Compact and finite‐type support in the homology of big mapping class groups

Abstract For any infinite‐type surface , a natural question is whether the homology of its mapping class group contains any non‐trivial classes that are supported on (i) a compact subsurface; or (ii) a finite‐type subsurface. Our purpose here is to study this question, in particular giving an almost‐complete answer when the genus of is positive (including infinite) and a partial answer when the genus of is zero. Our methods involve the notion of shiftable subsurfaces as well as homological stability for mapping class groups of finite‐type surfaces.