Exponential local p-torsion growth conjectures

Establish the exponential growth of p-primary torsion in the homotopy groups of spaces at the primes relevant to Conjecture 1.6 of Huang and Tanré and Conjecture 1.7 of Huang, thereby proving these partial strengthenings of the hyperbolic direction of Moore’s Conjecture.

Background

The authors state that their hyperbolicity result for the skeleton of a Poincaré duality complex provides further evidence for two external conjectures, cited as Conjecture 1.6 in Huang and Tanré and Conjecture 1.7 in Huang. These conjectures strengthen the hyperbolic direction of Moore’s Conjecture by requiring more than infinite homotopy exponent.

Specifically, the paper explains that the conjectures seek exponential growth of p-torsion in homotopy groups at the relevant primes. The paper proves such exponential growth only away from a finite exceptional set, so the cited conjectures remain unresolved in their stated generality.