#6750: [with spkg, needs review] New version of optional Group Cohomology spkg
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Reporter: SimonKing | Owner: SimonKing
Type: enhancement | Status: assigned
Priority: major | Milestone:
Component: optional packages | Keywords: cohomology ring p-group
Reviewer: | Author: Simon King
Merged: |
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Comment(by SimonKing):
Dear John,
Replying to [comment:42 jhpalmieri]:
> There is a theorem for it. Look in ''Complex cobordism and stable
homotopy groups of spheres'' by Ravenel, Appendix A1.4.
> According to A1.4.6, up to a sign, b<x1, x2, ...> is contained in <bx1,
x2, ...>, and similarly for the last position.
Thank you for your hint!
Ravenel says "In most cases the first page or two of the file is blank.
These files are in the process of being revised and should not be quoted
publicly."
So, it seems that I can not cite the theorems from his book. Or is the
numbering in the printed version the same?
> There is also an addition formula: see A1.4.5 in Ravenel. If you can
provide doctests for some of these in some examples, that would be great.
This theorem refers to the matric Massey product. If I understand
correctly (but I am not an expert) what I implemented is not the matric
version of the Massey products. So, can you explain how I can create an
example out of Addition Theorem?
But the Juggling Theorem works:
{{{
sage: from pGroupCohomology import CohomologyRing
sage: H = CohomologyRing.user_db(27,3)
sage: H.make()
sage: H.gens()
[1,
b_2_0, a 2-Cochain in H^*(E27; GF(3)),
b_2_1, a 2-Cochain in H^*(E27; GF(3)),
b_2_2, a 2-Cochain in H^*(E27; GF(3)),
b_2_3, a 2-Cochain in H^*(E27; GF(3)),
c_6_8, a 6-Cochain in H^*(E27; GF(3)),
a_1_0, a 1-Cochain in H^*(E27; GF(3)),
a_1_1, a 1-Cochain in H^*(E27; GF(3)),
a_3_4, a 3-Cochain in H^*(E27; GF(3)),
a_3_5, a 3-Cochain in H^*(E27; GF(3))]
sage: H.massey_products(H.6,H.6,H.6,all=False)
set([-b_2_0, a 2-Cochain in H^*(E27; GF(3))])
sage: H.massey_products(H.1*H.6,H.6,H.6,all=False)
set([-b_2_0^2, a 4-Cochain in H^*(E27; GF(3))])
sage: H.massey_products(H.2*H.6,H.6,H.6,all=False)
set([-b_2_0*b_2_1, a 4-Cochain in H^*(E27; GF(3))])
sage: H.massey_products(H.3*H.6,H.6,H.6,all=False)
set([-b_2_0*b_2_2, a 4-Cochain in H^*(E27; GF(3))])
sage: H.massey_products(H.4*H.6,H.6,H.6,all=False)
set([-b_2_0*b_2_1, a 4-Cochain in H^*(E27; GF(3))])
}}}
There is no sign change, since we multiplied by cocycles of even degree.
So far, so easy.
But now comes the really interesting example:
{{{
sage: H.massey_products(H.6,H.6,H.6*H.5,all=False)
set([-b_2_0^2*a_1_1*a_3_5+b_2_0^2*a_1_0*a_3_5-b_2_0*c_6_8, a 8-Cochain in
H^*(E27; GF(3))])
}}}
Is this fine?
Yes, it is!
Note that there is the summand {{{-b_2_0*c_6_8}}} that we would expect.
There remains to show that the other summands belong to the indeterminacy
of the Massey product. More precisely, I want to show that
{{{-b_2_0^2*a_1_1*a_3_5+b_2_0^2*a_1_0*a_3_5}}} is a multiple of {{{H.6}}}.
{{{
sage: H.6*(2*H.8*H.2*H.2+2*H.8*H.1*H.2 + H.1^2*H.9) ==
H('-b_2_0^2*a_1_1*a_3_5+b_2_0^2*a_1_0*a_3_5')
True
}}}
I think this would be a nice example for the doc string of
COHO.massey_products!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6750#comment:43>
Sage <http://sagemath.org/>
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