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<Chapter><Heading> Simplicial groups</Heading> <Section><Heading> </Heading>
<ManSection> <Func Name="NerveOfCatOneGroup" Arg="G,n"/> <Description> <P/> Inputs a cat-1-group <M>G</M> and a positive integer <M>n</M>. It returns the low-dimensional part of the nerve of <M>G</M> as a simplicial group of length <M>n</M>. <Br/> <Br/> This function applies both to cat-1-groups for which IsHapCatOneGroup(G) is true, and to cat-1-groups produced using the Xmod package. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap12.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>3</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="EilenbergMacLaneSimplicialGroup" Arg="G,n,dim"/> <Description> <P/> Inputs a group <M>G</M>, a positive integer <M>n</M>, and a positive integer <M>dim </M>. The function returns the first <M>1+dim</M> terms of a simplicial group with <M>n-1</M>st homotopy group equal to <M>G</M> and all other homotopy groups equal to zero. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap3.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../tutorial/chap12.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>5</LinkText></URL>
</Description> </ManSection>
<ManSection> <Var Name="EilenbergMacLaneSimplicialGroupMap"/> <Description> <P/> Inputs a group homomorphism <M>f:G\rightarrow Q</M>, a positive integer <M>n</M>, and a positive integer <M>dim </M>. The function returns the first <M>1+dim</M> terms of a simplicial group homomorphism <M>f:K(G,n) \rightarrow K(Q,n)</M> of Eilenberg-MacLane simplicial groups. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="MooreComplex" Arg="G"/> <Description> <P/> Inputs a simplicial group <M>G</M> and returns its Moore complex as a <M>G</M>-complex. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="ChainComplexOfSimplicialGroup" Arg="G"/> <Description> <P/> Inputs a simplicial group <M>G</M> and returns the cellular chain complex <M>C</M> of a CW-space <M>X</M> represented by the homotopy type of the simplicial group. Thus the homology groups of <M>C</M> are the integral homology groups of <M>X</M>. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap3.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../tutorial/chap12.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>5</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>6</LinkText></URL>
</Description> </ManSection>
<ManSection> <Var Name="SimplicialGroupMap"/> <Description> <P/> Inputs a homomorphism <M>f:G\rightarrow Q</M> of simplicial groups. The function returns an induced map <M>f:C(G) \rightarrow C(Q)</M> of chain complexes whose homology is the integral homology of the simplicial group G and Q respectively. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="HomotopyGroup" Arg="G,n"/> <Description> <P/> Inputs a simplicial group <M>G</M> and a positive integer <M>n</M>. The integer <M>n</M> must be less than the length of <M>G</M>. It returns, as a group, the (n)-th homology group of its Moore complex. Thus HomotopyGroup(G,0) returns the "fundamental group" of <M>G</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../tutorial/chap12.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutNonabelian.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCrossedMods.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>5</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>6</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>7</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutTensorSquare.html</Link><LinkText>8</LinkText></URL>
</Description> </ManSection>
<ManSection> <Var Name="Representation of elements in the bar resolution"/> <Description> <P/> For a group G we denote by <M>B_n(G)</M> the free <M>\mathbb ZG</M>-module with basis the lists <M>[g_1 | g_2 | ... | g_n]</M> where the <M>g_i</M> range over <M>G</M>. <Br/> <Br/> We represent a word <Br/> <Br/> <M>w = h_1.[g_{11} | g_{12} | ... | g_{1n}] - h_2.[g_{21} | g_{22} | ... | g_{2n}] + ... + h_k.[g_{k1} | g_{k2} | ... | g_{kn}] </M> <Br/> <Br/> in <M>B_n(G)</M> as a list of lists: <Br/> <Br/> <M> [ [+1,h_1,g_{11} , g_{12} , ... , g_{1n}] , [-1, h_2,g_{21} , g_{22} , ... | g_{2n}] + ... + [+1, h_k,g_{k1} , g_{k2} , ... , g_{kn}] </M>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Var Name="BarResolutionBoundary"/> <Description> <P/> This function inputs a word <M>w</M> in the bar resolution module <M>B_n(G)</M> and returns its image under the boundary homomorphism <M>d_n\colon B_n(G) \rightarrow B_{n-1}(G)</M> in the bar resolution. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Var Name="BarResolutionHomotopy"/> <Description> <P/> This function inputs a word <M>w</M> in the bar resolution module <M>B_n(G)</M> and returns its image under the contracting homotopy <M>h_n\colon B_n(G) \rightarrow B_{n+1}(G)</M> in the bar resolution. <Br/> <Br/> This function is currently being implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Var Name="Representation of elements in the bar complex"/> <Description> <P/> For a group G we denote by <M>BC_n(G)</M> the free abelian group with basis the lists <M>[g_1 | g_2 | ... | g_n]</M> where the <M>g_i</M> range over <M>G</M>. <Br/> <Br/> We represent a word <Br/> <Br/> <M>w = [g_{11} | g_{12} | ... | g_{1n}] - [g_{21} | g_{22} | ... | g_{2n}] + ... + [g_{k1} | g_{k2} | ... | g_{kn}] </M> <Br/> <Br/> in <M>BC_n(G)</M> as a list of lists: <Br/> <Br/> <M> [ [+1,g_{11} , g_{12} , ... , g_{1n}] , [-1, g_{21} , g_{22} , ... | g_{2n}] + ... + [+1, g_{k1} , g_{k2} , ... , g_{kn}] </M>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Var Name="BarComplexBoundary"/> <Description> <P/> This function inputs a word <M>w</M> in the n-th term of the bar complex <M>BC_n(G)</M> and returns its image under the boundary homomorphism <M>d_n\colon BC_n(G) \rightarrow BC_{n-1}(G)</M> in the bar complex. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="BarResolutionEquivalence" Arg="R"/> <Description> <P/> This function inputs a free <M>ZG</M>-resolution <M>R</M>. It returns a component object HE with components <List> <Item> HE!.phi(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>B_n(G)</M>. It returns the image of <M>w</M> in <M>R_n</M> under a chain equivalence <M>\phi\colon B_n(G) \rightarrow R_n</M>.</Item> <Item> HE!.psi(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>R_n</M>. It returns the image of <M>w</M> in <M>B_n(G)</M> under a chain equivalence <M>\psi\colon R_n \rightarrow B_n(G)</M>.</Item> <Item> HE!.equiv(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>B_n(G)</M>. It returns the image of <M>w</M> in <M>B_{n+1}(G)</M> under a <M>ZG</M>-equivariant homomorphism <Br/> <Br/> <M>equiv(n,-) \colon B_n(G) \rightarrow B_{n+1}(G)</M> <Br/> <Br/> satisfying <Display>w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . </Display> where <M>d(n,-)\colon B_n(G) \rightarrow B_{n-1}(G)</M> is the boundary homomorphism in the bar resolution. </Item> </List> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="BarComplexEquivalence" Arg="R"/> <Description> <P/> This function inputs a free <M>ZG</M>-resolution <M>R</M>. It first constructs the chain complex <M>T=TensorWithIntegerts(R)</M>. The function returns a component object HE with components <List> <Item> HE!.phi(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>BC_n(G)</M>. It returns the image of <M>w</M> in <M>T_n</M> under a chain equivalence <M>\phi\colon BC_n(G) \rightarrow T_n</M>.</Item> <Item> HE!.psi(n,w) is a function which inputs a non-negative integer <M>n</M> and an element <M>w</M> in <M>T_n</M>. It returns the image of <M>w</M> in <M>BC_n(G)</M> under a chain equivalence <M>\psi\colon T_n \rightarrow BC_n(G)</M>.</Item> <Item> HE!.equiv(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>BC_n(G)</M>. It returns the image of <M>w</M> in <M>BC_{n+1}(G)</M> under a homomorphism <Br/> <Br/> <M>equiv(n,-) \colon BC_n(G) \rightarrow BC_{n+1}(G)</M> <Br/> <Br/> satisfying <Display>w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . </Display> where <M>d(n,-)\colon BC_n(G) \rightarrow BC_{n-1}(G)</M> is the boundary homomorphism in the bar complex. </Item> </List> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Var Name="Representation of elements in the bar cocomplex"/> <Description> <P/> For a group G we denote by <M>BC^n(G)</M> the free abelian group with basis the lists <M>[g_1 | g_2 | ... | g_n]</M> where the <M>g_i</M> range over <M>G</M>. <Br/> <Br/> We represent a word <Br/> <Br/> <M>w = [g_{11} | g_{12} | ... | g_{1n}] - [g_{21} | g_{22} | ... | g_{2n}] + ... + [g_{k1} | g_{k2} | ... | g_{kn}] </M> <Br/> <Br/> in <M>BC^n(G)</M> as a list of lists: <Br/> <Br/> <M> [ [+1,g_{11} , g_{12} , ... , g_{1n}] , [-1, g_{21} , g_{22} , ... | g_{2n}] + ... + [+1, g_{k1} , g_{k2} , ... , g_{kn}] </M>. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Var Name="BarCocomplexCoboundary"/> <Description> <P/> This function inputs a word <M>w</M> in the n-th term of the bar cocomplex <M>BC^n(G)</M> and returns its image under the coboundary homomorphism <M>d^n\colon BC^n(G) \rightarrow BC^{n+1}(G)</M> in the bar cocomplex. <Br/> <Br/> This function was implemented by <B>Van Luyen Le</B>. <P/><B>Examples:</B>
</Description> </ManSection> </Section> </Chapter>
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