Equivariant topology of configuration spaces

Blagojević, Pavle V. M.; Lück, Wolfgang; Ziegler, Günter M.

doi:10.1112/jtopol/jtv002

Cited by 17 publications

(18 citation statements)

References 70 publications

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“…In the context of the topological Tverberg problem, index computations played a crucial role (see [15], [17], [6], [8]). For their uses in other geometric contexts see [5], [1], [10], [16], [13]. Our index computations yield the following conclusions for geometric problems which are inspired from Kakutani's Theorem and it's generalizations (see Corollary 5.5 and Theorems 5.6, 5.7).…”

Section: Introductionmentioning

confidence: 66%

The index of certain Stiefel manifolds

Basu,

Kundu

2021

Preprint

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Section: Introductionmentioning

confidence: 66%

The index of certain Stiefel manifolds

Basu,

Kundu

2021

Preprint

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“…where ∆ : X −→ X × X is the diagonal embedding and ∆ * the corresponding homomorphism in homology. From Proposition 2.3 (4)- (5) we have that…”

Section: Some Ingredientsmentioning

confidence: 92%

On Complex Highly Regular Embeddings and the Extended Vassiliev Conjecture:

Blagojević

Cohen

Lück

et al. 2015

Int Math Res Notices

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A continuous map C d −→ C N is a complex k-regular embedding if any k pairwise distinct points in C d are mapped by f into k complex linearly independent vectors in C N . The existence of such maps is closely connected with classical problems of algebraic/differential topology, such as embedding/immersion problems. Our central result on complex k-regular embeddings extends results of Cohen & Handel (1978), Chisholm (1979) and Blagojević, Lück & Ziegler (2013) on real k-regular embeddings. We give the following lower bounds for the existence of complex k-regular embeddingsLet p be an odd prime, k ≥ 1 and d = p t for t ≥ 1. If there exists a complex k-regular embeddingHere αp(k) denotes the sum of coefficients in the p-adic expansion of k. These lower bounds are obtained by modifying the framework of Cohen & Handel (1978) and a study of Chern classes of complex regular representations. As a main technical result we establish for this an extended Vassiliev conjecture, the following upper bound for the height of the cohomology of an unordered configuration space:If d ≥ 2 and k ≥ 2 are integers, and p is an odd prime. ThenFurthermore, we give similar lower bounds for the existence of complex ℓ-skew embeddings C d −→ C N , for which we require that the images of the tangent spaces at any ℓ distinct points are skew complex affine subspaces of C N . In addition we give improved lower bounds for the Lusternik-Schnirelmann category of F (C d , k)/S k as well as for the sectional category of the covering Some ingredientsIn this section we collect facts that will be used in the rest of the paper, in particular in the proofs of Theorems 3.1 and 3.2. For this we • recall the relationship between colim k≥0 F (R d , k)/S and Ω d 0 S d , Section 2.1; • summarize relevant properties of the Araki-Kudo-Dyer-Lashof homology operations, Section 2.2; • introduce the space Q(X) := colim d≥0 Ω d Σ d (X) and describe its homology in the language of Araki-Kudo-Dyer-Lashof operations, Section 2.3;

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“…In the case when

t = 1

the claim that

{prefixIndex}_{Z / r} false(prefixConf ({double-struckR}^{d + 1}, r); F_{r} false) = H^{⩾ d (r - 1) + 1} false(double-struckZ / r; F_{r} false)

is a content of [, Theorem 6.1]. This result can also be deduced from the Vanishing theorem of Cohen [, Theorem 8.2, p. 268].…”

Section: Proof Of Theorem and Theoremmentioning

confidence: 99%

“…For that we use the Künneth theorem for joins and have that

\begin{matrix} left {\overset{H}{true}}^{i + t - 1} false(prefixConf {false(R^{d + 1}, r false)}^{* t}; F_{r} false) ≅ \sum_{a_{1} + \dots + a_{t} = i} {\overset{H}{true}}^{a_{1}} false(prefixConf ({double-struckR}^{d + 1}, r); F_{r} false) \otimes \dots \otimes {\overset{H}{true}}^{a_{t}} false(prefixConf ({double-struckR}^{d + 1}, r); F_{r} false), \end{matrix}

where the action of

Z / r

on the tensor product is the diagonal action. The cohomology of the configuration space

H^{*} false(prefixConf ({double-struckR}^{d + 1}, r); F_{r} false)

, as a

{double-struckF}_{r} false[double-struckZ / r false]

‐module, was described in [, Proof of Theorem 8.5; , Theorem 3.1, Corollary 6.2]. In summary,

H^{q} false(prefixConf ({double-struckR}^{d + 1}, r); F_{r} false) = \{\begin{matrix} left F_{r}, & left for q = 0, \\ left F_{r} {false[double-struckZ / r false]}^{a_{q}}, & left for q = d j where 1 ⩽ j ⩽ r ... \end{matrix}

…”

Section: Proof Of Theorem and Theoremmentioning

confidence: 99%

Thieves can make sandwiches

Blagojević

Soberón

2017

Bull. London Math. Soc.

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We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of m measures in R d among r thieves using roughly mr/d convex pieces, even in the cases when m is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of R d into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces.

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Equivariant topology of configuration spaces

Cited by 17 publications

References 70 publications

The index of certain Stiefel manifolds

The index of certain Stiefel manifolds

On Complex Highly Regular Embeddings and the Extended Vassiliev Conjecture:

Thieves can make sandwiches

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