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 $S^2\times S^2$, 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.

10 pages. Comments welcome! v2: fixed an error in an important lemma

References in corpus (2)

• Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle
• 4-manifolds, surgery on loops and geometric realization of Tietze transformations

Cited by in corpus (1)

• Counterexamples in 4-manifold topology