$p$-hyperbolicity of homotopy groups via $K$-theory

Abstract

We show that S n ∨ S m is Z/p r -hyperbolic for all primes p and all r ∈ N, provided n, m ≥ 2, and consequently that various spaces containing S n ∨ S m as a p-local retract are Z/p r -hyperbolic. We then give a K-theory criterion for a suspension ΣX to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian ΣGr k,n is p-hyperbolic for all odd primes p when n ≥ 3 and 0 < k < n. We obtain similar results for some related spaces. * (S * ), V ) inherits the structure of a non-negatively graded associative algebra.Let ν : A → ΩΣX be a map. There is a compositionwhere the first map is the natural map which is the identity on K TF * (A) and satisfies [x, y] → xy − (−1) |x||y| yx, and the second map is obtained by applying the universal property of the tensor algebra to ν * . Let Φ K ν : Hom( K TF * (S * ), L( K TF * (A))) → Hom( K TF * (S * ), K TF * (ΩΣX)) be the pushforward along the above composite. It is then automatic that Φ K ν is a map of non-negatively graded Lie algebras over Z, where the structures are defined as above.We write deg :be the unique map which restricts to deg : π * (A) → Hom( K TF * (S * ), K TF * (A)) ⊂ Hom( K TF * (S * ), L( K TF * (A))) and carries brackets to brackets. The above maps are related as follows. Lemma 7.2. Let ν : A → ΩΣX, for spaces A and X having the homotopy type of finite CWcomplexes. The following diagram commutes: B(π * (A)) Φ π ν / / deg ′ π * (ΩΣX) deg Hom( K TF * (S * ), L( K TF * (A))) Φ K ν / / Hom( K TF * (S * ), K TF * (ΩΣX)).

$\mathbb{Z}/p^r$-hyperbolicity via homology

“…By Theorem 1.3 and the work of Huang-Wu [HW] and Zhu-Pan [ZP21], it follows that M is Z/p s -hyperbolic whenever p s−1 divides the order of the torsion part of H n (M ). In fact, if r ≥ 2 then ΩM contains Ω(S n ∨ S m ) as a retract, so is Z/p s -hyperbolic for all p and s by [Boy21]. Conversely, if M is not Z/p s hyperbolic for any p and s (and is not the sphere S 2n+1 ) then we must have H n (M ) ∼ = Z.…”

“…Together, Theorems 1.3 and 1.6 may be thought of as doing for Moore spaces what [Boy21] did for wedges of spheres. The main difference between the homological results of that paper and this is that the Hurewicz map is enough to detect p r -torsion in the homotopy groups of the Moore space P n (p r ).…”

“…The main difference between the homological results of that paper and this is that the Hurewicz map is enough to detect p r -torsion in the homotopy groups of the Moore space P n (p r ). In contrast, one needs more sophisticated machinery to see p r -torsion in a wedge of spheres; [Boy21] used Adams' e-invariant. This meant that the theorems of that paper had to be stated in terms of K-theory, rather than ordinary homology.…”