GAP (HAP) - Chapter 4: Functors

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4 Functors

### 4 Functors

`ExtendScalars(R,G,EltsG)` Inputs a ZH-resolution R, a group G containing H as a subgroup, and a list EltsG of elements of G. It returns the free ZG-resolution (R ⊗_ZH ZG). The returned resolution S has S!.elts:=EltsG. This is a resolution of the ZG-module (Z ⊗_ZH ZG). (Here ⊗_ZH means tensor over ZH.)
`HomToIntegers(X)` Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S). It returns the cochain complex or cochain map obtained by applying HomZG( _ , Z) where Z is the trivial module of integers (characteristic 0).
`HomToIntegersModP(R)` Inputs a ZG-resolution R and returns the cochain complex obtained by applying HomZG( _ , Z_p) where Z_p is the trivial module of integers mod p. (At present this functor does not handle equivariant chain maps.)
`HomToIntegralModule(R,f)` Inputs a ZG-resolution R and a group homomorphism f:G ⟶ GL_n(Z) to the group of n×n invertible integer matrices. Here Z must have characteristic 0. It returns the cochain complex obtained by applying HomZG( _ , A) where A is the ZG-module Z^n with G action via f. (At present this function does not handle equivariant chain maps.)
`TensorWithIntegralModule(R,f)` Inputs a ZG-resolution R and a group homomorphism f:G ⟶ GL_n(Z) to the group of n×n invertible integer matrices. Here Z must have characteristic 0. It returns the chain complex obtained by tensoring over ZG with the ZG-module A=Z^n with G action via f. (At present this function does not handle equivariant chain maps.)
`HomToGModule(R,A)` Inputs a ZG-resolution R and an abelian G-outer group A. It returns the G-cocomplex obtained by applying HomZG( _ , A). (At present this function does not handle equivariant chain maps.)
`InduceScalars(R,hom)` Inputs a ZQ-resolution R and a surjective group homomorphism hom:G→ Q. It returns the unduced non-free ZG-resolution.
`LowerCentralSeriesLieAlgebra(G)` `LowerCentralSeriesLieAlgebra(f)` Inputs a pcp group G. If each quotient G_c/G_c+1 of the lower central series is free abelian or p-elementary abelian (for fixed prime p) then a Lie algebra L(G) is returned. The abelian group underlying L(G) is the direct sum of the quotients G_c/G_c+1 . The Lie bracket on L(G) is induced by the commutator in G. (Here G_1=G, G_c+1=[G_c,G] .) The function can also be applied to a group homomorphism f: G ⟶ G' . In this case the induced homomorphism of Lie algebras L(f):L(G) ⟶ L(G') is returned. If the quotients of the lower central series are not all free or p-elementary abelian then the function returns fail. This function was written by Pablo Fernandez Ascariz
`TensorWithIntegers(X)` Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S). It returns the chain complex or chain map obtained by tensoring with the trivial module of integers (characteristic 0).
`FilteredTensorWithIntegers(R)` Inputs a ZG-resolution R for which "filteredDimension" lies in NamesOfComponents(R). (Such a resolution can be produced using TwisterTensorProduct(), ResolutionNormalSubgroups() or FreeGResolution().) It returns the filtered chain complex obtained by tensoring with the trivial module of integers (characteristic 0).
`TensorWithTwistedIntegers(X,rho)` Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S). It also inputs a function rho: G→ Z where the action of g ∈ G on Z is such that g.1 = rho(g). It returns the chain complex or chain map obtained by tensoring with the (twisted) module of integers (characteristic 0).
`TensorWithIntegersModP(X,p)` Inputs either a ZG-resolution X=R, or a characteristics 0 chain complex, or an equivariant chain map X = (F:R ⟶ S), or a chain map between characteristic 0 chain complexes, together with a prime p. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo p.
`TensorWithTwistedIntegersModP(X,p,rho)` Inputs either a ZG-resolution X=R, or an equivariant chain map X = (F:R ⟶ S), and a prime p. It also inputs a function rho: G→ Z where the action of g ∈ G on Z is such that g.1 = rho(g). It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo p.
`TensorWithRationals(R)` Inputs a ZG-resolution R and returns the chain complex obtained by tensoring with the trivial module of rational numbers.

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