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The abstract of the talk (see below) may be downloaded in one of the following formats:
Poincar\'{e} constructed the first example of a homological 3-sphere with a
nontrivial fundamental group.
The complement of an open 3-ball in this space is an acyclic finite
noncontractible polyhedron $P$. It
follows by the Mayer-Vietoris sequence and the van Kampen theorem that the
suspension $\Sigma P$ of
this polyhedron is an acyclic space with the trivial fundamental group. It
follows by the Hurewicz theorem
that the suspension $\Sigma P$ has all homotopy groups trivial and is hence a
contractible space.
Complex $P$ is an acyclic noncontractible compactum. Every cell-like space is
acyclic in \v{C}ech
cohomology and every contractible compactum is clearly cell-like. So there is a
natural question: Does
there exist a noncontractible cell-like compactum whose suspension is
contractible?
(M.Bestvina-R.D.Edwards, Problem D28 in J.van Mill and G.M.Reed, {\sl Open
Problems in Topology},
North-Holland, Amsterdam 1990).
Earlier we have proved (with U. H. Karimov) that there exists a noncontractible
cohomologically locally
connected (clc) 2-dimensional compact metric space $X$ of trivial (Borsuk)
shape whose reduced
suspension is a contractible absolute retract. However, the unreduced
suspension of $X$ turned out to be
noncontractible, so the question remained open. In this talk I shall present
our new result - we have proved
that the answer to the Bestvina-Edwards question is affirmative. I shall also
state some interesting
applications and related open problems.