Nonorientable 4 4 -manifolds with fundamental group of order 2 2

In this paper we classify nonorientable topological closed 4-manifolds with fundamental group Z/2\mathbb {Z}/2 up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the η\eta-invariant associated to a normal PincPin^c-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are stably homeomorphic (after connected sum with S2×S2{S^2} \times {S^2}) but not homeomorphic.