Equivariant methods in combinatorial geometry. (English) Zbl 1235.52015

Blagojević, Pavle V. M.

Stedman, Ched E. (ed.), Algebra and algebraic topology. New York, NY: Nova Science Publishers (ISBN 978-1-60021-187-4). 117-155 (2007).

Summary: Partition problems are classical problems of combinatorial geometry whose solutions often rely on the methods of the equivariant topology. The \(k\)-fan partition problems introduced by A. Kaneko and M. Kano [Comput. Geom. 13, No. 4, 253–261 (1999; Zbl 0948.68199)] and first discussed by equivariant methods by I. Bárány and J. Matoušek [Discrete Comput. Geom. 25, No. 3, 317–334 (2001; Zbl 0989.52009); Discrete Comput. Geom. 27, No. 3, 293–301 (2002; Zbl 1002.60006)] have forced some hard concrete combinatorial calculations in equivariant cohomology [P. V. M. Blagojević, “Topology of partition of measures by fans and the second obstruction”, arxiv:math/0402400; P. V. M. Blagojević, S. T. Vrećica and R. T. Živaljević, Trans. Am. Math. Soc. 361, No. 2, 1007–1038 (2009; Zbl 1159.52007), arxiv:math/0403161]. These problems can be reduced, by the beautiful scheme of Bárány and Matoušek [Zbl 0989.52009], to topological problems of the existence of \(\mathbb D_{2n}\) equivariant maps \(V_2(\mathbb R^3)\to W_n\setminus\bigcup\mathcal A(\alpha)\) from a Stiefel manifold of all orthonormal 2-frames in \(\mathbb R^3\) to complements of appropriate arrangements. ”In this paper we present a set of techniques, based on the equivariant obstruction theory, which can help in answering the question of the existence of an equivariant map to a complement of an arrangement. With the help of the target extension scheme, introduced in [arxiv:math/0403161], we are able to deal with problems where the existence of the map depends on more than one obstruction. The techniques introduced, with an emphasis on computation, are applied on the known results of the fan partition problems.
For the entire collection see [Zbl 1234.00017].

MSC:

52A37	Other problems of combinatorial convexity
55P91	Equivariant homotopy theory in algebraic topology
28A12	Contents, measures, outer measures, capacities

Citations:

Zbl 0948.68199; Zbl 0989.52009; Zbl 1002.60006; Zbl 1159.52007