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

TL;DR

This paper investigates the Seifert linking form for punctured n-manifolds embedded in (2n-1)-space, establishing its properties, relations to Stiefel-Whitney classes, and realization by embeddings, along with a survey of embedding classifications.

Contribution

It characterizes the Seifert linking form modulo two in terms of Stiefel-Whitney classes and proves that any such form can be realized by an embedding, extending understanding of manifold embeddings.

Findings

Values of the Seifert linking form relate to Stiefel-Whitney classes.

Any symmetric bilinear form satisfying certain conditions can be realized by an embedding.

Survey of existing results on embedding classification of manifolds with boundary.

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.