Classification of Closed Nonorientable 4-manifolds with Infinite Cyclic Fundamental Group

Closed, connected, nonorientable, topological 4-manifolds with infinite cyclic fundamental group are classified. The classification is an extension of results of Freedman and Quinn and of Kreck. The stable classification of such 4-manifolds is also obtained. In principle, it is a consequence of Freedman’s work on topological 4manifolds that one can classify topological 4-manifolds with “good” fundamental group. Roughly, a “good” group is a group for which topological surgery in dimension 4 works ([F2]). For example, by results of Freedman, all groups of polynomial growth are “good”. However, the full classification for closed, orientable 4-manifolds has only been obtained for manifolds with cyclic fundamental groups ([F1][FQ], [HK1][HK2], [FQ][K][SW][Wa]). For closed, nonorientable 4-manifolds the classification has only been given for fundamental group Z2 ([HKT]). In this note, we give the classification for closed, nonorientable 4-manifolds with infinite cyclic fundamental group. Our main result is the following theorem: Theorem 1. (1) Existence: Suppose (H, λ) is a nonsingular ω1-hermitian form on a finitely generated free Z[Z]-module, k ∈ Z2, and if λ is even then we assume k = [λ] ∈ L4(Z−). Then there is a closed, connected, nonorientable 4-manifold with π1 = Z, intersection form λ and KirbySiebenmann invariant k. (2) Uniqueness: Suppose M and N are closed 4-manifolds with π1 = Z, not orientable but locally oriented, h : H2(M ;Z[Z]) −→ H2(N ; Z[Z]) is a Z[Z]-isomorphism which preserves intersection forms, and ks(M) = ks(N). Then there is a homeomorphism f : M −→ N which induces the identification of fundamental groups, preserves local orientations, and with f∗ = h. Received March 29, 1995. 1991 Mathematics Subject Classification: Primary 57N13, Secondary 57N35, 57N75. Supported by a Sloan Dissertation Fellowship.