The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions on mapping spaces that are induced by an action on the source.
We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra.
Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded commutative structure in cohomology. In this work we introduce through explicit combinatorial methods an extension of this product to a full ${E_\infty}$-structure. As an application we prove that the Cartan-Serre map, which relates the cubical and simplicial singular cochains of spaces, is a quasi-isomorphism of ${E_\infty}$-algebras.
Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(T)/T|$ is invertible in the field $k$ then the algebra of cochains $C^*(BG;k)$ is formal as an $A_\infty$ algebra, or equivalently as a DG algebra.
In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod…
Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E-infinity algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E-infinity algebras is faithful but not full.
We construct a Frobenius algebra structure on the Hochschild cochains of a group ring k[G] that extends the known structure of a <1, 2> topological quantum field theory on HH^0(k[G]; k[G]), k a field and G a finite group. The convolution product extends to the homotopy commutative Gerstenhaber product on cochains, the Frobenius coproduct extends to a coproduct on the chain complex for Hochschild homology, and there is a pairing on Hochschild cocahins satisfying Frobenius associativity. The pairing, however, degenerates on a certain subcomplex of Hochschild cochains. The cochain complex for gro…
We study a naturally occurring $E_{\infty}$-subalgebra of the full $E_2$-Hochschild cochain complex arising from coherent cochains. For group rings and certain category algebras, these cochains detect $H^*(B {\cal{C}})$, the simplicial cohomology of the classifying space of the underlying group or category, $\cal{C}$. In this setting the simplicial cup product of cochains on $B{\cal{C}}$ agrees with the Gerstenhaber product and Steenrod's cup-one product of cochains agrees with the pre-Lie product. We extend the idea of coherent cochains to algebras more general than category algebras and dub …
We prove that the integral cohomology algebra of the moment-angle complex Z_K, or of the corresponding coordinate subspace arrangement complement U(K), is isomorphic to the Tor-algebra of the face ring Z[K] of simplicial complex K.
A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results …
The commutative differential graded algebra $A_{\mathrm{PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal{I}}(X)$ of $A_{\mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal{I}$ to model $E_{\infty}$ differential graded algebras by strictly commutative objects, called commutative $\mathcal{I}$-dgas. We define a functor $A^{\mathcal{I}}$ from simplicial sets to commutative $\mathcal{I}$-dgas and show that it is a…
Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are parametrised by props and thus include various kinds of bialgebras. We give a general formula to compute subsets of equivalences classes of realizations as quotients of automorphism groups, and determine the higher homotopy groups via the cohomology of deformation complexes. As a motivating example, we compute subsets of equivalences classes of realizations …
The Gerstenhaber and Schack cohomology comparison theorem asserts that there is a cochain equivalence between the Hochschild complex of a certain algebra and the usual singular cochain complex of a space. We show that this comparison theorem preserves the brace algebra structures. This result gives a structural reason for the recent results establishing fine topological structures on the Hochschild cohomology, and a simple way to derive them from the corresponding properties of cochain complexes.
We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The theory is sufficiently general to include also ringed spaces X. The construction is by itself interesting since it uses the difference calculus (instead of the differential calculus of Sullivan's theory) and a new type…
The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a 1-connected manifold. We regard it as a homomorphism from the algebra of differential forms on the structure group to the algebra of differential forms on the based loopspace of the base, represented by the (reduced) bar-complex of differential forms on it. Next we discuss the notion of "twisting cochains", or more generally "twisting maps", their equivalence re…
The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction. We apply our construction to cochain complexes of topol…
We introduce two coloured operads in sets -- the lattice path operad and a cyclic extension of it -- closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an $E_2$-action (resp. framed $E_2$-action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.
Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we als…
Early in the history of higher homotopy algebra, it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeisvili that showed they were but a shadow of an A-infinity-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim and K.T. (Chester) Chen. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.
The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen's iterated integrals are considered in a diffeological framework. As a consequ…
Using the $E_\infty-$structure on singular cochains, we construct a homotopy coherent map from the cyclic bar construction of the differential graded algebra of cochains on a space to a model for the cochains on its free loop space. This fills a gap in the paper "Cyclic homology and equivariant homology" by John D.S. Jones.
We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of $k$-algebras. Our main examples are of the form $R = C^*(X;k)$, the ring spectrum of cochains on a space $X$ for a field $k$. In particular, we establish local Gorenstein duality in characteristic $p$ for $p$-compact groups and $p$-local finite groups as well as for $k = \Q$ and $X$ a simply connected space which is Gorenstein in the s…
We use factorization homology and higher Hochschild (co)chains to study various problems in algebraic topology and homotopical algebra, notably brane topology, centralizers of $E_n$-algebras maps and iterated bar constructions. In particular, we obtain an $E_{n+1}$-algebra model on the shifted integral chains of the mapping space of the n-sphere into an orientable closed manifold $M$. We construct and use $E_\infty$-Poincaré duality to identify higher Hochschild cochains, modeled over the $n$-sphere, with the chains on the above mapping space, and then relate Hochschild cochains to the deforma…
We consider representation varieties in $SL_2$ for lattices in solvable Lie groups, and representation varieties in $sl_2$ for finite-dimensional Lie algebras. Inside them, we examine depth 1 characteristic varieties for solvmanifolds, respectively resonance varieties for cochain Differential Graded Algebras of Lie algebras. We prove a general result that leads, in both cases, to the complete description of the analytic germs at the origin, for the corresponding embedded rank 2 jump loci.
We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore's theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology. In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small,…
The Cartan formula encodes the relationship between the cup product and the action of the Steenrod algebra in $\mathbb F_p$-cohomology. In this work, we present an effective proof of the Cartan formula at the cochain level when the field is $\mathbb F_2$. More explicitly, for an arbitrary pair of cocycles and any non-negative integer, we construct a natural coboundary that descends to the associated instance of the Cartan formula. Our construction works for general algebras over the Barratt-Eccles operad, in particular, for the singular cochains of spaces.
The present paper is a continuation of author's paper arXiv:1909.00940 [math.AT] devoted to the lemmas of Alexander and Sperner, but is independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-…
Banagl's method of intersection spaces allows to modify certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincaré duality in analogy with Goresky-MacPherson's intersection homology. In the case of one isolated singularity, we show that the duality isomorphism comes from a nondegenerate intersection pairing which depends on the choice of a chain representative of the fundamental class of the regular stratum. On the technical side, we use piecewise linear polynomial differential forms due to…
The embedding Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaning…
We construct E-infinity cell algebra models for the cochain algebras of the free and based loop spaces on a simply-connected topological space. Techniques from rational homotopy theory are exploited throughout.
It is observed that Kaygun's Hopf-Hochschild cochain complex for a module-algebra is a brace algebra with multiplication. As a result, (i) an analogue of Deligne's Conjecture holds for module-algebras, and (ii) the Hopf-Hochschild cohomology of a module-algebra has a Gerstenhaber algebra structure.
We introduce a notion of Koszul A-infinity algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A-infinity algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.
In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by {\it Clifford cohomology.} We show that {\it Clifford cohomology} controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold \(M\) which admits a \(Spin^{c}\) structure.
In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and Grothendieck-Teichmüller groups". In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite…
Let g be a finite-dimensional complex semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra z g[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible representation of g at most once.
We construct for any algebra over an operad an Hochschild chain complex. In the case of the singular cochain complex of a topological space, considered as a commutative algebra up to homotopy, we show that this complex computes the singular cohomology of the free loop space over this topological space.
The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.
A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains, cobounds form an ideal of the ring of cocycles, and the quotient ring is a ring of cohomologies. If the complex is simplicial, then the algorithm reduces to the well known algorithm of ${\rm\breve{C}}$ech. Similar algorithms are used in geometry of polyhedra for multiplying of cocycles taking values in the exterior algebra of the ambient space. Therefore, we as…
The normalized cochain complex of a simplicial set N^*(Y) is endowed with the structure of an E_{infinity} algebra. More specifically, we prove in a previous article that N^*(Y) is an algebra over the Barratt-Eccles operad. According to M. Mandell, under reasonable completeness assumptions, this algebra structure determines the homotopy type of Y. In this article, we construct a model of the mapping space Map(X,Y). For that purpose, we extend the formalism of Lannes' T functor in the framework of E_{infinity} algebras. Precisely, in the category of algebras over the Barratt-Eccles operad, we h…
Let $X$ be a simply connected space and $\Bbb K$ be any field. The normalized singular cochains $N^*(X; {\Bbb K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $HH_* N^*X$ of the space $X$. We prove that, endowed with this product, $HH_*N^*X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\Bbb K$. We also show how to construct a simpler Hochschild complex which allows direct computation.
Elaborating on works by Abouzaid and Mescher, we prove that for a Morse function on a smooth compact manifold, its Morse cochain complex can be endowed with an $ΩB As$-algebra structure by counting moduli spaces of perturbed Morse gradient trees. This rich structure descends to its already known $A_\infty$-algebra structure. We then introduce the notion of $ΩB As$-morphism between two $ΩB As$-algebras and prove that given two Morse functions, one can construct an $ΩB As$-morphism between their associated $ΩB As$-algebras by counting moduli spaces of two-colored perturbed Morse gradient trees. …
A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $Φ$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1] \to \mathbb{R}^n$ and $Φ$ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized $Φ$ for the case where $\mathcal{X}$ is a space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show that th…
