<Chapter><Heading> Functors</Heading> <Table Align="|l|" > <Row> <Item> <Index>HomToIntegers </Index> <C> HomToIntegers(X) </C> <P/> Inputs either a <M>ZG</M>-resolution <M>X=R</M>, or an equivariant chain map <M>X = (F:R \longrightarrow S)</M>. It returns the cochain complex or cochain map obtained by applying <M>HomZG( _ , Z)</M> where <M>Z</M> is the trivial module of integers (characteristic 0). </Item> </Row> <Row> <Item> <Index> HomToIntegersModP </Index> <C> HomToIntegersModP(R) </C> <P/> Inputs a <M>ZG</M>-resolution <M>R</M> and returns the cochain complex obtained by applying <M>HomZG( _ , Z_p)</M> where <M>Z_p</M> is the trivial module of integers mod <M>p</M>. (At present this functor does not handle equivariant chain maps.) </Item> </Row> <Row> <Item> <Index> HomToIntegralModule </Index> <C> HomToIntegralModule(R,f) </C> <P/> Inputs a <M>ZG</M>-resolution <M>R</M> and a group homomorphism <M>f:G \longrightarrow GL_n(Z)</M> to the group of <M>n×n</M> invertible integer matrices. Here <M>Z</M> must have characteristic 0. It returns the cochain complex obtained by applying <M>HomZG( _ , A)</M> where <M>A</M> is the <M>ZG</M>-module <M>Z_n</M> with <M>G</M> action via <M>f</M>. (At present this function does not handle equivariant chain maps.) </Item> </Row> <Row> <Item> <Index> HomToGModule </Index> <C> HomToGModule(R,A) </C> <P/> Inputs a <M>ZG</M>-resolution <M>R</M> and an abelian G-outer group A. It returns the G-cocomplex obtained by applying <M>HomZG( _ , A)</M>. (At present this function does not handle equivariant chain maps.) </Item> </Row> <Row> <Item> <Index>LowerCentralSeriesLieAlgebra</Index> <C> LowerCentralSeriesLieAlgebra(G) </C> <C> LowerCentralSeriesLieAlgebra(f) </C> <P/> Inputs a pcp group <M>G</M>. If each quotient <M>G_c/G_{c+1}</M> of the lower central series is free abelian or p-elementary abelian (for fixed prime p) then a Lie algebra <M>L(G)</M> is returned. The abelian group underlying <M>L(G)</M> is the direct sum of the quotients <M>G_c/G_{c+1}</M> . The Lie bracket on <M>L(G)</M> is induced by the commutator in <M>G</M>. (Here <M>G_1=G</M>, <M>G_{c+1}=[G_c,G]</M> .) <P/> The function can also be applied to a group homomorphism <M>f: G \longrightarrow G'</M> . In this case the induced homomorphism of Lie algebras <M>L(f):L(G) \longrightarrow L(G')</M> is returned.<P/> If the quotients of the lower central series are not all free or p-elementary abelian then the function returns fail.<P/> This function was written by Pablo Fernandez Ascariz </Item> </Row> <Row> <Item> <Index> TensorWithIntegers </Index> <C> TensorWithIntegers(X) </C> <P/> Inputs either a <M>ZG</M>-resolution <M>X=R</M>, or an equivariant chain map <M>X = (F:R \longrightarrow S)</M>. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers (characteristic 0). </Item> </Row> <Row> <Item> <Index> TensorWithIntegersModP</Index> <C> TensorWithIntegersModP(X,p) </C> <P/> Inputs either a <M>ZG</M>-resolution <M>X=R</M>, or an equivariant chain map <M>X = (F:R \longrightarrow S)</M>, and a prime <M>p</M>. It returns the chain complex or chain map obtained by tensoring with the trivial module of integers modulo <M>p</M>. </Item> </Row> <Row> <Item> <Index> TensorWithRationals </Index> <C> TensorWithRationals(R) </C> <P/> Inputs a <M>ZG</M>-resolution <M>R</M> and returns the chain complex obtained by tensoring with the trivial module of rational numbers. </Item> </Row> </Table> </Chapter>