Abstract. We present a new approach to an equivariant version of Farber's topological complexity called invariant topological complexity. It seems that the presented approach is more adequate for the analysis of impact of a symmetry on a motion planning algorithm than the one introduced and studied by Colman and Grant. We show many bounds for the invariant topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the topological complexity of the orbit space. We define the Whitehead version of it.

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A necklace splitting theorem of Goldberg and West [5] asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erdős on strongly nonrepetitive sequences, Alon et al. [3] proved that there is a (t + 3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R d such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k ≥ (t + 4) d − (t + 3) d + (t + 2) d − 2 d + d(t + 2) + 3. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville andŽivaljević [7]. The question how many axes aligned hyperplanes are needed for a fair splitting of a d-dimensional k-colored cube remains open.