# Unstable homotopy groups of spheres

From the bottom of this page you can download Mathematica programs that know many results about the unstable homotopy groups of spheres up to the 19-stem. Most of the information is taken from Toda's book "Composition methods in homotopy groups of spheres"; 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 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:

SpherePi[14,2]
 Z2η5°ε6⊕Z2μ5⊕Z2ν5°ν8°ν11
HomotopySet[Sphere[7],Sphere[4]]
 Zν4⊕Z4ν′
GroupType[SpherePi[19,10]]
 Z2⊕Z2⊕Z2⊕Z
ShowMap[Hopf,8,4]
 π84
 →
 π87
 Z2ν4°η7⊕Z2(Σν′)°η7
 Z2η7
 ν4°η7
 →
 η7
 (Σν′)°η7
 →
 0
ShowMap[Sigma,To[sigmaprime]]
 π136
 →
 π147
 Z4σ"
 Z8σ′
 σ"
 →
 2 σ′
Explain[o[nuprime,nu[7]]]
 ν′°ν7 → 0
 This takes place in the trivial group π10S3 - see Toda′s Proposition 5.15

## Files

To use the program:
• Download at least the files Toda.m and TeXToMath.m, and save them in a directory (say /home/neil/toda).
• Start up Mathematica, then enter SetDirectory["/home/neil/toda"] (or equivalent).
• Enter <<Toda.m to read in the program.
• You can now enter questions using the syntax described in Toda.pdf
Here are the files:
• Toda.pdf - more detailed documentation
• Toda.m - the core of the program
• TeXToMath.m - used by Toda.m to convert help messages from LaTeX to Mathematica encoding. (This is rather crude, and will only handle a limited range of LaTeX constructs.)
• LambdaAlgebra.m - 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.m - some auxiliary functions for checking the completeness and consistency of the results encoded in Toda.m. (There are currently numerous failures of completeness, but none of consistency.)
• LambdaData.m - 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.