Generalized Rochlin invariants of fixed point sets
type: Item
instance of: scholarly article
title: Generalized Rochlin invariants of fixed point sets
zbMATH Open document ID: 0854.57032
author: Mikiya Masuda
author: Reinhard Schultz
published in: Osaka Journal of Mathematics
publication date: 1995-02-09T00:00:00Z
review text: 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.
reviewed by: K. H. Mayer
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 720171
zbMATH Keywords: mod 2 homology sphere
zbMATH Keywords: generalized Rochlin invariant
zbMATH Keywords: spin structure
zbMATH Keywords: signature
zbMATH Keywords: semifree \(S^ 3\)-action
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 720171