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A cocyclic construction of $S^1$-equivariant homology and application to string topology

Yi Wang

Abstract

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to $S^1$-equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) $S^1$-equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an $S^1$-equivariant version of operadic Deligne's conjecture.

A cocyclic construction of $S^1$-equivariant homology and application to string topology

Abstract

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to -equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) -equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an -equivariant version of operadic Deligne's conjecture.
Paper Structure (15 sections, 27 theorems, 119 equations)

This paper contains 15 sections, 27 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on cyclic homology
  3. A cocyclic complex and an $\infty$-quasi-isomorphism
  4. The story of differentiable spaces
  5. Differentiable spaces and de Rham chains
  6. $S^1$-equivariant homology of differentiable $S^1$-spaces
  7. Application to marked Moore loop spaces
  8. Preliminaries on operads and algebraic structures
  9. Cyclic brace operations
  10. Chain level structures in $S^1$-equivariant string topology
  11. Sign rules
  12. Koszul sign rule
  13. Sign change rule for (de)suspension
  14. Operadic suspension
  15. Cyclic permutation

Key Result

Theorem 1.1

Let $X$ be a topological space with an $S^1$-action. Then $\{S_*(X\times\Delta^k)\}_{k\geq0}$ can be made into a cocyclic chain complex, such that there are natural isomorphisms between three versions of cyclic homology of $\{S_*(X\times\Delta^k)\}_{k\geq0}$ and three versions of $S^1$-equivariant h

Theorems & Definitions (76)

  • Theorem 1.1: See Theorem \ref{['thm:cocyclic topological isom']}
  • Theorem 1.2: See Corollary \ref{['cor:homotopy gravity action']}
  • Theorem 1.3: See Theorem \ref{['thm:chain level gravity string topology']})
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 66 more