Involutions on 8 r+2-spin manifolds

Abstract

For any closed Spin manifold M 8 r+2 of dimension 8 r+2, there is an associated symmetric bilinear form. In a recent paper, Landweber and Stong showed that the rank of this form is given by the Stiefel-Whitney number w 4 w 8 r−2 [ M]. Here we consider the relationship between this number and the involutions on M, determining w 4 w 8 r−2 [ M] in terms of certain fixed point data. As one special case, if we let F 8∗+4 denote the fixed point components of dimension≡ 4 (mod8), we prove: If ( T,M) is an involution of odd-type, then w 4w 8r−2[M] = χ(F 8∗+4) .