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A spectral sequence for tangent cohomology of algebras over algebraic operads

José Moreno-Fernández, Pedro Tamaroff

Abstract

Operadic tangent cohomology generalizes the existing cohomology theories of Chevalley--Eilenberg, Hochschild, and Harrison to address the deformation theory of general types of algebras through gadgets known as deformation complexes. The cohomology of these is in general very non-trivial to compute, and in this paper we complement the existing computational techniques by producing a spectral sequence that converges to the operadic cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. We also consider relative Sullivan--de Rham models of a fibration $p$, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber-homotopy equivalences of $p$.

A spectral sequence for tangent cohomology of algebras over algebraic operads

Abstract

Operadic tangent cohomology generalizes the existing cohomology theories of Chevalley--Eilenberg, Hochschild, and Harrison to address the deformation theory of general types of algebras through gadgets known as deformation complexes. The cohomology of these is in general very non-trivial to compute, and in this paper we complement the existing computational techniques by producing a spectral sequence that converges to the operadic cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. We also consider relative Sullivan--de Rham models of a fibration , where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber-homotopy equivalences of .

Paper Structure

This paper contains 16 sections, 21 theorems, 40 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Operadic background
  3. $\Sigma$-modules and operads
  4. Algebras over operads
  5. Modules over algebras
  6. Cofibrations of ${\mathcal{P}}$-algebras
  7. A spectral sequence of derivations
  8. The complex of derivations
  9. Tangent cohomology
  10. The spectral sequence
  11. Collapse results
  12. Applications to rational homotopy theory
  13. Three flavours of homology
  14. The Adams-Hilton model
  15. Multiplicative structure
  16. ...and 1 more sections

Key Result

Theorem A

Let $Q$ be the colimit of a tower $\mathcal{T}$ of cofibrations of $\;{\mathcal{P}}$-algebras. There is a functorial right half-plane spectral sequence with first page

Figures (1)

  • Figure 1: A nice filtration and a truncated target.

Theorems & Definitions (40)

  • Definition
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 30 more