<Chapter><Heading> Chain complexes</Heading> <Section><Heading> &nbsp;</Heading> 
<ManSection> <Func Name="ChainComplex" Arg="T"/> <Description> <P/> Inputs a pure cubical complex, or cubical complex, or simplicial complex <M>T</M> and returns the (often very large) cellular chain complex of <M>T</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap1.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap3.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap4.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap10.html</Link><LinkText>4</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap12.html</Link><LinkText>5</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutMetrics.html</Link><LinkText>6</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutBredon.html</Link><LinkText>7</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutPersistent.html</Link><LinkText>8</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>9</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>10</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutCubical.html</Link><LinkText>11</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>12</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>13</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="ChainComplexOfPair" Arg="T,S"/> <Description> <P/> Inputs a pure cubical complex or cubical complex <M>T</M> and contractible subcomplex <M>S</M>. It returns the quotient <M>C(T)/C(S)</M> of cellular chain complexes. <P/><B>Examples:</B> <URL><Link>../tutorial/chap10.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutCubical.html</Link><LinkText>2</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="ChevalleyEilenbergComplex" Arg="X,n"/> <Description> <P/> Inputs either a Lie algebra <M>X=A</M> (over the ring of integers <M>Z</M> or over a field <M>K</M>) or a homomorphism of Lie algebras <M>X=(f:A \longrightarrow B)</M>, together with a positive integer <M>n</M>. It returns either the first <M>n</M> terms of the Chevalley-Eilenberg chain complex <M>C(A)</M>, or the induced map of Chevalley-Eilenberg complexes <M>C(f):C(A) \longrightarrow C(B)</M>. <P/> (The homology of the Chevalley-Eilenberg complex <M>C(A)</M> is by definition the homology of the Lie algebra <M>A</M> with trivial coefficients in <M>Z</M> or <M>K</M>). <P/> This function was written by <B>Pablo Fernandez Ascariz</B> <P/><B>Examples:</B> <URL><Link>../tutorial/chap7.html</Link><LinkText>1</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="LeibnizComplex" Arg="X,n"/> <Description> <P/> Inputs either a Lie or Leibniz algebra <M>X=A</M> (over the ring of integers <M>Z</M> or over a field <M>K</M>) or a homomorphism of Lie or Leibniz algebras <M>X=(f:A \longrightarrow B)</M>, together with a positive integer <M>n</M>. It returns either the first <M>n</M> terms of the Leibniz chain complex <M>C(A)</M>, or the induced map of Leibniz complexes <M>C(f):C(A) \longrightarrow C(B)</M>. <P/> (The Leibniz complex <M>C(A)</M> was defined by J.-L.Loday. Its homology is by definition the Leibniz homology of the algebra <M>A</M>). <P/> This function was written by <B>Pablo Fernandez Ascariz</B> <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="SuspendedChainComplex" Arg="C"/> <Description> <P/> Inputs a chain complex <M>C</M> and returns the chain complex <M>S</M> defined by applying the degree shift <M>S_n = C_{n-1}</M> to chain groups and boundary homomorphisms. <P/><B>Examples:</B> <URL><Link>../tutorial/chap10.html</Link><LinkText>1</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="ReducedSuspendedChainComplex" Arg="C"/> <Description> <P/> Inputs a chain complex <M>C</M> and returns the chain complex <M>S</M> defined by applying the degree shift <M>S_n = C_{n-1}</M> to chain groups and boundary homomorphisms for all <M>n > 0</M>. The chain complex <M>S</M> has trivial homology in degree <M>0</M> and <M>S_0=\mathbb Z</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap10.html</Link><LinkText>1</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="CoreducedChainComplex" Arg="C"/> <Func Name="CoreducedChainComplex" Arg="C,2"/> <Description> <P/> Inputs a chain complex <M>C</M> and returns a quasi-isomorphic chain complex <M>D</M>. In many cases the complex <M>D</M> should be smaller than <M>C</M>. If an optional second input argument is set equal to 2 then an alternative method is used for reducing the size of the chain complex. <P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>1</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="TensorProductOfChainComplexes" Arg="C,D"/> <Description> <P/> Inputs two chain complexes <M>C</M> and <M>D</M> of the same characteristic and returns their tensor product as a chain complex. <P/> This function was written by <B> Le Van Luyen</B>. <P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutSimplicialGroups.html</Link><LinkText>1</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="LefschetzNumber" Arg="F"/> <Description> <P/> Inputs a chain map <M>F\colon C\rightarrow C</M> with common source and target. It returns the Lefschetz number of the map (that is, the alternating sum of the traces of the homology maps in each degree). <P/><B>Examples:</B> 
</Description> </ManSection> </Section> </Chapter>