Nonorientable 4-manifolds with fundamental group of order 2

Hambleton, Ian; Kreck, Matthias; Teichner, Peter

doi:10.1090/S0002-9947-1994-1234481-2

Nonorientable $4$-manifolds with fundamental group of order $2$
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by Ian Hambleton, Matthias Kreck and Peter Teichner PDF

Trans. Amer. Math. Soc. 344 (1994), 649-665 Request permission

Abstract:

In this paper we classify nonorientable topological closed 4-manifolds with fundamental group $\mathbb {Z}/2$ 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 $\eta$-invariant associated to a normal $Pin^c$-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 ${S^2} \times {S^2}$) but not homeomorphic.

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Poincaré complexes

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