On coshuffle comultiplication on configuration spaces

TL;DR

This paper introduces a coshuffle comultiplication on the chain complex of configuration spaces, establishing a DGCoAlg structure, and explores its applications to graph configuration spaces and their homology.

Contribution

It defines a new coshuffle comultiplication structure on configuration spaces and proves its compatibility with external products, with applications to graph braid groups.

Findings

Configuration space chain complex has a DGCoAlg structure.

For graphs with circumference ≤ 1, the chain complex is formal as a DGCoAlg.

Complete classification of primitivity in homology for these graphs.

Abstract

We introduce a coshuffle comultiplication on the singular chain complex of configuration spaces, and we show that this structure endows the configuration space with the structure of a differential graded coalgebra (DGCoAlg). We then prove that the coshuffle comultiplication is compatible with the external product through a natural commutation relation. As an application, we investigate configuration spaces of graphs and the associated graph braid groups. In particular, for graphs of topological circumference at most 1, we prove that the singular chain complex of the configuration space is formal as a DGCoAlg. Moreover, we obtain a complete classification of the primitivity in the homology of configuration spaces of such graphs.