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6. Homology and cohomology groups

### 6. Homology and cohomology groups

`Cohomology(X,n)`

Inputs either a cochain complex X=C (or G-cocomplex C) or a cochain map X=(C --> D) over the integers Z together with a non-negative intereg n.

- If X=C then the torsion coefficients of H^n(C) are retuned.

- If X=(C --> D) then the induced homomorphism H^n(C)--> H^n(D) is returned as a homomorphism of finitely presented groups.

`CohomologyModule(C,n)`

Inputs a G-cocomplex C together with a non-negative integer n. It returns the cohomology H^n(C) as a G-outer group. If C was constructed from a resolution R by homing to an abelian G-outer group A then, for each x in H:=CohomologyModule(C,n), there is a function f:=H!.representativeCocycle(x) which is a standard n-cocycle corresponding to the cohomology class x. (At present this works only for n=1,2,3.)

`CohomologyPrimePart(C,n,p)` Inputs a cochain complex C in characteristic 0, a positive integer n, and a prime p. It returns a list of those torsion coefficients of H^n(C) that are positive powers of p. The function uses the EDIM package by Frank Luebeck.

`GroupCohomology(X,n)` `GroupCohomology(X,n,p)`

Inputs a positive integer n and either a finite group X=G or a Coxeter diagram X=D representing an infinite Artin group G. It returns the torsion coefficients of the integral cohomology H^n(G,Z).

There is an optional third argument which, when set equal to a prime p, causes the function to return the the mod p cohomology H^n(G,Z_p) as a list of length equal to its rank.

This function is a composite of more basic functions, and makes choices for a number of parameters. For a particular group you would almost certainly be better using the more basic functions and making the choices yourself!

`GroupHomology(X,n)`
`GroupHomology(X,n,p)`

Inputs a positive integer n and either a finite group X=G or a Coxeter diagram X=D representing an infinite Artin group G. It returns the torsion coefficients of the integral homology H_n(G,Z).

There is an optional third argument which, when set equal to a prime p, causes the function to return the mod p homology H_n(G,Z_p) as a list of lenth equal to its rank.

This function is a composite of more basic functions, and makes choices for a number of parameters. For a particular group you would almost certainly be better using the more basic functions and making the choices yourself!

`Homology(X,n)`

Inputs either a chain complex X=C or a chain map X=(C --> D).

- If X=C then the torsion coefficients of H_n(C) are retuned.

- If X=(C --> D) then the induced homomorphism H_n(C) --> H_n(D) is returned as a homomorphism of finitely presented groups.

`HomologyPb(C,n)`

This is a back-up function which might work in some instances where Homology(C,n) fails. It is most useful for chain complexes whose boundary homomorphisms are sparse.

It inputs a chain complex C in characteristic 0 and returns the torsion coefficients of H_n(C) . There is a small probability that an incorrect answer could be returned. The computation relies on probabilistic Smith Normal Form algorithms implemented in the Simplicial Homology GAP package. This package therefore needs to be loaded. The computation is stored as a component of C so, when called a second time for a given C and n, the calculation is recalled without rerunning the algorithm.

The choice of probabalistic algorithm can be changed using the command

SetHomologyAlgorithm(HomologyAlgorithm[i]);

where i = 1,2,3 or 4. The upper limit for the probability of an incorrect answer can be set to any rational number 0<e<= 1 using the following command.

SetUncertaintyTolerence(e);

See the Simplicial Homology package manual for further details.

`HomologyPrimePart(C,n,p)` Inputs a chain complex C in characteristic 0, a positive integer n, and a prime p. It returns a list of those torsion coefficients of H_n(C) that are positive powers of p. The function uses the EDIM GAP package by Frank Luebeck.
`LeibnizAlgebraHomology(A,n)` Inputs a Lie or Leibniz algebra X=A (over the ring of integers Z or over a field K), together with a positive integer n. It returns the n-dimensional Leibniz homology of A.
`LieAlgebraHomology(A,n)` Inputs a Lie algebra A (over the integers or a field) and a positive integer n. It returns the homology H_n(A,k) where k denotes the ground ring.

`PrimePartDerivedFunctor(G,R,F,n)`

Inputs a finite group G, a positive integer n, at least n+1 terms of a ZP-resolution for a Sylow subgroup P<G and a "mathematically suitable" covariant additive functor F such as TensorWithIntegers . It returns the abelian invariants of the p-component of the homology H_n(F(R)) .

Warning: All calculations are assumed to be in characteristic 0. The function should not be used if the coefficient module is over the field of p elements.

"Mathematically suitable" means that the Cartan-Eilenberg double coset formula must hold.

`RankHomologyPGroup(G,n)` `RankHomologyPGroup(R,n)` `RankHomologyPGroup(G,n,"empirical")`

Inputs a (smallish) p-group G, or n terms of a minimal Z_pG-resolution R of Z_p , together with a positive integer n. It returns the minimal number of generators of the integral homology group H_n(G,Z).

If an option third string argument "empirical" is included then an empirical algorithm will be used. This is one which always seems to yield the right answer but which we can't prove yields the correct answer.

`RankPrimeHomology(G,n)` Inputs a (smallish) p-group G together with a positive integer n. It returns a function dim(k) which gives the rank of the vector space H_k(G,Z_p) for all 0 <= k <= n.

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