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.
Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(ΩX;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(ΩX;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel.
This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory.
We study the twisted cohomology groups of $A_\infty$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-algebraic Massey products.
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.
`Loop-fusion cohomology' is defined on the continuous loop space of a manifold in terms of \vCech cochains satisfying two multiplicative conditions with respect to the fusion and figure-of-eight products on loops. The main result is that these cohomology groups, with coefficients in an abelian group, are isomorphic to those of the manifold and the transgression homomorphism factors through the isomorphism.
We construct a complex analytic version of an equivariant cohomology theory which appeared in a recent paper of Rezk, and which is roughly modeled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for …
We prove that the cohomology algebra of a conilpotent Lie coalgebra is generated in degree 1 as an A-infinity algebra. By dualizing, the same is true about cohomology algebras of finite dimensional nilpotent Lie algebras. In the process, we provide a proof of the conilpotent version of the dual Poincare-Birkhoff-Witt theorem.
Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(ΩX) is isomorphic to the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this isomorphism is compatible with both the cup product on HH^{*}S_*(ΩX) and on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}}) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac {1}{(p-1)!}\in k.
We apply the Transfer Algorithm introduced in arXiv:1106.5090 to transfer an A_\infty-algebra structure that cannot be computed using the classical Basic Perturbation Lemma. We construct a space X whose (base pointed) loop cohomology H = H^*(ΩX; Z_2) comes equipped with a nontrivial operation ω: H x H --> H x H.
We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.
For n>2, we prove the mod 2 cohomology of the finite Chevalley group Spin_n(F_q) is isomorphic to that of the classifying space of the loop group of the spin group Spin(n).
A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $Spin(4)/T^2$, $Spin(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of rank 2 complete flag manifolds.
In this paper we shall calculate the cohomology ring and Hochschild's cohomology ring of plactic monoid algebra via Anick's resolution using Groubner-Shirshov basis
We show that the Connes-Moscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a "cyclic operad with multiplication" is a cocyclic module whose cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree -2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra.
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.
We present a study of the homological algebra of bimodules over $A_\infty$-algebras endowed with an involution. Furthermore we introduce a derived description of Hochschild homology and cohomology for involutive $A_\infty$-algebras.
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.
Let PSigma_n denote the group that can be thought of either as the group of motions of the trivial n-component link or the group of symmetric automorphisms of a free group of rank n. The integral cohomology ring of PSigma_n is determined, establishing a conjecture of Brownstein and Lee.
In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type
In this paper, we compute the cohomology ring of all homology split polyhedral product spaces and the cohomology algebra over a field of all polyhedral product spaces. As an application, we give two polyhedral product spaces such that all the cohomology homomorphisms induced by inclusion map are the same, but the cohomology rings of the two polyhedral product spaces are not isomorphic.
The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connecti…
We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the cohomology algebra of the free loop spaces. We build new theory to allow for the computation of $H^*(Λ(SU(n+1)/T^{n});\mathbb{Z})$, a significantly more complicated structure than other known examples. In addition to our theoretical results, we explicitly implement a novel integral Gröbner basis procedure for computation. This procedure is applicable to any…
In this paper, we compute the homology coalgebra and cohomology algebra over a field of all generalized moment-angle complexes and give a duality theorem on complementary moment-angle complexes.
In this paper, we investigate the existence of free involutions on some Wall manifolds and we compute the mod 2 cohomology algebra of the correspondent orbit space.
We study the Mahowald operator $M = \langle g_2,h_0^3, - \rangle$ in the cohomology of the Steenrod algebra. We show that the operator interacts well with the cohomology of $A(2)$, in both the classical and $\mathbb{C}$-motivic contexts. This generalizes previous work of Margolis, Priddy, and Tangora.
In this paper we describe methods for computing rack and quandle cohomology. We illustrate these methods by completely determining the cohomology of prime dihedral quandles.
We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is the positive part L_1 of the Witt algebra $W$. We consider an associated graded algebra m_0 of L_1 with respect to the descending central series and prove that H*(m_0) is generated with respect to non-trivial Massey products by one cohomology H^1(m_0).
