Growth of homotopy groups

Abstract

This thesis studies the limiting behaviour of the torsion in the homotopy groups π_n(X) of a space X as n → ∞. It is a ‘three paper thesis’, the main body of which consists of the following papers:
[1] G. Boyde, Bounding size of homotopy groups of spheres. Proceedings of the Edinburgh Mathematical Society, 63(4):1100–1105, 2020.
[2] G. Boyde, p-hyperbolicity of homotopy groups via K-theory, preprint, available at arXiv:2101.04591 [math.AT], 2021.
[3] G. Boyde, Z/p^r-hyperbolicity via homology, preprint, available at arXiv:2106.03516 [math.AT], 2021.

In [1], we improve on the best known bound for the size of the homotopy group π_q(Sⁿ), using the combinatorics of the EHP sequence.

In [2], we study Huang and Wu’s p- and Z/p^r -hyperbolicity for spaces related to the wedge of two spheres Sⁿ ∨ S^m. We show that Sⁿ v S^mSn ∨ S m Sn ∨ S m Sⁿ ∨ S^{mSn V Sm}Sⁿ ∨ S^m. Sⁿ ∨ S^m.is Z/p^r -hyperbolic for all primes p and all r ∈ N, which implies that various spaces containing Sⁿ ∨ S^m as a retract are similarly hyperbolic. We then prove a K-theory criterion for p-hyperbolicity of a finite suspension ΣX, and deduce some examples.
In [3], we study p- and Z/p^r -hyperbolicity for spaces related to the Moore space Pⁿ (p^r). When p^s≠2, we show that Pⁿ (p^r) is Z/p^s -hyperbolic for s ≤ r. Combined with Huang and Wu’s work, and Neisendorfer’s results on homotopy exponents, this completely resolves the question of when such a Moore space is Z/p^s -hyperbolic for p ≥ 5. We then prove a homological criterion for Z/p^r -hyperbolicity of a space X, and deduce some examples.

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ORCID for Stephen Theriault: orcid.org/0000-0002-7729-5527

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