The genus and the category of configuration spaces

Карасев, Роман

doi:10.1016/j.topol.2009.06.012

Cited by 9 publications

(17 citation statements)

References 27 publications

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“…For n = p (i.e., a prime number) this theorem is valid even in Z/p-equivariant cohomology (if we embed Z/p ⊂ Σ p in the natural way). This is a particular case of [16,Lemma 5], and seems to be have been known much earlier, see [9, Theorem 3.4, Corollaries 3.5 and 3.6], for example.…”

mentioning

confidence: 69%

“…Borsuk-Ulam-type theorem for configuration spaces Theorem 1.10 was contained in [16], where its proof was sketched, based on previously known facts. In fact, the most important cases of this theorem were previously known.…”

mentioning

confidence: 99%

“…While Vassiliev's proof uses Euler class considerations, it seems hard to find a satisfying reference for Poincaré duality with twisted coefficients in the noncompact case. Hence we provide a version of the proof from [16] that bypasses the Euler class and is based just on homology with the usual transversality argument. We provide a proof with compact support homology, then we include a variation that assumes only standard singular homology.…”

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confidence: 99%

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Convex equipartitions: the spicy chicken theorem

2013

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Abstract. We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in R d , there exists a partition of K into n convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem for convex sets in the model spaces of constant curvature.

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confidence: 69%

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Convex equipartitions: the spicy chicken theorem

2013

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“…More general index calculations for configuration spaces can be found in Karasev [Kar09a] and Blagojević-Lück-Ziegler [BLZ12].…”

Section: Proof Of the Parametrized Volovikov Theoremmentioning

confidence: 99%

A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

Matschke

2014

J. Topol. Anal.

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We present a parametrized version of Volovikov's powerful Borsuk-Ulam-BourginYang type theorem, based on a new Fadell-Husseini type ideal-valued index of G-bundles which makes computations easy.As an application we provide a parametrized version of the following waist of the sphere theorem due to Gromov, Memarian, and Karasev-Volovikov: Any map f from an n-sphere to a k-manifold (n ≥ k) has a preimage f −1 (z) whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator S n−k (if n = k we further need that f has even degree).

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“…In this paper we generally use purely geometric methods, based on the subadditivity, the dimension upper bound, and other properties of the genus. The reader may compare this approach with the lower bounds for the genus (actually for the number i G ) in [12], made with computations in cohomology and spectral sequences.…”

Section: It Is Clear Thatmentioning

confidence: 99%

Configuration-like spaces and coincidences of maps on orbits

Карасев

Volovikov²

2011

Algebr. Geom. Topol.

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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

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The genus and the category of configuration spaces

Cited by 9 publications

References 27 publications

Convex equipartitions: the spicy chicken theorem

Convex equipartitions: the spicy chicken theorem

A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

Configuration-like spaces and coincidences of maps on orbits

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