Braids, configuration spaces and applications to multivalued maps

Abstract

Given a space X where a group G acts freely, consider the n-coordinated configuration space FGn(X), consisting of the n-tuples (x1,...,xn) in X^n such that the intersection between Gxi and Gxj is empty for every i different from j. The relevance of this space is well known. For example, if G is the trivial group, the fundamental group is the n-string braid group of X. Besides that, many questions, such as fixed point and coincidence ones, can be codificated using these spaces. One intends to compute the homotopy type of the fiber of the inclusion map. Many cases are known, but there is still much to be done. Non-elementary homotopy techniques may be useful. To see this, besides the well known results, one may check references [1], [2] and [3]. The space FGn(X) has been applied to the determination of the homotopy fiber for X=G and the computation of its fundamental group has risen as a generalization of the braid groups. So far, the only calculation made is the one where X is the cylinder and G=Z2, which one can find in [1]. For all pairs (X,G), where X is a surface and G a finite group, one wants to determine pi_1(FGn(X)). The classical group theory techniques, such as short exact sequences, presentation of a group given by a short exact sequence, plus surface geometry have a relevant role in the techniques used.By the use of configuration spaces, questions from Nielsen fixed point theory for multivalued maps are codificated. This is about determining formulas and tools which can allow the calculation of the Nielsen number of multivalued maps, specially generalizing the known cases for singlevaluated maps. Besides that, one wants to study fixed point questions related to the fixed point property, as well as the characterization of the fixed point free homotopy classes. The recent and known results can be seen in [4], [5] and [6].Finally, a better understanding of the braid groups uses, at some point, the analysis of its subgroups. This problem has a long story, beginning with the initial intention of knowing whether the quaternionic group of order 8, Q8, was a subgroup of the sphere braid group. In that sense, one considers the problem of determining the conjugacy classes of the finite subgroups of the braid groups of the sphere and the projective plane. We will consider the case of the projective plan, as well as infinite subgroups, specially the virtually cyclic ones. These groups are intimally related to the groups acting on the homotopy spheres and this connection has shown fruitful. Results which show the development of these questions can be seen in [7, 8, 9]. All the references above are listed in the research project PDF. (AU)