#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):
The problem is fixed and the package is updated at
http://sage.math.washington.edu/home/SimonKing/Cohomology/p_group_cohomology-1.1.spkg
Technical reason for the problem was:
If Y is the composition of two Yoneda cochains Y1, Y2, I thought that for
computing higher terms of the composition it suffices to compose the
lowest terms and then lift Y, i.e., extend a certain (anti-)commutative
diagram. This is in fact possible if one just wants to compute the cup
product. But for computing higher Massey products, one must lift Y1, Y2
sufficiently, and then compose the higher terms.
I extended the documentation, and thanks to the help of David Green I am
now able to present a non-trivial example, explicitly verifying D. Kraines
result for a cocycle of degree 3 of the Extraspecial 3-group of order 27
and exponent 3. It is also part of the doc tests.
I think that the example indicates that my implementation makes sense, so,
I hope that a positive review (after installation+doctests on a couple of
platforms) is now possible.
The example works as follows.
__First step:__ Study elementary abelian groups.
{{{
sage: tmp_root = tmp_filename()
sage: from pGroupCohomology import CohomologyRing
sage: CohomologyRing.set_user_db(tmp_root)
sage: H = CohomologyRing.user_db(9,2)
sage: H.make()
sage: H.gens()
[1,
c_2_1, a 2-Cochain in H^*(SmallGroup(9,2); GF(3)),
c_2_2, a 2-Cochain in H^*(SmallGroup(9,2); GF(3)),
a_1_0, a 1-Cochain in H^*(SmallGroup(9,2); GF(3)),
a_1_1, a 1-Cochain in H^*(SmallGroup(9,2); GF(3))]
}}}
Of course, there is a sign involved when choosing the generators, but in
fact {{{c_2_1, c_2_2}}} are the Bocksteins of {{{a_1_0, a_1_1}}}. So, the
following agrees with Kraines' theorem:
{{{
sage: H.cochain_to_polynomial(H.3.massey_power())
-c_2_1, a 2-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: H.cochain_to_polynomial(H.4.massey_power())
-c_2_2, a 2-Cochain in H^*(SmallGroup(9,2); GF(3))
}}}
Fortunately the cohomology rings of elementary abelian groups are very
simple, and thus allow for a direct computation of Steenrod powers and
Bocksteins in higher degree. We consider C = a_1_0*c_2_1_. By Cartan
formula and since P^0^ is the identity, we have
P^1^(C) = P^1^(a_1_0) c_2_1 + a_1_0 P(c_2_1).
Since P^1^ vanishes in degree one and acts as the p-th power in degree
two, we get
P^1^(C) = a_1_0 c_2_1^3^.
Applying the Bockstein operator \beta, we get
\beta P^1^(C) = c_2_1^4^,
since \beta(c_2_1) = \beta^2^(a_1_0) = 0 and since \beta(xy)=\beta(x)y +
(-1)^deg x^x\beta(y).
Hence, according to Kraines, the first restricted Massey power of C should
be
<C; 1> = - c_2_1^4^.
And indeed:
{{{
sage: (H.1*H.3).massey_power()
<(c_2_1)*(a_1_0); 1>, a 8-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: H.cochain_to_polynomial(_)
-c_2_1^4, a 8-Cochain in H^*(SmallGroup(9,2); GF(3))
}}}
__Second Step:__ Apply Kraines formula to the restrictions to maximal
elementary abelian subgroups
This is the cohomology ring of the extraspecial 3-group of order 27 and
exponent 3:
{{{
sage: H = CohomologyRing.user_db(27,3)
sage: H.make()
}}}
We want to compute the 1st Massey power of a generator in degree 3:
{{{
sage: C = H.8
sage: C
a_3_4, a 3-Cochain in H^*(E27; GF(3))
}}}
There are 4 maximal elementary abelian subgroups (all of order 9). We get
the restriction maps and compute the restrictions of C:
{{{
sage: r1 = H.restriction_maps()[2][1]
sage: r1
Induced homomorphism of degree 0 from H^*(E27; GF(3)) to
H^*(SmallGroup(9,2); GF(3))
sage: r2 = H.restriction_maps()[3][1]
sage: r3 = H.restriction_maps()[4][1]
sage: r4 = H.restriction_maps()[5][1]
sage: U = r1.codomain()
sage: U.cochain_to_polynomial(r1(C))
-c_2_2*a_1_0+c_2_1*a_1_1, a 3-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: U.cochain_to_polynomial(r2(C))
0, a 3-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: U.cochain_to_polynomial(r3(C))
-c_2_2*a_1_0+c_2_1*a_1_1, a 3-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: U.cochain_to_polynomial(r4(C))
c_2_2*a_1_1+c_2_2*a_1_0-c_2_1*a_1_1, a 3-Cochain in H^*(SmallGroup(9,2);
GF(3))
}}}
Hence, after computing Bockstein and Steenrod power in the maximal
elementary abelian subgroups as above, and since Steenrod power and
Bockstein commute with restriction maps, the theorem of Kraines tells us
that <C; 1> should restrict to
* c_2_1 c_2_2^3^ - c_2_1^3^c_2_2,
* 0,
* c_2_1 c_2_2^3^ - c_2_1^3^c_2_2, and
* -c_2_2^4^ - c_2_1 c_2_2^3^ + c_2_1^3^c_2_2.
This is indeed the case:
{{{
sage: CP = C.massey_power()
sage: U.cochain_to_polynomial(r1(CP))
c_2_1*c_2_2^3-c_2_1^3*c_2_2, a 8-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: U.cochain_to_polynomial(r2(CP))
0, a 8-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: U.cochain_to_polynomial(r3(CP))
c_2_1*c_2_2^3-c_2_1^3*c_2_2, a 8-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: U.cochain_to_polynomial(r4(CP))
-c_2_2^4-c_2_1*c_2_2^3+c_2_1^3*c_2_2, a 8-Cochain in H^*(SmallGroup(9,2);
GF(3))
}}}
It is known that for this group, a cocycle is uniquely determined by its
restrictions to the maximal elementary abelian subgroups. Hence, we have
verified the computation of the first restricted Massey power of C.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6750#comment:11>
Sage <http://sagemath.org/>
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