Miguel A. Xicoténcatl

On orbit configuration spaces associated to the Gaussian integers: Homotopy and homology groups

Abstract The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from the standard integral lattice Z + i Z in the complex numbers. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to the Lie algebra obtained from the higher homotopy groups of “higher dimensional arrangements” modulo torsion. The resulting Lie algebras are similar to those studied by T. Kohno associated to elliptic KZ systems [Topology Appl. 78 (1997) 79–94]. A question about the generality of this behavior is posed.

Topological complexity of motion planning in projective product spaces

We study Farber’s topological complexity (TC) of Davis’ projective product spaces (PPS’s). We show that, in many nontrivial instances, the TC of PPS’s coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in marked contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of nonsimply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension at most 2, and now in this work for infinite families of PPS’s. We discuss general bounds for the TC (and the Lusternik‐Schnirelmann category) of PPS’s, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS’s through a generalized concept of axial maps or, alternatively (in an appendix), nonsingular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS’s. 55M30, 57R42; 68T40

Product decomposition of loop spaces of configuration spaces

Abstract The configuration space of k points in R P n , C P n and H P n are studied. In this article we show that after looping once, they split as a product of spheres and the loop space of certain orbit configuration spaces.