## Complements of connected hypersurfaces in $$S^4$$

### Jonathan A. Hillman

#### Abstract

Let $$X$$ and $$Y$$ be the complementary regions of a closed hypersurface $$M$$ in $$S^4=X\cup_MY$$. We use the Massey product structure in $$H^*(M;\mathbb{Z})$$ to limit the possibilities for $$\chi(X)$$ and $$\chi(Y)$$. We show also that if $$\pi_1(X)\not=1$$ then it may be modified by a 2-knot satellite construction, while if $$\chi(X)\leq1$$ and $$\pi_1(X)$$ is abelian then $$\beta_1(M)\leq4$$ or $$\beta_1(M)=6$$. Finally we use TOP surgery to propose a characterization of the simplest embeddings of $$F\times{S^1}$$.

Keywords: embedding, Euler characteristic, lower central series, Massey product, satellite, Seifert manifold, surgery.

: Primary 57N13.

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 Monday, March 16, 2015