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Eight flavours of cyclic homology

K. Cieliebak, E. Volkov

TL;DR

The paper develops a comprehensive framework of eight cyclic-homology flavours arising from mixed complexes and investigates their relations to loop-space topology via Chen's iterated integral. It demonstrates how δ_u=δ+uD organizes these flavours, proves invariance results for the classical three flavours, and establishes precise links between de Rham and singular-cochain models, S^1-spaces, and their equivariant cohomology. A central achievement is showing that, for simply connected X, Chen's iterated integral induces an isomorphism between the reduced Connes version HC_λ(Ω^*(X)) and the S^1-equivariant cohomology of LX relative to a base point, with dual results for cyclic cohomology. The paper also provides explicit computations for spheres using minimal models, develops duality theory for mixed complexes, and highlights implications for string topology by clarifying which loop-space invariants carry the natural algebraic structures predicted by theory.

Abstract

We introduce eight versions of cyclic homology of a mixed complex and study their properties. In particular, we determine their behaviour with respect to Chen iterated integrals.

Eight flavours of cyclic homology

TL;DR

The paper develops a comprehensive framework of eight cyclic-homology flavours arising from mixed complexes and investigates their relations to loop-space topology via Chen's iterated integral. It demonstrates how δ_u=δ+uD organizes these flavours, proves invariance results for the classical three flavours, and establishes precise links between de Rham and singular-cochain models, S^1-spaces, and their equivariant cohomology. A central achievement is showing that, for simply connected X, Chen's iterated integral induces an isomorphism between the reduced Connes version HC_λ(Ω^*(X)) and the S^1-equivariant cohomology of LX relative to a base point, with dual results for cyclic cohomology. The paper also provides explicit computations for spheres using minimal models, develops duality theory for mixed complexes, and highlights implications for string topology by clarifying which loop-space invariants carry the natural algebraic structures predicted by theory.

Abstract

We introduce eight versions of cyclic homology of a mixed complex and study their properties. In particular, we determine their behaviour with respect to Chen iterated integrals.

Paper Structure

This paper contains 20 sections, 34 theorems, 163 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cyclic homology of mixed complexes
  3. Mixed complexes
  4. Cocyclic and cyclic objects
  5. From cyclic cochain complexes to mixed complexes
  6. From differential graded algebras to mixed complexes
  7. From $S^1$-spaces to mixed complexes
  8. Examples
  9. Chen's iterated integral
  10. Chen's iterated integral as a map of mixed complexes
  11. Chen's iterated integral on Connes' version
  12. Chen's iterated integral for simply connected manifolds
  13. Computations for spheres
  14. Duality
  15. Generalities
  16. ...and 5 more sections

Key Result

Proposition 2.3

The $3$ classical versions $HC^{[[u]]}$, $HC^{[[u,u^{-1}]}$ and $HC^{[u^{-1}]}$ of cyclic homology are quasi-isomorphism invariants of mixed complexes, whereas the other $5$ versions are not.

Figures (3)

  • Figure 1: The double complex $C[[\theta,\theta^{-1}]]$
  • Figure 2: Cyclic complex of the trivial dga $(A={\mathbb{R}},d=0)$
  • Figure 3: Cyclic complex of singular cochains on $ES^1$

Theorems & Definitions (70)

  • Definition 1
  • Remark 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5: Hood and Jones Hood-Jones
  • proof
  • Example 2.6
  • Example 2.7
  • ...and 60 more