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 TCn(X), n=2,3,… , such that TC2(X)=TC(X) and TCn(X)⩽TCn+1(X). For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.