  We prove a version of the Cohen--Lenstra conjecture over function fields
(completing the results of our prior paper). This is deduced from two more
general theorems, one topological, one arithmetic: We compute the direct limit
of homology, over puncture-stabilization, of spaces of maps from a punctured
manifold to a fixed target; and we compute the Galois action on the set of
stable components of Hurwitz schemes.
