#### Volume 17, issue 6 (2017)

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$3$–manifolds built from injective handlebodies

### James Coffey and Hyam Rubinstein

Algebraic & Geometric Topology 17 (2017) 3213–3257
##### Abstract

This paper studies a class of closed orientable $3$–manifolds constructed from a gluing of three handlebodies, such that the inclusion of each handlebody is ${\pi }_{1}$–injective. This construction is the generalisation to handlebodies of the construction for gluing three solid tori to produce non-Haken Seifert fibred $3$–manifolds with infinite fundamental group. It is shown that there is an efficient algorithm to decide if a gluing of handlebodies satisfies the disk-condition. Also, an outline for the construction of the characteristic variety (JSJ decomposition) in such manifolds is given. Some non-Haken and atoroidal examples are given.

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