Cancellation for 4-manifolds with virtually abelian fundamental group

摘要

Suppose X and Y are compact connected topological 4-manifolds with fundamental group pi. For any r >= 0, X is r-stably homeomorphic to Y if X#r(S-2 x S-2) is homeomorphic to Y#r(S-2 x S-2). How close is stable homeomorphism to homeomorphism? When the common fundamental group pi is virtually abelian, we show that large r can be diminished to n + 2, where pi has a finite-index subgroup that is free-abelian of rank n. In particular, if pi is finite then n = 0, hence X and Y are 2-stably homeomorphic, which is one S-2 x S-2 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]. (C) 2017 Elsevier B.V. All rights reserved.