Configuration-like spaces and coincidences of maps on orbits

Abstract

In this paper we study the spaces of q -tuples of points in a Euclidean space, without k -wise coincidences (configuration-like spaces). A transitive group action by permuting these points is considered, and some new upper bounds on the genus (in the sense of Krasnosel'skii-Schwarz and Clapp-Puppe) for this action are given. Some theorems of Cohen-Lusk type for coincidence points of continuous maps to Euclidean spaces are deduced. 55R80; 55M20, 55M30, 55M35, 57S17 IntroductionIn this paper we address the question of finding some sufficient conditions that guarantee that a continuous map f W X ! R m has a certain number of self-coincidences on an orbit of a G -action on X , where G is a finite group. The most famous result of this kind is the Borsuk-Ulam theorem [5], where X is the m-dimensional sphere and G D Z=2 acts on X by the antipodal action. Partial solutions of the Knaster problem by Makeev [18] and the second author [24] provide another application.In order to simplify the statements we need some definitions. Definition 1.1 Let G be a finite group, and X be a G -space, that is, a topological space with continuous left G -action. For a given continuous map f W X ! Y we denote by A.f; k/ the coincidence set A.f; k/ D fx 2 X W 9 distinct g 1 ; : : : ;Generally, to deduce existence theorems for coincidences, we have to define the complexity of the action of G on the space X . The following definition was made for G D Z=2 by Krasnosel'skii [15; 17] and Yang [28;29], and for arbitrary finite G by Krasnosel'skii in [16], as noted by Schwarz [20]. It is usually called the Krasnosel'skiiSchwarz genus. Definition 1.2 The free genus of a free G -space X is the least number n such that X can be covered by n open subsets X 1 ; : : : ; X n so that for every i there exists a

Equivariant topology of configuration spaces

“…The results on the Fadell-Husseini index are used in Section 8 as the main ingredient to give a new proof of the Nandakumar & Ramana-Rao conjecture for n = p and give a proof of the second conjecture for n = p k , where p is a prime. The partial calculation of the Fadell-Husseini index in the case n = p k , Section 7, allows us to extend and improve Borsuk-Ulam type coincidence results by Cohen & Connett [17], Cohen & Lusk [19], and Karasev & Volovikov [41].…”