$$PD_3$$-groups and HNN extensions

Jonathan A. Hillman

Abstract

We show that if a $$PD_3$$-group $$G$$ splits as an HNN extension $$A*_C\varphi$$ where $$C$$ is a $$PD_2$$-group then the Poincaré dual in $$H^1(G;\mathbb{Z})= \mathrm{Hom}(G,\mathbb{Z})$$ of the homology class $$[C]$$ is the epimorphism $$f:G\to\mathbb{Z}$$ with kernel the normal closure of $$A$$. We also make several other observations about $$PD_3$$-groups which split over $$PD_2$$-groups.

Keywords: fundamental class, HNN extension, $$PD_3$$-group, surface group.

: Primary 57N13.

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 Tuesday, March 24, 2020