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Nerve complexes of circular arcs. (English) Zbl 1354.05149
The main result of this paper, Theorem 5.4, states that both the nerve complex and clique complex of a finite collection of arcs in the circle are homotopy equivalent to either a point, a sphere of odd dimension, or a wedge sum of spheres of the same even dimension. This is an interesting first step away from the hypotheses of the nerve theorem.
The proof builds up through the special case of evenly spaced arcs, for which there is a concise summary in Sections 3 and 4. The general case is tackled in Section 5.
Section 6 contains a nice application to a result of Lovász that the chromatic number of a graph is at least 3 more than the connectivity number of its neighbourhood complex. A certain circulant graph connected with the circular chromatic number is shown to exceed this bound by at most one.
The exposition is very clear and is aided with some helpful diagrams.

MSC:
05E45 Combinatorial aspects of simplicial complexes
05C40 Connectivity
05C15 Coloring of graphs and hypergraphs
55U10 Simplicial sets and complexes in algebraic topology
55P15 Classification of homotopy type
52B15 Symmetry properties of polytopes
68R05 Combinatorics in computer science
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