A description of values of Seifert form for punctured n-manifolds in (2n-1)-space

Abstract

We study Seifert linking form which is an invariant of embeddings of punctured $n$-manifolds in ${\mathbb{R}}^{2n-1}$. For punctured $n$-manifold $N_0$ the values of this invariant are integer valued bilinear symmetric forms on $H_{n-1}(N_0;\mathbb{Z})$. We prove that value modulo two of this invariant at $x,y\in H_{n-1}(N_0;\mathbb{Z})$ equals $\mathrm{PD}{\bar{w}}_{n-2}(N_0)\cap {\rho }_{2}x\cap {\rho }_{2}y$, where $\mathrm{PD}{\bar{w}}_{n-2}(N_0)$ is Poincare dual to Steifel-Whitney class. We also prove that any such form can be realized by some embedding $N_0\to {\mathbb{R}}^{2n-1}$. Also, we survey known results on classification of embeddings of connected manifolds with non-empty boundary.

Based on D. Tonkonog's drafts, 10 pages, 2 figures

References in corpus (2)

• A short exposition of the Levine-Lidman example of spineless 4-manifolds
• Embedding punctured n-manifolds in Euclidean (2n-1)-space