On multiplicative spectral sequences for nerves and the free loop spaces

TL;DR

This paper develops a multiplicative spectral sequence framework for computing cohomology algebras of nerves, classifying spaces, and free loop spaces, with explicit calculations for real projective spaces and applications to diffeological spaces.

Contribution

It introduces a new spectral sequence construction that converges to cohomology algebras of complex spaces, including free loop spaces, and applies it to explicit topological examples.

Findings

Computed mod p cohomology algebra of free loop space of real projective space

Constructed a spectral sequence for classifying spaces of topological categories

Represented generators in de Rham cohomology via Chen's iterated integrals

Abstract

We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a topological category. By applying the machinery to a Borel construction, we determine explicitly the mod $p$ cohomology algebra of the free loop space of the real projective space for each odd prime $p$. This is highlighted as an important computational example of such a spectral sequence. Moreover, we try to represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold $M$ with differential forms on the universal cover of $M$ via Chen's iterated integral map.