On equivariant and invariant topological complexity of smooth ℤ/_{𝕡}-spheres

Abstract

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.

Effective topological complexity of spaces with symmetries

“…As a result, they can be arbitrarily larger than T C(X). (See [2,Proposition 4.5] for what we believe to be an interesting family of examples of this sort of behaviour.) In particular, if X G is disconnected, the invariants are infinite, a phenomenon for which we do not know a satisfactory explanation from the point of view of robotics.…”

Equivariant topological complexities

“…Remark 4.22. B laszczyk and Kaluba in [8] investigate the equivariant and invariant topological complexities of C p -spheres, where C p is the cyclic group of order p, a prime. They show that if the action is linear or semi-linear, then…”

“…Problem 7.5 (Z. B laszczyk, M. Kaluba [8]). Is it true that if S n is a smooth C psphere with non-empty and path connected fixed point set, then S n is equivariantly equivalent to a linear C p -sphere if and only if TC Cp (S n ) ≤ 2?…”