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

TL;DR

This paper demonstrates exponential growth in the rational homology of free loop spaces and in torsion homotopy groups, revealing new connections and generalizations in algebraic topology.

Contribution

It generalizes previous results on homology growth and introduces new spaces with exponential torsion homotopy group growth, supporting a conjectured link between rational and torsion homotopy groups.

Findings

Rational homology groups of free loop spaces grow exponentially.

Constructs new spaces with exponential p-torsion in homotopy groups.

Supports conjecture linking rational and p-torsion homotopy growth.

Abstract

Using integral methods we recover and generalize some results by F\'{e}lix, 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 $p$.