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<Chapter><Heading> Functors</Heading>

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<Table Align="|l|" >

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<Row>

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<Item>

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<Index>ExtendScalars</Index>

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<C>ExtendScalars(R,G,EltsG)</C>

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<P/>

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Inputs  a <M>ZH</M>-resolution <M>R</M>,

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a group <M>G</M> containing <M>H</M> as a subgroup, and a list <M>EltsG</M>

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of elements of <M>G</M>. It returns the free <M>ZG</M>-resolution

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 <M>(R  \otimes_{ZH} ZG)</M>. The returned resolution <M>S</M> has S!.elts:=EltsG. This is a resolution of the <M>ZG</M>-module

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<M>(Z \otimes_{ZH} ZG)</M>. (Here <M>\otimes_{ZH}</M> means tensor over <M>ZH</M>.)

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>HomToIntegers</Index>

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<C>HomToIntegers(X) </C>

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<P/>

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Inputs either a <M>ZG</M>-resolution <M>X=R</M>, 

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or an equivariant chain map <M>X = (F:R

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\longrightarrow 

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S)</M>. 

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It returns the cochain complex or cochain map obtained by applying 

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<M>HomZG( _ , Z)</M> where <M>Z</M> is the 

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trivial module of integers (characteristic 0).

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> HomToIntegersModP </Index>

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<C>HomToIntegersModP(R) </C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> and 

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returns the cochain complex obtained by applying 

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<M>HomZG( _ , Z_p)</M> where <M>Z_p</M> is the trivial module of 

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integers mod <M>p</M>. 

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(At present this functor does not handle equivariant chain maps.)

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> HomToIntegralModule </Index>

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<C>HomToIntegralModule(R,f) </C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> and a 

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group homomorphism <M>f:G \longrightarrow

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GL_n(Z)</M>  

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to the group of <M>n�n</M> invertible integer matrices. Here 

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<M>Z</M> must have characteristic 0. 

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It returns the cochain complex obtained by applying 

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<M>HomZG( _ , A)</M> where <M>A</M> is the <M>ZG</M>-module <M>Z^n</M> 

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with <M>G</M> action via <M>f</M>. 

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(At present this function does not handle equivariant chain maps.)

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TensorWithIntegralModule </Index>

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<C>TensorWithIntegralModule(R,f) </C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> and a

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group homomorphism <M>f:G \longrightarrow

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GL_n(Z)</M>

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to the group of <M>n�n</M> invertible integer matrices. Here

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<M>Z</M> must have characteristic 0.

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It returns the chain complex obtained by tensoring over <M>ZG</M>  

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with  the <M>ZG</M>-module <M>A=Z^n</M>

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with <M>G</M> action via <M>f</M>.

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(At present this function does not handle equivariant chain maps.)

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> HomToGModule </Index>

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<C>HomToGModule(R,A) </C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> and an abelian

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G-outer group A.

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It returns the G-cocomplex obtained by applying

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<M>HomZG( _ , A)</M>. 

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(At present this function does not handle equivariant chain maps.)

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>InduceScalars </Index>

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<C>InduceScalars(R,hom) </C>

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<P/>

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Inputs  a <M>ZQ</M>-resolution <M>R</M> and a surjective group homomorphism

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<M>hom:G\rightarrow Q</M>.

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 It returns the unduced non-free  <M>ZG</M>-resolution.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>LowerCentralSeriesLieAlgebra</Index>

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<C>LowerCentralSeriesLieAlgebra(G) </C>

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<C>LowerCentralSeriesLieAlgebra(f) </C>

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<P/>

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Inputs a pcp group <M>G</M>. 

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If each quotient <M>G_c/G_{c+1}</M> 

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of the lower central series is free abelian or p-elementary abelian 

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(for fixed prime p) then a Lie algebra  <M>L(G)</M> is returned. 

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The abelian group underlying <M>L(G)</M> is the 

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direct sum of the quotients <M>G_c/G_{c+1}</M> . 

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The Lie bracket on <M>L(G)</M> is induced by the commutator in 

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<M>G</M>. (Here <M>G_1=G</M>, <M>G_{c+1}=[G_c,G]</M> .)

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<P/>

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The function can also be applied to a group homomorphism <M>f: G

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\longrightarrow

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G'</M> . In this case the induced homomorphism of Lie algebras 

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<M>L(f):L(G) \longrightarrow

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L(G')</M> is returned.<P/>

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If the quotients of the lower central series are not all free or p-elementary abelian then the function returns fail.<P/>

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This function was written by Pablo Fernandez Ascariz

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TensorWithIntegers </Index>

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<C>TensorWithIntegers(X) </C>

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<P/>

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Inputs either a <M>ZG</M>-resolution <M>X=R</M>, 

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or an equivariant chain map <M>X = (F:R

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\longrightarrow S)</M>. It returns the 

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chain complex or chain map obtained by tensoring with the 

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trivial module of integers (characteristic 0).

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> FilteredTensorWithIntegers </Index>

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<C>FilteredTensorWithIntegers(R) </C>

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<P/>

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Inputs  a <M>ZG</M>-resolution <M>R</M> for which "filteredDimension" lies in NamesOfComponents(R). (Such a resolution can be produced using TwisterTensorProduct(), ResolutionNormalSubgroups() or FreeGResolution().) 

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It returns the

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filtered chain complex  obtained by tensoring with the

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trivial module of integers (characteristic 0).

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TensorWithTwistedIntegers </Index>

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<C>TensorWithTwistedIntegers(X,rho) </C>

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<P/>

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Inputs either a <M>ZG</M>-resolution <M>X=R</M>,

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or an equivariant chain map <M>X = (F:R

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\longrightarrow S)</M>. It also inputs a function <M>rho\colon G\rightarrow \mathbb Z</M> where the action of <M>g \in G</M> on <M>\mathbb Z</M> is such that <M>g.1 = rho(g)</M>. It returns the

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chain complex or chain map obtained by tensoring with the

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(twisted) module of integers (characteristic 0).

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TensorWithIntegersModP</Index>

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<C>TensorWithIntegersModP(X,p) </C>

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 <P/>

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Inputs either a <M>ZG</M>-resolution <M>X=R</M>, or a characteristics 0 chain complex, or 

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an equivariant chain map <M>X = (F:R

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\longrightarrow

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S)</M>, or a chain map between characteristic 0 chain complexes, together with

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 a prime <M>p</M>. 

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 It returns the chain 

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complex or chain map obtained by tensoring with the 

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trivial module of integers modulo <M>p</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TensorWithTwistedIntegersModP</Index>

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<C>TensorWithTwistedIntegersModP(X,p,rho)</C>

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<P/>

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Inputs either a <M>ZG</M>-resolution <M>X=R</M>, or

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an equivariant chain map <M>X = (F:R

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\longrightarrow

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S)</M>, and a prime <M>p</M>. It also inputs a function <M>rho\colon G\rightarrow \mathbb Z</M> where the action of <M>g \in G</M> on <M>\mathbb Z</M> is such that <M>g.1 = rho(g)</M>. It returns the chain

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complex or chain map obtained by tensoring with the

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trivial module of integers modulo <M>p</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TensorWithRationals </Index>

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<C>TensorWithRationals(R)</C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> and returns the chain 

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complex obtained by tensoring with the trivial module of 

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rational numbers. 

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</Item>

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</Row>

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</Table>

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</Chapter>

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