The intersection form on the homology of a surface acted on by a finite group

TL;DR

This paper proves the existence of a G-invariant Lagrangian subspace in the first homology of a surface under a free finite group action, revealing structural insights into the homology influenced by symmetry.

Contribution

It establishes the existence of a G-invariant Lagrangian subspace in the homology of surfaces with free finite group actions, a new result in surface topology.

Findings

Existence of G-invariant Lagrangian subspace proven

Structural understanding of homology under group actions enhanced

Implications for surface topology and symmetry studied

Abstract

Let G be a finite group acting freely on a compact oriented surface S by homeomorphisms preserving the orientation. Then, there exists a G-invariant Lagrangian subspace in the first homology group of S.