The Bing-Borsuk and the Busemann Conjectures

Duˇsan Repovˇs Faculty of Mathematics and Physics University of Ljubljana Ljubljana, Slovenia Email address: dusan.repovs@guest.arnes.si Homepage: www.fmf.uni-lj.si/∼repovs/ We shall present a survey of two classical conjectures concerning the characterization of topological n-manifolds: the Bing-Borsuk Conjec- ture asserts that every n-dimensional homogeneous absolute neighbor- hood retract (ANR), n ≥ 3, is a topological n-manifold, whereas the Busemann Conjecture asserts that every n-dimensional Busemann G- space, n ≥ 3, is a topological n-manifold. The key object in both cases are so-called generalized n-manifolds, i.e. Euclidean neighbor- hood retracts (ENR) which are also Z-homology n-manifolds. We shall look at their history, from the early beginnings to the present day, concentrating on those geometric properties of these spaces which are particular for dimensions 3 and 4, in comparison with generalized (n ≥ 5)-manifolds. In the second part of the talk we shall present the current state of the two conjectures. We shall also list open problems and related conjectures.

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