Cancellation for 4-manifolds with virtually abelian fundamental group
  Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $\pi$. For any $r \geqslant 0$, $Y$ 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. 
Qayum Khan
math.GT math.AT