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<p><a id="X7B0C706A848A2542" name="X7B0C706A848A2542"></a></p>
<div class="ChapSects"><a href="chap8_mj.html#X7B0C706A848A2542">8 <span class="Heading">(Co-)Homology of simplicial complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7DCD23807E4DD2B2">8.1 <span class="Heading">Homology computation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X8762B952823BDEF3">8.1-1 SCBoundaryOperatorMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7E5983748742026D">8.1-2 SCBoundarySimplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7F9B47A97F5FA005">8.1-3 SCHomologyBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7C569D9A7868C697">8.1-4 SCHomologyBasisAsSimplices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X858989CE797A8366">8.1-5 SCHomologyInternal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X86A940277A35331B">8.2 <span class="Heading">Cohomology computation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X8527CC347F606648">8.2-1 SCCoboundaryOperatorMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7926D71A8169629B">8.2-2 SCCohomology</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X85910A6B824D63A4">8.2-3 SCCohomologyBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7A5047E57E150716">8.2-4 SCCohomologyBasisAsSimplices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7A9137847BF96DF3">8.2-5 SCCupProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X835E7016826D14A8">8.2-6 SCIntersectionForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X791C2B45872A71CA">8.2-7 SCIntersectionFormParity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X7B5F948B84F5E316">8.2-8 SCIntersectionFormDimensionality</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8_mj.html#X87375A6978F3C8CD">8.2-9 SCIntersectionFormSignature</a></span>
</div></div>
</div>

<h3>8 <span class="Heading">(Co-)Homology of simplicial complexes</span></h3>

<p>By default, <strong class="pkg">simpcomp</strong> uses an algorithm based on discrete Morse theory (see Chapter <a href="chap12_mj.html#X7E9FD84F822A58D6"><span class="RefLink">12</span></a>, <code class="func">SCHomology</code> (<a href="chap7_mj.html#X78D66254858CE901"><span class="RefLink">7.3-9</span></a>)) for its homology computations. However, some additional (co-)homology related functionality cannot be realised using this algorithm. For this, <strong class="pkg">simpcomp</strong> contains an additional (co-)homology algorithm (cf. <code class="func">SCHomologyInternal</code> (<a href="chap8_mj.html#X858989CE797A8366"><span class="RefLink">8.1-5</span></a>)), which will be presented in this chapter.</p>

<p>Furthermore, whenever possible <strong class="pkg">simpcomp</strong> makes use of the <strong class="pkg">GAP</strong> package ''homology'' <a href="chapBib_mj.html#biBDumas04Homology">[DHSW11]</a>, for an alternative method to calculate homology groups (cf. <code class="func">SCHomologyClassic</code> (<a href="chap6_mj.html#X864978877E7D4DA0"><span class="RefLink">6.9-31</span></a>)) which sometimes is much faster than the built-in discrete Morse theory algorithm.</p>

<p><a id="X7DCD23807E4DD2B2" name="X7DCD23807E4DD2B2"></a></p>

<h4>8.1 <span class="Heading">Homology computation</span></h4>

<p>Apart from calculating boundaries of simplices, boundary matrices or the simplicial homology of a given complex, <strong class="pkg">simpcomp</strong> is also able to compute a basis of the homology groups.</p>

<p><a id="X8762B952823BDEF3" name="X8762B952823BDEF3"></a></p>

<h5>8.1-1 SCBoundaryOperatorMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCBoundaryOperatorMatrix</code>( <var class="Arg">complex</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a rectangular matrix upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates the matrix of the boundary operator <span class="SimpleMath">\(\partial_{\textit{k+1}}\)</span> of a simplicial complex <var class="Arg">complex</var>. Note that each column contains the boundaries of a <var class="Arg">k</var><span class="SimpleMath">\(+1\)</span>-simplex as a list of oriented <var class="Arg">k</var>-simplices and that the matrix is stored as a list of row vectors (as usual in GAP).</p>

<div class="example"><pre>
gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],\
                       [2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);;
gap> mat:=SCBoundaryOperatorMatrix(c,1);
[ [ 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
   [ -1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0 ],
   [ 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0 ],
   [ 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, 0 ],
   [ 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 1 ],
   [ 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, -1 ] ]
</pre></div>

<p><a id="X7E5983748742026D" name="X7E5983748742026D"></a></p>

<h5>8.1-2 SCBoundarySimplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCBoundarySimplex</code>( <var class="Arg">simplex</var>, <var class="Arg">orientation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates the boundary of a given <var class="Arg">simplex</var>. If the flag <var class="Arg">orientation</var> is set to <code class="keyw">true</code>, the function returns the boundary as a list of oriented simplices of the form [ ORIENTATION, SIMPLEX ], where ORIENTATION is either +1 or -1 and a value of +1 means that SIMPLEX is positively oriented and a value of -1 that SIMPLEX is negatively oriented. If <var class="Arg">orientation</var> is set to <code class="keyw">false</code>, an unoriented list of simplices is returned.</p>

<div class="example"><pre>
gap> SCBoundarySimplex([1..5],true);
[ [ -1, [ 2, 3, 4, 5 ] ], [ 1, [ 1, 3, 4, 5 ] ], [ -1, [ 1, 2, 4, 5 ] ],
   [ 1, [ 1, 2, 3, 5 ] ], [ -1, [ 1, 2, 3, 4 ] ] ]
gap> SCBoundarySimplex([1..5],false);
[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 5 ], [ 1, 2, 4, 5 ], [ 1, 2, 3, 5 ],
   [ 1, 2, 3, 4 ] ]
</pre></div>

<p><a id="X7F9B47A97F5FA005" name="X7F9B47A97F5FA005"></a></p>

<h5>8.1-3 SCHomologyBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCHomologyBasis</code>( <var class="Arg">complex</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of pairs of the form <code class="code">[ integer, list of linear combinations of simplices ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates a set of basis elements for the <var class="Arg">k</var>-dimensional homology group (with integer coefficients) of a simplicial complex <var class="Arg">complex</var>. The entries of the returned list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ], where the value MODULUS is 1 for the basis elements of the free part of the <var class="Arg">k</var>-th homology group and <span class="SimpleMath">\(q\geq 2\)</span> for the basis elements of the <span class="SimpleMath">\(q\)</span>-torsion part. In contrast to the function <code class="func">SCHomologyBasisAsSimplices</code> (<a href="chap8_mj.html#X7C569D9A7868C697"><span class="RefLink">8.1-4</span></a>) the basis elements are stored as lists of coefficient-index pairs referring to the simplices of the complex, i.e. a basis element of the form <span class="SimpleMath">\([ [ \lambda_1, i], [\lambda_2, j], \dots ] \dots\)</span> encodes the linear combination of simplices of the form <span class="SimpleMath">\(\lambda_1*\Delta_1+\lambda_2*\Delta_2\)</span> with <span class="SimpleMath">\(\Delta_1\)</span>=<code class="code">SCSkel(complex,k)[i]</code>, <span class="SimpleMath">\(\Delta_2\)</span>=<code class="code">SCSkel(complex,k)[j]</code> and so on.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("(S^2xS^1)#RP^3");
[ [ 237, "(S^2xS^1)#RP^3" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> SCHomologyBasis(c,1);
[ [ 1, [ [ 1, 12 ], [ -1, 7 ], [ 1, 1 ] ] ],
   [ 2, [ [ 1, 68 ], [ -1, 69 ], [ -1, 71 ], [ 2, 72 ], [ -2, 73 ] ] ] ]
</pre></div>

<p><a id="X7C569D9A7868C697" name="X7C569D9A7868C697"></a></p>

<h5>8.1-4 SCHomologyBasisAsSimplices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCHomologyBasisAsSimplices</code>( <var class="Arg">complex</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of pairs of the form <code class="code">[ integer, list of linear combinations of simplices ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates a set of basis elements for the <var class="Arg">k</var>-dimensional homology group (with integer coefficients) of a simplicial complex <var class="Arg">complex</var>. The entries of the returned list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ], where the value MODULUS is 1 for the basis elements of the free part of the <var class="Arg">k</var>-th homology group and <span class="SimpleMath">\(q\geq 2\)</span> for the basis elements of the <span class="SimpleMath">\(q\)</span>-torsion part. In contrast to the function <code class="func">SCHomologyBasis</code> (<a href="chap8_mj.html#X7F9B47A97F5FA005"><span class="RefLink">8.1-3</span></a>) the basis elements are stored as lists of coefficient-simplex pairs, i.e. a basis element of the form <span class="SimpleMath">\([ [ \lambda_1, \Delta_1], [\lambda_2, \Delta_2], \dots ]\)</span> encodes the linear combination of simplices of the form <span class="SimpleMath">\(\lambda_1*\Delta_1+\lambda_2*\Delta_2 + \dots\)</span>.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("(S^2xS^1)#RP^3");
[ [ 237, "(S^2xS^1)#RP^3" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> SCHomologyBasisAsSimplices(c,1);
[ [ 1, [ [ 1, [ 2, 8 ] ], [ -1, [ 1, 8 ] ], [ 1, [ 1, 2 ] ] ] ],
   [ 2,
       [ [ 1, [ 11, 12 ] ], [ -1, [ 11, 13 ] ], [ -1, [ 12, 13 ] ],
           [ 2, [ 12, 14 ] ], [ -2, [ 13, 14 ] ] ] ] ]
</pre></div>

<p><a id="X858989CE797A8366" name="X858989CE797A8366"></a></p>

<h5>8.1-5 SCHomologyInternal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCHomologyInternal</code>( <var class="Arg">complex</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of pairs of the form <code class="code">[ integer, list ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>This function computes the reduced simplicial homology with integer coefficients of a given simplicial complex <var class="Arg">complex</var> with integer coefficients. It uses the algorithm described in <a href="chapBib_mj.html#biBDesbrun08DiscDiffFormCompModel">[DKT08]</a>.</p>

<p>The output is a list of homology groups of the form <span class="SimpleMath">\([H_0,....,H_d]\)</span>, where <span class="SimpleMath">\(d\)</span> is the dimension of <var class="Arg">complex</var>. The format of the homology groups <span class="SimpleMath">\(H_i\)</span> is given in terms of their maximal cyclic subgroups, i.e. a homology group <span class="SimpleMath">\(H_i\cong \mathbb{Z}^f + \mathbb{Z} / t_1 \mathbb{Z} \times \dots \times \mathbb{Z} / t_n \mathbb{Z}\)</span> is returned in form of a list <span class="SimpleMath">\([ f, [t_1,...,t_n] ]\)</span>, where <span class="SimpleMath">\(f\)</span> is the (integer) free part of <span class="SimpleMath">\(H_i\)</span> and <span class="SimpleMath">\(t_i\)</span> denotes the torsion parts of <span class="SimpleMath">\(H_i\)</span> ordered in weakly incresing size. See also <code class="func">SCHomology</code> (<a href="chap7_mj.html#X78D66254858CE901"><span class="RefLink">7.3-9</span></a>) and <code class="func">SCHomologyClassic</code> (<a href="chap6_mj.html#X864978877E7D4DA0"><span class="RefLink">6.9-31</span></a>).</p>

<div class="example"><pre>
gap> c:=SCSurface(1,false);;
gap> SCHomologyInternal(c);
[ [ 0, [  ] ], [ 0, [ 2 ] ], [ 0, [  ] ] ]
</pre></div>

<p><a id="X86A940277A35331B" name="X86A940277A35331B"></a></p>

<h4>8.2 <span class="Heading">Cohomology computation</span></h4>

<p><strong class="pkg">simpcomp</strong> can also compute the cohomology groups of simplicial complexes, bases of these cohomology groups, the cup product of two cocycles and the intersection form of (orientable) 4-manifolds.</p>

<p><a id="X8527CC347F606648" name="X8527CC347F606648"></a></p>

<h5>8.2-1 SCCoboundaryOperatorMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCCoboundaryOperatorMatrix</code>( <var class="Arg">complex</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a rectangular matrix upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates the matrix of the coboundary operator <span class="SimpleMath">\(d^{\textit{k+1}}\)</span> as a list of row vectors.</p>

<div class="example"><pre>
gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],\
                       [2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);
> <SimplicialComplex: unnamed complex 2 | dim = 2 | n = 6>
gap> mat:=SCCoboundaryOperatorMatrix(c,1);
[ [ -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
   [ -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 ],
   [ 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ],
   [ 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0 ],
   [ 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ],
   [ 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 0, 0 ],
   [ 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0 ],
   [ 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1 ],
   [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0 ],
   [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1 ] ]
</pre></div>

<p><a id="X7926D71A8169629B" name="X7926D71A8169629B"></a></p>

<h5>8.2-2 SCCohomology</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCCohomology</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of pairs of the form <code class="code">[ integer, list ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>This function computes the simplicial cohomology groups of a given simplicial complex <var class="Arg">complex</var> with integer coefficients. It uses the algorithm described in <a href="chapBib_mj.html#biBDesbrun08DiscDiffFormCompModel">[DKT08]</a>.</p>

<p>The output is a list of cohomology groups of the form <span class="SimpleMath">\([H^0,....,H^d]\)</span>, where <span class="SimpleMath">\(d\)</span> is the dimension of <var class="Arg">complex</var>. The format of the cohomology groups <span class="SimpleMath">\(H^i\)</span> is given in terms of their maximal cyclic subgroups, i.e. a cohomology group <span class="SimpleMath">\(H^i\cong \mathbb{Z}^f + \mathbb{Z} / t_1 \mathbb{Z} \times \dots \times \mathbb{Z} / t_n \mathbb{Z}\)</span> is returned in form of a list <span class="SimpleMath">\([ f, [t_1,...,t_n] ]\)</span>, where <span class="SimpleMath">\(f\)</span> is the (integer) free part of <span class="SimpleMath">\(H^i\)</span> and <span class="SimpleMath">\(t_i\)</span> denotes the torsion parts of <span class="SimpleMath">\(H^i\)</span> ordered in weakly increasing size.</p>

<div class="example"><pre>
gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],
                       [2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);
> <SimplicialComplex: unnamed complex 4 | dim = 2 | n = 6>
gap> SCCohomology(c);
[ [ 1, [  ] ], [ 0, [  ] ], [ 0, [ 2 ] ] ]
</pre></div>

<p><a id="X85910A6B824D63A4" name="X85910A6B824D63A4"></a></p>

<h5>8.2-3 SCCohomologyBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCCohomologyBasis</code>( <var class="Arg">complex</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of pairs of the form <code class="code">[ integer, list of linear combinations of simplices ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates a set of basis elements for the <var class="Arg">k</var>-dimensional cohomology group (with integer coefficients) of a simplicial complex <var class="Arg">complex</var>. The entries of the returned list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ], where the value MODULUS is 1 for the basis elements of the free part of the <var class="Arg">k</var>-th homology group and <span class="SimpleMath">\(q\geq 2\)</span> for the basis elements of the <span class="SimpleMath">\(q\)</span>-torsion part. In contrast to the function <code class="func">SCCohomologyBasisAsSimplices</code> (<a href="chap8_mj.html#X7A5047E57E150716"><span class="RefLink">8.2-4</span></a>) the basis elements are stored as lists of coefficient-index pairs referring to the linear forms dual to the simplices in the <span class="SimpleMath">\(k\)</span>-th cochain complex of <var class="Arg">complex</var>, i.e. a basis element of the form <span class="SimpleMath">\([ [ \lambda_1, i], [\lambda_2, j], \dots ] \dots\)</span> encodes the linear combination of simplices (or their dual linear forms in the corresponding cochain complex) of the form <span class="SimpleMath">\(\lambda_1*\Delta_1+\lambda_2*\Delta_2\)</span> with <span class="SimpleMath">\(\Delta_1\)</span>=<code class="code">SCSkel(complex,k)[i]</code>, <span class="SimpleMath">\(\Delta_2\)</span>=<code class="code">SCSkel(complex,k)[j]</code> and so on.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("SU(3)/SO(3)");
[ [ 219, "SU(3)/SO(3) (VT)" ], [ 477, "SU(3)/SO(3) (VT)" ],
   [ 484, "SU(3)/SO(3) (VT)" ], [ 486, "SU(3)/SO(3) (VT)" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> SCCohomologyBasis(c,3);
[ [ 2, [ [ -9, 259 ], [ 9, 262 ], [ 9, 263 ], [ -9, 270 ], [ 9, 271 ],
           [ -9, 273 ], [ -9, 274 ], [ -18, 275 ], [ -9, 276 ], [ 9, 278 ],
           [ -9, 279 ], [ -9, 280 ], [ 3, 283 ], [ -3, 285 ], [ 3, 289 ],
           [ -3, 294 ], [ 3, 310 ], [ -3, 313 ], [ 3, 316 ], [ -1, 317 ],
           [ -6, 318 ], [ 3, 319 ], [ -6, 320 ], [ 6, 321 ], [ 1, 322 ],
           [ 3, 325 ], [ -1, 328 ], [ 6, 330 ], [ -2, 331 ], [ 12, 332 ],
           [ 7, 333 ], [ -5, 334 ], [ 1, 345 ], [ 3, 355 ], [ -9, 357 ],
           [ 9, 358 ], [ 1, 363 ], [ 12, 365 ], [ -9, 366 ], [ -3, 370 ],
           [ -1, 371 ], [ -3, 372 ], [ 8, 373 ], [ -1, 374 ], [ 6, 375 ],
           [ 9, 376 ], [ 3, 377 ], [ 1, 380 ], [ 3, 383 ], [ -8, 385 ],
           [ -9, 386 ], [ -9, 388 ], [ -18, 404 ], [ 9, 410 ], [ -9, 425 ],
           [ -18, 426 ], [ -9, 427 ], [ 9, 428 ], [ -9, 429 ], [ 3, 433 ],
           [ -3, 435 ], [ -9, 437 ], [ 10, 442 ], [ 12, 445 ], [ 1, 447 ],
           [ -19, 448 ], [ 2, 449 ], [ -1, 450 ], [ -9, 451 ], [ 3, 453 ],
           [ 1, 455 ], [ 1, 457 ], [ -11, 458 ], [ -9, 459 ], [ 9, 461 ],
           [ 9, 462 ], [ -9, 468 ], [ 9, 469 ], [ -18, 471 ], [ -9, 472 ],
           [ 9, 474 ], [ -9, 475 ], [ 9, 488 ], [ 9, 495 ], [ -9, 500 ],
           [ -3, 504 ], [ 9, 505 ], [ 9, 512 ], [ 9, 515 ], [ 6, 519 ],
           [ 18, 521 ], [ -15, 523 ], [ 9, 524 ], [ -3, 525 ], [ 18, 527 ],
           [ -18, 528 ], [ 6, 529 ], [ 6, 531 ], [ 12, 532 ] ] ] ]
</pre></div>

<p><a id="X7A5047E57E150716" name="X7A5047E57E150716"></a></p>

<h5>8.2-4 SCCohomologyBasisAsSimplices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCCohomologyBasisAsSimplices</code>( <var class="Arg">complex</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of pars of the form <code class="code">[ integer, linear combination of simplices ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Calculates a set of basis elements for the <var class="Arg">k</var>-dimensional cohomology group (with integer coefficients) of a simplicial complex <var class="Arg">complex</var>. The entries of the returned list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ], where the value MODULUS is 1 for the basis elements of the free part of the <var class="Arg">k</var>-th homology group and <span class="SimpleMath">\(q\geq 2\)</span> for the basis elements of the <span class="SimpleMath">\(q\)</span>-torsion part. In contrast to the function <code class="func">SCCohomologyBasis</code> (<a href="chap8_mj.html#X85910A6B824D63A4"><span class="RefLink">8.2-3</span></a>) the basis elements are stored as lists of coefficient-simplex pairs referring to the linear forms dual to the simplices in the <span class="SimpleMath">\(k\)</span>-th cochain complex of <var class="Arg">complex</var>, i.e. a basis element of the form <span class="SimpleMath">\([ [ \lambda_1, \Delta_i], [\lambda_2, \Delta_j], \dots ] \dots\)</span> encodes the linear combination of simplices (or their dual linear forms in the corresponding cochain complex) of the form <span class="SimpleMath">\(\lambda_1*\Delta_1+\lambda_2*\Delta_2 + \dots\)</span>.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("SU(3)/SO(3)");
[ [ 219, "SU(3)/SO(3) (VT)" ], [ 477, "SU(3)/SO(3) (VT)" ],
   [ 484, "SU(3)/SO(3) (VT)" ], [ 486, "SU(3)/SO(3) (VT)" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> SCCohomologyBasisAsSimplices(c,3);
[ [ 2,
       [ [ -9, [ 2, 7, 8, 9 ] ], [ 9, [ 2, 7, 8, 12 ] ],
           [ 9, [ 2, 7, 8, 13 ] ], [ -9, [ 2, 7, 11, 12 ] ],
           [ 9, [ 2, 7, 11, 13 ] ], [ -9, [ 2, 8, 9, 10 ] ],
           [ -9, [ 2, 8, 9, 11 ] ], [ -18, [ 2, 8, 9, 12 ] ],
           [ -9, [ 2, 8, 9, 13 ] ], [ 9, [ 2, 8, 10, 12 ] ],
           [ -9, [ 2, 8, 10, 13 ] ], [ -9, [ 2, 8, 11, 12 ] ],
           [ 3, [ 2, 9, 10, 12 ] ], [ -3, [ 2, 9, 11, 12 ] ],
           [ 3, [ 3, 4, 5, 7 ] ], [ -3, [ 3, 4, 5, 12 ] ],
           [ 3, [ 3, 4, 10, 12 ] ], [ -3, [ 3, 5, 6, 7 ] ],
           [ 3, [ 3, 5, 6, 11 ] ], [ -1, [ 3, 5, 6, 13 ] ],
           [ -6, [ 3, 5, 7, 8 ] ], [ 3, [ 3, 5, 7, 10 ] ],
           [ -6, [ 3, 5, 7, 11 ] ], [ 6, [ 3, 5, 7, 12 ] ],
           [ 1, [ 3, 5, 7, 13 ] ], [ 3, [ 3, 5, 8, 12 ] ],
           [ -1, [ 3, 5, 9, 13 ] ], [ 6, [ 3, 5, 10, 12 ] ],
           [ -2, [ 3, 5, 10, 13 ] ], [ 12, [ 3, 5, 11, 12 ] ],
           [ 7, [ 3, 5, 11, 13 ] ], [ -5, [ 3, 5, 12, 13 ] ],
           [ 1, [ 3, 6, 9, 13 ] ], [ 3, [ 3, 7, 10, 12 ] ],
           [ -9, [ 3, 7, 11, 12 ] ], [ 9, [ 3, 7, 11, 13 ] ],
           [ 1, [ 3, 8, 9, 13 ] ], [ 12, [ 3, 8, 10, 12 ] ],
           [ -9, [ 3, 8, 10, 13 ] ], [ -3, [ 3, 9, 10, 12 ] ],
           [ -1, [ 3, 9, 10, 13 ] ], [ -3, [ 3, 9, 11, 12 ] ],
           [ 8, [ 3, 9, 11, 13 ] ], [ -1, [ 3, 9, 12, 13 ] ],
           [ 6, [ 3, 10, 11, 12 ] ], [ 9, [ 3, 10, 11, 13 ] ],
           [ 3, [ 3, 10, 12, 13 ] ], [ 1, [ 4, 5, 6, 8 ] ],
           [ 3, [ 4, 5, 6, 11 ] ], [ -8, [ 4, 5, 6, 13 ] ],
           [ -9, [ 4, 5, 7, 8 ] ], [ -9, [ 4, 5, 7, 11 ] ],
           [ -18, [ 4, 6, 8, 9 ] ], [ 9, [ 4, 6, 9, 13 ] ],
           [ -9, [ 4, 8, 9, 10 ] ], [ -18, [ 4, 8, 9, 12 ] ],
           [ -9, [ 4, 8, 9, 13 ] ], [ 9, [ 4, 8, 10, 12 ] ],
           [ -9, [ 4, 8, 10, 13 ] ], [ 3, [ 4, 9, 10, 12 ] ],
           [ -3, [ 4, 9, 11, 12 ] ], [ -9, [ 4, 9, 12, 13 ] ],
           [ 10, [ 5, 6, 7, 8 ] ], [ 12, [ 5, 6, 7, 11 ] ],
           [ 1, [ 5, 6, 7, 13 ] ], [ -19, [ 5, 6, 8, 9 ] ],
           [ 2, [ 5, 6, 8, 11 ] ], [ -1, [ 5, 6, 8, 12 ] ],
           [ -9, [ 5, 6, 8, 13 ] ], [ 3, [ 5, 6, 9, 11 ] ],
           [ 1, [ 5, 6, 9, 13 ] ], [ 1, [ 5, 6, 10, 13 ] ],
           [ -11, [ 5, 6, 11, 13 ] ], [ -9, [ 5, 7, 8, 9 ] ],
           [ 9, [ 5, 7, 8, 12 ] ], [ 9, [ 5, 7, 8, 13 ] ],
           [ -9, [ 5, 7, 11, 12 ] ], [ 9, [ 5, 7, 11, 13 ] ],
           [ -18, [ 5, 8, 9, 12 ] ], [ -9, [ 5, 8, 9, 13 ] ],
           [ 9, [ 5, 8, 10, 12 ] ], [ -9, [ 5, 8, 11, 12 ] ],
           [ 9, [ 6, 7, 8, 13 ] ], [ 9, [ 6, 7, 11, 13 ] ],
           [ -9, [ 6, 8, 10, 13 ] ], [ -3, [ 6, 9, 11, 12 ] ],
           [ 9, [ 6, 9, 11, 13 ] ], [ 9, [ 7, 8, 9, 13 ] ],
           [ 9, [ 7, 8, 11, 12 ] ], [ 6, [ 7, 9, 11, 12 ] ],
           [ 18, [ 7, 11, 12, 13 ] ], [ -15, [ 8, 9, 10, 12 ] ],
           [ 9, [ 8, 9, 10, 13 ] ], [ -3, [ 8, 9, 11, 12 ] ],
           [ 18, [ 8, 10, 11, 12 ] ], [ -18, [ 8, 10, 12, 13 ] ],
           [ 6, [ 9, 10, 11, 12 ] ], [ 6, [ 9, 10, 12, 13 ] ],
           [ 12, [ 9, 11, 12, 13 ] ] ] ] ]
</pre></div>

<p><a id="X7A9137847BF96DF3" name="X7A9137847BF96DF3"></a></p>

<h5>8.2-5 SCCupProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCCupProduct</code>( <var class="Arg">complex</var>, <var class="Arg">cocycle1</var>, <var class="Arg">cocycle2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of pairs of the form <code class="code">[ ORIENTATION, SIMPLEX ]</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>The cup product is a method of adjoining two cocycles of degree <span class="SimpleMath">\(p\)</span> and <span class="SimpleMath">\(q\)</span> to form a composite cocycle of degree <span class="SimpleMath">\(p + q\)</span>. It endows the cohomology groups of a simplicial complex with the structure of a ring.</p>

<p>The construction of the cup product starts with a product of cochains: if <var class="Arg">cocycle1</var> is a p-cochain and <var class="Arg">cocylce2</var> is a q-cochain of a simplicial complex <var class="Arg">complex</var> (given as list of oriented p- (q-)simplices), then</p>

<p><var class="Arg">cocycle1</var> <span class="SimpleMath">\(\smile\)</span> <var class="Arg">cocycle2</var><span class="SimpleMath">\((\sigma) = \)</span><var class="Arg">cocycle1</var><span class="SimpleMath">\((\sigma \circ \iota_{0,1, ... p}) \cdot\)</span> <var class="Arg">cocycle2</var><span class="SimpleMath">\((\sigma \circ \iota_{p, p+1 ,..., p + q})\)</span></p>

<p>where <span class="SimpleMath">\(\sigma\)</span> is a <span class="SimpleMath">\(p + q\)</span>-simplex and <span class="SimpleMath">\(\iota_S\)</span>, <span class="SimpleMath">\(S \subset \{0,1,...,p+q \}\)</span> is the canonical embedding of the simplex spanned by <span class="SimpleMath">\(S\)</span> into the <span class="SimpleMath">\((p + q)\)</span>-standard simplex.</p>

<p><span class="SimpleMath">\(\sigma \circ \iota_{0,1, ..., p}\)</span> is called the <span class="SimpleMath">\(p\)</span>-th front face and <span class="SimpleMath">\(\sigma \circ \iota_{p, p+1, ..., p + q}\)</span> is the <span class="SimpleMath">\(q\)</span>-th back face of <span class="SimpleMath">\(\sigma\)</span>, respectively.</p>

<p>Note that this function only computes the cup product in the case that <var class="Arg">complex</var> is an orientable weak pseudomanifold of dimension <span class="SimpleMath">\(2k\)</span> and <span class="SimpleMath">\(p = q = k\)</span>. Furthermore, <var class="Arg">complex</var> must be given in standard labeling, with sorted facet list and <var class="Arg">cocylce1</var> and <var class="Arg">cocylce2</var> must be given in simplex notation and labeled accordingly. Note that the latter condition is usually fulfilled in case the cocycles were computed using <code class="func">SCCohomologyBasisAsSimplices</code> (<a href="chap8_mj.html#X7A5047E57E150716"><span class="RefLink">8.2-4</span></a>).</p>

<div class="example"><pre>
gap> SCLib.SearchByName("K3");
[ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> basis:=SCCohomologyBasisAsSimplices(c,2);;
gap> SCCupProduct(c,basis[1][2],basis[1][2]);
[ [ 1, [ 1, 2, 4, 7, 11 ] ], [ 1, [ 2, 3, 4, 5, 9 ] ] ]
gap> SCCupProduct(c,basis[1][2],basis[2][2]);
[ [ -1, [ 1, 2, 4, 7, 11 ] ], [ -1, [ 1, 2, 4, 7, 15 ] ],
   [ -1, [ 2, 3, 4, 5, 9 ] ] ]
</pre></div>

<p><a id="X835E7016826D14A8" name="X835E7016826D14A8"></a></p>

<h5>8.2-6 SCIntersectionForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCIntersectionForm</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a square matrix of integer values upon success, <code class="keyw">fail</code> otherwise.</p>

<p>For <span class="SimpleMath">\(2k\)</span>-dimensional orientable manifolds <span class="SimpleMath">\(M\)</span> the cup product (see <code class="func">SCCupProduct</code> (<a href="chap8_mj.html#X7A9137847BF96DF3"><span class="RefLink">8.2-5</span></a>)) defines a bilinear form</p>

<p>H<span class="SimpleMath">\(^k ( M ) \times \)</span>H<span class="SimpleMath">\(^k ( M ) \to \)</span>H<span class="SimpleMath">\(^{2k} (M), (a,b) \mapsto a \cup b \)</span></p>

<p>called the intersection form of <span class="SimpleMath">\(M\)</span>. This function returns the intersection form of an orientable combinatorial <span class="SimpleMath">\(2k\)</span>-manifold <var class="Arg">complex</var> in form of a matrix <code class="code">mat</code> with respect to the basis of H<span class="SimpleMath">\(^k ( \)</span><var class="Arg">complex</var>M<span class="SimpleMath">\()\)</span> computed by <code class="func">SCCohomologyBasisAsSimplices</code> (<a href="chap8_mj.html#X7A5047E57E150716"><span class="RefLink">8.2-4</span></a>). The matrix entry <code class="code">mat[i][j]</code> equals the intersection number of the <code class="code">i</code>-th base element with the <code class="code">j</code>-th base element of H<span class="SimpleMath">\(^k ( \)</span><var class="Arg">complex</var>M<span class="SimpleMath">\()\)</span>.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("CP^2");
[ [ 16, "CP^2 (VT)" ], [ 96, "CP^2#-CP^2" ], [ 97, "CP^2#CP^2" ],
   [ 185, "CP^2#(S^2xS^2)" ], [ 397, "Gaifullin CP^2" ],
   [ 457, "(S^3~S^1)#(CP^2)^{#5} (VT)" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> c1:=SCConnectedSum(c,c);;
gap> c2:=SCConnectedSumMinus(c,c);;
gap> q1:=SCIntersectionForm(c1);;
gap> q2:=SCIntersectionForm(c2);;
gap> PrintArray(q1);
[ [  1,  0 ],
   [  0,  1 ] ]
gap> PrintArray(q2);
[ [   1,   0 ],
   [   0,  -1 ] ]
</pre></div>

<p><a id="X791C2B45872A71CA" name="X791C2B45872A71CA"></a></p>

<h5>8.2-7 SCIntersectionFormParity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCIntersectionFormParity</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: <code class="code">0</code> or <code class="code">1</code> upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Computes the parity of the intersection form of a combinatorial manifold <var class="Arg">complex</var> (see <code class="func">SCIntersectionForm</code> (<a href="chap8_mj.html#X835E7016826D14A8"><span class="RefLink">8.2-6</span></a>)). If the intersection for is even (i. e. all diagonal entries are even numbers) <code class="code">0</code> is returned, otherwise <code class="code">1</code> is returned.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("S^2xS^2");;
gap> c:=SCLib.Load(last[1][1]);;
gap> SCIntersectionFormParity(c);
0
gap> SCLib.SearchByName("CP^2");;
gap> c:=SCLib.Load(last[1][1]);;
gap> SCIntersectionFormParity(c);
1
</pre></div>

<p><a id="X7B5F948B84F5E316" name="X7B5F948B84F5E316"></a></p>

<h5>8.2-8 SCIntersectionFormDimensionality</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCIntersectionFormDimensionality</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: an integer upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Returns the dimensionality of the intersection form of a combinatorial manifold <var class="Arg">complex</var>, i. e. the length of a minimal generating set of H<span class="SimpleMath">\(^k (M)\)</span> (where <span class="SimpleMath">\(2k\)</span> is the dimension of <var class="Arg">complex</var>). See <code class="func">SCIntersectionForm</code> (<a href="chap8_mj.html#X835E7016826D14A8"><span class="RefLink">8.2-6</span></a>) for further details.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("CP^2");;
gap> c:=SCLib.Load(last[1][1]);;
gap> SCIntersectionFormParity(c);
1
gap> SCCohomology(c);
[ [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
gap> SCIntersectionFormDimensionality(c);
1
gap> d:=SCConnectedProduct(c,10);;
gap> SCIntersectionFormDimensionality(d);
10
</pre></div>

<p><a id="X87375A6978F3C8CD" name="X87375A6978F3C8CD"></a></p>

<h5>8.2-9 SCIntersectionFormSignature</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCIntersectionFormSignature</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a triple of integers upon success, <code class="keyw">fail</code> otherwise.</p>

<p>Computes the dimensionality (see <code class="func">SCIntersectionFormDimensionality</code> (<a href="chap8_mj.html#X7B5F948B84F5E316"><span class="RefLink">8.2-8</span></a>)) and the signature of the intersection form of a combinatorial manifold <var class="Arg">complex</var> as a <span class="SimpleMath">\(3\)</span>-tuple that contains the dimensionality in the first entry and the number of positive / negative eigenvalues in the second and third entry. See <code class="func">SCIntersectionForm</code> (<a href="chap8_mj.html#X835E7016826D14A8"><span class="RefLink">8.2-6</span></a>) for further details.</p>

<p>Internally calls the <strong class="pkg">GAP</strong>-functions <code class="code">Matrix_CharacteristicPolynomialSameField</code> and <code class="code">CoefficientsOfLaurentPolynomial</code> to compute the number of positive / negative eigenvalues of the intersection form.</p>

<div class="example"><pre>
gap> SCLib.SearchByName("CP^2");;
gap> c:=SCLib.Load(last[1][1]);;
gap> SCIntersectionFormParity(c);
1
gap> SCCohomology(c);
[ [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
gap> SCIntersectionFormSignature(c);
[ 1, 0, 1 ]
gap> d:=SCConnectedSum(c,c);
<SimplicialComplex: CP^2 (VT)#+-CP^2 (VT) | dim = 4 | n = 13>
gap> SCIntersectionFormSignature(d);
[ 2, 2, 0 ]
gap> d:=SCConnectedSumMinus(c,c);;
gap> SCIntersectionFormSignature(d);
[ 2, 1, 1 ]
</pre></div>

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