Unstable homotopy groups of spheres

This repository contains Mathematica programs you can download that know many results about the unstable homotopy groups of spheres up to the 19-stem. Most of the information is taken from Hiroshi Toda's book "Composition methods in homotopy groups of spheres [1]. A few additional facts are proved in an accompanying note or quoted from elsewhere. The program does not do any serious calculations for itself; it is merely a convenient way of organizing and accessing results obtained by traditional methods. One can also do some automated consistency checking to detect any errors. I have tried to use a fairly general framework so that other computations in unstable homotopy can be included later if people are interested.

This program is not really finished, but I (Neil P Strickland) think it is already useful, and I have moved on to other things, so I have decided to release it.

Here are some examples of things one can type, and the program's response:

Example 1

SpherePi[14,2]

$$\mathbb{Z}_2\eta_5^\circ\varepsilon_6 \oplus \mathbb{Z}_2\mu_5 \oplus \mathbb{Z}_2\nu_5^\circ\nu_8^\circ\nu_{11}$$

Example 2

HomotopySet[Sphere[7],Sphere[4]]

$$\mathbb{Z}\nu_4 \oplus \mathbb{Z}_4\nu'$$

Example 3

GroupType[SpherePi[19,10]]

$$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}$$

Example 4

ShowMap[Hopf,8,4]

$\pi_8^4$	$\to$	$\pi_8^7$
$\mathbb{Z}_2\nu_4^\circ\eta_7 \oplus \mathbb{Z}_2(\Sigma\nu')^\circ\eta_7$		$\mathbb{Z}_2\eta_7$
$\nu_4^\circ\eta_7$	$\to$	$\eta_7$
$(\Sigma\nu')^\circ\eta_7$	$\to$	$0$

Example 5

ShowMap[Sigma,To[sigmaprime]]

$\pi_{13}^6$	$\to$	$\pi_{14}^7$
$\mathbb{Z}_4\sigma''$		$\mathbb{Z}_8\sigma'$
$\sigma''$	$\to$	$2\sigma'$

Example 6

Explain[o[nuprime,nu[7]]]

$$\nu'^\circ\nu_7 \to 0$$

This takes place in the trivial group $\pi_{10}S^3$ — see Toda's Proposition 5.15

Using

The code currently works on Mathics3 10.0.2dev0 or later. Once that is installed:

$ git clone https://github.com/Mathics3/Mathics3-Packages-Toda toda
$ cd toda
$ mathics3
<<Toda.wl  (* This takes several minutes to load. To see loading progress use: Get["Toda.wl", Trace->True] *)

Files

Here are the files:

• Toda.pdf - more detailed documentation
• Toda.wl - the core of the program
• TeXToMath.wl - used by Toda.wl to convert help messages from LaTeX to Mathematica encoding. (This is rather crude, and will only handle a limited range of LaTeX constructs.)
• LambdaAlgebra.wl - definitions related to the lambda algebra, including lambda algebra representatives for some of Toda's generators. This program can calculate the cohomology of the lambda algebra, but it uses a very naive and inefficient algorithm, and takes a ridiculous amount of space to represent the result.
• TodaChecks.wl - some auxiliary functions for checking the completeness and consistency of the results encoded in Toda.wl. (There are currently numerous failures of completeness, but none of consistency.)
• LambdaData.wl - 10MB of data on bases for the lambda algebra that put the differential in canonical form. You probably do not want or need to download this file.

See also

https://strickland1.org/toda/

References

[1] H. Toda, Composition methods in homotopy groups of spheres, Princeton University Press, Annals of Mathematics Studies (No. 49), 1962, 193 pp. ISBN: 978-069109586.