Martin Palmer

Mapping class groups of infinite-type surfaces ("big mapping class groups") have recently become the subject of intensive study. However, their homology above degree one has only very recently begun to be understood.

I will describe joint work with Xiaolei Wu, in which we prove that there is an uncountable family of infinite type surfaces S such that H_*(MCG(S);Z) = 0 in all positive degrees, and another uncountable family of infinite type surfaces S such that H_*(MCG(S);Z) is uncountable in each positive degree. An example of the first family of surfaces is the disc minus a Cantor set and an example of the second is the plane minus a countably infinite discrete subset.

If time permits, I will also discuss an ongoing joint project with Xiaolei Wu where we study the question of whether H_*(MCG(S);Z) contains non-trivial elements having support on a compact subsurface of S. This question turns out to be especially subtle to answer when S has genus 0.