Cancellation for 4-manifolds with virtually abelian fundamental group

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Suppose X and Y are compact connected topological 4-manifolds with fundamental group π. For any r 0, Y is r-stably homeomorphic to X if Y # r(S² × S²) is homeomorphic to X # 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. The last section is a case-study investigation of the homeomorphism classification of closed manifolds in the tangential homotopy type of X = X- # X+, where X± are closed nonorientable topological 4-manifolds whose fundamental groups have order two.