Coarse Alexander duality and duality groups. (English) Zbl 1086.57019

Kapovich, Michael; Kleiner, Bruce

J. Differ. Geom. 69, No. 2, 279-352 (2005).

Summary: The Scott-Shalen Core Theorem asserts that if the fundamental group \(G\) of a 3-manifold \(M\) is finitely generated then there is a compact codimension-\(0\) submanifold \(N\) for which the inclusion \(N\subseteq M\) is a homotopy equivalence. Hence \(G\) is finitely presentable, and it follows easily that all 3-manifold groups are coherent.
The main result of this paper is an algebraic analogue of the Core Theorem, for \((n- 1)\)-dimensional duality groups acting freely on “coarse \(PD_n\)-spaces”. (This class of spaces includes the universal covering spaces of finite \(PD_n\)-complexes.) In particular, if \(G\) is a one-ended \(FP_2\) subgroup of infinite index in a \(PD_3\)-group then \(G\) is the ambient group of a \(PD_3\)-group pair \((G, \{H_i\})\), and so \(G\) contains surface subgroups \(H_i\) . Many interesting constraints on the possible subgroups of \(PD_3\)-groups follow from this Algebraic Core Theorem. (Note however that it remains unknown whether \(PD_3\)-groups are coherent.) The main technique is an extension of Alexander duality to subcomplexes of coarse \(PD_n\)-complexes. The arguments are essentially homological, and the paper is self-contained. As there are \(PD_n\)-groups which are not finitely presentable and so which do not act freely and cocompactly on contractible simplicial complexes the argument is extended in the Appendix to cover such situations.

MSC:

57P10	Poincaré duality spaces
20J05	Homological methods in group theory

Keywords:

Alexander duality; coarse topology; Core Theorem; \(PD_n\)-space