# Effective topological complexity of orientable-surface groups

## TL;DR

This paper uses rewriting systems to analyze the cohomology of surface groups, providing bounds on the effective topological complexity of orientable surfaces with antipodal symmetry, closely related to non-orientable surface complexity.

## Contribution

It introduces a novel method to compute cup-products in cohomology of surface groups, establishing bounds on effective topological complexity with respect to involutions.

## Key findings

- Detected non-trivial obstructions to bounding effective topological complexity.
- Provided estimates close to optimal for orientable surfaces.
- Linked effective topological complexity to the regular topological complexity of non-orientable surfaces.

## Abstract

We use rewriting systems to spell out cup-products in the (twisted) cohomology groups of a product of surface groups. This allows us to detect a non-trivial obstruction bounding from below the effective topological complexity of an orientable surface with respect to its antipodal involution. Our estimates are at most one unit from being optimal, and are closely related to the (regular) topological complexity of non-orientable surfaces.