Stable homotopy invariance of Teichner's sec invariant
Clark, Gregory Sterling
Doctor of Philosophy
Given a notion of equivalence of 4-manifolds, there is a corresponding notion of stable equivalence: M is stably equivalent to N if M#rS2 x S2 is equivalent to N# sS2 x S2 for some non-negative integers r, s. Any equivalence relation which extends over an S2 x S2 summand gives a well-defined equivalence relation, and homotopy equivalence is such a relation. In this paper, we examine how the invariant sec of a 4-manifold M with finite fundamental group and spin universal cover relates to the stable homotopy type of M. The sec invariant of a manifold M may be defined in terms of a characteristic 3-dimensional homology class w2 + w on a null-cobordism of M. In the case where sec = 0, we are able to conclude some geometric information about w2 + w; namely, that w 2 + w is represented by S3 . This allows us to prove that sec (M) determines the stable homotopy type of M, or more generally, that manifolds M and N for which sec (M - N) is defined and equal to 0, are stably homotopy equivalent. We also prove a partial converse to this theorem. If M and N are homotopy equivalent, and there exists a homeomorphism M# C P2 → N# C P2 which preserves the homotopy classes of the core 2-spheres of the C P2, then sec (M - N) = 0.