<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivariantChainMap</code>( <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> of <span class="SimpleMath">Z</span>, a free <span class="SimpleMath">ZQ</span>-resolution <span class="SimpleMath">S</span> of <span class="SimpleMath">Z</span>, and a group homomorphism <span class="SimpleMath">f: G → Q</span>. It returns the induced <span class="SimpleMath">f</span>-equivariant chain map <span class="SimpleMath">F: R → S</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGResolution</code>( <var class="Arg">P</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">P_∗</span> and a positive integer <span class="SimpleMath">n</span>. It attempts to return <span class="SimpleMath">n</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span>. However, the stabilizer groups in the non-free resolution must be such that HAP can construct free resolutions with contracting homotopies for them.</p>
<p>The contracting homotopy on the resolution was implemented by Bui Anh Tuan.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionBieberbachGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionBieberbachGroup</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a torsion free crystallographic group <span class="SimpleMath">G</span>, also known as a Bieberbach group, represented using <strong class="button">AffineCrystGroupOnRight</strong> as in the GAP package Cryst. It also optionally inputs a choice of vector <span class="SimpleMath">v</span> in the Euclidean space <span class="SimpleMath">R^n</span> on which <span class="SimpleMath">G</span> acts freely. The function returns <span class="SimpleMath">n+1</span> terms of the free ZG-resolution of <span class="SimpleMath">Z</span> arising as the cellular chain complex of the tessellation of <span class="SimpleMath">R^n</span> by the Dirichlet-Voronoi fundamental domain determined by <span class="SimpleMath">v</span>. No contracting homotopy is returned with the resolution.</p>
<p>This function is part of the HAPcryst package written by Marc Roeder and thus requires the HAPcryst package to be loaded.</p>
<p>The function requires the use of Polymake software.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionCubicalCrystGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a crystallographic group <span class="SimpleMath">G</span> represented using <strong class="button">AffineCrystGroupOnRight</strong> as in the GAP package <span class="SimpleMath">Cryst</span> together with an integer <span class="SimpleMath">k ≥ 1</span>. The function tries to find a cubical fundamental domain in the Euclidean space <span class="SimpleMath">R^n</span> on which <span class="SimpleMath">G</span> acts. If it succeeds it uses this domain to return <span class="SimpleMath">k+1</span> terms of a free ZG-resolution of <span class="SimpleMath">Z</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionFiniteGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">G</span> and an integer <span class="SimpleMath">k ≥ 1</span>. It returns <span class="SimpleMath">k+1</span> terms of a free ZG-resolution of <span class="SimpleMath">Z</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNilpotentGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a nilpotent group <span class="SimpleMath">G</span> (which can be infinite) and an integer <span class="SimpleMath">k ≥ 1</span>. It returns <span class="SimpleMath">k+1</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNormalSeries</code>( <var class="Arg">L</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a a list <span class="SimpleMath">L</span> consisting of a chain $<span class="SimpleMath">1=N_1 ≤ N_2 ≤ ⋯ ≤ N_n =G</span> of normal subgroups of <span class="SimpleMath">G</span>, together with an integer <span class="SimpleMath">k ≥ 1</span>. It returns <span class="SimpleMath">k+1</span> terms of a free ZG-resolution of <span class="SimpleMath">Z</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionPrimePowerGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and an integer <span class="SimpleMath">k ≥ 1</span>. It returns <span class="SimpleMath">k+1</span> terms of a minimal free <span class="SimpleMath">FG</span>-resolution of the field <span class="SimpleMath">F</span> of <span class="SimpleMath">p</span> elements.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSL2Z</code>( <var class="Arg">m</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs positive integers <span class="SimpleMath">m, n</span> and returns <span class="SimpleMath">n</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span> for the group <span class="SimpleMath">G=SL_2( Z[1/m])</span>.</p>
<p>This function is joint work with Bui Anh Tuan.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a small group <span class="SimpleMath">G</span> and an integer <span class="SimpleMath">k ≥ 1</span>. It returns <span class="SimpleMath">k+1</span> terms of a free ZG-resolution of <span class="SimpleMath">Z</span>.</p>
<p>If <span class="SimpleMath">G</span> is a finitely presented group then up to degree <span class="SimpleMath">2</span> the resolution coincides with cellular chain complex of the universal cover of the <span class="SimpleMath">2</span> complex associated to the presentation of <span class="SimpleMath">G</span>. Thus the boundaries of the generators in degree <span class="SimpleMath">3</span> provide a generating set for the module of identities of the presentation.</p>
<p>This function was written by Irina Kholodna.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSubgroup</code>( <var class="Arg">R</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free ZG-resolution of <span class="SimpleMath">Z</span> and a finite index subgroup <span class="SimpleMath">H ≤ G</span>. It returns a free ZH-resolution of <span class="SimpleMath">Z</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeibnizComplex</code>( <var class="Arg">g</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Leibniz algebra, or Lie algebra, <span class="SimpleMath">mathfrakg</span> over a ring <span class="SimpleMath">K</span> together with an integer <span class="SimpleMath">n≥ 0</span>. It returns the first <span class="SimpleMath">n</span> terms of the Leibniz chain complex over <spanclass="SimpleMath">K</span>. The complex was implemented by Pablo Fernandez Ascariz.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegers</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegers</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegers</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain complex <span class="SimpleMath">C</span> of free abelian groups and returns the cochain complex <span class="SimpleMath">Hom_ Z(C, Z)</span>.</p>
<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> in characteristic <span class="SimpleMath">0</span> and returns the cochain complex <span class="SimpleMath">Hom_ ZG(R, Z)</span>.</p>
<p>Inputs an equivariant chain map <span class="SimpleMath">F: R→ S</span> of resolutions and returns the induced cochain map <span class="SimpleMath">Hom_ ZG(S, Z) ⟶ Hom_ ZG(R, Z)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegralModule</code>( <var class="Arg">R</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> in characteristic <span class="SimpleMath">0</span> and a group homomorphism <span class="SimpleMath">A: G → GL_n( Z)</span>. The homomorphism <span class="SimpleMath">A</span> can be viewed as the <span class="SimpleMath">ZG</span>-module with underlying abelian group <span class="SimpleMath">Z^n</span> on which <span class="SimpleMath">G</span> acts via the homomorphism <span class="SimpleMath">A</span>. It returns the cochain complex <span class="SimpleMath">Hom_ ZG(R,A)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegers</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegers</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> of characteristic <span class="SimpleMath">0</span> and returns the chain complex <span class="SimpleMath">R ⊗_ ZG Z</span>.</p>
<p>Inputs an equivariant chain map <span class="SimpleMath">F: R → S</span> in characteristic <span class="SimpleMath">0</span> and returns the induced chain map <span class="SimpleMath">F⊗_ ZG Z : R ⊗_ ZG Z ⟶ S ⊗_ ZG Z</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><codespan="ContSS"<br />span="nocss"spanahref.#X796632C585D47245>.- ModPCohomologyRingspan
derivation classS":E →E/> on SimpleMath"=/I/> apolynomial < class=""R Fx_1,..,]/span> over fieldspan ="><span returns <span="impleMath">SJh</spanwhere< class"SimpleMath">S</span is a ring <spanclassSimpleMathJ<spanis list of foranidealin< class=""S<spansuch thatthere isan <spanclassSimpleMath>α SJ →ker/im/span This liftsto thering <spanclass="SimpleMath">h:S → d/>.This was written by Paul Smith. It uses the Singular commutative algebra package.</p>
<< class>xamples:strongp>
p< id="" name""><a><pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<h5>2.4-5 IntegralCohomologyGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntegralCohomologyGenerators</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs at least <span class="SimpleMath">n+1</span> terms of a free <span class="SimpleMath">ZG</span>-resolution of/>
=""="func" width0>tr ="tdleft">code=func27 /> var=ArgG/>,< =ArgN/> < =Argr/>)/>td=tdright(&;functionnbsp</d</tr/able/div
<p>Inputs a finite <span class="SimpleMath">2</span>-group <span class="SimpleMath">G</span>, and normal subgroup <span class="SimpleMath">N</span> and an integer <span class="SimpleMath">r</span>. It returns a list of length <span class="SimpleMath">r</span> whose <span class="SimpleMath">i</span>-th term is a presentation for the <span class="SimpleMath">i</span>-th page of the Lyndon-Hochschild-Serre spectral sequence. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>
class"unc>>&8227; LHSSpectralSequenceLastSheet</code(<var class=ArgG/>, <varclass="Arg>N >( &;)</></tr>/table>/>
<p>Inputs a finite <span class="SimpleMath">2</span>-group <span class="SimpleMath">G</span> and normal subgroup <span class="SimpleMath">N</span>. It returns presentation for the <span class="SimpleMath"<p>Inputs a non-free <span class"SimpleMath"ZG<span-resolution < class="SimpleMath">P_∗/pananda integer< class"impleMath"n/span> It attemptsto return < class="SimpleMath>n
‣ ResolutionBieberbachGroup( G, (nbsp )<td>/></table<div divclass"">< class"unc width=10">tr""< =func>&27 ModPCohomologyGenerators < class/>)td">&bspfunction&bsp;)
<p>Inputs either a <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and positive integer <span class="SimpleMath">n</span>, or else <span class="SimpleMath">n+1</span> terms of a minimal <span class="SimpleMath">FG</span>-resolution <span class="SimpleMath">R</span> of the field <span class="SimpleMath">F</span> of <span class="SimpleMath">p</span> elements. It returns a pair whose first entry is a minimal list of homogeneous
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">R</var> )</td><td class="tdright">( function > a <span=SimpleMath</pan < =""><> an< class"k /pan. < class""+/> of free< class=S"<>resolution field=> of class"">/>.</>
<p>When applied toafinite <span="SimpleMath"></>-group<spanclass=SimpleMath>G/> this returnsapresentation mod-<panclass"SimpleMath"2</>cohomology <span=SimpleMath>H∗,F</span Lyndon-Hochschild-Serrespectralsequenceisusedtothatthepresentationis . When functionisappliedto spanclassSimpleMath></span- Gandpositiveinteger<span class"impleMath"><span the functionfirst constructsspan class"">n1span terms a freespan class=SimpleMath>FG/>-resolutionspanclass""><span> then thefinite-dimensionalgradedalgebra < class"">=^( ≤n)G, F)</>,and finallyuses<pan="SimpleMath> to approximate a presentation SimpleMath>^G </>. sufficientlylarge" spanclass"">n the approximation will a correct presentation for H^(G F</ thefunctioncan applied resolutions =">R or graded algebra A. This function was written by Paul Smith. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.
>(nbsp )</d><tr/table>>
></panItreturns quotientof span=""fx)P()/Q(x</>whose="xk"H_kG, ) forall< ="SimpleMathk/span inthe <span="">1 k ≤n<span>.Thesecond variablecanbeomitted inwhichcasethefunction to choose a `reasonable valuefor< =""n/pan For2groups function <strong="">oincareSeriesLHSG)<strongcanbeusedtoproduce spanclass"SimpleMath">(x/>that correctinalldegrees thegroups classSimpleMath>G</span thefunction also(tleastspanclass"impleMath"</> of mod-spanclassSimpleMathp</panresolution <spanclass"SimpleMath"R<span for <spanclassG>.Alternatively,thefirst input can bea list<span class"SimpleMath"L</span of. this the of<span class"SimpleMath"xk/span> < class=SimpleMath>(x)/span>is to the <span class="SimpleMath">(k+1)</span>st term in the list.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns a quotient of polynomials <span class="SimpleMath">f(x)=P(x)/Q(x)</span> whose expansion has coefficient of <span class="SimpleMath">x^k</span> equal to the rank of the vector space <span class="SimpleMath">H_k(G, F_p)</span> for all <span class="SimpleMath">k</span> in the range <span class="SimpleMath">1 ≤ k ≤ n</span>. (The second input variable can be omitted, in which case the function tries to choose a `reasonable' value for n. For 2-groups the function PoincareSeriesLHS(G) can be used to produce an f(x) that is correct in all degrees.) In place of the
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeriesII.html">7</a></span> , <span class=<>When to <pan=""></span>groupspan="SimpleMath>G/span>thisfunctionreturns presentationthe mod- =""<>cohomology class">∗G )span Lyndon-Hochschild-Serre that .When spanclassSimpleMath<- integer =<>the constructs ="n<> s =""FG/>resolution spanclassSR/, the finite-dimensionalgraded spanclassSimpleMath>=^ n(,F, uses< =SimpleMath>/>toapproximatea span="impleMathH,/> sufficiently ="the presentation SimpleMathH∗,<> function resolution=SimpleMath<spangradedclass></> It handle /java.lang.StringIndexOutOfBoundsException: Index 1338 out of bounds for length 1338
<div="">< class"width"10%>tr class"tdleft"< classfunc#27 RankHomologyPGroupcode < classArg">G, Pvar">< class="tdright&<>t>/<>
<p>Inputs a <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span>, a rational function <span class="SimpleMath">P</span> representing the Poincaré series of the mod-<span class="SimpleMath">p</span> cohomology of <span class="SimpleMath">G</span> and a positive integer <span class="SimpleMath">n</span>. It returns the minimum number
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupAlgebraAsFpGModule</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">p</span>-group <span class="SimpleMath">G</span> and returns the modular group algebra <span class="SimpleMath">F_pG</span> in the form of an <span class="SimpleMath">F_pG</span>-module.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalSeries</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalSeries</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an <span class="SimpleMath">F_pG</span>-module <span class="SimpleMath">M</span> and returns its radical series as a list of <span class="SimpleMath">F_pG</span>-modules.</p>
<p>Inputs a free <span class="SimpleMath">F_pG</span>-resolution R and returns the filtered chain complex <span class="SimpleMath">⋯ Rad_2( F_pG)R ≤ Rad_1( F_pG)R ≤ R</span>.</p>
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