# Symmetrized topological complexity

## TL;DR

This paper investigates symmetrized topological complexity, providing bounds and exact calculations for odd spheres, and shows its equivalence to the monoidal version using algebraic topology tools.

## Contribution

It introduces bounds for $TC^
abla$ using equivariant obstruction theory and cohomology, and proves the equivalence with the monoidal version, with explicit calculations for odd spheres.

## Key findings

- Upper bounds from equivariant obstruction theory
- Lower bounds from cohomology of symmetric square
- Symmetrized topological complexity equals its monoidal version

## Abstract

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.