GAP (HAP commands) - Chapter 13: Cohomology ring structure

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13 Cohomology ring structure
13.1
13.1-1 IntegralCupProduct
13.1-2 IntegralRingGenerators
13.1-3 ModPCohomologyGenerators
13.1-4 ModPCohomologyRing
13.1-5 ModPRingGenerators
13.1-6 Mod2CohomologyRingPresentation

### 13 Cohomology ring structure

#### 13.1

##### 13.1-1 IntegralCupProduct

`‣ IntegralCupProduct`( R, u, v, p, q ) | ( function )

`‣ IntegralCupProduct`( R, u, v, p, q, P, Q, N ) | ( function )

(Various functions used to construct the cup product are also available.)

Inputs a \(ZG\)-resolution \(R\), a vector \(u\) representing an element in \(H^p(G,Z)\), a vector \(v\) representing an element in \(H^q(G,Z)\) and the two integers \(p,q\) >\( 0\). It returns a vector \(w\) representing the cup product \(u\cdot v\) in \(H^{p+q}(G,Z)\). This product is associative and \(u\cdot v = (-1)pqv\cdot u\) . It provides \(H^\ast(G,Z)\) with the structure of an anti-commutative graded ring. This function implements the cup product for characteristic 0 only.

The resolution \(R\) needs a contracting homotopy.

To save the function from having to calculate the abelian groups \(H^n(G,Z)\) additional input variables can be used in the form \(IntegralCupProduct(R,u,v,p,q,P,Q,N)\) , where

- \(P\) is the output of the command \(CR_CocyclesAndCoboundaries(R,p,true)\)

- \(Q\) is the output of the command \(CR_CocyclesAndCoboundaries(R,q,true)\)

- \(N\) is the output of the command \(CR_CocyclesAndCoboundaries(R,p+q,true)\) .

Examples: 1 , 2

##### 13.1-2 IntegralRingGenerators

`‣ IntegralRingGenerators`( R, n ) | ( function )

Inputs at least \(n+1\) terms of a \(ZG\)-resolution and integer \(n\)> \(0\). It returns a minimal list of cohomology classes in \(H^n(G,Z)\) which, together with all cup products of lower degree classes, generate the group \(H^n(G,Z)\) .

(Let \(a_i\) be the \(i\)-th canonical generator of the \(d\)-generator abelian group \(H^n(G,Z)\). The cohomology class \(n_1a_1 + ... +n_da_d\) is represented by the integer vector \(u=(n_1, ..., n_d)\). )

Examples: 1 , 2

##### 13.1-3 ModPCohomologyGenerators

`‣ ModPCohomologyGenerators`( G, n ) | ( function )

`‣ ModPCohomologyGenerators`( R ) | ( function )

Inputs either a \(p\)-group \(G\) and positive integer \(n\), or else \(n\) terms of a minimal \(Z_pG\)-resolution \(R\) of \(Z_p\). It returns a pair whose first entry is a minimal set of homogeneous generators for the cohomology ring \(A=H^*(G,Z_p)\) modulo all elements in degree greater than \(n\). The second entry of the pair is a function \(deg\) which, when applied to a minimal generator, yields its degree.

WARNING: the following rule must be applied when multiplying generators \(x_i\) together. Only products of the form \(x_1*(x_2*(x_3*(x_4*...)))\) with \(deg(x_i) \le deg(x_{i+1})\) should be computed (since the \(x_i\) belong to a structure constant algebra with only a partially defined structure constants table).

Examples: 1

##### 13.1-4 ModPCohomologyRing

`‣ ModPCohomologyRing`( G, n ) | ( function )

`‣ ModPCohomologyRing`( G, n, level ) | ( function )

`‣ ModPCohomologyRing`( R ) | ( function )

`‣ ModPCohomologyRing`( R, level ) | ( function )

Inputs either a \(p\)-group \(G\) and positive integer \(n\), or else \(n\) terms of a minimal \(Z_pG\)-resolution \(R\) of \(Z_p\). It returns the cohomology ring \(A=H^*(G,Z_p)\) modulo all elements in degree greater than \(n\).

The ring is returned as a structure constant algebra \(A\).

The ring \(A\) is graded. It has a component \(A!.degree(x)\) which is a function returning the degree of each (homogeneous) element \(x\) in \(GeneratorsOfAlgebra(A)\).

An optional input variable "level" can be set to one of the strings "medium" or "high". These settings determine parameters in the algorithm. The default setting is "medium".

When "level" is set to "high" the ring \(A\) is returned with a component \(A!.niceBasis\). This component is a pair \([Coeff,Bas]\). Here \(Bas\) is a list of integer lists; a "nice" basis for the vector space \(A\) can be constructed using the command \(List(Bas,x->Product(List(x,i->Basis(A)[i]))\). The coefficients of the canonical basis element \(Basis(A)[i]\) are stored as \(Coeff[i]\).

If the ring \(A\) is computed using the setting "level"="medium" then the component \(A!.niceBasis\) can be added to \(A\) using the command \( A:=ModPCohomologyRing_part_2(A) \).

Examples: 1 , 2

##### 13.1-5 ModPRingGenerators

`‣ ModPRingGenerators`( A ) | ( function )

Inputs a mod \(p\) cohomology ring \(A\) (created using the preceeding function). It returns a minimal generating set for the ring \(A\). Each generator is homogeneous.

Examples: 1 , 2

##### 13.1-6 Mod2CohomologyRingPresentation

`‣ Mod2CohomologyRingPresentation`( G ) | ( function )

`‣ Mod2CohomologyRingPresentation`( G, n ) | ( function )

`‣ Mod2CohomologyRingPresentation`( A ) | ( function )

`‣ Mod2CohomologyRingPresentation`( R ) | ( function )

When applied to a finite \(2\)-group \(G\) this function returns a presentation for the mod 2 cohomology ring \(H^*(G,Z_2)\). The Lyndon-Hochschild-Serre spectral sequence is used to prove that the presentation is correct.

When the function is applied to a \(2\)-group \(G\) and positive integer \(n\) the function first constructs \(n\) terms of a free \(Z_2G\)-resolution \(R\), then constructs the finite-dimensional graded algebra \(A=H^(*\le n)(G,Z_2)\), and finally uses \(A\) to approximate a presentation for \(H^*(G,Z_2)\). For "sufficiently large" the approximation will be a correct presentation for \(H^*(G,Z_2)\).

Alternatively, the function can be applied directly to either the resolution \(R\) or graded algebra \(A\).

This function was written by Paul Smith. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.

Examples: 1 , 2

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