Stable diffeomorphism classification of some unorientable 4-manifolds

TL;DR

This paper classifies certain unorientable 4-manifolds up to stable diffeomorphism using modified surgery theory, simplifying computations for specific fundamental groups and identifying distinct classes based on pin structures.

Contribution

It applies Kreck's modified surgery theory to unorientable 4-manifolds with specific fundamental groups, providing explicit classification results.

Findings

Nine stable diffeomorphism classes for pin$^+$ manifolds

One stable diffeomorphism class for pin$^-$ manifolds

Four stable diffeomorphism classes for neither pin$^+$ nor pin$^-$

Abstract

Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of $S^2\times S^2$, to a series of bordism questions. We implement this in the case of unorientable 4-manifolds M and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which $\pi_1(M)$ is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which M is pin$^+$, one stable diffeomorphism class for which M is pin$^-$, and four stable diffeomorphism classes for which M is neither. We also determine the corresponding stable homeomorphism classes.