New version of optional Group Cohomology spkg

[simon-king-jena]

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/Cohomology/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))])
sage: H.massey_products(H.3,H.2,H.3,H.2)

set([c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_1^2+c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_0^2+c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_1^2+b_1_0^2+c_2_2, a 2-Cochain in H^*(D8; GF(2))])
sage: H.massey_products(H.2,H.3,H.2,H.3)

set([c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_0^2+c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_1^2+b_1_0^2+c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_1^2+c_2_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!

CC: david.green@uni-jena.de graham.ellis@nuigalway.ie mik@math.stanford.edu

Component: packages: optional

Keywords: cohomology ring p-group

Author: Simon King, David J. Green

Reviewer: John Palmieri, David Joyner, Mikael Vejdemo Johansson

Merged: Sage 4.1.2.alpha2

Issue created by migration from https://trac.sagemath.org/ticket/6750

[7c09a680-e216-4024-bb8e-9bfd4aa7f313]

Merged: Sage 4.1.2.alpha2

[7c09a680-e216-4024-bb8e-9bfd4aa7f313]

comment:49

Merged in the optional packages repository at

http://www.sagemath.org/packages/optional/

The updated optional package can be found at

http://www.sagemath.org/packages/optional/p_group_cohomology-1.1.spkg