Illinois Journal of Mathematics

Four-dimensional Haken cobordism theory

Bell Foozwell and Hyam Rubinstein

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Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_{1}(W)$ is injective. In this paper, we prove that there are $4$-dimensional Haken cobordisms whose boundary consists of any two closed Haken $3$-manifolds. In particular, each closed Haken $3$-manifold is the $\pi_{1}$-injective boundary of some Haken $4$-manifold.

Article information

Illinois J. Math., Volume 60, Number 1 (2016), 1-17.

Received: 14 April 2015
Revised: 18 August 2015
First available in Project Euclid: 21 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 57N10: Topology of general 3-manifolds [See also 57Mxx] 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx] 57Q20: Cobordism


Foozwell, Bell; Rubinstein, Hyam. Four-dimensional Haken cobordism theory. Illinois J. Math. 60 (2016), no. 1, 1--17. doi:10.1215/ijm/1498032021.

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