Symmetrized topological complexity

We present upper and lower bounds for symmetrized topological complexity TC Sigma(X) in the sense of Basabe-Gonzalez-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square SP2(X). We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Using these results, we calculate the symmetrized topological complexity of all odd spheres.