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8. Cohomology ring structure

IntegralCupProduct(R,u,v,p,q)
IntegralCupProduct(R,u,v,p,q,P,Q,N)

(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* v in H^p+q(G,Z). This product is associative and u* v = (-1)pqv* u . It provides H^*(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) .

IntegralRingGenerators(R,n)

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). )

ModPCohomologyRing(G,n)
ModPCohomologyRing(R)

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).

ModPRingGenerators(A)

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


 





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