Braid groups and discrete diffeomorphisms of the punctured disk

TL;DR

This paper explores the relationship between the cohomology of diffeomorphism groups of punctured disks and braid groups, revealing structural similarities and implications for group embeddings.

Contribution

It demonstrates that the cohomology of diffeomorphism groups includes braid group cohomology and shows no obstruction to certain group embeddings.

Findings

Cohomology of diffeomorphism groups contains braid group cohomology as a summand.

No cohomological obstruction exists for lifting specific braid group embeddings to diffeomorphism groups.

Establishes a connection between braid groups and diffeomorphism groups of punctured disks.

Abstract

We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.