The complementary regions of embeddings of 3-manifolds in S^4

Jonathan Hillman (University of Sydney)

Friday 26th September 15:00-16:00
Maths 311B

Abstract

If a closed 3-manifold $M$ embeds in $S^4$ and is not an homology sphere it has infinitely many embeddings, distinguished by the fundamental groups of the complementary regions. In order to apply surgery to study embeddings of 3-manifolds in $S^4$ we must restrict the groups that arise. If $M$ has an embedding in $S^4$ such that each of the complementary regions has abelian fundamental group then either $\beta=\beta_1(M)=1,3,4$ or $6$ and $H_1(M;\Z)\cong\Z^\beta$, or $H_1(M;\Z)\cong{C_n^2}$ or $\Z^2\oplus{C_n^2}$, for some $n>0$. If $H_1(M;\Z)=0$ then $M$ is a homology 3-sphere, and has an essentially unique abelian embedding (Freedman), while if $H_1(M;\Z)\cong\Z$ then it has at most one such embedding. On the other hand, if $\beta=2,3,4$ or 6 there are examples with several distinct such ``abelian" embeddings.

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