Topological coHochschild Homology and the Homology of Free Loop Spaces

Abstract

We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.

Computations of relative topological coHochschild homology

“…Bohmann-Gerhardt-Shipley show that under appropriate coflatness conditions the coBökstedt spectral sequence for a cocommutative coalgebra spectrum has what is called a -Hopf algebra structure, an analog of a Hopf algebra structure working over a coalgebra [4], where the in this notation is the cotensor product. Recall that for an R-coalgebra C, a right C-comodule M with γ : M → M ⊗ C, and a left C-comodule N with ψ : N → C ⊗ N , the cotensor of M and N over C is defined to be the following equalizer in R-modules:…”

“…In order to define a C -Hopf algebra for a coalgebra C over a field k, we first recall the definitions of a C -algebra, a C -coalgebra, and a C -bialgebra from [4]. See Definition 2.10, Definition 2.11, Definition 2.12, and Definition 2.13 of [4] for the coassociativity and counitality diagrams as well as those specifying the interactions between the algebra and coalgebra structures. In [4], they further extend these definitions to that of a differential bigraded C -Hopf algebra (Definition 6.8) and a spectral sequence of C -Hopf algebras (Definition 6.9).…”

“…Thus coTHH is relevant for studying the homology of free loop spaces, LX, the main topic of the field of string topology [8,9]. In fact, Bohmann-Gerhardt-Shipley use topological coHochschild homology to compute the homology of free loop spaces in [4]. Further, because THH(Σ ∞ + (ΩX)) is directly related to the algebraic Ktheory of the space X via trace methods, coTHH also has applications for algebraic K-theory of spaces.…”

“…By work of Angeltveit-Rognes, the classical Bökstedt spectral sequence for a commutative ring spectrum has the structure of a spectral sequence of Hopf algebras under certain flatness conditions [1]. Bohmann-Gerhardt-Shipley show that under appropriate coflatness conditions, the coBökstedt spectral sequence for a cocommutative coalgebra spectrum has what is called a -Hopf algebra structure, an analog of a Hopf algebra structure for working over a coalgebra [4]. We show that the relative coBökstedt spectral sequence has a similar structure and use it to prove the following results.…”

“…In Section 3 we construct the relative coBökstedt spectral sequence. Section 4 examines the algebraic structures of this spectral sequence based on work of [4], and Section 5 discusses some explicit calculations of topological coHochschild homology.…”