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 $\mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y$, 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.