${\mathbb{CP}}^2$-stable classification of $4$-manifolds with finite fundamental group

Abstract

We show that two closed, connected $4$-manifolds with finite fundamental groups are ${\mathbb{CP}}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.

16 pages, minor changes following a referee report, to appear in Pacific Journal of Math