Spaces with high topological complexity

TL;DR

This paper investigates spaces with topological complexity near the maximum predicted by Farber's formula, showing that in such cases the bounds are tight and TC(X) often matches the lower estimates.

Contribution

It demonstrates that for spaces with high topological complexity, the upper and lower bounds are close, and TC(X) frequently equals the lower bounds, clarifying the structure of these spaces.

Findings

Spaces with high topological complexity have narrow bounds.

TC(X) often coincides with lower bounds in these spaces.

The gap between bounds can be arbitrarily large in general.

Abstract

By a formula of Farber 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 $H^*(X\times X,\Delta(X))$, and the weak and stable topological compexity wTC(X) and $\sigma\TC(X)$. 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.