Stable diffeomorphism classification of some unorientable 4-manifolds

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 × 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 completely solve the case when π 1 (M ) is finite of order 2 mod 4: for each such choice of π 1 (M ), 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.