Symmetrized topological complexity

We present upper and lower bounds for symmetrized topological complexity $TC^\Sigma(X)$ in the sense of Basabe-Gonz\'alez-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square $SP^2(X)$. 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.