide a generalisation of this result to moduli spaces with more general submanifold decorations and specialise in the case of decorations being unparametrised unlinked circles.

We prove definable versions of the Universal Coefficient Theorems of Eilenberg--Mac Lane expressing the (Steenrod) homology groups of a compact metrizable space in terms of its integral cohomology groups, and the (Čech) cohomology groups of a polyhedron in terms of its integral homology groups. Precisely, we show that, given a compact metrizable space X , a (not necessarily compact) polyhedron Y , and an abelian Polish group G with the division closure property, there are natural definable exact sequences 0→Ext( H n+1 (X),G)→ H n (X;G)→Hom( H n (X),G)→0 and 0→Ext( H n−1 (Y),G)→ H n (Y;G)→Hom( H n (Y),G)→0 which definably split, where H n (X;G) is the n -dimensional definable homology group of X with coefficients in G and H n (Y;G) is the n -dimensional definable cohomology group of Y with coefficients in G . Both of these results are obtained as corollaries of a general algebraic Universal Coefficient Theorem relating the cohomology of a cochain complex of countable free abelian groups to the definable homology of its G -dual chain complex of Polish groups.

We extend a classical fact about deformations of groups of units of commutative rings to E ∞ -ring spectra, and we use this result to provide a map of spectra generalizing the ordinary logarithmic derivative induced by an R -module derivation.

The aim of this article is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and identify the homotopy cofibers in terms of configuration spaces of simpler graphs. The construction's main benefit lies in making the operations functorial - in particular, graph minors give rise to compatible maps at the level of fundamental groups as well as generalized (co)homology theories. As applications we provide a long exact sequence for half-edge deletion in any generalized cohomology theory, compatible with cohomology operations such as the Steenrod and Adams operations, allowing for inductive calculations in this general context. We also show that the generalized homology of unordered configuration spaces is finitely generated as a representation of the opposite graph minor category.