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<title>GAP (HAP) - Chapter 8: Cohomology ring structure</title>
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<div class="ChapSects"><a href="chap8.html#X7A9561E47A4994F5">8. <span class="Heading"> Cohomology ring structure</span></a>
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<h3>8. <span class="Heading"> Cohomology ring structure</span></h3>
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<td class="tdleft"><code class="code">IntegralCupProduct(R,u,v,p,q) </code> <br /> <code class="code"> IntegralCupProduct(R,u,v,p,q,P,Q,N) </code></p>
<p>(Various functions used to construct the cup product are also <span class="URL"><a href=" CR_functions.html">available</a></span>.)</p>
<p>Inputs a ZG-resolution R, a vector u representing an element in H^p(G,Z), a vector v representing an element in H^q(G,Z) and the two integers p,q >0. It returns a vector w representing the cup product u* v in H^p+q(G,Z). This product is associative and u* v = (-1)pqv* u . It provides H^*(G,Z) with the structure of an anti-commutative graded ring. This function implements the cup product for characteristic 0 only.</p>
<p>The resolution R needs a contracting homotopy.</p>
<p>To save the function from having to calculate the abelian groups H^n(G,Z) additional input variables can be used in the form IntegralCupProduct(R,u,v,p,q,P,Q,N) , where</p>
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<li><p>P is the output of the command CR_CocyclesAndCoboundaries(R,p,true)</p>
</li>
<li><p>Q is the output of the command CR_CocyclesAndCoboundaries(R,q,true)</p>
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<li><p>N is the output of the command CR_CocyclesAndCoboundaries(R,p+q,true) .</p>
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<td class="tdleft"><code class="code"> IntegralRingGenerators(R,n) </code></p>
<p>Inputs at least n+1 terms of a ZG-resolution and integer n> 0. It returns a minimal list of cohomology classes in H^n(G,Z) which, together with all cup products of lower degree classes, generate the group H^n(G,Z) .</p>
<p>(Let a_i be the i-th canonical generator of the d-generator abelian group H^n(G,Z). The cohomology class n_1a_1 + ... +n_da_d is represented by the integer vector u=(n_1, ..., n_d). )</td>
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<td class="tdleft"><code class="code"> ModPCohomologyGenerators(G,n) </code> <br /> <code class="code"> ModPCohomologyGenerators(R) </code></p>
<p>Inputs either a p-group G and positive integer n, or else n terms of a minimal Z_pG-resolution R of Z_p. It returns a pair whose first entry is a minimal set of homogeneous generators for the cohomology ring A=H^*(G,Z_p) modulo all elements in degree greater than n. The second entry of the pair is a function deg which, when applied to a minimal generator, yields its degree.</p>
<p>WARNING: the following rule must be applied when multiplying generators x_i together. Only products of the form x_1*(x_2*(x_3*(x_4*...))) with deg(x_i) le deg(x_i+1) should be computed (since the x_i belong to a structure constant algebra with only a partially defined structure constants table).</td>
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<td class="tdleft"><code class="code"> ModPCohomologyRing(G,n) </code> <br /> <code class="code"> ModPCohomologyRing(G,n,level) </code> <br /> <code class="code"> ModPCohomologyRing(R) </code> <br /> <code class="code"> ModPCohomologyRing(R,level) </code></p>
<p>Inputs either a p-group G and positive integer n, or else n terms of a minimal Z_pG-resolution R of Z_p. It returns the cohomology ring A=H^*(G,Z_p) modulo all elements in degree greater than n.</p>
<p>The ring is returned as a structure constant algebra A.</p>
<p>The ring A is graded. It has a component A!.degree(x) which is a function returning the degree of each (homogeneous) element x in GeneratorsOfAlgebra(A).</p>
<p>An optional input variable "level" can be set to one of the strings "medium" or "high". These settings determine parameters in the algorithm. The default setting is "medium".</p>
<p>When "level" is set to "high" the ring A is returned with a component A!.niceBasis. This component is a pair [Coeff,Bas]. Here Bas is a list of integer lists; a "nice" basis for the vector space A can be constructed using the command List(Bas,x->Product(List(x,i->Basis(A)[i])). The coefficients of the canonical basis element Basis(A)[i] are stored as Coeff[i].</p>
<p>If the ring A is computed using the setting "level"="medium" then the component A!.niceBasis can be added to A using the command A:=ModPCohomologyRing_part_2(A).</td>
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<td class="tdleft"><code class="code"> ModPRingGenerators(A) </code></p>
<p>Inputs a mod p cohomology ring A (created using the preceeding function). It returns a minimal generating set for the ring A. Each generator is homogeneous.</td>
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