A construction of complex analytic elliptic cohomology from double free loop spaces

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled 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 any given complex elliptic curve $\mathcal {C}$ , the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$ .

Hodge decomposition of string topology

Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

Representation Homology of Topological Spaces

In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, ${{\mathbb{S}}}^1$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in ${\mathbb{R}}^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in ${\mathbb{R}}^3$.

On an Algebra Structure on the Hochschild Homology of Simply Connected Topological Space

In this note, we use the pairing induced by the interchange map in conjunction with the strongly homotopy commutative algebra structure to define products on the Eilenberg–Moore differential Tor and give a simplified proof of an improved outcome of Jones’s result due to Ndombol and Thomas. As a result, we establish an isomorphism of graded algebras between the Hochschild homology and the free loop space cohomology of a simply connected topological space.

Koszul A∞-algebras and free loop space homology

We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-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.

On vector bundles for a Morse decomposition of $L\mathbb{C}\mathrm{P}^n$

We give a description of the negative bundles for the energy integral on the free loop space $L\mathbb{C}\mathrm{P}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles.

Free loop space homology of highly connected manifolds

We calculate the homology of the free loop space of