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<div class="ChapSects"><a href="chap25.html#X7D818E5F80F4CF63">25 <span class="Heading"> Simplicial groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X7CFDEEC07F15CF82">25.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X8624D2DF8433E4BF">25.1-1 NerveOfCatOneGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X78C5BCC37F629F5D">25.1-2 EilenbergMacLaneSimplicialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X7D7FCFA37DE30855">25.1-3 EilenbergMacLaneSimplicialGroupMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X80CF81187D3A7C0A">25.1-4 MooreComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X7EC0E55F818D234C">25.1-5 ChainComplexOfSimplicialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X84CCE7557D779553">25.1-6 SimplicialGroupMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X7F2E058F7AF17E82">25.1-7 HomotopyGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X805075A27E048EAE">25.1-8 Representation of elements in the bar resolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X823D749782FFFE8B">25.1-9 BarResolutionBoundary</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X80843E4E79F4A64B">25.1-10 BarResolutionHomotopy</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X7B3D23AB78BA441F">25.1-11 Representation of elements in the bar complex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X82DF5A738004E574">25.1-12 BarComplexBoundary</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X8713DCF68409B0F3">25.1-13 BarResolutionEquivalence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X8133375F7E120A6F">25.1-14 BarComplexEquivalence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X87D4504181AD3006">25.1-15 Representation of elements in the bar cocomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap25.html#X7C5E8E197E9331F1">25.1-16 BarCocomplexCoboundary</a></span>
</div></div>
</div>
<h3>25 <span class="Heading"> Simplicial groups</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>25.1 <span class="Heading"> </span></h4>
<p><a id="X8624D2DF8433E4BF" name="X8624D2DF8433E4BF"></a></p>
<h5>25.1-1 NerveOfCatOneGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NerveOfCatOneGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns the low-dimensional part of the nerve of <span class="SimpleMath">G</span> as a simplicial group of length <span class="SimpleMath">n</span>. <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 <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap12.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">3</a></span> </p>
<p><a id="X78C5BCC37F629F5D" name="X78C5BCC37F629F5D"></a></p>
<h5>25.1-2 EilenbergMacLaneSimplicialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EilenbergMacLaneSimplicialGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">dim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span>, a positive integer <span class="SimpleMath">n</span>, and a positive integer <span class="SimpleMath">dim</span>. The function returns the first <span class="SimpleMath">1+dim</span> terms of a simplicial group with <span class="SimpleMath">n-1</span>st homotopy group equal to <span class="SimpleMath">G</span> and all other homotopy groups equal to zero. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap3.html">1</a></span> , <span class="URL"><a href="../tutorial/chap12.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">5</a></span> </p>
<p><a id="X7D7FCFA37DE30855" name="X7D7FCFA37DE30855"></a></p>
<h5>25.1-3 EilenbergMacLaneSimplicialGroupMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EilenbergMacLaneSimplicialGroupMap</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a group homomorphism <span class="SimpleMath">f:G→ Q</span>, a positive integer <span class="SimpleMath">n</span>, and a positive integer <span class="SimpleMath">dim</span>. The function returns the first <span class="SimpleMath">1+dim</span> terms of a simplicial group homomorphism <span class="SimpleMath">f:K(G,n) → K(Q,n)</span> of Eilenberg-MacLane simplicial groups. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X80CF81187D3A7C0A" name="X80CF81187D3A7C0A"></a></p>
<h5>25.1-4 MooreComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MooreComplex</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a simplicial group <span class="SimpleMath">G</span> and returns its Moore complex as a <span class="SimpleMath">G</span>-complex. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7EC0E55F818D234C" name="X7EC0E55F818D234C"></a></p>
<h5>25.1-5 ChainComplexOfSimplicialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChainComplexOfSimplicialGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a simplicial group <span class="SimpleMath">G</span> and returns the cellular chain complex <span class="SimpleMath">C</span> of a CW-space <span class="SimpleMath">X</span> represented by the homotopy type of the simplicial group. Thus the homology groups of <span class="SimpleMath">C</span> are the integral homology groups of <span class="SimpleMath">X</span>. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap3.html">1</a></span> , <span class="URL"><a href="../tutorial/chap12.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">6</a></span> </p>
<p><a id="X84CCE7557D779553" name="X84CCE7557D779553"></a></p>
<h5>25.1-6 SimplicialGroupMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimplicialGroupMap</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a homomorphism <span class="SimpleMath">f:G→ Q</span> of simplicial groups. The function returns an induced map <span class="SimpleMath">f:C(G) → C(Q)</span> of chain complexes whose homology is the integral homology of the simplicial group G and Q respectively. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7F2E058F7AF17E82" name="X7F2E058F7AF17E82"></a></p>
<h5>25.1-7 HomotopyGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomotopyGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a simplicial group <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. The integer <span class="SimpleMath">n</span> must be less than the length of <span class="SimpleMath">G</span>. It returns, as a group, the (n)-th homology group of its Moore complex. Thus HomotopyGroup(G,0) returns the "fundamental group" of <span class="SimpleMath">G</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap12.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutNonabelian.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCrossedMods.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutquasi.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTensorSquare.html">8</a></span> </p>
<p><a id="X805075A27E048EAE" name="X805075A27E048EAE"></a></p>
<h5>25.1-8 Representation of elements in the bar resolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Representation of elements in the bar resolution</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>For a group G we denote by <span class="SimpleMath">B_n(G)</span> the free <span class="SimpleMath">ZG</span>-module with basis the lists <span class="SimpleMath">[g_1 | g_2 | ... | g_n]</span> where the <span class="SimpleMath">g_i</span> range over <span class="SimpleMath">G</span>. <br /> <br /> We represent a word <br /> <br /> <span class="SimpleMath">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]</span> <br /> <br /> in <span class="SimpleMath">B_n(G)</span> as a list of lists: <br /> <br /> <span class="SimpleMath">[ [+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]</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X823D749782FFFE8B" name="X823D749782FFFE8B"></a></p>
<h5>25.1-9 BarResolutionBoundary</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BarResolutionBoundary</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This function inputs a word <span class="SimpleMath">w</span> in the bar resolution module <span class="SimpleMath">B_n(G)</span> and returns its image under the boundary homomorphism <span class="SimpleMath">d_n: B_n(G) → B_n-1(G)</span> in the bar resolution. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X80843E4E79F4A64B" name="X80843E4E79F4A64B"></a></p>
<h5>25.1-10 BarResolutionHomotopy</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BarResolutionHomotopy</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This function inputs a word <span class="SimpleMath">w</span> in the bar resolution module <span class="SimpleMath">B_n(G)</span> and returns its image under the contracting homotopy <span class="SimpleMath">h_n: B_n(G) → B_n+1(G)</span> in the bar resolution. <br /> <br /> This function is currently being implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7B3D23AB78BA441F" name="X7B3D23AB78BA441F"></a></p>
<h5>25.1-11 Representation of elements in the bar complex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Representation of elements in the bar complex</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>For a group G we denote by <span class="SimpleMath">BC_n(G)</span> the free abelian group with basis the lists <span class="SimpleMath">[g_1 | g_2 | ... | g_n]</span> where the <span class="SimpleMath">g_i</span> range over <span class="SimpleMath">G</span>. <br /> <br /> We represent a word <br /> <br /> <span class="SimpleMath">w = [g_11 | g_12 | ... | g_1n] - [g_21 | g_22 | ... | g_2n] + ... + [g_k1 | g_k2 | ... | g_kn]</span> <br /> <br /> in <span class="SimpleMath">BC_n(G)</span> as a list of lists: <br /> <br /> <span class="SimpleMath">[ [+1,g_11 , g_12 , ... , g_1n] , [-1, g_21 , g_22 , ... | g_2n] + ... + [+1, g_k1 , g_k2 , ... , g_kn]</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X82DF5A738004E574" name="X82DF5A738004E574"></a></p>
<h5>25.1-12 BarComplexBoundary</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BarComplexBoundary</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This function inputs a word <span class="SimpleMath">w</span> in the n-th term of the bar complex <span class="SimpleMath">BC_n(G)</span> and returns its image under the boundary homomorphism <span class="SimpleMath">d_n: BC_n(G) → BC_n-1(G)</span> in the bar complex. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8713DCF68409B0F3" name="X8713DCF68409B0F3"></a></p>
<h5>25.1-13 BarResolutionEquivalence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BarResolutionEquivalence</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span>. It returns a component object HE with components</p>
<ul>
<li><p>HE!.phi(n,w) is a function which inputs a non-negative integer <span class="SimpleMath">n</span> and a word <span class="SimpleMath">w</span> in <span class="SimpleMath">B_n(G)</span>. It returns the image of <span class="SimpleMath">w</span> in <span class="SimpleMath">R_n</span> under a chain equivalence <span class="SimpleMath">ϕ: B_n(G) → R_n</span>.</p>
</li>
<li><p>HE!.psi(n,w) is a function which inputs a non-negative integer <span class="SimpleMath">n</span> and a word <span class="SimpleMath">w</span> in <span class="SimpleMath">R_n</span>. It returns the image of <span class="SimpleMath">w</span> in <span class="SimpleMath">B_n(G)</span> under a chain equivalence <span class="SimpleMath">ψ: R_n → B_n(G)</span>.</p>
</li>
<li><p>HE!.equiv(n,w) is a function which inputs a non-negative integer <span class="SimpleMath">n</span> and a word <span class="SimpleMath">w</span> in <span class="SimpleMath">B_n(G)</span>. It returns the image of <span class="SimpleMath">w</span> in <span class="SimpleMath">B_n+1(G)</span> under a <span class="SimpleMath">ZG</span>-equivariant homomorphism <br /> <br /> <span class="SimpleMath">equiv(n,-) : B_n(G) → B_n+1(G)</span> <br /> <br /> satisfying</p>
<p class="pcenter">w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . </p>
<p>where <span class="SimpleMath">d(n,-): B_n(G) → B_n-1(G)</span> is the boundary homomorphism in the bar resolution.</p>
</li>
</ul>
<p>This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X8133375F7E120A6F" name="X8133375F7E120A6F"></a></p>
<h5>25.1-14 BarComplexEquivalence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BarComplexEquivalence</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span>. It first constructs the chain complex <span class="SimpleMath">T=TensorWithIntegerts(R)</span>. The function returns a component object HE with components</p>
<ul>
<li><p>HE!.phi(n,w) is a function which inputs a non-negative integer <span class="SimpleMath">n</span> and a word <span class="SimpleMath">w</span> in <span class="SimpleMath">BC_n(G)</span>. It returns the image of <span class="SimpleMath">w</span> in <span class="SimpleMath">T_n</span> under a chain equivalence <span class="SimpleMath">ϕ: BC_n(G) → T_n</span>.</p>
</li>
<li><p>HE!.psi(n,w) is a function which inputs a non-negative integer <span class="SimpleMath">n</span> and an element <span class="SimpleMath">w</span> in <span class="SimpleMath">T_n</span>. It returns the image of <span class="SimpleMath">w</span> in <span class="SimpleMath">BC_n(G)</span> under a chain equivalence <span class="SimpleMath">ψ: T_n → BC_n(G)</span>.</p>
</li>
<li><p>HE!.equiv(n,w) is a function which inputs a non-negative integer <span class="SimpleMath">n</span> and a word <span class="SimpleMath">w</span> in <span class="SimpleMath">BC_n(G)</span>. It returns the image of <span class="SimpleMath">w</span> in <span class="SimpleMath">BC_n+1(G)</span> under a homomorphism <br /> <br /> <span class="SimpleMath">equiv(n,-) : BC_n(G) → BC_n+1(G)</span> <br /> <br /> satisfying</p>
<p class="pcenter">w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . </p>
<p>where <span class="SimpleMath">d(n,-): BC_n(G) → BC_n-1(G)</span> is the boundary homomorphism in the bar complex.</p>
</li>
</ul>
<p>This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X87D4504181AD3006" name="X87D4504181AD3006"></a></p>
<h5>25.1-15 Representation of elements in the bar cocomplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Representation of elements in the bar cocomplex</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>For a group G we denote by <span class="SimpleMath">BC^n(G)</span> the free abelian group with basis the lists <span class="SimpleMath">[g_1 | g_2 | ... | g_n]</span> where the <span class="SimpleMath">g_i</span> range over <span class="SimpleMath">G</span>. <br /> <br /> We represent a word <br /> <br /> <span class="SimpleMath">w = [g_11 | g_12 | ... | g_1n] - [g_21 | g_22 | ... | g_2n] + ... + [g_k1 | g_k2 | ... | g_kn]</span> <br /> <br /> in <span class="SimpleMath">BC^n(G)</span> as a list of lists: <br /> <br /> <span class="SimpleMath">[ [+1,g_11 , g_12 , ... , g_1n] , [-1, g_21 , g_22 , ... | g_2n] + ... + [+1, g_k1 , g_k2 , ... , g_kn]</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7C5E8E197E9331F1" name="X7C5E8E197E9331F1"></a></p>
<h5>25.1-16 BarCocomplexCoboundary</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BarCocomplexCoboundary</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This function inputs a word <span class="SimpleMath">w</span> in the n-th term of the bar cocomplex <span class="SimpleMath">BC^n(G)</span> and returns its image under the coboundary homomorphism <span class="SimpleMath">d^n: BC^n(G) → BC^n+1(G)</span> in the bar cocomplex. <br /> <br /> This function was implemented by <strong class="button">Van Luyen Le</strong>.</p>
<p><strong class="button">Examples:</strong></p>
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