Special Issue for Tim Cochran; Guest Editors: J. E. Grigsby, S. Harvey, K. Orr and D. RubermanNo Access

Complements of connected hypersurfaces in $S^{4}$ $S^{4}$

Jonathan A. Hillman

School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

E-mail Address: johnathanhillman47@gmail.com

https://doi.org/10.1142/S0218216517400144Cited by:1 (Source: Crossref)

Abstract

Let $X$ $X$ and $Y$ $Y$ be the complementary regions of a closed hypersurface $M$ $M$ in $S^{4} = X \cup_{M} Y$ $S^{4} = X \cup_{M} Y$ . We use the Massey product structure in $H^{*} (M; ℤ)$ to limit the possibilities for $χ (X)$ and $χ (Y)$ . We show also that if $π_{1} (X) \neq 1$ then it may be modified by a 2-knot satellite construction, while if $χ (X) \leq 1$ and $π_{1} (X)$ is abelian then $β_{1} (M) \leq 4$ or $β_{1} (M) = 6$ . Finally we use TOP surgery to propose a characterization of the simplest embeddings of $F \times S^{1}$ .

Keywords:

AMSC: 57N13

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