Generalized Rochlin invariants of fixed point sets
Let \(\Sigma\) be a smooth mod 2 homology sphere of dimension \(n\). If \(n\) is not congruent to 3 mod 8 then the generalized Rochlin invariant is defined to be zero. When \(n\) is congruent to 3 mod 8 then \(\Sigma\) has a unique spin structure and \(\Sigma\) is the boundary of a spin manifold \(W\). The generalized Rochlin invariant \(R(\Sigma)\) of \(\Sigma\) is then defined as the signature of \(W \bmod 16\). In this case \(R(\Sigma)\) is well defined by the theorem of Ochanine which says that the signature of a closed spin manifold of dimension \(8s + 4\) is divisible by 16. Now assume that \(\Sigma\) admits a smooth semifree \(S^3\)-action \(\Phi\), i.e. \(\Phi\) is free outside the fixed point set \(F\). The fixed point set \(F\) is a differentiable submanifold of \(\Sigma\) with codimension divisible by 4, and \(F\) is known to be a mod 2 homology sphere. The main result of the paper is that in this case the equality \(R(\Sigma) = R(F)\) holds.
scientific article; zbMATH DE number 720171