Invariants of embeddings of 2-surfaces in 3-space

Abstract

Let $M$ be a sphere with handles and holes, $f:M\to {\mathbb{R}}^3$ an embedding, and $H_1=H_1(M;\mathbb{Z})$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to \mathbb{Z}$. Let $\cap :H_1\times H_1\to \mathbb{Z}$ be the intersection form of $M$. Then the Seifert form is $\cap$-symmetric, i.e., $L(f)(\beta ,\gamma )-L(f)(\gamma ,\beta )=\beta \cap \gamma$ for any $\beta ,\gamma \in H_1$. If $M$ has non-empty boundary, then any $\cap$-symmetric bilinear form $H_1\times H_1\to \mathbb{Z}$ is realizable as $L(f)$ for some embedding $f$. We present a characterization of realizable forms for the torus $M$. The results are simple and presumably known in folklore. We present a simplified exposition accessible to non-specialists.

arXiv comment4 pages, 1 figure