The positive fundamental group of ${\rm Sp}(2n)$

TL;DR

This paper investigates the homotopy classes of positive loops in symplectic groups, establishing conditions for homotopy equivalence and extending known results to higher-dimensional symplectic manifolds.

Contribution

It proves that positive loops are homotopic if and only if they are homotopic through positive loops, extending previous results to higher dimensions.

Findings

Positive loops in ${ m Sp}(2n)$ are homotopic iff homotopic through positive loops.

Results of McDuff and Chance are extended to higher-dimensional symplectic manifolds.

The work removes dimensional restrictions in the study of positive loops.

Abstract

In this paper, we examine the homotopy classes of positive loops in ${\rm Sp}(2n)$. We demonstrate that two positive loops are homotopic if and only if they are homotopic through positive loops. As consequences, we can extend several results of McDuff \cite{McD} and Chance \cite{Cha} to higher dimensional symplectic manifolds without dimensional restrictions.