On coshuffle comultiplication on configuration spaces

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.