Equivariant topology of configuration spaces

2010

In this paper we deal with the action of the symmetric group on the cohomology of the configuration space C n ðdÞ of n points in R d. This topic has been studied by several authors and it is well-known that for d even H Ã ðC n ðdÞ; CÞ G 2 Ind Sn S2 1 while, for d odd, H Ã ðC n ðdÞ; CÞ G CS n. On the cohomology algebra H Ã ðC n ðdÞ; CÞ there is, in addition to the natural S n-action, an extended action of S nþ1 ; this was shown for the case when d is even by Mathieu, Robinson and Whitehouse and the second author using three di¤erent methods. For the case when d is odd it was shown by Mathieu (anyway we will give an elementary algebraic construction of the extended action for this case). The purpose of this article is to present some results that can be obtained, in an elementary way, exploiting the interplay between the extended action and the standard action. Among these we will recall a quick proof for the formula cited above for the case when d is even and show how to extend this proof to the case when d is odd. We will also show how to locate among the homogeneous components of the graded algebra H Ã ðC n ðdÞ; CÞ the copies of the standard, sign and standard tensor sign representations and we will give explicit formulas for both the extended and the canonical actions on the low-degree cohomology modules.

Configuration Spaces, 2012

In the first part we review some topological and algebraic aspects in the theory of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated). In the following parts, among other things, we compute the Schwartz genus of the covering associated to the orbit space for all affine Artin groups. We also give a partial computation of the cohomology of the braid group with non-abelian coefficients coming from geometric representations. We introduce an interesting class of "sheaves over posets", which we call "weighted sheaves over posets", and use them for explicit computations.

Topology and its Applications, 2015

Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space Conf(RP m , k) of k ordered points in the m-dimensional real projective space RP m. The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2 k-fold covering of Conf(RP m , k). This implies that, for odd m, the Leray spectral sequence for the inclusion Conf(RP m , k) ⊂ (RP m) k collapses after its first non-trivial differential, just as it does when RP m is replaced by a complex projective variety. The method also allows us to handle the R-cohomology ring of the configuration space of k ordered points in the punctured manifold RP m − ⋆. Lastly, we compute the Lusternik-Schnirelmann category and all of the higher topological complexities of some of the auxiliary orbit configuration spaces.

Geometry and Topology Monographs, 2008

We discuss various aspects of "braid spaces" or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomological dimension of braid spaces. This is related to some sharp and useful connectivity bounds that we establish for the reduced symmetric products of any simplicial complex. Our methods are geometric and exploit a dual version of configuration spaces given in terms of truncated symmetric products. We finally refine and then apply a theorem of McDuff on the homological connectivity of a map from braid spaces to some spaces of "vector fields". 55R80; 55S15, 18G20 To Fred Cohen on his 60th birthday

Following ideas of Graeme Segal we construct an equivariant configuration space that is a representation space of equivariant connective K-homology for group actions of finite groups. We describe explicitly the homology with complex coefficients for the fixed points of this configuration space as a Hopf algebra. As a consequence of this work we obtain new models of representing spaces for equivariant connective K-theory.

2011

Abstract. The main thrust of these notes is 3-fold: (1) An analysis of certain K(π,1)’s that arise from the connections between configuration spaces, braid groups, and mapping class groups, (2) a function space interpretation of these results, and (3) a homological analysis of the cohomology of some of these groups for genus zero, one, and two surfaces possibly with marked points, as well as the cohomology of certain associated function spaces. An example of the type of results given here is an analysis of the space k particles moving on a punctured torus up to equivalence by the natural SL(2, Z) action. 1.

2004

We study the combinatorics and topology of general arrangements of subspaces of the form D + SP (X) in symmetric products SP (X) where D ∈ SP (X). Symmetric products SP (X) := X/Sm, also known as the spaces of effective “divisors” of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [8] and [22] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [35] and [38], we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP (Mg,k), where Mg,k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative (m + k, m)-groups [33]. 1 Arrangements of symmetric products The study of homotopy types of...

Cornell University - arXiv, 2022

In this follow-up to [16], we continue developing the notion of a lego category and its many applications to stratifiable spaces and the computation of their Grothendieck classes. We illustrate the effectiveness of this construction by giving very short derivations of the class of a quotient by the "stratified action" of a discrete group [1], the class of a crystallographic quotient, the class of both a polyhedral product and a polyhedral (or simplicial) configuration space [8], the class of a permutation product [19] and, foremost, the class of spaces of 0-cycles [11].

2003

We study the combinatorics and topology of general arrangements of subspaces of the form D + SP (X) in symmetric products SP (X) where D ∈ SP (X). Symmetric products SP (X) := X/Sm, also known as the spaces of effective “divisors” of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [7] and [21] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [34] and [37], we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP (Mg,k), where Mg,k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative (m + k, m)-groups [32]. 1 Arrangements of symmetric products The study of homotopy types of...

Inventiones Mathematicae, 1982