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<p><a id="X86A9B6F87E619FFF" name="X86A9B6F87E619FFF"></a></p>
<div class="ChapSects"><a href="chap2.html#X86A9B6F87E619FFF">2. <span class="Heading">Usage</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X85BF21B17BF76907">2.1 <span class="Heading">Cohomology Objects</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8716E7657F9683B7">2.1-1 CohomologyObject</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7FEB30657DDEFE44">2.2 <span class="Heading">Minimal Projective Resolutions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8273999C7B352F22">2.2-1 ProjectiveResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7AF313D387F6BA22">2.2-2 BoundaryMap</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X78C37B2E7D297FC5">2.3 <span class="Heading">Cohomology Generators and Relators</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81F141187B0EC653">2.3-1 CohomologyGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X812BE8E983DC20BC">2.3-2 CohomologyRelators</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X87C9251687CD4247">2.4 <span class="Heading">Tests for Completion</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X80B6849C835B7F19">2.5 <span class="Heading">Cohomology Rings</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B77337486380998">2.5-1 CohomologyRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D8EA4528082A74D">2.5-2 IsHomogeneous</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X826A861E7E7D944E">2.5-3 Degree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X813A8E4C7D4C9FBD">2.5-4 LocateGeneratorsInCohomologyRing</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7E21853583D6D86D">2.6 <span class="Heading">What Happens if <code class="keyw">n</code> Isn't
Big Enough?</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X837261007B39795C">2.7 <span class="Heading">Induced Maps</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8608BD148307390B">2.7-1 InducedHomomorphismOnCohomology</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87D180E2801B1EA8">2.7-2 Inclusion</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7B541EB97BAB36D7">2.8 <span class="Heading">Massey Products</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E3023CE86641F9E">2.8-1 MasseyProduct</a></span>
</div>
</div>
<h3>2. <span class="Heading">Usage</span></h3>
<p>Unless otherwise specified, all the functions described below taking an argument <code class="keyw">n</code> do whatever the manual says they do up to homological degree <code class="keyw">n</code>. These functions are idempotent in the sense that called a second time with the same argument <code class="keyw">n</code>, they do nothing, but called with a bigger <code class="keyw">n</code>, they continue computing from where the previous calculations left off.</p>
<p><a id="X85BF21B17BF76907" name="X85BF21B17BF76907"></a></p>
<h4>2.1 <span class="Heading">Cohomology Objects</span></h4>
<p>The computation of group cohomology involves several calculations, the results of which are reused in later calculations, and are thus collected in an object of type <code class="keyw">CObject</code>, which is created with the following command.</p>
<p><a id="X8716E7657F9683B7" name="X8716E7657F9683B7"></a></p>
<h5>2.1-1 CohomologyObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CohomologyObject</code>( <var class="Arg">G, M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CohomologyObject</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>a <code class="keyw">CObject</code>.</p>
<p>This function creates a <code class="keyw">CObject</code> having components the p-group G and the <strong class="pkg">MeatAxe</strong> module M, which should be a kG-module with G the same group and k a field of characteristic p. Note that <strong class="pkg">MeatAxe</strong> modules record k but not G, which is why this operation requires the user to specify G but not k.</p>
<p>Fortunately, most users don't need to know anything about <strong class="pkg">MeatAxe</strong> modules, being interested primarily in the case where k=GF(p), and M=k, the trivial kG-module. In this situation, the second invocation creates a cohomology object having components the p-group G and the trivial <strong class="pkg">MeatAxe</strong> kG-module k=GF(p).</p>
<p>We emphasize that in the first invocation, k can be any field of characteristic p and <code class="keyw">M</code> can be any <strong class="pkg">MeatAxe</strong> module over kG, and that <code class="keyw">ProjectiveResolution</code> works when <code class="keyw">M</code> is an arbitrary <strong class="pkg">MeatAxe</strong> module, but that all the functions dealing with the ring-structure of H*(G,k) require that <code class="keyw">M</code> be the trivial module.</p>
<p>The cohomology object is used to store, in addition to the items mentioned above, the boundary maps, the Betti numbers, the multiplication table, etc.</p>
<p><a id="X7FEB30657DDEFE44" name="X7FEB30657DDEFE44"></a></p>
<h4>2.2 <span class="Heading">Minimal Projective Resolutions</span></h4>
<p>Given a p-group G, a field k of characteristic p, and a kG-module M, the function below computes the first n terms of the minimal projective resolution of M</p>
<p class="pcenter"> P_n -> ... -> P_2 -> P_1 -> P_0 -> M -> 0</p>
<p>where P_i is the direct sum (kG)^(b_i) for certain numbers b_i, the <em>Betti numbers</em> of the resolution. The minimal kG-projective resolution of M is unique up to chain isomorphism. Because of the minimality of P, the groups Ext^i(M,N) are simply Hom(P_i,N), and if M and N are both the trivial kG-module k, then H^i(G,k)=Ext^i(k,k)=k^(b_i).</p>
<p><a id="X8273999C7B352F22" name="X8273999C7B352F22"></a></p>
<h5>2.2-1 ProjectiveResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ProjectiveResolution</code>( <var class="Arg">C, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>a list containing the Betti numbers b_0, b_1,..., b_n.</p>
<p>Given a cohomology object <code class="keyw">C</code> having components G and M, this function computes the first <code class="keyw">n</code>+1 terms of the minimal projective resolution P of M of the form P_i=(kG)^(b_i) for i=0,1,...,n, and returns the numbers b_i as a list.</p>
<p><a id="X7AF313D387F6BA22" name="X7AF313D387F6BA22"></a></p>
<h5>2.2-2 BoundaryMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BoundaryMap</code>( <var class="Arg">C, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>the <code class="keyw">n</code>th boundary map.</p>
<p>Given the cohomology object <code class="keyw">C</code>, this function computes a projective resolution to degree <code class="keyw">n</code> if it hasn't been computed already, and returns the <code class="keyw">n</code>th boundary map.</p>
<p>The map returned is a b_n x b_n-1|G| matrix, having in the ith row the image of the element 1_G from the ith direct summand of P_n.</p>
<p>See the file <code class="keyw">doc/example.*</code> for an example of the usage and interpretation of the result of this function.</p>
<p><a id="X78C37B2E7D297FC5" name="X78C37B2E7D297FC5"></a></p>
<h4>2.3 <span class="Heading">Cohomology Generators and Relators</span></h4>
<p>See <a href="chapBib.html#biBcarlson">[CTVZ03]</a> for the details of the calculation of cohomology products using composition of chain maps. See also the file <code class="keyw">doc/explanation.*</code> for an explanation of the implementation.</p>
<p><a id="X81F141187B0EC653" name="X81F141187B0EC653"></a></p>
<h5>2.3-1 CohomologyGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CohomologyGenerators</code>( <var class="Arg">C, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>a list containing the degrees of the elements of a set of generators of the cohomology ring.</p>
<p>Given a cohomology object <code class="keyw">C</code> having group component G and module component the trivial kG-module, this function computes a set of generators of H*(G,k) having degree <code class="keyw">n</code> or less, and stores them in <code class="keyw">C</code>. The function returns a list of the degrees of these generators.</p>
<p>The actual cohomology generators are represented by maps P_n -> k and are stored in <code class="keyw">C</code> as matrices. Only their degrees are returned.</p>
<p><a id="X812BE8E983DC20BC" name="X812BE8E983DC20BC"></a></p>
<h5>2.3-2 CohomologyRelators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CohomologyRelators</code>( <var class="Arg">C, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>a list of generators and a list of relators.</p>
<p>Given a cohomology object <code class="keyw">C</code> having group component G and module component k, this function computes a set of generators of the ideal of relators in H*(G,k), all having multidegree n or less.</p>
<p>More specifically, the function returns two lists, the first list containing the variables <code class="keyw">z</code>, <code class="keyw">y</code>, <code class="keyw">x</code>, ... corresponding to the generators of H*(G,k) if there are fewer than 12 generators and containing the variables <code class="keyw">x_1</code>, <code class="keyw">x_2</code>, <code class="keyw">x_3</code>, ... otherwise. The second list is a list of polynomials in the variables from the first list.</p>
<p>These two lists should be interpreted as follows. A degree <code class="keyw">n</code> approximation of the cohomology ring H*(G,k) is given by the polynomial ring over k in the non-commuting variables from the first list, (having degrees given by the list returned by <code class="keyw">CohomologyGenerators</code> in section <a href="chap2.html#X81F141187B0EC653"><b>2.3-1</b></a> ) and subject to the relators in the second list. See section <a href="chap2.html#X7E21853583D6D86D"><b>2.6</b></a> for more details still.</p>
<p>For example, consider the following commands.</p>
<table class="example">
<tr><td><pre>
gap> C:=CohomologyObject(DihedralGroup(8));
<object>
gap> CohomologyGenerators(C,10);
[ 1, 1, 2 ]
gap> CohomologyRelators(C,10);
[ [ z, y, x ], [ z*y+y^2 ] ]
</pre></td></tr></table>
<p>This tells us that for G=D_8, the cohomology ring H*(G,k) is the graded-commutative polynomial ring in the variables z, y, and x of degrees 1, 1, and 2, subject to the relation zy+y^2. But since H*(G,k) is commutative, k being of characteristic 2, we have H*(G,k)=k[z,y,x]/(zy+y^2). This result can be further improved by taking z=z+y, giving H*(G,k)=k[z,y,x]/(zy).</p>
<p>Observe that in this case, we knew in advance that there was a set of generators for H*(G,k) all having degree less than 10, and that there was a set of generators of the ideal of relators all having multidegree less than 10. See see section <a href="chap2.html#X7E21853583D6D86D"><b>2.6</b></a> for details.</p>
<p>While this isn't likely to occur, we point out that if there are 12 or more generators and some of the indeterminates <code class="keyw">x_1</code>, <code class="keyw">x_2</code>, <code class="keyw">x_3</code>, ... have already been named, say by a previous call to <code class="keyw">CohomologyRelators</code>, then these variables will retain their old names. If this is confusing, you could restart <strong class="pkg">GAP</strong> and do it again.</p>
<p>Finally, <code class="keyw">CohomologyRelators</code> is <em>not</em> idempotent for efficiency reasons, so sadly, if you don't uncover all the relators the first time, you will have to start all over from the beginning.</p>
<p><a id="X87C9251687CD4247" name="X87C9251687CD4247"></a></p>
<h4>2.4 <span class="Heading">Tests for Completion</span></h4>
<p>A test or series of tests for completion of the calculation will hopefully be implemented soon. See <a href="chapBib.html#biBcarlson">[CTVZ03]</a> for the details.</p>
<p><a id="X80B6849C835B7F19" name="X80B6849C835B7F19"></a></p>
<h4>2.5 <span class="Heading">Cohomology Rings</span></h4>
<p>Whereas the operations in sections <a href="chap2.html#X81F141187B0EC653"><b>2.3-1</b></a> and <a href="chap2.html#X812BE8E983DC20BC"><b>2.3-2</b></a> calculate a presentation for the cohomology ring, the operation below creates the ring in <strong class="pkg">GAP</strong> as a structure constant algebra.</p>
<p>See <a href="chapBib.html#biBcarlson">[CTVZ03]</a> for the details of the calculation of cohomology products using composition of chain maps. See also the file <code class="keyw">doc/explanation.*</code> for an explanation of the implementation.</p>
<p><a id="X7B77337486380998" name="X7B77337486380998"></a></p>
<h5>2.5-1 CohomologyRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CohomologyRing</code>( <var class="Arg">C, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CohomologyRing</code>( <var class="Arg">G, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>the cohomology ring of G.</p>
<p>Given a cohomology object <code class="keyw">C</code> with group component G and module component the trivial kG-module, this function returns the degree <code class="keyw">n</code> truncation of the cohomology ring H*(G,k). See <a href="chap2.html#X7E21853583D6D86D"><b>2.6</b></a> for what this means exactly. The object returned is a structure constant algebra.</p>
<p>Users interested only in working with the cohomology ring of a group as a <strong class="pkg">GAP</strong> object, and not in calculating generators, relators, induced maps, etc, can use the second invocation of this function, which returns the cohomology ring of the group <code class="keyw">G</code> immediately, throwing away all intermediate calculations.</p>
<p>Observe that the object returned is a degree <code class="keyw">n</code> truncation of the infinite-dimensional cohomology ring. A consequence of this is that multiplying two elements whose product has degree greater than <code class="keyw">n</code> results in zero, whether or not the product is really zero.</p>
<p>Observe also that calling <code class="keyw">CohomologyRing</code> a second time with a bigger <code class="keyw">n</code> does <em>not</em> extend the previous ring, but rather, recalculates the entire ring from the beginning. Extending the previous ring appears not to be worth the effort for technical reasons, since almost everything would need to be recalculated again anyway.</p>
<p>Recall that H*(G,k) is a graded structure constant algebra, the components being the cohomology groups H^i(G,k). The following functions were intended to be used for cohomology rings, but in principle, they work for any graded structure constant algebra.</p>
<p><a id="X7D8EA4528082A74D" name="X7D8EA4528082A74D"></a></p>
<h5>2.5-2 IsHomogeneous</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsHomogeneous</code>( <var class="Arg">e</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>Given an element <code class="keyw">e</code> of a cohomology ring H*(G,k), this operation determines whether or not <code class="keyw">e</code> is homogeneous, that is, whether <code class="keyw">e</code> is contained in H^i(G,k) for some <code class="keyw">i</code>.</p>
<p><a id="X826A861E7E7D944E" name="X826A861E7E7D944E"></a></p>
<h5>2.5-3 Degree</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> Degree</code>( <var class="Arg">e</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><b>Returns: </b>the degree of <code class="keyw">e</code>.</p>
<p>This function returns the degree of the possibly non-homogeneous element <code class="keyw">e</code> of a cohomology ring H*(G,k). Specifically, if H*(G,k) = A_0 + A_1 + A_2 + ... where A_i = H^i(G,k), then this function returns the minimum n such that <code class="keyw">e</code> is in A_0 + A_1 + ... + A_n.</p>
<table class="example">
<tr><td><pre>
gap> A:=CohomologyRing(DihedralGroup(8),10);
<algebra of dimension 66 over GF(2)>
gap> b:=Basis(A);
CanonicalBasis( <algebra of dimension 66 over GF(2)> )
gap> x:=b[2]+b[4];
v.2+v.4
gap> IsHomogeneous(x);
false
gap> Degree(x);
2
</pre></td></tr></table>
<p><a id="X813A8E4C7D4C9FBD" name="X813A8E4C7D4C9FBD"></a></p>
<h5>2.5-4 LocateGeneratorsInCohomologyRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> LocateGeneratorsInCohomologyRing</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a list containing the cohomology generators.</p>
<p>Having already called <code class="keyw">CohomologyRing</code> (see <a href="chap2.html#X7B77337486380998"><b>2.5-1</b></a>), this function returns a list of elements of the cohomology ring which together with the identity element generate the cohomology ring.</p>
<p>This function is a wrapper for <code class="keyw">CohomologyGenerators</code> (see <a href="chap2.html#X81F141187B0EC653"><b>2.3-1</b></a>). It points out which elements of the cohomology ring correspond with the generators found by <code class="keyw">CohomologyGenerators</code>.</p>
<table class="example">
<tr><td><pre>
gap> C:=CohomologyObject(SmallGroup(8,4));
<object>
gap> A:=CohomologyRing(C,10);
<algebra of dimension 17 over GF(2)>
gap> L:=LocateGeneratorsInCohomologyRing(C);
[ v.2, v.3, v.7 ]
gap> A=Subalgebra(A,Concatenation(L,[One(A)]));
true
</pre></td></tr></table>
<p><a id="X7E21853583D6D86D" name="X7E21853583D6D86D"></a></p>
<h4>2.6 <span class="Heading">What Happens if <code class="keyw">n</code> Isn't
Big Enough?</span></h4>
<p>Since P is a <em>minimal</em> projective resolution, we have H^i(G,k) = Hom_{kG}(P_i,k) where P_i = (kG)^b_i so that H^i(G,k) has a natural basis consisting of the maps sending the element 1_G of the jth direct summand of P_i to 1_k and all other direct summands to 0, for j=1,2,...,b_i, where b_i is the kG-rank of P_i.</p>
<p>The command <code class="keyw">CohomologyRing(C,n)</code> forms the vector space whose basis is the concatenation of the natural bases of H^i(G,k) for i=1,2,...,n and computes all products of basis elements x and y for which deg x+deg y <= n. Thinking of H*(G,k) in terms of it's multiplication table, this means that the function computes the upper left-hand corner of the multiplication table. If deg x + deg y > n, the product xy is taken to be zero. Therefore, the ring returned by <code class="keyw">CohomologyRing</code> is H*(G,k)/J where J is the ideal of all elements of degree >n.</p>
<p>The ring determined by <code class="keyw">CohomologyGenerators</code> and <code class="keyw">CohomologyRelators</code> is somewhat different. <code class="keyw">CohomologyGenerators</code> proceeds inductively, taking all natural basis elements of H^1(G,k) as generators, and for i=2... n, taking all natural basis elements of H^i(G,k) which are <em>not</em> products of lower-degree elements as generators. Therefore, unless you know that there is an n for which there exists a generating set of H*(G,k) consisting of elements of degree n or less, then you are <em>not</em> guaranteed that the elements returned by the <code class="keyw">CohomologyGenerators</code> generate H*(G,k) as a ring. The knowledge of such an n is the subject of section <a href="chap2.html#X87C9251687CD4247"><b>2.4</b></a>.</p>
<p>Similarly, <code class="keyw">CohomologyRelators</code> proceeds inductively until degree n, returning a list of polynomials which generate the ideal of relators of multidegree n or less. Again, you have to already know how big n should be.</p>
<p>The result of the preceding information is that there is a homomorphism k< x_1,x_2,..., x_m >/ I -> H*(G,k), where k< x_1,x_2,... x_m > is the graded polynomial ring over k in the non-commuting variables x_1,x_2,...,x_m, having degrees the numbers in the list returned by <code class="keyw">CohomologyGenerators</code>, and I is the ideal in k< x_1,x_2,..., x_m > generated by the elements returned by <code class="keyw">CohomologyRelators(C,n)</code>.</p>
<p>Therefore, if there is a generator of degree greater than n, then f won't be surjective. Similarly, if there is a relator of multidegree greater than n which is not a consequence of lower degree relators, then f won't be injective. See section <a href="chap2.html#X87C9251687CD4247"><b>2.4</b></a> for a discussion on how big n needs to be to ensure that f be an isomorphism.</p>
<p><a id="X837261007B39795C" name="X837261007B39795C"></a></p>
<h4>2.7 <span class="Heading">Induced Maps</span></h4>
<p>Let f: H -> G be a group homomorphism. Then f induces a homomorphism on cohomology H*(G,k) -> H*(H,k) which is returned by the following function.</p>
<p><a id="X8608BD148307390B" name="X8608BD148307390B"></a></p>
<h5>2.7-1 InducedHomomorphismOnCohomology</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> InducedHomomorphismOnCohomology</code>( <var class="Arg">C, D, f, n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>the induced homomorphism on cohomology.</p>
<p>This function returns the induced homomorphism H*(G,k) -> H*(H,k), where the groups H and G are the components of the cohomology objects <code class="keyw">C</code> and <code class="keyw">D</code> and f: H -> G is a group homomorphism. If the cohomology rings have not yet been calculated, they will be computed to degree n, and in this case, they can then be accessed by calling <code class="keyw">CohomologyRing</code> (see <a href="chap2.html#X7B77337486380998"><b>2.5-1</b></a>).</p>
<p><a id="X87D180E2801B1EA8" name="X87D180E2801B1EA8"></a></p>
<h5>2.7-2 Inclusion</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> Inclusion</code>( <var class="Arg">H, G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>the inclusion H-> G</p>
<p>This function returns the group homomorphism H-> G when H is a subgroup of G. The returned map can be used as the <code class="keyw">f</code> argument of <code class="keyw">InducedHomomorphismOnCohomology</code>, in which case the induced homomorphism is the restriction map Res: H*(G,k) -> H*(H,k).</p>
<p>The following example calculates the homomorphism on cohomology induced by the inclusion of the cyclic group of size 4 into the dihedral group of size 8.</p>
<table class="example">
<tr><td><pre>
gap> G:=DihedralGroup(8);H:=Subgroup(G,[G.2]);
<pc group of size 8 with 3 generators>
Group([ f2 ])
gap> C:=CohomologyObject(H);D:=CohomologyObject(G);
<object>
<object>
gap> i:=Inclusion(H,G);
[ f2 ] -> [ f2 ]
gap> Res:=InducedHomomorphismOnCohomology(C,D,i,10);;
gap> A:=CohomologyRing(D,10);
<algebra of dimension 66 over GF(2)>
gap> LocateGeneratorsInCohomologyRing(D);
[ v.2, v.3, v.6 ]
gap> A.1^Res; A.2^Res; A.3^Res; A.6^Res;
v.1
0*v.1
v.2
v.3
</pre></td></tr></table>
<p><a id="X7B541EB97BAB36D7" name="X7B541EB97BAB36D7"></a></p>
<h4>2.8 <span class="Heading">Massey Products</span></h4>
<p>See <a href="chapBib.html#biBkraines">[Kra66]</a> for the definitions and <a href="chapBib.html#biBborge">[Bor01]</a> for the details of the calculation using the Yoneda cocomplex. See also the file <code class="keyw">doc/explanation.*</code> for an explanation of the implementation.</p>
<p><a id="X7E3023CE86641F9E" name="X7E3023CE86641F9E"></a></p>
<h5>2.8-1 MasseyProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MasseyProduct</code>( <var class="Arg">x1, x2, ..., xn</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>the Massey product < x1, x2, ... , xn>.</p>
<p>Given elements x1, x2, ... , xn of the ring returned by <code class="keyw">CohomologyRing</code> (see <a href="chap2.html#X80B6849C835B7F19"><b>2.5</b></a>), this function computes the n-fold Massey product < x1, x2, ... , xn > provided that the lower-degree Massey products < xi, x{i+1}, ... , xj > vanish for all 1 <= i < j <= n, and returns <code class="keyw">fail</code> otherwise.</p>
<p>As an example, recall that the cohomology rings of the cyclic groups C_3 and C_9 of sizes 3 and 9 over k=GF(3) are both given by k< z,y >/(z^2), so they are isomorphic as rings. However, the following example shows that < z, z, z > is non-zero in H*(C_3,k) but is zero in H*(C_9,k).</p>
<table class="example">
<tr><td><pre>
gap> A:=CohomologyRing(CyclicGroup(3),10);
<algebra of dimension 11 over GF(3)>
gap> z:=Basis(A)[2];
v.2
gap> MasseyProduct(z,z);
0*v.1
gap> MasseyProduct(z,z,z);
v.3
gap> A:=CohomologyRing(CyclicGroup(9),10);
<algebra of dimension 11 over GF(3)>
gap> z:=Basis(A)[2];
v.2
gap> MasseyProduct(z,z);
0*v.1
gap> MasseyProduct(z,z,z);
0*v.1
gap> MasseyProduct(z,z,z,z,z,z,z,z,z);
v.3
</pre></td></tr></table>
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