Secondary Stiefel-Whitney numbers and corresponding cobordism groups

Abstract

For every relation $R$ between Stiefel-Whitney numbers of closed $(n+1)$-manifolds we consider an associated invariant ${\varkappa }_R$ of null-cobordant $n$-manifolds with a certain additional structure. For $n=2k-1$ and $R=w_{n+1}+v_k^2$ the invariant ${\varkappa }_R$ equals the Kervaire semi-characteristic. In addition, we construct the cobordism group ${\Omega }_n^R$, which extends the unoriented cobordism group ${\Omega }_n^O$. We show that ${\varkappa }_R$ is a complete invariant of $R$-cobordism classes of null-cobordant $n$-manifolds. We prove that our invariant ${\varkappa }_R$ and $R$-cobordism class of manifold are quadratic in the sense of Gusarov-Vassiliev-Podkorytov.