Using equivariant obstruction theory in combinatorial geometry

Abstract

A significant group of problems coming from the realm of combinatorial geometry can be approached fruitfully through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lovász [L. Lovász, Knester's conjecture, chromatic number of distance graphs on the sphere, Acta. Sci. Math (Szeged) 45 (1983) 317-323] through the solution of the Lovász conjecture [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication; C. Schultz, A short proof of w1n (Hom (C2 r + 1, Kn + 2)) = 0 for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2], many methods from Algebraic Topology have been developed. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk-Ulam and Dold theorem to the integer/ideal-valued index theories. Recently equivariant obstruction theory has provided answers where the previous methods failed. For example, in papers [R.T. Živaljević, Equipartitions of measures in R4, arXiv: math.0412483, Trans. Amer. Math. Soc., submitted for publication] and [P. Blagojević, A. Dimitrijević Blagojević, Topology of partition of measures by fans and the second obstruction, arXiv: math.CO/0402400, 2004] obstruction theory was used to prove the existence of different mass partitions. In this paper we extract the essence of the equivariant obstruction theory in order to obtain an effective general position map scheme for analyzing the problem of existence of equivariant maps. The fact that this scheme is useful is demonstrated in this paper with three applications: (A)a "half-page" proof of the Lovász conjecture due to Babson and Kozlov [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication] (one of two key ingredients is Schultz' map [C. Schultz, A short proof of w1n (Hom (C2 r + 1, Kn + 2)) = 0 for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2]),(B)a generalization of the result of V. Makeev [V.V. Makeev, Equipartitions of continuous mass distributions on the sphere an in the space, Zap. Nauchn. Sem. S.-Petersburg (POMI) 252 (1998) 187-196 (in Russian)] about the sphere S2 measure partition by 3-planes (Section 3), and(C)the new (a, b, a), class of 3-fan 2-measures partitions (Section 3). These three results, sorted by complexity, share the spirit of analyzing equivariant maps from spheres to complements of arrangements of subspaces.