Homology and Cohomology of Finite Spaces

On the Axiomatic Systems of Singular Cohomology Theory

On the category of pairs of topological spaces having a homotopy type of $CW$ complexes the singular (co)homology theory was axiomatically studied by J.Milnor. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomorphic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support \cite{3}. In the paper \cite{6}, L. Mdzinarishvili proposed {\it partially compact support property} for a cohomology theory and gave another axiomatic characterization of the sin…

Higher Hochschild cohomology of the Lubin-Tate ring spectrum

We give a method for computing factorization homology of $\oper{E}_n$-algebra using as an input an algebraic version of higher Hochschild homology due to Pirashvili. We then show how to compute higher Hochschild homology and cohomology when the algebra is étale in a sense that we make precise. As an application, we compute higher Hochschild cohomology of the Lubin-Tate ring spectrum.

The homology of simplicial complement and the cohomology of the moment-angle complexes

A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,σ}(Λ^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(Λ[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}σ$, $star_{K}σ$ and the cohomology of the generalized moment-angle complexes.

Homology and Cohomology of Topological Quandles

A homology and cohomology theory for topological quandles are introduced. The relation between these (co)homology groups and quandle (co)homology groups are studied. The 1 - topological quandle cocycles are used to compute state sum invariants corresponding to knot diagrams.

Some geometric equivariant cohomology theories

In this paper we give a geometric construction of the Borel equivariant (co)homology for spaces with a $G$-action, where $G$ is a compact Lie group with the property that the adjoint representation is orientable. A nice feature of these constructions is that there are corresponding Poincaré dual (co)homology theories called backwards (co)homology. This gives rise to a third relative (co)homology theory which we call stratifold Tate (co)homology. These Tate groups agree with the original definition of Tate cohomology for finite groups given by Swan. All constructions in this paper are geometric…

Topological Hochschild homology and cohomology of A_\infty ring spectra

Let A be an A_\infty ring spectrum. We use the description from [2] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the A_\infty structure into THH(A), and allows us to study how THH(A) varies over the moduli space of A_\infty structures on A. As an example, we study how topological Hochschild cohomology of Morava K-theory varies over the moduli space of A_\infty structures and show that in the…

利用对合持久同调快速计算持久同调代表元

Fast computation of persistent homology representatives with involuted persistent homology

针对持久上同调计算无法直接提取同调代表元的问题，提出一种利用上同调信息加速直接持久同调计算的新方法，在Rips复形上能显著快于标准方法地提取代表元。

Homology and Cohomology of E-infinity Ring Spectra

Every homology or cohomology theory on a category of E-infinity ring spectra is Topological Andre-Quillen homology or cohomology with appropriate coefficients. Analogous results hold for the category of A-infinity ring spectra and for categories of algebras over many other operads.

模 $p$ 上同调的模 $τ^{p-1}$ 动机 Hochschild 同调

Modulo $τ^{p-1}$ motivic Hochschild homology of modulo $p$ motivic cohomology

本文利用 motivic Greenlees 谱序列计算代数闭域上稳定 motivic 同伦范畴中模 $p$ motivic 同调的模 $τ^{p-1}$ Hochschild 同调。

Geometric Objects and Cohomology Operations

Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology o…

Hodge Theory for Intersection Space Cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincaré duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering met…

Homology decompositions and groups inducing fusion systems

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.

以对象系为系数的方体集合的同调与上同调

Homology and cohomology of cubical sets with coefficients in systems of objects

本文证明方体集合以反变系为系数的同调同构于余极限函子的左卫星，并由此建立同调不变性、谱序列及与范畴同调和 Baues-Wirsching 上同调的联系。

A new class of homology and cohomology 3-manifolds

We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology 3-manifold with coefficients in $\mathbb{Z}_q$ for $q\not\in \mathcal{ P}$. Moreover, $M_{\mathcal{P}}$ is neither a homology nor cohomology 3-manifold with coefficients in $\mathbb{Z}$ or $\mathbb{Q}$.

On Axiomatic Characterization of Alexander-Spanier Normal Homology Theory of General Topological Spaces

The Alexandroff-Čech normal cohomology theory [Mor$_1$], [Bar], [Ba$_1$],[Ba$_2$] is the unique continuous extension \cite{Wat} of the additive cohomology theory [Mil], [Ber-Mdz$_1$] from the category of polyhedral pairs $\mathcal{K}^2_{Pol}$ to the category of closed normally embedded, the so called, $P$-pairs of general topological spaces $\mathcal{K}^2_{Top}$. In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-Čech normal cohomology. Using this fact and methods developed in [Ber-Mdz$_3$] we c…

Intersection Space Cohomology of Three-Strata Pseudomanifolds

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically fla…

Deformation of Singularities and the Homology of Intersection Spaces

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This…

Bredon Homology of Partition Complexes

We prove that the Bredon homology or cohomology of the partition complex with fairly general coefficients is either trivial or computable in terms of constructions with the Steinberg module. The argument involves developing a theory of Bredon homology and cohomology approximation.

Cohomology of Polychromatic Configuration Spaces of Euclidean Space

Recently, the homology and cohomology of non-k-overlapping discs, or, equivalently, no k-equal subspaces of Euclidean space, were calculated by Dobrinskaya and Turchin. We calculate the homology and cohomology of two classes of more general spaces: decreasing polychromatic configuration spaces and bicolored configuration spaces. Instead of all points behaving similarly, we allow for varying behavior between points.

Homology and cohomology via enriched bifunctors

We show that the category of numerically generated pointed spaces is complete, cocomplete, and monoidally closed with respect to the smash product, and then utilize these features to establish a simple but flexible method for constructing generalized homology and cohomology theories by using the notion of enriched bifunctors.

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a gamma-equivariant G-module A, when a separate group "gamma" acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.

The loop homology algebra of discrete torsion

We show that Lupercio-Uribe-Xicoténcatl's orbifold loop product and coproduct can be described by a group cohomology class in some cases. By computing this cohomology class, we show that in some cases the orbifold loop product is trivial.

Equivariant ordinary homology and cohomology

Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of…

An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson

This paper provides an introduction to equivariant cohomology and homology using the approach of Goresky, Kottwitz, and MacPherson. When a group G acts suitably on a variety X, the equivariant cohomology of X can be computed using the combinatorial data of a skeleton of G-orbits on X. We give both a geometric definition and the traditional definition of equivariant cohomology. We include a discussion of the moment map and an algorithm for finding a set of generators for the equivariant cohomology of X. Many examples and explicit calculations are provided.

The homology of real subspace arrangements

Associated to any subspace arrangement is a "De Concini-Procesi model", a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne-Mumford compactification of the moduli space of genus 0 curves with marked points. In the present work, we calculate the integral homology of real De Concini-Procesi models, extending earlier work of Etingof, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology of the real locus of the moduli space. To be precise, we show that the integral homology of a real De Concini-Procesi model is iso…

A note on the Thom isomorphism in geometric (co)homology

Using geometric homology and cohomology we give a simple and conceptual proof of the Thom isomorphism theorem.

尺度同调与拓扑熵

Scaled Homology and Topological Entropy

提出适用于度量空间的 $lc$-同调理论，说明其公理性性质，并借此放宽光滑流形限制，证明熵猜想对第一 $lc$-同调群成立。

The homology of the little disks operad

In this expository paper we give an elementary, hands-on computation of the homology of the little disks operad, showing that the homology of a $d-fold loop space is a Poisson algebra. One aim is to familiarize a greater audience with Euclidean configuration spaces, using tools accessible to second-year graduate students. We also give a brief introduction to the theory of operads. New results include identifying the pairing between homology and cohomology of these spaces as a pairing of graphs and trees, and treating the cooperad structure on cohomology.

A(1) 的 Hochschild 同调与上同调

The Hochschild homology and cohomology of A(1)

本文利用若干 May 型谱序列，计算了由 $Sq^1$、$Sq^2$ 生成的 2-素 Steenrod 代数子代数 $A(1)$ 的 Hochschild 同调与上同调。

加权多面体的加权同调与上同调

Weighted Homology and Cohomology of Weighted Polyhedra

该研究定义了加权多面体的概念，构建了其加权奇异同调理论并证明其与加权单纯同调同构，还推广了相关上同调乘积并阐释了orbifold理论。

Cohomology with local coefficients and knotted manifolds

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings $N\hookrightarrow M$ of one topological manifold into another. More specifically, we describe an algorithm for computing the homology $H_n(X,A)$ and cohomology $H^n(X,A)$ of a finite connected CW-complex X with local coefficients in a $\mathbb Zπ_1X$-module $A$ when $A$ is finitely generated over $\math…

Relative measure homology and continuous bounded cohomology of topological pairs

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loeh's result plays a fundamental…

Hochschild homology of mod-$p$ motivic cohomology over algebraically closed fields

We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.

单位切丛同调中映射类群的稳定扭曲上同调

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

本研究计算了带边紧致可定向曲面映射类群以单位切丛同调为系数的稳定扭曲上同调，发现其为协变函子超出传统框架，还计算了符合传统框架的一阶上同调系数的稳定上同调群。

Some new homology and cohomology theories of manifolds and orbifolds

For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying relations, where $V$ is an oriented manifold with corners, $n\in\mathbb N$, and $s:V\to{\mathbb R}^n$, $t:V\to Y$ are smooth with $s$ proper near 0 in ${\mathbb R}^n$. We show that $MH_*(Y;R),MH^*(Y;R)$ satisfy the Eilenberg-Steenrod axioms, and so are canonically isomorphic to conventional (co)homolog…

E_n-cohomology with coefficients as functor cohomology

Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.

The Galois action and cohomology of a relative homology group of Fermat Curves

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.

计算同调与上同调基的特征--技术报告

Characterization of the computed homology and cohomology bases -- technical report

本文提出了一种计算同调基的特征方法，利用同调离散向量场，改进了组合同调理论并提升了同调生成器计算的控制能力。

Pattern-Equivariant Homology

Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. In this paper we consider homology groups of PE infinite chains. We establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example, we show how through this formalism one may give highly approachable geometric descriptions of the generators of the Čech cohomology of their tiling space. These invariants are also considered in the context of rotational symmet…

The cohomology of Bestvina-Brady groups

For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals.

The stable cohomology of automorphisms of free groups with coefficients in the homology representation

We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules \wedge^q H, \wedge H^*, Sym^q H or Sym^q H^*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. For reasons which are not conceptually clear, taking coefficients in H and its related modules behaves in a far less trivial way than taking coefficients in H^* and its related modules. Based on a conjectural homology stability theorem for spaces of graphs labeled by a simply connected background space, we give a stable integral calculation of these groups in low degrees, and modulo a further …

Isomorphisms in l^1-homology

Taking the l^1-completion and the topological dual of the singular chain complex gives rise to l^1-homology and bounded cohomology respectively. In contrast to l^1-homology, major structural properties of bounded cohomology are well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to l^1-homology…

从Floer同调恢复广义同调：复定向情形

Recovering generalized homology from Floer homology: the complex oriented case

本文通过完备Tate上同调将滤过圆等变谱与复定向上同调理论关联，证明对Liouville流形仅依赖稳定同伦型，并利用Morava K-理论及复K-理论计算恢复整数同调与复K-理论。

Persistent cohomology for data with multicomponent heterogeneous information

Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and reveal certain interaction patterns in terms of topological invariants. However, in addition to the geometric information, there is a wide variety of non-geometric information of molecular structures, such as element types, atomic partial charges, atomic pairwise interactions, and electrostatic potential function, that is not described by persistent homolog…

拓扑 Hochschild 同调中的等变运算

Equivariant operations in topological Hochschild homology

本文发现拓扑 Hochschild 同调与上同调之间的新等变关系，并通过计算有限素域的拓扑 Hochschild 上同调的拓扑 Hochschild 同调，将其解释为结构化运算环。

Cellular cosheaf homology are cosheaf homology

A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of a CW complex to the category of abelian groups. We show that given a cellular cosheaf $F$, there is a natural way to associate a cosheaf $\widehat{F}$, for which we can define homology as the formal dual of sheaf cohomology, such that the Borel-Moore homology of $F$ is isomorphic to the homology of $\widehat{F}$ whenever the underlying CW complex of $F$ is …

合成上同调与实拓扑循环同调

Syntomic cohomology and real topological cyclic homology

该研究定义实拓扑霍赫希尔德同调及其相关对象的动机滤层，证明实拓扑循环同调有自然完备滤层，其分次部分是合成上同调的等变悬架，并在相关定理假设下计算了特定对象的同伦群和等变切片。

Factorization homology and calculus à la Kontsevich Soibelman

Using factorization homology techniques, we give a simple proof of the action of Kontsevich and Soibelman operad on the pair consisting of Hochschild cohomology and Hochschild homology. The proof obviously extends to higher Hochschild cohomology and factorization (also called chiral) homology. Note that our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum.

On homology of complements of compact sets in Hilbert Cube

We introduce the notion of spaces with weak relative cohomology and show the acyclicity of the complement $Q\setminus X$ in the Hilbert cube $Q$ of a compactum $X$ with weak relative cohomology. As a corollary we obtain the acyclicity of the complement results when (a) $X$ is weakly infinite dimensional; (b) $X$ has finite cohomological dimension.

Quandle Homology Groups, Their Betti Numbers, and Virtual Knots

Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The results can be applied to knot theory as the first step towards evaluationg the state-sum invariants defined from quandle cohomology.

A generalization of Kostant theorem to integral cohomology

In this paper, we find weight decomposition and rank of a weight in the integral (co)homology of the positive system of a semi-simple Lie algebra over $\Bbb C$ and prove that the (co)homology of the weight subcomplex over a field of characteristic p is 0 if the rank of the weight is not divisible by p. This generalizes Kostant theorem to the integral cohomology of the positive system.

An algebraic model for mod 2 topological cyclic homology

For any space X with the homotopy type of simply-connected, finite-type CW-complex, we construct an associative cochain algebra fls(X) whose cohomology algebra is isomorphic to that of LX, the free loop space on X. For certain X, we define a cochain map from fls(X) to itself that is a good model of the pth-power operation on LX. Under additional conditions on X, e.g., when X is a wedge of spheres, we define a cochain complex hos(X) by twisting together fls(X) and H*(BS^1) and prove that the cohomology of hos(X) is isomorphic to the Borel cohomology of LX. Finally, we define tc(X) to be the map…

Topology of two-connected graphs and homology of spaces of knots

We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration.

André spectral sequences for Baues-Wirsching cohomology of categories

We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical cohomology and homology theories including Hochschild-Mitchell cohomology and those studied before by Watts, Roos, Quillen and others.