Cancellation for 4-manifolds with virtually abelian fundamental group

TL;DR

This paper investigates the relationship between stable homeomorphism and homeomorphism of 4-manifolds with virtually abelian fundamental groups, showing that the stable homeomorphism can often be reduced to a finite number of stabilizations, especially for finite groups.

Contribution

It establishes bounds on the number of stabilizations needed for homeomorphism of 4-manifolds with virtually abelian fundamental groups, extending the classical cancellation results.

Findings

For finite fundamental groups, 2-stable homeomorphism implies actual homeomorphism.

Large stabilization number can be reduced to n+2 for virtually abelian groups.

Case study on manifolds with fundamental groups of order two.

Abstract

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.