# zbMATH — the first resource for mathematics

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
##### Keywords:
nerve complex; clique complex; circular arc; cyclic polytope
Full Text:
##### References:
  Adamaszek, M, Clique complexes and graph powers, Isr. J. Math., 196, 295-319, (2013) · Zbl 1275.05041  Adamaszek, M., Adams, H.: The Vietoris-Rips complex of the circle. Preprint, arXiv:1503.03669 · Zbl 1366.05124  Adamaszek, M., Adams, H., Motta, F.: Random cyclic dynamical systems. Preprint, arXiv:1511.07832 · Zbl 1379.37011  Attali, D; Lieutier, A, Geometry driven collapses for converting a čech complex into a triangulation of a shape, Discrete Comput. Geom., 54, 798-825, (2014) · Zbl 1336.68258  Attali, D; Lieutier, A; Salinas, D, Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes, Comput. Geom., 46, 448-465, (2013) · Zbl 1262.68171  Babenko, AG, An extremal problem for polynomials, Math. Notes, 35, 181-186, (1984) · Zbl 0601.41028  Babson, E; Kozlov, DN, Complexes of graph homomorphisms, Isr. J. Math., 152, 285-312, (2006) · Zbl 1205.52009  Bagchi, B; Datta, B, Minimal triangulations of sphere bundles over the circle, J. Comb. Theory, Ser. A, 115, 737-752, (2008) · Zbl 1146.52007  Barmak, JA, On quillen’s theorem A for posets, J. Comb. Theory, Ser. A, 118, 2445-2453, (2011) · Zbl 1234.05237  Barmak, JA; Minian, EG, Strong homotopy types, nerves and collapses, Discrete Comput. Geom., 47, 301-328, (2012) · Zbl 1242.57019  Björner, A.: Topological Methods. Handbook of Combinatorics, vol. 2. Elsevier, Amsterdam (1995)  Borsuk, K, On the imbedding of systems of compacta in simplicial complexes, Fundam. Math., 35, 217-234, (1948) · Zbl 0032.12303  Carlsson, G, Topology and data, Bull. Am. Math. Soc., 46, 255-308, (2009) · Zbl 1172.62002  Chazal, F; Silva, V; Oudot, S, Persistence stability for geometric complexes, Geom. Dedicata, 173, 193-214, (2013) · Zbl 1320.55003  Chazal, F., Oudot, S.: Towards persistence-based reconstruction in Euclidean spaces. In: Proceedings of the 24th Annual Symposium on Computational Geometry, pp. 232-241. ACM, New York (2008) · Zbl 1271.57058  Colin de Verdière, É., Ginot, G., Goaoc, X.: Multinerves and Helly numbers of acyclic families. In: Proceedings of the 28th Annual Symposium on Computational Geometry, pp. 209-218. ACM, New York (2012) · Zbl 1293.68286  Edelsbrunner, H., Harer, J.L.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010) · Zbl 1193.55001  Gale, D, Neighborly and cyclic polytopes, Proc. Symp. Pure Math., 7, 225-232, (1963) · Zbl 0137.41801  Gilbert, AD; Smyth, CJ, Zero-Mean cosine polynomials which are non-negative for as long as possible, J. Lond. Math. Soc., 62, 489-504, (2000) · Zbl 1032.42002  Golumbic, MC; Hammer, PL, Stability in circular arc graphs, J. Algorithms, 9, 314-320, (1988) · Zbl 0651.68083  Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001  Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004) · Zbl 1062.05139  Kozlov, D.N.: Combinatorial Algebraic Topology. Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin (2008) · Zbl 1130.55001  Kozma, G; Oravecz, F, On the gaps between zeros of trigonometric polynomials, Real Anal. Exch., 28, 447-454, (2002) · Zbl 1050.42001  Kühnel, W, Higherdimensional analogues of Császár’s torus, Result. Math., 9, 95-106, (1986) · Zbl 0552.52003  Kühnel, W; Lassmann, G, Permuted difference cycles and triangulated sphere bundles, Discrete Math., 162, 215-227, (1996) · Zbl 0866.52011  Latschev, J, Vietoris-rips complexes of metric spaces near a closed Riemannian manifold, Arch. Math., 77, 522-528, (2001) · Zbl 1001.53026  Lovász, L, Kneser’s conjecture, chromatic number, and homotopy, J. Comb. Theory, Ser. A, 25, 319-324, (1978) · Zbl 0418.05028  Matoušek, J, LC reductions yield isomorphic simplicial complexes, Contrib. Discrete Math., 3, 37-39, (2008) · Zbl 1191.52011  Montgomery, HL; Ulrike, MA, Biased trigonometric polynomials, Am. Math. Mon., 114, 804-809, (2007) · Zbl 1268.42001  Previte-Johnson, C.: The $$D$$-Neighborhood Complex of a Graph. PhD thesis, Colorado State University, Fort Collins (2014)  Taylan, D.: Matching trees for simplicial complexes and homotopy type of devoid complexes of graphs. Order (2015). doi:10.1007/s11083-015-9379-3 · Zbl 1353.05133  Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995) · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.