Involutions on \(8r+2\)-spin manifolds (Q1103233)

For a closed spin manifold M of dimension \(8r+2\), the Stiefel-Whitney number \(w_ 4w_{8r-2}[M]\) has been shown to give the rank of a bilinear form on M. This paper studies the relationship between this number and the fixed point set of an involution on M. The most attractive result is that if (T,M) is an involution of odd type then \(w_ 4w_{8r- 2}[M]=\chi (F^{8r+4})\) is the mod 2 Euler characteristic of the part of the fixed set having dimension 4 mod 8.