Secondary Stiefel-Whitney numbers and corresponding cobordism groups

2025/12/06 by Lavrukhin, Viktor
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.06371

openalex publication_date 2025/12/06 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

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_k² the invariant \varkappa_R equals the Kervaire semi-characteristic. In addition, we construct the cobordism group Ω_n^R, which extends the unoriented cobordism group Ω_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.

Citations

• On homotopy invariants of finite degree
• Order of a homotopy invariant in the stable case

Related

• Cobordism of maps on Z₂-Witt spaces
• On the homology description of equivalent unoriented bordism groups
• On the cobordism classification of manifolds with Z/p-action
• Cobordism of G-manifolds
• Cobordism of fold maps, stably framed manifolds and immersions