The first integral cohomology of pure mapping class groups

TL;DR

This paper investigates the first integral cohomology of pure mapping class groups, revealing nontrivial homomorphisms for infinite-genus surfaces and providing a computation method based on surface homology.

Contribution

It demonstrates that pure mapping class groups of infinite-genus surfaces split as semi-direct products and computes their first integral cohomology groups.

Findings

Pure mapping class groups of infinite-genus surfaces admit nontrivial homomorphisms.

The first integral cohomology group is computed in terms of surface simplicial homology.

Pure mapping class groups of finite genus at least three are perfect, contrasting with the infinite-genus case.

Abstract

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.