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<title>GAP (HAP) - Chapter 1: Resolutions of the ground ring</title>
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<div class="ChapSects"><a href="chap1.html#X8735FC5E7BB5CE3A">1. <span class="Heading">Resolutions of the ground ring</span></a>
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<h3>1. <span class="Heading">Resolutions of the ground ring</span></h3>
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<td class="tdleft"><code class="code"> ResolutionAbelianGroup(L,n)</code> <code class="code"> ResolutionAbelianGroup(G,n)</code></p>
<p>Inputs a list L:=[m_1,m_2, ..., m_d] of nonnegative integers, and a positive integer n. It returns n terms of a ZG-resolution for the abelian group G=Z_L[1]+Z_L[2]+···+Z_L[d] .</p>
<p>If G is finite then the first argument can also be the abelian group G itself.</td>
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<td class="tdleft"><code class="code"> ResolutionAlmostCrystalGroup(G,n)</code></p>
<p>Inputs a positive integer n and an almost crystallographic pcp group G. It returns n terms of a free ZG-resolution. (A group is almost crystallographic if it is nilpotent-by-finite and has no non-trivial finite normal subgroup. Such groups can be constructed using the ACLIB package.)</td>
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<td class="tdleft"><code class="code"> ResolutionAlmostCrystalQuotient(G,n,c)</code> <code class="code"> ResolutionAlmostCrystalQuotient(G,n,c,false)</code></p>
<p>An almost crystallographic group G is an extension of a finite group P by a nilpotent group T, and has no non-trivial finite normal subgroup. We define the relative lower central series by setting T_1=T and T_i+1=[T_i,G].</p>
<p>This function inputs an almost crystallographic group G together with positive integers n and c. It returns n terms of a free ZQ-resolution R for the group Q=G/T_c .</p>
<p>In addition to the usual components, the resolution R has the component R.quotientHomomorphism which gives the quotient homomorphism G --> Q.</p>
<p>If a fourth optional variable is set equal to "false" then the function omits to test whether Q is finite and a "more canonical" resolution is constructed.</td>
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<td class="tdleft"><code class="code"> ResolutionArtinGroup(D,n)</code></p>
<p>Inputs a Coxeter diagram D and an integer n>1. It returns n terms of a free ZG-resolution R where G is the Artin monoid associated to D. It is conjectured that R is also a free resolution for the Artin group G. The conjecture is known to hold in <span class="URL"><a href="About/aboutArtinGroups.html">certain cases</a></span>.</p>
<p>G=R.group is infinite and returned as a finitely presented group. The list R.elts is a partial listing of the elements of G which grows as R is used. Initially R.elts is empty and then, any time the boundary of a resolution generator is called, R.elts is updated to include elements of G involved in the boundary.</p>
<p>The contracting homotopy on R has not yet been implemented! Furthermore, the group G is currently returned only as a finitely presented group (without any method for solving the word problem).</td>
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<td class="tdleft"><code class="code"> ResolutionAsphericalPresentation(F,R,n)</code></p>
<p>Inputs a free group F, a set R of words in F which constitute an aspherical presentation for a group G, and a positive integer n. (Asphericity can be a difficult property to verify. The function IsAspherical(F,R) could be of help.)</p>
<p>The function returns n terms of a free ZG-resolution R which has generators in dimensions < 3 only. No contracting homotopy on R will be returned.</td>
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<td class="tdleft"><code class="code"> ResolutionBieberbachGroup( G ) </code> <br /> <code class="code"> ResolutionBieberbachGroup( G, v ) </code></p>
<p>Inputs a Bieberbach group G (represented using AffineCrystGroupOnRight as in the GAP package Cryst). It also optionally inputs a choice of vector v in the euclidean space R^n on which G acts freely. The function returns n+1 terms of the free ZG-resolution of Z arising as the cellular chain complex of the tesselation of R^n by the Dirichlet-Voronoi fundamental domain determined by v.</p>
<p>This is a HAPcryst function and is thus only available if HAPcryst is loaded.</p>
<p>The function requires the use of Polymake software.</td>
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<td class="tdleft"><code class="code"> ResolutionDirectProduct(R,S) </code> <code class="code"> ResolutionDirectProduct(R,S,"internal")</code></p>
<p>Inputs a ZG-resolution R and ZH-resolution S. It outputs a ZD-resolution for the direct product D=G x H.</p>
<p>If G and H lie in a common group K, and if they commute and have trivial intersection, then an optional third variable "internal" can be used. This will force D to be the subgroup GH in K.</td>
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<td class="tdleft"><code class="code"> ResolutionExtension(g,R,S) </code> <code class="code"> ResolutionExtension(g,R, S,"TestFiniteness")</code> <code class="code"> ResolutionExtension(g,R,S,"NoTest",GmapE)</code></p>
<p>Inputs a surjective group homomorphism g:E --> G with kernel N. It also inputs a ZN-resolution R and a ZG-resolution S. It returns a ZE-resolution. The groups E and G can be infinite.</p>
<p>If an optional fourth argument is set equal to "TestFiniteness" then the groups N and G will be tested to see if they are finite. If they are finite then some speed saving routines will be invoked.</p>
<p>If the homomorphism g is such that the GAP function PreImagesElement(g,x) doesn't work, then a function GmapE() should be included as a fifth input. For any x in G this function should return an element GmapE(x) in E which gets mapped onto x by g.</p>
<p>The contracting homotopy on the ZE-resolution has not yet been fully implemented for infinite groups!</td>
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<td class="tdleft"><code class="code"> ResolutionFiniteDirectProduct(R,S) </code> <code class="code"> ResolutionFiniteDirectProduct(R,S, "internal")</code></p>
<p>Inputs a ZG-resolution R and ZH-resolution S where G and H are finite groups. It outputs a ZD-resolution for the direct product D=G×H.</p>
<p>If G and H lie in a common group K, and if they commute and have trivial intersection, then an optional third variable "internal" can be used. This will force D to be the subgroup GH in K.</td>
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<td class="tdleft"><code class="code"> ResolutionFiniteExtension(gensE,gensG,R,n)</code> <code class="code"> ResolutionFiniteExtension(gensE,gensG,R,n,true) </code> <code class="code"> ResolutionFiniteExtension(gensE,gensG,R,n,false,S) </code></p>
<p>Inputs: a set gensE of generators for a finite group E; a set gensG equal to the image of gensE in a quotient group G of E; a ZG-resolution R up to dimension at least n; a positive integer n. It uses the TwistedTensorProduct() construction to return n terms of a ZE-resolution.</p>
<p>The function has an optional fourth argument which, when set equal to "true", invokes tietze reductions in the construction of a resolution for the kernel of E --> G.</p>
<p>If a ZN-resolution S is available, where N is the kernel of the quotient E --> G, then this can be incorporated into the computations using an optional fifth argument.</td>
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<td class="tdleft"><code class="code"> ResolutionFiniteGroup(gens,n)</code> <code class="code"> ResolutionFiniteGroup(gens,n,true)</code> <code class="code"> ResolutionFiniteGroup(gens,n,false,p) </code></p>
<p>Inputs a set gens of generators for a finite group G and a positive integer n. It outputs n terms of a ZG-resolution.</p>
<p>The function has an optional third argument which, when set equal to true, invokes tietze reductions in the construction of the resolution.</p>
<p>The function has an optional fourth argument which, when set equal to a prime p, records the fact that the resolution will only be used for mod p calculations. This could speed up subsequent constructions.</td>
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<td class="tdleft"><code class="code"> ResolutionFiniteSubgroup(R,K)</code> <code class="code"> ResolutionFiniteSubgroup(R,gensG,gensK)</code></p>
<p>Inputs a ZG-resolution for a finite group G and a subgroup K of index |G:K|. It returns a free ZK-resolution whose ZK-rank is |G:K| times the ZG-rank in each dimension.</p>
<p>Generating sets gensG, gensK for G and K can also be input to the function (though the method does not depend on a choice of generators).</p>
<p>This ZK-resolution is not reduced. ie. it has more than one generator in dimension 0.</td>
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<td class="tdleft"><code class="code"> ResolutionGraphOfGroups(D,n) </code> <code class="code"> ResolutionGraphOfGroups(D,n,L) </code></p>
<p>Inputs a graph of groups D and a positive integer n. It returns n terms of a free ZG-resolution for the fundamental group G of D.</p>
<p>An optional third argument L=[R_1 , ... , R_t] can be used to list (in any order) free resolutions for some/all of the vertex and edge groups in D. If for some vertex or edge group no resolution is listed in L then the function ResolutionFiniteGroup() will be used to try to construct the resolution.</p>
<p>The ZG-resolution is usually not reduced. i.e. it has more than one generator in dimension 0.</p>
<p>The contracting homotopy on the ZG-resolution has not yet been implemented! Furthermore, the group G is currently returned only as a finitely presented group (without any method for solving the word problem).</td>
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<td class="tdleft"><code class="code"> ResolutionNilpotentGroup(G,n) </code> <code class="code"> ResolutionNilpotentGroup(G,n,"TestFiniteness")</code></p>
<p>Inputs a nilpotent group G and positive integer n. It returns n terms of a free ZG-resolution. The resolution is computed using a divide-and-conquer technique involving the lower central series.</p>
<p>This function can be applied to infinite groups G. For finite groups the function ResolutionNormalSeries() probably gives better results.</p>
<p>If an optional third argument is set equal to "TestFiniteness" then the groups N and G will be tested to see if they are finite. If they are finite then some speed saving routines will be invoked.</p>
<p>The contracting homotopy on the ZE-resolution has not yet been fully implemented for infinite groups.</td>
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<td class="tdleft"><code class="code"> ResolutionNormalSeries(L,n) </code> <code class="code"> ResolutionNormalSeries(L,n,true)</code> <code class="code"> ResolutionNormalSeries(L,n,false,p)</code></p>
<p>Inputs a positive integer n and a list L = [L_1 , ..., L_k] of normal subgroups L_i of a finite group G satisfying G = L_1 > L2 >... >L_k. Alternatively, L = [gensL_1, ... gensL_k] can be a list of generating sets for the L_i (and these particular generators will be used in the construction of resolutions). It returns a ZG-resolution by repeatedly using the function ResolutionFiniteExtension().</p>
<p>The function has an optional third argument which, if set equal to true, invokes tietze reductions in the construction of resolutions.</p>
<p>The function has an optional fourth argument which, if set equal to p > 0, produces a resolution which is only valid for mod p calculations.</td>
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<td class="tdleft"><code class="code"> ResolutionPrimePowerGroup(P,n) </code> <code class="code"> ResolutionPrimePowerGroup(G,n,p)</code></p>
<p>Inputs a p-group P and integer n>0. It uses GAP's standard linear algebra functions over the field F of p elements to construct a free FP-resolution for mod p calculations only. The resolution is minimal - meaning that the number of generators of R_n equals the rank of H_n(P,F).</p>
<p>The function can also be used to obtain a free non-minimal FG-resolution of a small group G of non-prime-power order. In this case the prime p must be entered as the third input variable. (In the non-prime-power case the algorithm is naive and not very good.)</td>
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<td class="tdleft"><code class="code"> ResolutionSmallFpGroup(G,n) </code> <code class="code"> ResolutionSmallFpGroup(G,n,p) </code></p>
<p>Inputs a small finitely presented group G and an integer n>0. It returns n terms of a ZG-resolution which, in dimensions 1 and 2, corresponds to the given presentation for G. The method returns no contracting homotopy for the resolution.</p>
<p>The function has an optional fourth argument which, when set equal to a prime p, records the fact that the resolution will only be used for mod p calculations. This could speed up subsequent constructions.</p>
<p>This function was written by Irina Kholodna.</td>
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<td class="tdleft"><code class="code"> ResolutionSubgroup(R,K)</code></p>
<p>Inputs a ZG-resolution for an (infinite) group G and a subgroup K of finite index |G:K|. It returns a free ZK-resolution whose ZK-rank is |G:K| times the ZG-rank in each dimension.</p>
<p>If G is finite then the function ResolutionFiniteSubgroup(R,G,K) will probably work better. In particular, resolutions from this function probably won't work with the function EquivariantChainMap(). This ZK-resolution is not reduced. i.e. it has more than one generator in dimension 0.</td>
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<td class="tdleft"><code class="code"> ResolutionSubnormalSeries(L,n) </code></p>
<p>Inputs a positive integer n and a list L = [L_1 , ... , L_k] of subgroups L_i of a finite group G=L_1 such that L_1 > L2 ... > L_k is a subnormal series in G (meaning that each L_i+1 must be normal in L_i). It returns a ZG-resolution by repeatedly using the function ResolutionFiniteExtension().</p>
<p>If L is a series of normal subgroups in G then the function ResolutionNormalSeries(L,n) will possibly work more efficiently.</td>
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<td class="tdleft"><code class="code"> TwistedTensorProduct(R,S,EhomG,GmapE,NhomE,NEhomN,EltsE,Mult,InvE)</code></p>
<p>Inputs a ZG-resolution R, a ZN-resolution S, and other data relating to a short exact sequence 1 --> N --> E --> G --> 1. It uses a perturbation technique of CTC Wall to construct a ZE-resolution F. Both G and N could be infinite. The "length" of F is equal to the minimum of the "length"s of R and S. The resolution R needs no contracting homotopy if no such homotopy is requied for F.</td>
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