Boyde, Guy
(2021)
Growth of homotopy groups.
*University of Southampton, Doctoral Thesis*, 151pp.

## 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*, preprint, available at arXiv:2106.03516 [math.AT], 2021.

^{r}-hyperbolicity via homologyIn [1], we improve on the best known bound for the size of the homotopy group

*π*, using the combinatorics of the EHP sequence.

_{q}(S^{n})In [2], we study Huang and Wu’s

*p-*and

*Z/p*-hyperbolicity for spaces related to the wedge of two spheres

^{r}*S*. We show that

^{n}∨ S^{m}*S*Sn ∨ S m Sn ∨ S m

^{n}v S^{m}*S*

^{n}∨ S^{mSn V Sm}*S*.

^{n}∨ S^{m}*S*.is

^{n}∨ S^{m}*Z/p*-hyperbolic for all primes

^{r}*p*and all

*r ∈ N*, which implies that various spaces containing S

^{n}∨ 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*-hyperbolicity for spaces related to the Moore space P

^{r}^{n}(p

^{r}). When

*p*≠2, we show that

^{s}*P*is

^{n}(p^{r})*Z/p*-hyperbolic for

^{s}*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*-hyperbolic for

^{s}*p ≥*5. We then prove a homological criterion for

*Z/p*

^{r }-hyperbolicity of a space X, and deduce some examples.

**Thesis_with_corrections - Version of Record**

**Permission to deposit thesis - form - Version of Record**

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