Relative Rochlin invariants

Abstract

The Rochlin invariant of a compact 3-manifold with a fixed spin structure can be regarded as the signature (mod 16) of any solution to a certain surgery problem. This paper explores this remark in some detail. The relative Rochlin invariants arise from consideration of other surgery problems. We work out the general theory and apply it with RP³ replacing S³ to study free involutions on 3-manifolds. The Morgan-Sullivan linking cycle theory gives new insight into the relation between spin structures on the 3-manifold and how circles in the manifold link. From the algebra which expresses this relation one can calculate the relative Rochlin invariants mod 8, and can often recover the spin structure on the manifold.