On the intersection form of surfaces

TL;DR

This paper investigates the properties of the algebraic intersection form on the homology of surfaces, focusing on its norm relative to the stable norm induced by a Riemannian metric.

Contribution

It analyzes the interaction between the intersection form and the stable norm on surface homology, providing new insights into their relationship.

Findings

Characterization of the intersection form's norm

Relationships between intersection form and stable norm

Implications for surface topology and geometry

Abstract

Given a closed, oriented surface M, the algebraic intersection of closed curves induces a symplectic form Int(.,.) on the first homology group of M. If M is equipped with a Riemannian metric g, the first homology group of M inherits a norm, called the stable norm. We study the norm of the bilinear form Int(.,.), with respect to the stable norm.