Higher topological complexity and homotopy dimension of configuration spaces on spheres

Details Higher topological complexity and homotopy dimension of configuration spaces on spheres Higher topological complexity and homotopy dimension of configuration spaces on spheres

2010-09-09

arxiv.org/abs/1009.1851v4

Mathematics

Algebraic Topology

Corrected typos. Version submitted for publication. 50 pages

Scientific paper

Yu. Rudyak has recently extended Farber's notion of topological complexity by defining, for n>1, the n-th topological complexity TC_n(X) of a path-connected space X---Farber's original notion is recovered for n=2. In this paper we develop further the properties of this extended concept, relating it to the Lusternik-Schnirelmann category of cartesian powers of X, as well as to the cup-length of the diagonal embedding of X into X^n. We compute the numerical values of TC_n for products of spheres, closed 1-connected symplectic manifolds (e.g. complex projective spaces), and quaternionic projective spaces. We explore the symmetrized version of the concept (TC^S_n(X)) and introduce a new symmetrization (STC_n(X)) which is a homotopy invariant of X. We obtain a (conjecturally sharp) upper bound for TC^S_n(X) when X is a sphere. This is attained by introducing and studying the idea of cellular stratified spaces, a new concept that allows us to import techniques from the theory of hyperplane arrangements in order to construct finite CW complexes of the lowest possible dimension modelling, up to equivariant homotopy, configuration spaces of ordered distinct points on spheres---our models are in fact simplicial complexes. In particular, we show that the configuration space of n points (either ordered or unordered) in the k-dimensional sphere has homotopy dimension (n-1)(k-1)+1.

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