This page covers the functions used in chapter 4 of the book An Invitation to Computational Homotopy.
‣ CcGroup( N, f ) | ( function ) |
Inputs a \(G\)-outer group \(N\) with nonabelian cocycle describing some extension \(N \rightarrowtail E \twoheadrightarrow G\) together with standard 2-cocycle \(f\colon G \times G \rightarrow A\) where \(A=Z(N)\). It returns the extension group determined by the cocycle \(f\). The group is returned as a cocyclic group.
This function is part of the HAPcocyclic package of functions implemented by Robert F. Morse.
‣ CocycleCondition( R, n ) | ( function ) |
Inputs a free \(\mathbb ZG\)-resolution \(R\) of \(\mathbb Z\) and an integer \(n \ge 1\). It returns an integer matrix \(M\) with the following property. Let \(d\) be the \(\mathbb ZG\)-rank of \(R_n\). An integer vector \(f=[f_1, ... , f_d]\) then represents a \(\mathbb ZG\)-homomorphism \(R_n \rightarrow \mathbb Z_q\) which sends the \(i\)th generator of \(R_n\) to the integer \(f_i\) in the trivial \(\mathbb ZG\)-module \(\mathbb Z_q=\mathbb Z/q{\mathbb Z}\) (where possibly \(q=0\)). The homomorphism \(f\) is a cocycle if and only if \(M^tf=0\) mod \(q\).
‣ StandardCocycle( R, f, n ) | ( function ) |
‣ StandardCocycle( R, f, n, q ) | ( function ) |
Inputs a free \(\mathbb ZG\)-resolution \(R\) (with contracting homotopy), a positive integer \(n\) and an integer vector \(f\) representing an \(n\)-cocycle \(R_n \rightarrow \mathbb Z_q=\mathbb Z/q\mathbb Z\) where \(G\) acts trivially on \(\mathbb Z_q\). It is assumed \(q=0\) unless a value for \(q\) is entered. The command returns a function \(F(g_1, ..., g_n)\) which is the standard cocycle \(G^n \rightarrow \mathbb Z_q\) corresponding to \(f\). At present the command is implemented only for \(n=2\) or \(3\).
‣ ActedGroup( M ) | ( function ) |
Inputs a \(G\)-outer group \(M\) corresponding to a homomorphism \(\alpha\colon G\rightarrow {\rm Out}(N)\) and returns the group \(N\).
‣ ActingGroup( M ) | ( function ) |
Inputs a \(G\)-outer group \(M\) corresponding to a homomorphism \(\alpha\colon G\rightarrow {\rm Out}(N)\) and returns the group \(G\).
‣ Centre( M ) | ( function ) |
Inputs a \(G\)-outer group \(M\) and returns its group-theoretic centre as a \(G\)-outer group.
Examples: 1 , 2 , 3 , 4 , 5 , 6
‣ GOuterGroup( E, N ) | ( function ) |
‣ GOuterGroup( ) | ( function ) |
Inputs a group \(E\) and normal subgroup \(N\). It returns \(N\) as a \(G\)-outer group where \(G=E/N\). A nonabelian cocycle \(f\colon G\times G\rightarrow N\) is attached as a component of the \(G\)-Outer group.
The function can be used without an argument. In this case an empty outer group \(C\) is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).
‣ CohomologyModule( C, n ) | ( function ) |
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 \(\mathbb ZG\)-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 is implemented only for \(n=1,2,3\).)
‣ HomToGModule( R, A ) | ( function ) |
Inputs a \(\mathbb ZG\)-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.)
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