#6750: [with spkg, needs review] New version of optional Group Cohomology spkg
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Reporter: SimonKing | Owner: Simon King
Type: enhancement | Status: new
Priority: major | Milestone:
Component: optional packages | Keywords: cohomology ring p-group
Reviewer: | Author: Simon King
Merged: |
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There is a new version 1.1 of our optional spkg for the computation of
modular cohomology rings of finite p-groups. It can be installed by
{{{
> sage -i
http://sage.math.washington.edu/home/SimonKing/p_group_cohomology-1.1.spkg
}}}
As usual, if you did {{{export SAGE_CHECK=1}}} before installation, a test
suite is automatically executed.
__News and Changes__
There is now a basic implementation of the ''Yoneda cocomplex''. This
enables us to compute '''Massey products'''. These are higher structures
on cohomology rings and related with Steenrod powers and Bockstein
operation. See examples below.
Our repository of cohomology rings now also provides the cohomology rings
of the Sylow 2-subgroup of the Higman-Sims group (order 512) and of the
Sylow 2-subgroup of the third Conway group (order 1024). They can be
retrieved by
{{{
sage: H = CohomologyRing.web_db('Syl2HS')
}}}
or
{{{
sage: H = CohomologyRing.web_db('Syl2Co3')
}}}
I tested downloading the Conway example, and it took more than 30 minutes;
but this is certainly faster than a computation from scratch, which would
be about 3 days.
__Massey products__
The Massey product of cohomology ring elements {{{c1,c2,...,cn}}} is a
''set'' of cohomology ring elements; it may be empty or may contain
different cocycles.
D. Kraines modified this notion in the case of the {{{p^i}}} fold Massey
product of a cocylce with itself. I refer to this as the i-th restricted
Massey power. It is either not defined or is a single cocycle.
The restricted Massey powers can be expressed in terms of a composition of
Steenrod powers and Bockstein operation, and they can be used to
distinguish isomorphic cohomology rings. In particular, on degree one
cocycles, the 1st restricted Massey power is the same as minus the
Bockstein operation.
Example:
{{{
sage: from pGroupCohomology import CohomologyRing
sage: H3 = CohomologyRing(3,1)
sage: H3.make()
sage: H9 = CohomologyRing(9,1)
sage: H9.make()
sage: print H3
Cohomology ring of Small Group number 1 of order 3 with coefficients in
GF(3)
Computation complete
Minimal list of generators:
[c_2_0, a 2-Cochain in H^*(SmallGroup(3,1); GF(3)),
a_1_0, a 1-Cochain in H^*(SmallGroup(3,1); GF(3))]
Minimal list of algebraic relations:
[]
sage: print H9
Cohomology ring of Small Group number 1 of order 9 with coefficients in
GF(3)
Computation complete
Minimal list of generators:
[c_2_0, a 2-Cochain in H^*(SmallGroup(9,1); GF(3)),
a_1_0, a 1-Cochain in H^*(SmallGroup(9,1); GF(3))]
Minimal list of algebraic relations:
[]
}}}
So, the cohomology rings of the cyclic groups of order 3 and order 9
coincide. Note that for p>2, any element in odd degree squares to zero (by
graded commutativity). At some point in the past, I decided to not list
such ''obvious'' relations, but I might change my mind...
Now, we compute the 1st restricted Massey powers of the degree one
generators:
{{{
sage: H3.cochain_to_polynomial(H3.2.massey_power())
-c_2_0, a 2-Cochain in H^*(SmallGroup(3,1); GF(3))
sage: H9.cochain_to_polynomial(H9.2.massey_power())
0, a 2-Cochain in H^*(SmallGroup(9,1); GF(3))
}}}
They are different! Note that the 2nd restricted Massey power of the
degree one generator is non-trivial for the cyclic group of order 9:
{{{
sage: H9.cochain_to_polynomial(H9.2.massey_power(2))
-c_2_0, a 2-Cochain in H^*(SmallGroup(9,1); GF(3))
}}}
As I mentioned, the non-restricted Massey product is set valued. Indeed,
for the cohomology ring of the elementary abelian group of order 9, we
obtain:
{{{
sage: H3_3 = CohomologyRing(9,2)
sage: H3_3.make()
sage: H3_3.3
a_1_0, a 1-Cochain in H^*(SmallGroup(9,2); GF(3))
sage: H3_3.massey_products(H3_3.3,H3_3.3,H3_3.3)
set([-c_2_1, a 2-Cochain in H^*(SmallGroup(9,2); GF(3)),
a_1_0*a_1_1-c_2_1, a 2-Cochain in H^*(SmallGroup(9,2); GF(3)),
-a_1_0*a_1_1-c_2_1, a 2-Cochain in H^*(SmallGroup(9,2); GF(3))])
}}}
Or, with our default example, the Dihedral Group of order 8:
{{{
sage: H = CohomologyRing(8,3)
sage: H.make()
sage: print H
Cohomology ring of Dihedral group of order 8 with coefficients in GF(2)
Computation complete
Minimal list of generators:
[c_2_2, a 2-Cochain in H^*(D8; GF(2)),
b_1_0, a 1-Cochain in H^*(D8; GF(2)),
b_1_1, a 1-Cochain in H^*(D8; GF(2))]
Minimal list of algebraic relations:
[b_1_0*b_1_1]
sage: H.massey_products(H.2,H.3,H.2)
set([0, a 2-Cochain in H^*(D8; GF(2)), b_1_0^2, a 2-Cochain in H^*(D8;
GF(2))])
sage: H.massey_products(H.3,H.2,H.3)
set([0, a 2-Cochain in H^*(D8; GF(2)), b_1_1^2, a 2-Cochain in H^*(D8;
GF(2))])
}}}
__Notes for the reviewer(s)__
The new stuff is documented at
[http://sage.math.washington.edu/home/SimonKing/Cohomology/cochain.html#pGroupCohomology.cochain.YCOCH],
[http://sage.math.washington.edu/home/SimonKing/Cohomology/resolution.html#pGroupCohomology.resolution.MasseyDefiningSystems],
[http://sage.math.washington.edu/home/SimonKing/Cohomology/cochain.html#pGroupCohomology.cochain.COCH.massey_power]
and
[http://sage.math.washington.edu/home/SimonKing/Cohomology/cohomology.html#pGroupCohomology.cohomology.COHO.massey_products]
If you know about Steenrod powers and Bockstein operation and those
things, you might be able to cook up some interesting examples, and in
particular to do verifications. I would appreciate it!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6750>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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