Non-compact 3-manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper 3-realizability of groups

Details Non-compact 3-manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper 3-realizability of groups Non-compact 3-manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper 3-realizability of groups

2007-09-11

arxiv.org/abs/0709.1576v2

Mathematics

Geometric Topology

revised version, 18p, 1 figure

Scientific paper

A group is properly 3-realizable if it is the fundamental group of a compact 2-polyhedron whose universal covering is proper homotopy equivalent to a 3-manifold. Our main result states that a properly 3-realizable group which is also quasi-simply filtered has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. This result follows from a homotopy criterion for detecting the tameness of non-compact 3-manifolds which extends the one worked out in our ealier work for open 3-manifolds. As an application we find examples of finitely presented groups which are not properly 3-realizable.

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