2,140 research outputs found
Manifolds with non-stable fundamental groups at infinity, II
In this paper we continue an earlier study of ends non-compact manifolds. The
over-arching goal is to investigate and obtain generalizations of Siebenmann's
famous collaring theorem that may be applied to manifolds having non-stable
fundamental group systems at infinity. In this paper we show that, for
manifolds with compact boundary, the condition of inward tameness has
substatial implications for the algebraic topology at infinity. In particular,
every inward tame manifold with compact boundary has stable homology (in all
dimensions) and semistable fundamental group at each of its ends. In contrast,
we also construct examples of this sort which fail to have perfectly semistable
fundamental group at infinity. In doing so, we exhibit the first known examples
of open manifolds that are inward tame and have vanishing Wall finiteness
obstruction at infinity, but are not pseudo-collarable.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper7.abs.htm
Cell-Like Equivalences and Boundaries of CAT(0) Groups
In 2000, Croke and Kleiner showed that a CAT(0) group G can admit more than
one boundary. This contrasted with the situation for word hyperbolic groups,
where it was well-known that each such group admitted a unique boundary---in a
very stong sense. Prior to Croke and Kleiner's discovery, it had been observed
by Geoghegan and Bestvina that a weaker sort of uniquness does hold for
boundaries of torsion free CAT(0) groups; in particular, any two such
boundaries always have the same shape. Hence, the boundary really does carry
significant information about the group itself. In an attempt to strengthen the
correspondence between group and boundary, Bestvina asked whether boundaries of
CAT(0) groups are unique up to cell-like equivalence. For the types of space
that arise as boundaries of CAT(0) groups, this is a notion that is weaker than
topological equivalence and stronger than shape equivalence. In this paper we
explore the Bestvina Cell-like Equivalence Question. We describe a
straightforward strategy with the potential for providing a fully general
positive answer. We apply that strategy to a number of test cases and show that
it succeeds---often in unexpectedly interesting ways.Comment: 21 pages, 5 figure
Topological properties of spaces admitting free group actions
In 1992, David Wright proved a remarkable theorem about which contractible
open manifolds are covering spaces. He showed that if a one-ended open manifold
M has pro-monomorphic fundamental group at infinity which is not pro-trivial
and is not stably Z, then M does not cover any manifold (except itself). In the
non-manifold case, Wright's method showed that when a one-ended, simply
connected, locally compact ANR X with pro-monomorphic fundamental group at
infinity admits an action of Z by covering transformations then the fundamental
group at infinity of X is (up to pro-isomorphism) an inverse sequence of
finitely generated free groups. We improve upon this latter result, by showing
that X must have a stable finitely generated free fundamental group at
infinity. Simple examples show that a free group of any finite rank is
possible. We also prove that if X (as above), admits a non-cocompact action of
Z+Z by covering transformations, then X is simply connected at infinity.
Corollary: Every finitely presented one-ended group G which contains an element
of infinite order satisfies exactly one of the following: 1) G is simply
connected at infinity; 2) G is virtually a surface group; 3) The fundamental
group at infinity of G is not pro-monomorphic. Our methods also provide a quick
new proof of Wright's open manifold theorem.Comment: Revised version with a shorter proof of the main theorem, plus
numerous small corrections. To appear in the Journal of Topology. 31 pages, 4
figure
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