#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):
I just had a chat with Mikael, and if I understood him correctly then the
results of the current package version on the cohomology of C_4 times C_4
are what he expected.
Mikael, please correct me if the following contains too much nonsense.
{{{
sage: from pGroupCohomology import CohomologyRing
sage: H = CohomologyRing.user_db(16,2)
sage: H.make()
sage: x,a,b,c,d = H.gens()
sage: x,a,b,c,d
(1,
c_2_1, a 2-Cochain in H^*(SmallGroup(16,2); GF(2)),
c_2_2, a 2-Cochain in H^*(SmallGroup(16,2); GF(2)),
c_1_0, a 1-Cochain in H^*(SmallGroup(16,2); GF(2)),
c_1_1, a 1-Cochain in H^*(SmallGroup(16,2); GF(2)))
}}}
The Massey products allow to create the whole cohomology ring out of the
degree-1-generators, c and d:
{{{
sage: H.massey_products(c,c,c,c,all=False)
set([c_2_1, a 2-Cochain in H^*(SmallGroup(16,2); GF(2))])
sage: H.massey_products(d,d,d,d,all=False)
set([c_2_2, a 2-Cochain in H^*(SmallGroup(16,2); GF(2))])
}}}
After fixing the previous bug, massey_power yields the same result:
{{{
sage: H.cochain_to_polynomial(c.massey_power(2))
c_2_1, a 2-Cochain in H^*(SmallGroup(16,2); GF(2))
sage: H.cochain_to_polynomial(d.massey_power(2))
c_2_2, a 2-Cochain in H^*(SmallGroup(16,2); GF(2))
}}}
Usually, the Massey products behave multiplicatively in the first and last
position:
{{{
sage: H.massey_products(c*d,c,c,c,all=False)
set([c_2_1*c_1_1, a 3-Cochain in H^*(SmallGroup(16,2); GF(2))])
sage: H.massey_products(c,c,c,c*d,all=False)
set([c_2_1*c_1_1, a 3-Cochain in H^*(SmallGroup(16,2); GF(2))])
}}}
In this example, it is actually multiplicative in ''all'' four positions:
{{{
sage: H.massey_products(c,c*d,c,c,all=False)
set([c_2_1*c_1_1, a 3-Cochain in H^*(SmallGroup(16,2); GF(2))])
sage: H.massey_products(c,c,c*d,c,all=False)
set([c_2_1*c_1_1, a 3-Cochain in H^*(SmallGroup(16,2); GF(2))])
}}}
By consequence, higher Massey products involving at least two factors
{{{c*d}}} are either not defined or vanish:
{{{
sage: H.massey_products(c*d,c,c,c*d,c,all=False)
set()
}}}
This makes sense since {{{H.massey_products(c,c,c*d,c)}}} is non-zero,
thus, the above fivefold product is not defined.
{{{
sage: H.massey_products(c*d,c*d,c*d,c*d,c*d,c*d,c*d,c*d,c*d,c,all=False)
set([0, a 11-Cochain in H^*(SmallGroup(16,2); GF(2))])
}}}
This makes sense since we can extract the {{{d}}} factors, but {{{d*d}}}
is zero, so, the whole Massey product is zero.
David and John, do you think that this now seems stable enough for being
an optional package? AFAIK the build problems on Intel Mac are resolved:
Meanwhile William provided me with access to such machines, and it works
both 32 and 64 bit. Actually, I did the above examples on Intel Mac 64
bit, since on my computer in Galway and on sage.math I meanwhile have a
package version 2.0 minus epsilon.
Best regards,
Simon
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6750#comment:40>
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