A spectral sequence for tangent cohomology of algebras over algebraic operads

TL;DR

This paper introduces a spectral sequence for operadic tangent cohomology, providing a new computational tool that connects algebraic and topological invariants, with applications to rational homotopy theory and spectral sequences.

Contribution

It develops a spectral sequence for operadic tangent cohomology using filtrations from cofibration towers, linking algebraic deformation theory with topological spectral sequences.

Findings

Spectral sequence converges to operadic cohomology of algebras.

Provides an algebraic description of the Serre spectral sequence.

Shows the spectral sequence is multiplicative and converges to the loop product.

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 $p$.