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Problems on homology manifolds. (English) Zbl 1108.57017

Quinn, Frank (ed.) et al., Exotic homology manifolds. Proceedings of the mini-workshop, Oberwolfach, Germany, June 29–July 5, 2003. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 9, 87-103 (2006).
From the introduction: This is a compilation of questions for the proceedings of the Workshop on Exotic Homology Manifolds, Oberwolfach June 29–July 5, 2003.
In these notes “homology manifold” means ENR (Euclidean neighborhood retract) \(\mathbb{Z}\)-coefficient homology manifold, unless otherwise specified, and “exotic” means not a manifold factor (i.e., local or “Quinn” index \(\neq 1\). We use the multiplicative version of the local index, taking values in \(1+8\mathbb{Z})\). In the last decade exotic homology manifolds have been shown to exist and quite a bit of structure theory has been developed. However they have not yet appeared in other areas of mathematics. The first groups of questions suggest ways this might happen. Later questions are more internal to the subject. Section 1, Section 2, and Section 3 concern possible “natural” appearances of homology manifolds: as aspherical geometric objects; as Gromov-Hausdorff limits; and as boundaries of compactifications. Section 4 discusses group actions, where the use of homology manifold fixed sets may give simpler classification results. Section 5 and Section 6 consider possible generalizations to non-ANR and “approximate” homology manifolds. Section 7 concerns spaces with special metric structures. Section 8 describes still-open low dimensional cases of the current theory. Section 9 collects problems related to homeomorphisms and the “disjoint disk property” for exotic homology manifolds.
For the entire collection see [Zbl 1104.57001].

MSC:

57P99 Generalized manifolds
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