We develop a homology theory for directed spaces, based on the semi-abelian category of (non-unital) associative algebras. The major ingredient is a simplicial algebra constructed from convolution algebras of certain trace categories of a directed space. We show that this directed homology HA is invariant under directed homeomorphisms, and is computable as a simple algebra quotient for $HA_1$. We also show that the algebra structure for $HA_n$, $n\geq 2$ is degenerate, through a Eckmann-Hilton argument. We hint at some relationships between this homology theory and natural homology, another homology theory designed for directed spaces. Finally we pave the way towards some interesting long exact sequences.
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy invariance of the introduced path homology groups. We provide examples of computation of these homology groups.
While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a construction of homology of digraphs due to Grigor'yan, Lin, Muranov and Yau, and extend this construction to the persistent framework. The result, which we call persistent path homology, can provide information about the digraph structure of a directed network at varying resolutions. Moreover, this method encodes a rich level of detail about the asymmetric structu…
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the introduced notions.
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source an…
We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser [Bau18a], but is optimized specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelization by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an A…
In this note some recent developments in the study of homology in semi-abelian categories are briefly presented. In particular the role of protoadditive functors in the study of Hopf formulae for homology is explained.
We determine the BP-module structure, mod higher filtration, of the main part of the BP-homology of elementary abelian 2-groups. The action is related to symmetric polynomials and to Dickson invariants.
In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculations are more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some…
Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph $\mathcal{G}$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of $\mathcal{G}$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing g…
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery …
This paper generalizes the Max-Flow Min-Cut (MFMC) theorem from the setting of numerical capacities to sheaves of partial semimodules over semirings on directed graphs. Motivating examples of partial semimodules include probability distributions, multicommodity capacity constraints, and logical propositions. Directed (co)homology theories for such sheaves describes familar constructs on networks. First homology classifies locally decomposable flows, an orientation sheaf over a semiring generalizes directions, connecting maps for homology assign values to flows, connecting maps for cohomology a…
It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong Λ(A/p)\otimes Γ({}_pA)$ and for finitely generated abelian groups there is the following description of the cohomology algebra of $H^*(A,\mathbb F_p)\cong Λ((A/p)^\vee)\otimes {\sf Sym}(({}_pA)^\vee).$ We prove that there are no such descriptions for $p=2$ that `depend' only on $A/2$ and ${}_2A$ but we provide natural descriptions of $H_*(A,\mathbb F_2)$ and $H^*(A,\mathbb F_2)$ that `depend' on $A/2,$ ${}_2A$ and a linear map $\tilde β:{}_2A\to …
We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{…
Stanford University（斯坦福大学）；University of Minnesota（明尼苏达大学）；BAE Systems FAST Labs（英国宇航系统快速实验室）；UC Berkeley（加州大学伯克利分校）
We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces to computing the simplicial homology of the underlying undirected graph, which is explicitly given by Euler characteristic computations. We also show that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the layers withi…
We prove several new theorems regarding first homology. Some dichotomies for first homology of Peano continua are presented, as well as a notion of strong abelianization for arbitrary path connected metric spaces. We also show that the fundamental group of the Hawaiian earring has Borel subgroups of almost all multiplicative and additive types.
We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instanton-dyons. Signatures of well-separa…
The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category A. It is known that the category of bicommutative (i.e. commutative and cocommutative) Hopf algebras over a field k is an abelian category. Denote the category by H. In this paper, we give some ways to construct H-valued homology theories. As a main result, we give H-valued homology theories whose coefficients are neither group Hopf algebras nor function Hopf…
We give a description of the factorization homology and $E_n$ topological Hochschild cohomology of Thom spectra arising from $n$-fold loop maps $f: A \to BO$, where $A = Ω^n X$ is an $n$-fold loop space. We describe the factorization homology $\int_M Th(f)$ as the Thom spectrum associated to a certain map $\int _M A \to BO$. When $M$ is framed and $X$ is $(n-1)$-connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space $Map_c(M,X)$; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted fo…
The notion of a homotopy flow on a directed space was introduced in \cite{Raussen:07} as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all parameter directed maps preserve the homotopy type of path spaces, such a flow (and these parameter maps) are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points and whose morphisms may be induced by these inessential d-maps. Localization with respect to subcategories with these inessential d-maps as morphisms can be combi…
Let $K$ be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space $\vec{P}(K)_v^w$ of directed paths between two given vertices $v,w$ of $K$. In many cases, this construction is minimal: the cells of the constructed CW-complex are in 1--1 correspondence with the generators of the homology of $\vec{P}(K)_v^w$.
We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space $X$. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of $X$ which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good can…
The category of topological spaces endowed with two marked points is equipped with two families $\mathbf F_n$ and $\mathbf H_n$ of functors to the category of abelian groups, indexed by a nonnegative integer $n$: namely, the functor $\mathbf F_n$ takes the object $(X,x,y)$ to the quotient of $\mathbb Zπ_1(X,x,y)$ by an abelian subgroup associated with the $n+1$-st power of the augmentation ideal of the group algebra $\mathbb Zπ_1(X,x)$, and the functor $\mathbf H_n$ takes the same object to the $n$-th singular homology group of $X^n$ relative to a subspace defined in terms of partial diagonals…
Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory.
We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author which establishes the consistency, relative to a weakly compact cardinal, of $\mathrm{lim}^s \mathbf{A} = 0$ for all $s \geq …
