We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant, Dranishnikov and Lubawski-Marzantowicz. In particular, it is bounded from above by Farber's topological complexity.Comment: New title; a short section with open problems included at the end of the paper. Numerous minor improvements throughout the text. Final version, to appear in Publ. Mat. 19 pages, 2 figure

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We describe a unified approach to estimating the dimension of f −1 (A) for any G-equivariant map f : X → Y and any closed G-invariant subset A ⊆ Y in terms of connectivity of X and dimension of Y , where G is either a cyclic group of order p k , a p-torus (p a prime), or a torus.

We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear Z/p-spheres have both invariants either 2 or 3 and calculate exact values in all but two cases. On the other hand, we exhibit examples which show that these invariants can be arbitrarily large in the class of smooth Z/p-spheres.2000 Mathematics Subject Classification. Primary 57S17, 57S25; Secondary 55M30.

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