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Notes on nonpositively curved polyhedra. (English) Zbl 0961.53022

Böröczky, Károly jun. (ed.) et al., Low dimensional topology. Proceedings of the summer school, Budapest, Hungary, August 3-14, 1998. Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 8, 11-94 (1999).
This survey paper contains an extensive introduction to the main properties of nonpositively curved spaces, as well as a much more detailed description of some of the fields which have been of special interest to the authors.
The first lecture describes the main properties of nonpositively curved manifolds or polyhedral spaces, mainly the piecewise Euclidean or piecewise hyperbolic ones.
The second lecture focuses on cubical complexes, and on the related right angled Coxeter groups. The main point is then the description of some functorial procedures converting a cell complex into a non-positively curved one, along the ideas of M. Gromov [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)].
The next lecture contains material on some non-cubical but geometrically interesting polyhedral structures; a leading idea is that one way of hyperbolizing a 3-manifold would be to obtain an equivariant convex embedding of it into the 4-dimensional Minkowski space.
Finally, the last lecture is concerned with what the authors call the “Chern-Hopf-Thurston” conjecture, namely that the sign of the Euler characteristic of an even-dimensional closed manifold is determined by its dimension; some related conjectures are also described. The paper describes the \(L_2\)-homology tools that can be used to tackle those conjectures, as well as the partial results obtained by the first author and Boris Okun on those themes.
This paper contains many illuminating examples and most proofs are very clear.
For the entire collection see [Zbl 0938.57002].

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M50 General geometric structures on low-dimensional manifolds
20F55 Reflection and Coxeter groups (group-theoretic aspects)

Citations:

Zbl 0634.20015
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