On free loop spaces of toric spaces

TL;DR

This paper investigates the growth patterns of the rational homology of free loop spaces in toric spaces, linking fan structures to exponential growth and implications for periodic geodesics.

Contribution

It establishes a connection between the fan structure of toric manifolds and the exponential growth of their free loop space homology.

Findings

Hilbert-Poincaré series growth is exponential for certain toric manifolds

Fan structure determines the homological growth rate

Applications to existence of infinitely many periodic geodesics

Abstract

Growth of the Hilbert-Poincar\"e series for the rational homology of the free loop space of a toric space is addressed. In case the toric space is a manifold, the structure of the fan dictates whether the Hilbert-Poincar\"e series has exponential growth. Applications are made to the existence of infinitely many geometrically distinct periodic geodesics.