Contractible configuration spaces

Journal of Topology, 2015

We study the Fadell-Husseini index of the configuration space F (R d , n) with respect to various subgroups of the symmetric group Sn. For p prime and k 1, we compute Index Z/p (F (R d , p); Fp) and partially describe Index (Z/p) k (F (R d , p k); Fp). In this process, we obtain results of independent interest, including: (1) an extended equivariant Goresky-MacPherson formula, (2) a complete description of the top homology of the partition lattice Πp as an Fp[Zp]-module, and (3) a generalized Dold theorem for elementary abelian groups. The results on the Fadell-Husseini index yield a new proof of the Nandakumar and Ramana Rao conjecture for primes. For n = p k a prime power, we compute the Lusternik-Schnirelmann category cat(F (R d , n)/Sn) = (d − 1)(n − 1). Moreover, we extend coincidence results related to the Borsuk-Ulam theorem, as obtained by Cohen and Connett, Cohen and Lusk, and Karasev and Volovikov.

European Journal of Combinatorics, 2009

We show that topological (n 4 ) point-line configurations exist for all n ≥ 17. It has been proved earlier that they do not exist for n ≤ 16.

2018

Homological stability for unordered configuration spaces of connected manifolds was discovered by Th. Church and extended by O. Randal-Williams and B. Knudsen: $H_i(C_k(M);\mathbb{Q})$ is constant for $k\geq f(i)$. We characterize the manifolds satisfying strong stability: $H^*(C_k(M);\mathbb{Q})$ is constant for $k\gg 0$. We give few examples of manifolds whose top Betti numbers are stable after a shift of degree.

Pacific Journal of Mathematics, 1983

Let F(R n , k) be the configuration space of ordered sets of k distinct points in R". F(R n , k) is acted upon freely by the symmetric group on k letters, Σ k. In this paper we calculate the order of the vector bundles {",*: F(R", k) X Σt R k-F(R", k)/Σ k. Applications to the study of iterated loop spaces of spheres are also discussed.

Geometry and Topology of Caustics – Caustics '02, 2003

We discuss some approaches to the topological study of real quadratic mappings. Two effective methods of computing the Euler characteristics of fibers are presented which enable one to obtain comprehensive results for quadratic mappings with two-dimensional fibers. As an illustration we obtain a complete topological classification of configuration spaces of planar pentagons.

Topology and its Applications, 2002

Let F be a family of convex sets in R n and let T m (F ) be the space of m-transversals to F as subspace of the Grassmannian manifold. The purpose of this paper is to study the topology of T m (F ) through the polyhedron of configurations of (r + 1) points in R n . This configuration space has a natural polyhedral structure with faces corresponding to what has been called order types. In particular, if r = m + 1 and T m−1 (F ) is nonempty, we prove that the homotopy type of T m (F ) is ruled by the set of all possible order types achieved by the m-transversals of F . We shall also prove that the set of all m-transversals that intersect F with a prescribed order type is a contractible space. (J. Bracho), [email protected] (L. Montejano), [email protected] (D. Oliveros). 0166-8641/02/$ -see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 -8 6 4 1 ( 0 1 ) 0 0 0 1 1 -6

Journal of Mathematical Sciences, 2007

The structure set of a given manifold fits into a surgery exact sequence, which is the main tool for classification of manifolds. In the present paper, we describe relations between various structure sets and groups of obstructions which naturally arise for triples of manifolds. The main results are given by commutative braids and diagrams of exact sequences.

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily C 1-small variation of the initial submanifold, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide attractive proofs of the existence of a number of "special inscribed configurations" inside families of spheres embedded in R n using differential topology. For instance, there is a C 1dense family of smooth embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a C 1-dense family of smooth embedded (n − 1)-spheres in R n where each sphere has a family of inscribed regular n-simplices with the homology of O(n).

arXiv: Group Theory, 2013

Soit F i h (k, n) le ième espace de configuration ordonné de tous les points distincts H 1 ,. .. , H h dans la Grassmannienne Gr(k, n) de sousespaces de dimension k de C n , dont la somme est un sous-espace de dimension i. Nous prouvons que F i h (k, n) est (si non vide) une sousvariété complexe de Gr(k, n) h de dimension i(n − i) + hk(i − k) et son groupe fondamental est trivial si i = min(n, hk), hk = n et n > 2 et egal au groupe de tresses de la sphère CP 1 si n = 2. Finalement, nous calculons le groupe fondamental dans le cas particulier des arrangements d'hyperplans, c'està dire k = n − 1.

Proceedings of the American Mathematical Society, 1996

We obtain a general coincidence theorem for multifunctions in very large classes defined on contractible spaces. Our theorem generalizes a recent result due to Tarafdar and Yuan (1994) and many other earlier works including the Fan-Browder fixed point theorem.