For a finite group G and a finite G-CW-complex X, we construct groups H_\bullet(G,X) as the homology groups of the G-invariants of the cellular chain complex C_\bullet(X). These groups are related to the homology of the quotient space X/G via a norm map, and therefore provide a mechanism for calculating H_\bullet(X/G). We compute several examples and provide a new proof of ``Smith theory": if G=Z/p and X is a mod p homology sphere on which G acts, then the subcomplex X^G is empty or a mod $p$ homology sphere. We also get a new proof of the Conner conjecture: If G=Z/p acts on a Z-acyclic space X, then X/G is Z-acyclic.
We determine the ring structure of the loop homology of some global quotient orbifolds. We can compute by our theorem the loop homology ring with suitable coefficients of the global quotient orbifolds of the form $[M/G]$ for $M$ being some kinds of homogeneous manifolds, and $G$ being a finte subgroup of a path connected topological group $\mathcal{G}$ acting on $M$. It is shown that these homology rings split into the tensor product of the loop homology ring $\mathbb{H}_{*}(LM)$ of the manifold $M$ and that of the classifying space of the finite group, which coincides with the center of the g…
We study quotients of mapping class groups (Γ_{g,1}) of oriented surfaces with one boundary component by terms of their Johnson filtrations, and we show that the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity. We also compute the stable (co)homology with constant rational coefficients for one family of such quotients.
We introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of tensor excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group acti…
A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the f…
We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. For $s>1$, we easily deduce that the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction map $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras.
We give natural descriptions of the homology and cohomology algebras of regular quotient ring spectra of even E-infinity ring spectra. We show that the homology is a Clifford algebra with respect to a certain bilinear form naturally associated to the quotient ring spectrum F. To identify the cohomology algebra, we first determine the derivations of F and then prove that the cohomology is isomorphic to the exterior algebra on the module of derivations. We treat the example of the Morava K-theories in detail.
We compute the abelianisations of the mapping class groups of the manifolds $W_g^{2n} = g(S^n \times S^n)$ for $n \geq 3$ and $g \geq 5$. The answer is a direct sum of two parts. The first part arises from the action of the mapping class group on the middle homology, and takes values in the abelianisation of the automorphism group of the middle homology. The second part arises from bordism classes of mapping cylinders and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable $J$-homomorphism. We relat…
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 …
Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a homotopy…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain complex, or equivalently, the homology of the supremum chain complex. In this paper, we generalize the discrete Morse theory for simplicial complexes by R. Forman [5-7] and give a discrete Morse theory for hypergraphs. We use the critical simplices of the associated simplicial complex to construct a sub-chain complex of the infimum chain complex and a sub-ch…
We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces.
We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW deco…
The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diag…
We introduce a new homology theory of uniform spaces, provisionally called $μ$-homology theory. Our homology theory is based on hyperfinite chains of microsimplices. This idea is due to McCord. We prove that $μ$-homology theory satisfies the Eilenberg-Steenrod axioms. The characterization of chain-connectedness in terms of $μ$-homology is provided. We also introduce the notion of S-homotopy, which is weaker than uniform homotopy. We prove that $μ$-homology theory satisfies the S-homotopy axiom, and that every uniform space can be S-deformation retracted to a dense subset. It follows that for e…
If $Q$ is a group acting as a group of automorphisms of another group $G$ (with finite orbits), denote by $C_*(G)^Q$ the subcomplex of $Q$-invariant chains in the bar complex $C_*(G)$. In this paper, we study the homology of the complex $C_*(G)^Q$ and compute several examples.
Given a chain complex with the only modification that each cell of the complex has a probability distribution assigned. We will call this complex - a random complex and what should be understood in practice, is that we have a classical chain complex whose cells appear and disappear according to some probability distributions. In this paper, we will try to find the stochastic homology of random complex, whose simplices have independent discrete distributions.
In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we states our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.
In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order …
We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superperversities (those satisfying p(2)>0). For the case p(2)=1, these latter perversitie were introduced by Cappell and Shaneson and play a key role in their superduality theorem for embeddings. We further describe the sheafification of this singular chain complex and its adaptabilit…
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.
We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger-Gromov rho-invariant, we obtain new real-valued homology cobordism invariants rho_n for closed (4k-1)-dimensional manifolds. For 3-dimensional manifolds, we show that {rho_n} is a linearly independent set and for each n, the image of rho_n is an infinitely generated and dense subset of R. In their seminal work on knot concordance, T. Cochran, K. Orr, and P. Teichner define a filtration F_(n)^m of the m-component (string) link…
This paper proves that homology equivalences of cogenerating complexes induce homology equivalences of the cofree coalgebras in many interesting cases. We show that the underlying chain complex of any cofree coalgebra is naturally a direct summand of the underlying chain-complex of a cofree coalgebra over a free operad. This is combined with the previous result to prove the homology invariance of all cofree coalgebras.
We study the string topology of a closed oriented Riemannian manifold M. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space LM of M. These operations are well-defined on the homology of a quotient of this moduli space, which has the homotopy type of a compactification of the moduli space of Riemann surfaces. In particular, our action of the 0-dimensional homology of the quotient space on the homology of the free loop space of M recovers the Cohen-Godin positive boundary T…
Described the algebraic structure on the space of homotopy classes of cycles with marked topological flags of disks. This space is a non-commutative monoid, with an Abelian quotient corresponding to the group of singular homologies $H_k(M)$. For the marked flag contracted to a point the multiplication becomes commutative and the subgroup of spherical cycles corresponds to the usual homotopy group $π_k(M)$.
We show for g > 6 that the second homology group of the Torelli group of a surface of genus g and 1 boundary component is generated as an Sp(2g,Z)-module by the image under the stabilization map of the second homology group of the Torelli group of a surface of genus 6 and 1 boundary component. In the process we also show that the quotient of the complex of arcs with identity permutation by the Torelli group is (g-2)-connected, for one or two boundary components.
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem.
We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups $H_n(X)$ as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in earlier work of the authors. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.
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 give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R).The study has connections with spectral geometry and combinatorics.
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…
We compute the rank of the first homology group and we study the higher Betti numbers of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R)(after Kapranov).
In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the function to a Morse function, the second approach is to use moduli spaces of cascades, and the third approach is to use the Morse-Bott multicomplex. With respect to an explicit perturbation (which can be used to derived the Morse-Bott inequalities), the first two approaches yield the same chain complex up to sign. The third approach is fundamentally differe…
We consider groups G which have a cocompact, 3-manifold model for the classifying space \underline{E}G. We provide an algorithm for computing the rationalized equivariant K-homology of \underline{E}G. Under the additional hypothesis that the quotient 3-orbifold \underline{E}G/G is geometrizable, the rationalized K-homology groups coincide with the rationalized K-theory of the reduced C*-algebra of G. We illustrate our algorithm on some concrete examples.
We study rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces $X_{λ,μ}$. We present a combinatorial description of the cell structure of $X_{λ,μ}$ using the language of marked forests. As applications we obtain a new proof of a theorem of Arnold and a counterexample to a conjecture of Sundaram and Welker, along with a few other smaller results.
We construct Morse-Smale-Witten complex for an effective orientable orbifold. For a global quotient orbifold, we also construct a Morse-Bott complex. We show that certain type of critical points of a Morse function has to be discarded to construct such a complex, and gradient flows should be counted with suitable weights. The homology of these complexes are shown to be isomorphic to the singular homology of the quotient spaces under the self-indexing assumptions.
We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension.
Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $\tilde X$ with compact quotient $X$. Fix $n \geq 1$, assume $H_n(\tilde X, \mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the integral homology in dimension $n$ of $\tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$ is not compact, there are solvable groups of derived length 3 for which torsion in homology can grow faster than any given function.
Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy c…
We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.
We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover prov…
