First cohomology of pure mapping class groups of big genus one and zero surfaces (Q2181496)

It is known that the first integral cohomology of a surface (without boundary) of finite type and of genus at least one is trivial; for a sphere with finitely many punctures instead the rank of the first cohomology is a function of the number of punctures (non-trivial if there are at least four punctures). Recent work of \textit{J. Aramayona} et al. [``The first integral cohomology of pure mapping class groups'', Preprint, \url{arXiv:1711.03132}] describes the first integral cohomology of pure mapping class groups of infinite-type surfaces of genus at least 2. In the present paper, the authors investigate the integral cohomology of infinite-type surfaces of genus one and zero. They prove that, for a genus one surface \(S\) without boundary and with a nonempty closed set of marked points representing punctures, the first integral cohomology of the pure mapping class group of \(S\) is again trivial (i.e., of mapping classes acting trivially on the space of ends of the surface). For an infinite type surface \(S\) of genus zero instead, they prove that the first integral cohomology of the pure mapping class group of \(S\) (identified with the set of homomorphisms of the group to the integers \(\mathbb Z\)) contains cohomology classes which do not come from forgetful maps to finite type surfaces of genus zero (i.e., with only a finite set of punctures remaining); furthermore, there are uncountably many such cohomology classes (by varying the construction to encode a Cantor set of such cohomology classes). What remains open is an explicit description of the cohomology of the pure mapping class group of an infinite-type surface of genus zero.