Spaces with high topological complexity

Abstract

Abstract. By a formula of Farber [7, Theorem 5.2] the topological complexity TC(X) of a (p − 1)-connected, m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. There are also various lower estimates for TC(X) such as the nilpotency of the ring H * (X × X, ∆(X)), and the weak and stable topological compexity wTC(X) and σTC(X) (see [10]). In general the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces whose topological complexity is close to the maximal value given by Farber's formula and show that in these cases the gap between the lower and upper bounds is narrow and that TC(X) often coincides with the lower bounds.

Monotonicity of the Schwarz genus

“…Spaces X for which TC(X) is equal or close to that upper bound were called spaces with high topological complexity in [5]. Examples include all closed surfaces with the exception of the torus, all complex and quaternionic projective spaces, most 3-dimensional lens spaces, configuration spaces, and many other (cf.…”

Monotonicity of the Schwarz genus

“…Spaces X for which TC(X) is equal or close to that upper bound were called spaces with high topological complexity [5]. Examples include all closed surfaces with the exception of the torus, all complex and quaternionic projective spaces, most 3-dimensional lens spaces, configuration spaces, and many other (cf.…”