<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChainComplex</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex, or cubical complex, or simplicial complex <span class="SimpleMath">\(T\)</span> and returns the (often very large) cellular chain complex of <span class="SimpleMath">\(T\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChainComplexOfPair</code>( <var class="Arg">T</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex or cubical complex <span class="SimpleMath">\(T\)</span> and contractible subcomplex <span class="SimpleMath">\(S\)</span>. It returns the quotient <span class="SimpleMath">\(C(T)/C(S)\)</span> of cellular chain complexes.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChevalleyEilenbergComplex</code>( <var class="Arg">X</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a Lie algebra <span class="SimpleMath">\(X=A\)</span> (over the ring of integers <span class="SimpleMath">\(Z\)</span> or over a field <span class="SimpleMath">\(K\)</span>) or a homomorphism of Lie algebras <span class="SimpleMath">\(X=(f:A \longrightarrow B)\)</span>, together with a positive integer <span class="SimpleMath">\(n\)</span>. It returns either the first <span class="SimpleMath">\(n\)</span> terms of the Chevalley-Eilenberg chain complex <span class="SimpleMath">\(C(A)\)</span>, or the induced map of Chevalley-Eilenberg complexes <span class="SimpleMath">\(C(f):C(A) \longrightarrow C(B)\)</span>.</p>
<p>(The homology of the Chevalley-Eilenberg complex <span class="SimpleMath">\(C(A)\)</span> is by definition the homology of the Lie algebra <span class="SimpleMath">\(A\)</span> with trivial coefficients in <span class="SimpleMath">\(Z\)</span> or <span class="SimpleMath">\(K\)</span>).</p>
<p>This function was written by <strong class="button">Pablo Fernandez Ascariz</strong></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeibnizComplex</code>( <var class="Arg">X</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a Lie or Leibniz algebra <span class="SimpleMath">\(X=A\)</span> (over the ring of integers <span class="SimpleMath">\(Z\)</span> or over a field <span class="SimpleMath">\(K\)</span>) or a homomorphism of Lie or Leibniz algebras <span class="SimpleMath">\(X=(f:A \longrightarrow B)\)</span>, together with a positive integer <span class="SimpleMath">\(n\)</span>. It returns either the first <span class="SimpleMath">\(n\)</span> terms of the Leibniz chain complex <span class="SimpleMath">\(C(A)\)</span>, or the induced map of Leibniz complexes <span class="SimpleMath">\(C(f):C(A) \longrightarrow C(B)\)</span>.</p>
<p>(The Leibniz complex <span class="SimpleMath">\(C(A)\)</span> was defined by J.-L.Loday. Its homology is by definition the Leibniz homology of the algebra <span class="SimpleMath">\(A\)</span>).</p>
<p>This function was written by <strong class="button">Pablo Fernandez Ascariz</strong></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SuspendedChainComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain complex <span class="SimpleMath">\(C\)</span> and returns the chain complex <span class="SimpleMath">\(S\)</span> defined by applying the degree shift <span class="SimpleMath">\(S_n = C_{n-1}\)</span> to chain groups and boundary homomorphisms.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReducedSuspendedChainComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain complex <span class="SimpleMath">\(C\)</span> and returns the chain complex <span class="SimpleMath">\(S\)</span> defined by applying the degree shift <span class="SimpleMath">\(S_n = C_{n-1}\)</span> to chain groups and boundary homomorphisms for all <span class="SimpleMath">\(n > 0\)</span>. The chain complex <span class="SimpleMath">\(S\)</span> has trivial homology in degree <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(S_0=\mathbb Z\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoreducedChainComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoreducedChainComplex</code>( <var class="Arg">C</var>, <var class="Arg">2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain complex <span class="SimpleMath">\(C\)</span> and returns a quasi-isomorphic chain complex <span class="SimpleMath">\(D\)</span>. In many cases the complex <span class="SimpleMath">\(D\)</span> should be smaller than <span class="SimpleMath">\(C\)</span>. 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductOfChainComplexes</code>( <var class="Arg">C</var>, <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two chain complexes <span class="SimpleMath">\(C\)</span> and <span class="SimpleMath">\(D\)</span> of the same characteristic and returns their tensor product as a chain complex.</p>
<p>This function was written by <strong class="button"> Le Van Luyen</strong>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LefschetzNumber</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain map <span class="SimpleMath">\(F\colon C\rightarrow C\)</span> 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>
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