Compact and finite-type support in homology for infinite-genus, infinitely punctured surfaces without mixed ends

Ascertain whether, for any connected, orientable infinite-type surface S with g_S = p_S = \infty that has no mixed end, the inclusions Map_c(S) -> Map(S) and Map_f(S) -> Map(S) induce non-zero maps on homology (with field coefficients); equivalently, determine whether H_*(Map(S)) contains any non-zero classes supported on compact subsurfaces or properly embedded finite-type subsurfaces in this no–mixed-end setting.

Background

Theorem \ref{mainthm-infinite-genus} provides vanishing and non-vanishing results for compactly supported and finite-type supported classes in the homology of mapping class groups of infinite-type surfaces in several cases, including when there is a mixed end. However, their methods rely on properties that fail when g_S = p_S = \infty but S has no mixed end.

In this specific case (infinite genus and infinitely many punctures without mixed ends), the authors explicitly state that the status of both Question \ref{q-finite-type-pure} (pure finite-type support via Map_c(S)) and Question \ref{q-finite-type} (finite-type support via Map_f(S)) remains open.