Stable homotopy invariance of Teichner's sec invariant

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 × S2 is equivalent to N# sS2 × S2 for some non-negative integers r, s. Any equivalence relation which extends over an S2 × 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.