A Kuenneth theorem for configuration spaces

TL;DR

This paper develops a spectral sequence for the homology of configuration spaces of product manifolds, utilizing algebraic operad techniques, and demonstrates its collapse in characteristic zero due to formality results.

Contribution

It introduces a new spectral sequence for configuration space homology and a novel algebraic tensor product for operadic modules, advancing the understanding of these spaces.

Findings

Spectral sequence converges to configuration space homology.

Sequence collapses in characteristic zero due to formality.

Operadic tensor product is a key algebraic tool.

Abstract

We construct a spectral sequence converging to the homology of the ordered configuration spaces of a product of parallelizable manifolds. To identify the second page of this spectral sequence, we introduce a version of the Boardman--Vogt tensor product for linear operadic modules, a purely algebraic operation. Using the rational formality of the little cubes operads, we show that our spectral sequence collapses in characteristic zero.