Uniform polyhedra: old and new

Details Uniform polyhedra: old and new Uniform polyhedra: old and new

2011-09-02

arxiv.org/abs/1109.0346v1

Mathematics

Geometric Topology

52 pages. Some material was formerly a part of arXiv:1106.3249. Hyperlinks to the current version of that paper should work if

Scientific paper

We develop a theory of metric polyhedra, including locally infinite dimensional ones. Motivated by algebraic topology, we focus on their uniform properties (i.e., those preserved by homeomorphisms that are uniformly continuous in both directions) but in doing so we also study their metric and Lipschitz properties. On the combinatorial side, (the face posets of) simplicial or cubical complexes do not suffice for this, and we have to rework some basic PL topology into a purely combinatorial machinery (with all homeomorphisms eliminated in favor of combinatorial isomorphisms) based on posets and their canonical subdivision (which is just the poset of all order intervals of the given poset, ordered by inclusion). Antecedents of this approach to PL topology are found in van Kampen's 1929 dissertation and in modern Topological Combinatorics. Our main results establish, in particular, close but troubled relations between uniform polyhedra and uniform ANRs, and appear to provide a satisfactory solution to an open-ended problem raised by J. R. Isbell in a series of publications in 1959-64.

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