Cancellation for 4-manifolds with virtually abelian fundamental group

Abstract

Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $\pi$. For any $r⩾0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y\#r(S^2\times S^2)$ is homeomorphic to $X\#r(S^2\times S^2)$. 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 $S^2\times S^2$ 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 $X_{\pm }$ are closed nonorientable topological 4-manifolds whose fundamental groups have order two.

14 pages

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