Spaces with high topological complexity

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.