  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.
