[GAP Forum] Compute Homotopy Groups with GAP?

Jeffrey Rolland rollandj at uwm.edu
Fri Feb 8 17:39:48 GMT 2008

Hello, all!

I am a newbie to GAP.

I have used the module HAP to compute group homologies for a certain 
group - the binary icosahedral group, P, aka SL(2,5) and the Poincare 
group - to great success. HAP correctly tells me H_n(P) is Z_120 for n 
congruent to 3 mod 4.

I am interested, however, not in the space BP, but the space BP+, the 
result of applying Quillen's Plus Construction to BP with respect to P. 
Of course, this makes BP+ simply connected; moreover, it leaves the 
homology groups of BP+ unchanged from those of BP.

But now, by Hurewitz Theorem, pi_3(BP+) = H_3(BP+) = H_3(BP) = Z_120. 
So, BP+ is no longer aspherical.

What I would like is a way of computing pi_4(BP+) through pi_8(BP+) - 
without "building my own nails" (as one of the professors at my school 
accuses me of doing all to often), that is, by getting GAP or some other 
source to do it for me (I have bigger fish to fry).

If this beyond GAP's present capabilities, knowing that would be a plus.

If anyone knows of a reference for this in the literature, I would be 
eternally grateful.

Thank you in advance for any assistance you can provide.

Jeffrey Rolland
<rollandj at uwm.edu>

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