Cancellation for 4-manifolds with virtually abelian fundamental group

Abstract

Suppose X and Y are compact connected topological 4-manifolds with fundamental group π. For any r⩾0, Xisr-stably homeomorphic toY if X#r(S2×S2) is homeomorphic to Y#r(S2×S2). How close is stable homeomorphism to homeomorphism?When the common fundamental group π is virtually abelian, we show that large r can be diminished to n+2, where π has a finite-index subgroup that is free-abelian of rank n. In particular, if π is finite then n=0, hence X and Y are 2-stably homeomorphic, which is one S2×S2 summand in excess of the cancellation theorem of Hambleton–Kreck [12].The last section is a case study of the homeomorphism classification of closed manifolds in the tangential homotopy type of X=X−#X+, where X± are closed nonorientable topological 4-manifolds with order-two fundamental groups [13].