Configuration spaces of products

TL;DR

This paper demonstrates how the configuration spaces of product manifolds can be reconstructed from their factors using operadic tensor products, introducing a new operad for non-parallelizable cases.

Contribution

It introduces a novel operadic approach to understanding configuration spaces of product manifolds, including a new operad for non-parallelizable manifolds.

Findings

Configuration spaces of product manifolds are recoverable via operadic tensor products.

A new operad of skew little cubes is proposed for non-parallelizable manifolds.

The approach generalizes previous results for parallelizable manifolds.

Abstract

We show that the configuration spaces of a product of parallelizable manifolds may be recovered from those of the factors as the Boardman-Vogt tensor product of right modules over the operads of little cubes of the appropriate dimension. We also discuss an analogue of this result for manifolds that are not necessarily parallelizable, which involves a new operad of skew little cubes.