Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

TL;DR

The paper addresses how exponential growth phenomena in the rational homology of free loop spaces relate to torsion growth in homotopy groups, proposing a robust integral-method framework to generalize prior rational results. By developing cofibration-based criteria and introducing inert/strongly inert cofibrations, it establishes multiple cases (Cases I–III) where $H_*(\mathcal{L}Y;\mathbb{Q})$ exhibits good exponential growth and connects this growth to rational hyperbolicity. It also develops an accompanying theory for torsion growth, obtaining new families where the $p$-torsion in $\pi_*(X)$ grows exponentially and Moore's Conjecture holds for all but finitely many primes, with a p-local wedge-of-spheres approximation away from a finite prime set. Overall, the work strengthens the link between rational and $p$-torsion growth, proposes a stronger Moore-type conjecture, and provides a versatile framework for predicting loop-space growth and torsion phenomena via cofibrations and Hilbert-series techniques.

Abstract

Using integral methods we recover and generalize some results by Félix, Halperin and Thomas on the growth of the rational homology groups of free loop spaces, and obtain a new family of spaces whose $p$-torsion in homotopy groups grows exponentially and satisfies Moore's Conjecture for all but finitely many primes. In view of the results, we conjecture that there should be a strong connection between exponential growth in the rational homotopy groups and the $p$-torsion homotopy groups for any prime $p$.

Theorems & Definitions (37)

• Definition 1.1
• Definition 1.2
• Conjecture 1.3: Moore
• Definition 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 2.1: Theorem 1 in FHT4
• Theorem 2.2: Theorem 3 in FHT4
• Theorem 2.3: Proposition 3.5 in BT2
• Lemma 2.4