On higher analogs of topological complexity

Farber introduced a notion of topological complexity TC ( X ) that is related to robotics. Here we introduce a series of numerical invariants TC n ( X ) , n = 2 , 3 , …  , such that TC 2 ( X ) = TC ( X ) and TC n ( X ) ⩽ TC n + 1 ( X ) . For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.