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<div class="ChapSects"><a href="chap4.html#X7BFA4D1587D8DF49">4 <span class="Heading">Three Manifolds</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82D1348C79238C2D">4.1 <span class="Heading">Dehn Surgery</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X837811BB8181666E">4.10-2 <span class="Heading">Platonic cube manifolds</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F31DFDA846E8E75">4.17 <span class="Heading">Bipyramid manifolds</span></a>
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<h3>4 <span class="Heading">Three Manifolds</span></h3>
<p><a id="X82D1348C79238C2D" name="X82D1348C79238C2D"></a></p>
<h4>4.1 <span class="Heading">Dehn Surgery</span></h4>
<p>The following example constructs, as a regular CW-complex, a closed orientable 3-manifold <span class="SimpleMath">W</span> obtained from the 3-sphere by drilling out a tubular neighbourhood of a trefoil knot and then gluing a solid torus to the boundary of the cavity via a homeomorphism corresponding to a Dehn surgery coefficient <span class="SimpleMath">p/q=17/16</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ap:=ArcPresentation(PureCubicalKnot(3,1));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=17;;q:=16;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=ThreeManifoldViaDehnSurgery(ap,p,q);</span>
Regular CW-complex of dimension 3
</pre></div>
<p>The next commands show that this <span class="SimpleMath">3</span>-manifold <span class="SimpleMath">W</span> has integral homology</p>
<p><span class="SimpleMath">H_0(W, Z)= Z</span>, <span class="SimpleMath">H_1(W, Z)= Z_16</span>, <span class="SimpleMath">H_2(W, Z)=0</span>, <span class="SimpleMath">H_3(W, Z)= Z</span></p>
<p>and that the fundamental group <span class="SimpleMath">π_1(W)</span> is non-abelian.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,0);Homology(W,1);Homology(W,2);Homology(W,3);</span>
[ 0 ]
[ 16 ]
[ ]
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FundamentalGroup(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=LowIndexSubgroupsFpGroup(F,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(L,AbelianInvariants);</span>
[ [ 16 ], [ 3, 8 ], [ 3, 4 ], [ 2, 3 ], [ 16, 43 ], [ 8, 43, 43 ] ]
</pre></div>
<p>The following famous result of Lickorish and (independently) Wallace shows that Dehn surgery on knots leads to an interesting range of spaces.</p>
<p><strong class="button">Theorem:</strong> <em> Every closed, orientable, connected <span class="SimpleMath">3</span>-manifold can be obtained by surgery on a link in <span class="SimpleMath">S^3</span>. (Moreover, one can always perform the surgery with surgery coefficients <span class="SimpleMath">± 1</span> and with each individual component of the link unknotted.) </em></p>
<p><a id="X848EDEE882B36F6C" name="X848EDEE882B36F6C"></a></p>
<h4>4.2 <span class="Heading">Connected Sums</span></h4>
<p>The following example constructs the connected sum <span class="SimpleMath">W=A#B</span> of two <span class="SimpleMath">3</span>-manifolds, where <span class="SimpleMath">A</span> is obtained from a <span class="SimpleMath">5/1</span> Dehn surgery on the complement of the first prime knot on 11 crossings and <span class="SimpleMath">B</span> is obtained by a <span class="SimpleMath">5/1</span> Dehn surgery on the complement of the second prime knot on 11 crossings. The homology groups</p>
<p><span class="SimpleMath">H_1(W, Z) = Z_2⊕ Z_594</span>, <span class="SimpleMath">H_2(W, Z) = 0</span>, <span class="SimpleMath">H_3(W, Z) = Z</span></p>
<p>are computed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ap1:=ArcPresentation(PureCubicalKnot(11,1));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ThreeManifoldViaDehnSurgery(ap1,5,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ap2:=ArcPresentation(PureCubicalKnot(11,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=ThreeManifoldViaDehnSurgery(ap2,5,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=ConnectedSum(A,B); #W:=ConnectedSum(A,B,-1) would yield A#-B where -B has the opposite orientation</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,1);</span>
[ 2, 594 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,2);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,3);</span>
[ 0 ]
</pre></div>
<p><a id="X78AE684C7DBD7C70" name="X78AE684C7DBD7C70"></a></p>
<h4>4.3 <span class="Heading">Dijkgraaf-Witten Invariant</span></h4>
<p>Given a closed connected orientable <span class="SimpleMath">3</span>-manifold <span class="SimpleMath">W</span>, a finite group <span class="SimpleMath">G</span> and a 3-cocycle <span class="SimpleMath">α∈ H^3(BG,U(1))</span> Dijkgraaf and Witten define the complex number</p>
<p>$$ Z^{G,\alpha}(W) = \frac{1}{|G|}\sum_{\gamma\in {\rm Hom}(\pi_1W, G)} \langle \gamma^\ast[\alpha], [M]\rangle \ \in\ \mathbb C\ $$ where <span class="SimpleMath">γ</span> ranges over all group homomorphisms <span class="SimpleMath">γ: π_1W → G</span>. This complex number is an invariant of the homotopy type of <span class="SimpleMath">W</span> and is useful for distinguishing between certain homotopically distinct <span class="SimpleMath">3</span>-manifolds.</p>
<p>A homology version of the Dijkgraaf-Witten invariant can be defined as the set of homology homomorphisms $$D_G(W) =\{ \gamma_\ast\colon H_3(W,\mathbb Z) \longrightarrow H_3(BG,\mathbb Z) \}_{\gamma\in {\rm Hom}(\pi_1W, G)}.$$ Since <span class="SimpleMath">H_3(W, Z)≅ Z</span> we represent <span class="SimpleMath">D_G(W)</span> by the set <span class="SimpleMath">D_G(W)={ γ_∗(1) }_γ∈ Hom(π_1W, G)</span> where <span class="SimpleMath">1</span> denotes one of the two possible generators of <span class="SimpleMath">H_3(W, Z)</span>.</p>
<p>For any coprime integers <span class="SimpleMath">p,q≥ 1</span> the <em>lens space</em> <span class="SimpleMath">L(p,q)</span> is obtained from the 3-sphere by drilling out a tubular neighbourhood of the trivial knot and then gluing a solid torus to the boundary of the cavity via a homeomorphism corresponding to a Dehn surgery coefficient <span class="SimpleMath">p/q</span>. Lens spaces have cyclic fundamental group <span class="SimpleMath">π_1(L(p,q))=C_p</span> and homology <span class="SimpleMath">H_1(L(p,q), Z)≅ Z_p</span>, <span class="SimpleMath">H_2(L(p,q), Z)≅ 0</span>, <span class="SimpleMath">H_3(L(p,q), Z)≅ Z</span>. It was proved by J.H.C. Whitehead that two lens spaces <span class="SimpleMath">L(p,q)</span> and <span class="SimpleMath">L(p',q')</span> are homotopy equivalent if and only if <span class="SimpleMath">p=p'</span> and <span class="SimpleMath">qq'≡ ± n^2 mod p</span> for some integer <span class="SimpleMath">n</span>.</p>
<p>The following session constructs the two lens spaces <span class="SimpleMath">L(5,1)</span> and <span class="SimpleMath">L(5,2)</span>. The homology version of the Dijkgraaf-Witten invariant is used with <span class="SimpleMath">G=C_5</span> to demonstrate that the two lens spaces are not homotopy equivalent.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ap:=[[2,1],[2,1]];; #Arc presentation for the trivial knot</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L51:=ThreeManifoldViaDehnSurgery(ap,5,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=DijkgraafWittenInvariant(L51,CyclicGroup(5));</span>
[ g1^4, g1^4, g1, g1, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L52:=ThreeManifoldViaDehnSurgery(ap,5,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=DijkgraafWittenInvariant(L52,CyclicGroup(5));</span>
[ g1^3, g1^3, g1^2, g1^2, id ]
</pre></div>
<p>A theorem of Fermat and Euler states that if a prime <span class="SimpleMath">p</span> is congruent to 3 modulo 4, then for any <span class="SimpleMath">q</span> exactly one of <span class="SimpleMath">± q</span> is a quadratic residue mod p. For all other primes <span class="SimpleMath">p</span> either both or neither of <span class="SimpleMath">± q</span> is a quadratic residue mod <span class="SimpleMath">p</span>. Thus for fixed <span class="SimpleMath">p ≡ 3 mod 4</span> the lens spaces <span class="SimpleMath">L(p,q)</span> form a single homotopy class. There are precisely two homotopy classes of lens spaces for other <span class="SimpleMath">p</span>.</p>
<p>The following commands confirm that <span class="SimpleMath">L(13,1) ≄ L(13,2)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L13_1:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DijkgraafWittenInvariant(L13_1,CyclicGroup(13));</span>
[ g1^12, g1^12, g1^10, g1^10, g1^9, g1^9, g1^4, g1^4, g1^3, g1^3, g1, g1, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L13_2:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DijkgraafWittenInvariant(L13_2,CyclicGroup(13));</span>
[ g1^11, g1^11, g1^8, g1^8, g1^7, g1^7, g1^6, g1^6, g1^5, g1^5, g1^2, g1^2,
id ]
</pre></div>
<p><a id="X80B6849C835B7F19" name="X80B6849C835B7F19"></a></p>
<h4>4.4 <span class="Heading">Cohomology rings</span></h4>
<p>The following commands construct the multiplication table (with respect to some basis) for the cohomology rings <span class="SimpleMath">H^∗(L(13,1), Z_13)</span> and <span class="SimpleMath">H^∗(L(13,2), Z_13)</span>. These rings are isomorphic and so fail to distinguish between the homotopy types of the lens spaces <span class="SimpleMath">L(13,1)</span> and <span class="SimpleMath">L(13,2)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L13_1:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L13_2:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L13_1:=BarycentricSubdivision(L13_1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L13_2:=BarycentricSubdivision(L13_2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A13_1:=CohomologyRing(L13_1,13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A13_2:=CohomologyRing(L13_2,13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M13_1:=List([1..4],i->[]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B13_1:=CanonicalBasis(A13_1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M13_2:=List([1..4],i->[]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B13_2:=CanonicalBasis(A13_2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..4] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for j in [1..4] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">M13_1[i][j]:=B13_1[i]*B13_1[j];</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..4] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for j in [1..4] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">M13_2[i][j]:=B13_2[i]*B13_2[j];</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(M13_1);</span>
[ [ v.1, v.2, v.3, v.4 ],
[ v.2, 0*v.1, (Z(13)^6)*v.4, 0*v.1 ],
[ v.3, (Z(13)^6)*v.4, 0*v.1, 0*v.1 ],
[ v.4, 0*v.1, 0*v.1, 0*v.1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(M13_2);</span>
[ [ v.1, v.2, v.3, v.4 ],
[ v.2, 0*v.1, (Z(13))*v.4, 0*v.1 ],
[ v.3, (Z(13))*v.4, 0*v.1, 0*v.1 ],
[ v.4, 0*v.1, 0*v.1, 0*v.1 ] ]
</pre></div>
<p><a id="X7F56BB4C801AB894" name="X7F56BB4C801AB894"></a></p>
<h4>4.5 <span class="Heading">Linking Form</span></h4>
<p>Given a closed connected <strong class="button">oriented</strong> <span class="SimpleMath">3</span>-manifold <span class="SimpleMath">W</span> let <span class="SimpleMath">τ H_1(W, Z)</span> denote the torsion subgroup of the first integral homology. The <em>linking form</em> is a bilinear mapping</p>
<p><span class="SimpleMath">Lk_W: τ H_1(W, Z) × τ H_1(W, Z) ⟶ Q/ Z</span>.</p>
<p>To construct this form note that we have a Poincare duality isomorphism</p>
<p><span class="SimpleMath">ρ: H^2(W, Z) stackrel≅⟶ H_1(W, Z), z ↦ z∩ [W]</span></p>
<p>involving the cap product with the fundamental class <span class="SimpleMath">[W]∈ H^3(W, Z)</span>. That is, <span class="SimpleMath">[M]</span> is the generator of <span class="SimpleMath">H^3(W, Z)≅ Z</span> determining the orientation. The short exact sequence <span class="SimpleMath">Z ↣ Q ↠ Q/ Z</span> gives rise to a cohomology exact sequence</p>
<p><span class="SimpleMath">→ H^1(W, Q) → H^1(W, Q/ Z) stackrelβ⟶ H^2(W, Z) → H^2(W, Q) →</span></p>
<p>from which we obtain the isomorphism <span class="SimpleMath">β : τ H^1(W, Q/ Z) stackrel≅⟶ τ H^2(W, Z)</span>. The linking form <span class="SimpleMath">Lk_W</span> can be defined as the composite</p>
<p><span class="SimpleMath">Lk_W: τ H_1(W, Z) × τ H_1(W, Z) stackrel1× ρ^-1}⟶ τ H_1(W, Z) × τ H^2(W, Z) stackrel1× β^-1}⟶ τ H_1(W, Z) × τ H^1(W, Q/ Z) stackrelev⟶ Q/ Z</span></p>
<p>where <span class="SimpleMath">ev(x,α)</span> evaluates a <span class="SimpleMath">1</span>-cocycle <span class="SimpleMath">α</span> on a <span class="SimpleMath">1</span>-cycle <span class="SimpleMath">x</span>.</p>
<p>The linking form can be used to define the set</p>
<p><span class="SimpleMath">I^O(W) = {Lk_W(g,g) : g∈ τ H_1(W, Z)}</span></p>
<p>which is an oriented-homotopy invariant of <span class="SimpleMath">W</span>. Letting <span class="SimpleMath">W^+</span> and <span class="SimpleMath">W^-</span> denote the two possible orientations on the manifold, the set</p>
<p><span class="SimpleMath">I(W) ={I^O(W^+), I^O(W^-)}</span></p>
<p>is a homotopy invariant of <span class="SimpleMath">W</span> which in this manual we refer to as the <em>linking form homotopy invariant</em>.</p>
<p>The following commands compute the linking form homotopy invariant for the lens spaces <span class="SimpleMath">L(13,q)</span> with <span class="SimpleMath">1≤ q≤ 12</span>. This invariant distinguishes between the two homotopy types that arise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LensSpaces:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for q in [1..12] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(LensSpaces,ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,q));</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(List(LensSpaces,LinkingFormHomotopyInvariant));;</span>
[ [ [ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ],
[ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ] ],
[ [ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ], [ 0, 2/13,
2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ],
[ [ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ],
[ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ] ],
[ [ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ],
[ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ] ],
[ [ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ],
[ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ],
[ [ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ],
[ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ],
[ [ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ],
[ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ],
[ [ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ],
[ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ],
[ [ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ],
[ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ] ],
[ [ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ],
[ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ] ],
[ [ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ],
[ 0, 2/13, 2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ],
[ [ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ],
[ 0, 1/13, 1/13, 3/13, 3/13, 4/13, 4/13, 9/13, 9/13, 10/13, 10/13, 12/13, 12/13 ] ] ]
</pre></div>
<p><a id="X850C76697A6A1654" name="X850C76697A6A1654"></a></p>
<h4>4.6 <span class="Heading">Determining the homeomorphism type of a lens space</span></h4>
<p>In 1935 K. Reidemeister <a href="chapBib.html#biBreidemeister">[Rei35]</a> classified lens spaces up to orientation preserving PL-homeomorphism. This was generalized by E. Moise <a href="chapBib.html#biBmoise">[Moi52]</a> in 1952 to a classification up to homeomorphism -- his method requred the proof of the Hauptvermutung for <span class="SimpleMath">3</span>-dimensional manifolds. In 1960, following a suggestion of R. Fox, a proof was given by E.J. Brody <a href="chapBib.html#biBbrody">[Bro60]</a> that avoided the need for the Hauptvermutung. Reidemeister's method, using what is know termed <em>Reidermeister torsion</em>, and Brody's method, using tubular neighbourhoods of <span class="SimpleMath">1</span>-cycles, both require identifying a suitable "preferred" generator of <span class="SimpleMath">H_1(L(p,q), Z)</span>. In 2003 J. Przytycki and A. Yasukhara <a href="chapBib.html#biBprzytycki">[PY03]</a> provided an alternative method for classifying lens spaces, which uses the linking form and again requires the identification of a "preferred" generator of <span class="SimpleMath">H_1(L(p,q), Z)</span>.</p>
<p>Przytycki and Yasukhara proved the following.</p>
<p><strong class="button">Theorem.</strong> <em>Let <span class="SimpleMath">ρ: S^ 3 → L(p, q)</span> be the <span class="SimpleMath">p</span>-fold cyclic cover and <span class="SimpleMath">K</span> a knot in <span class="SimpleMath">L(p, q)</span> that represents a generator of <span class="SimpleMath">H_1 (L(p, q), Z)</span>. If <span class="SimpleMath">ρ ^-1 (K)</span> is the trivial knot, then <span class="SimpleMath">Lk_ L(p,q) ([K], [K]) = q/p</span> or <span class="SimpleMath">= overline q/p ∈ Q/ Z</span> where <span class="SimpleMath">qoverline q ≡ 1 mod p</span>. </em></p>
<p>The ingredients of this theorem can be applied in HAP, but at present only to small examples of lens spaces. The obstruction to handling large examples is that the current default method for computing the linking form involves barycentric subdivision to produce a simplicial complex from a regular CW-complex, and then a homotopy equivalence from this typically large simplicial complex to a smaller non-regular CW-complex. However, for homeomorphism invariants that are not homotopy invariants there is a need to avoid homotopy equivalences. In the current version of HAP this means that in order to obtain delicate homeomorphism invariants we have to perform homology computations on typically large simplicial complexes. In a future version of HAP we hope to avoid the obstruction by implementing cup products, cap products and linking forms entirely within the category of regular CW-complexes.</p>
<p>The following commands construct a small lens space <span class="SimpleMath">L=L(p,q)</span> with unknown values of <span class="SimpleMath">p,q</span>. Subsequent commands will determine the homeomorphism type of <span class="SimpleMath">L</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=Random([2,3,5,7,11,13,17]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">q:=Random([1..p-1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],p,q);</span>
Regular CW-complex of dimension 3
</pre></div>
<p>We can readily determine the value of <span class="SimpleMath">p=11</span> by calculating the order of <span class="SimpleMath">π_1(L)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FundamentalGroupWithPathReps(L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(F);</span>
"C11"
</pre></div>
<p>The next commands take the default edge path <span class="SimpleMath">γ: S^1→ L</span> representing a generator of the cyclic group <span class="SimpleMath">π_1(L)</span> and lift it to an edge path <span class="SimpleMath">tildeγ: S^1→ tilde L</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=UniversalCover(L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=U!.group;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=EquivariantCWComplexToRegularCWMap(U,Group(One(G)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=Source(p);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gamma:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gamma[2]:=F!.loops[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gamma[2]:=List(gamma[2],AbsInt);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gamma[1]:=List(gamma[2],k->L!.boundaries[2][k]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gamma[1]:=SSortedList(Flat(gamma[1]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"></span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gammatilde:=[[],[],[],[]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for k in [1..U!.nrCells(0)] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">if p!.mapping(0,k) in gamma[1] then Add(gammatilde[1],k); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for k in [1..U!.nrCells(1)] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">if p!.mapping(1,k) in gamma[2] then Add(gammatilde[2],k); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gammatilde:=CWSubcomplexToRegularCWMap([U,gammatilde]);</span>
Map of regular CW-complexes
</pre></div>
<p>The next commands check that the path <span class="SimpleMath">tildeγ</span> is unknotted in <span class="SimpleMath">tilde L≅ S^3</span> by checking that <span class="SimpleMath">π_1(tilde L∖ image(tildeγ))</span> is infinite cyclic.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=RegularCWComplexComplement(gammatilde);</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=FundamentalGroup(C);</span>
<fp group of size infinity on the generators [ f2 ]>
</pre></div>
<p>Since <span class="SimpleMath">tildeγ</span> is unkotted the cycle <span class="SimpleMath">γ</span> represents the preferred generator <span class="SimpleMath">[γ]∈ H_1(L, Z)</span>. The next commands compute <span class="SimpleMath">Lk_L([γ],[γ])= 7/11</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LinkingFormHomeomorphismInvariant(L);</span>
[ 7/11 ]
</pre></div>
<p>The classification of Moise/Brody states that <span class="SimpleMath">L(p,q)≅ L(p,q')</span> if and only if <span class="SimpleMath">qq'≡ ± 1 mod p</span>. Hence the lens space <span class="SimpleMath">L</span> has the homeomorphism type</p>
<p><span class="SimpleMath">L≅ L(11,7) ≅ L(11,8) ≅ L(11,4) ≅ L(11,3)</span>.</p>
<p><a id="X7EC6B008878CC77E" name="X7EC6B008878CC77E"></a></p>
<h4>4.7 <span class="Heading">Surgeries on distinct knots can yield homeomorphic manifolds</span></h4>
<p>The lens space <span class="SimpleMath">L(5,1)</span> is a quotient of the <span class="SimpleMath">3</span>-sphere <span class="SimpleMath">S^3</span> by a certain action of the cyclic group <span class="SimpleMath">C_5</span>. It can be realized by a <span class="SimpleMath">p/q=5/1</span> Dehn filling of the complement of the trivial knot. It can also be realized by Dehn fillings of other knots. To see this, the following commands compute the manifold <span class="SimpleMath">W</span> obtained from a <span class="SimpleMath">p/q=1/5</span> Dehn filling of the complement of the trefoil and show that <span class="SimpleMath">W</span> at least has the same integral homology and same fundamental group as <span class="SimpleMath">L(5,1)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ap:=ArcPresentation(PureCubicalKnot(3,1));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=ThreeManifoldViaDehnSurgery(ap,1,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,1);</span>
[ 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,2);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(W,3);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FundamentalGroup(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(F);</span>
"C5"
</pre></div>
<p>The next commands construct the universal cover <span class="SimpleMath">widetilde W</span> and show that it has the same homology as <span class="SimpleMath">S^3</span> and trivivial fundamental group <span class="SimpleMath">π_1(widetilde W)=0</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=UniversalCover(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=U!.group;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Wtilde:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G)));</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Wtilde,1);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Wtilde,2);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(Wtilde,3);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FundamentalGroup(Wtilde);</span>
<fp group on the generators [ ]>
</pre></div>
<p>By construction the space <span class="SimpleMath">widetilde W</span> is a manifold. Had we not known how the regular CW-complex <span class="SimpleMath">widetilde W</span> had been constructed then we could prove that it is a closed <span class="SimpleMath">3</span>-manifold by creating its barycentric subdivision <span class="SimpleMath">K=sdwidetilde W</span>, which is homeomorphic to <span class="SimpleMath">widetilde W</span>, and verifying that the link of each vertex in the simplicial complex <span class="SimpleMath">sdwidetilde W</span> is a <span class="SimpleMath">2</span>-sphere. The following command carries out this proof.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Wtilde);</span>
true
</pre></div>
<p>The Poincare conjecture (now proven) implies that <span class="SimpleMath">widetilde W</span> is homeomorphic to <span class="SimpleMath">S^3</span>. Hence <span class="SimpleMath">W=S^3/C_5</span> is a quotient of the <span class="SimpleMath">3</span>-sphere by an action of <span class="SimpleMath">C_5</span> and is hence a lens space <span class="SimpleMath">L(5,q)</span> for some <span class="SimpleMath">q</span>.</p>
<p>The next commands determine that <span class="SimpleMath">W</span> is homeomorphic to <span class="SimpleMath">L(5,4)≅ L(5,1)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Lk:=LinkingFormHomeomorphismInvariant(W);</span>
[ 0, 4/5 ]
</pre></div>
<p>Moser <a href="chapBib.html#biBlmoser">[Mos71]</a> gives a precise decription of the lens spaces arising from surgery on the trefoil knot and more generally from surgery on torus knots. Greene <a href="chapBib.html#biBgreene">[Gre13]</a> determines the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere</p>
<p><a id="X7B425A3280A2AF07" name="X7B425A3280A2AF07"></a></p>
<h4>4.8 <span class="Heading">Finite fundamental groups of <span class="SimpleMath">3</span>-manifolds</span></h4>
<p>Lens spaces are examples of <span class="SimpleMath">3</span>-manifolds with finite fundamental groups. The complete list of finite groups <span class="SimpleMath">G</span> arising as fundamental groups of closed connected <span class="SimpleMath">3</span>-manifolds is recalled in <a href="chap7.html#X79B1406C803FF178"><span class="RefLink">7.12</span></a> where one method for computing their cohomology rings is presented. Their cohomology could also be computed from explicit <span class="SimpleMath">3</span>-manifolds <span class="SimpleMath">W</span> with <span class="SimpleMath">π_1W=G</span>. For instance, the following commands realize a closed connected <span class="SimpleMath">3</span>-manifold <span class="SimpleMath">W</span> with <span class="SimpleMath">π_1W = C_11× SL_2( Z_5)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ap:=ArcPresentation(PureCubicalKnot(3,1));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=ThreeManifoldViaDehnSurgery(ap,1,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FundamentalGroup(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(F);</span>
1320
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(F);</span>
[ 11 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(F);</span>
"C11 x SL(2,5)"
</pre></div>
<p>Hence the group <span class="SimpleMath">G=C_11× SL_2( Z_5)</span> of order <span class="SimpleMath">1320</span> acts freely on the <span class="SimpleMath">3</span>-sphere <span class="SimpleMath">widetilde W</span>. It thus has periodic cohomology with</p>
<p class="pcenter">
H_n(G,\mathbb Z) = \left\{ \begin{array}{ll}
\mathbb Z_{11} & n\equiv 1 \bmod 4 \\
0 & n\equiv 2 \bmod 4 \\
\mathbb Z_{1320} & n \equiv 3\bmod 4\\
\mathbb 0 & n\equiv 0 \bmod 4 \\
\end{array}\right.
</p>
<p>for <span class="SimpleMath">n > 0</span>.</p>
<p><a id="X78912D227D753167" name="X78912D227D753167"></a></p>
<h4>4.9 <span class="Heading">Poincare's cube manifolds</span></h4>
<p>In his seminal paper on "Analysis situs", published in 1895, Poincare constructed a series of closed 3-manifolds which played an important role in the development of his theory. A good account of these manifolds is given in the online <span class="URL"><a href="http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_cube_manifolds">Manifold Atlas Project (MAP)</a></span>. Most of his examples are constructed by identifications on the faces of a (solid) cube. The function <code class="code">PoincareCubeCWComplex()</code> can be used to construct any 3-dimensional CW-complex arising from a cube by identifying the six faces pairwise; the vertices and faces of the cube are numbered as follows</p>
<p><img src="images/pcube.png" align="center" height="200" alt="cube"/></p>
<p>and barycentric subdivision is used to ensure that the quotient is represented as a regular CW-complex.</p>
<p>Examples 3 and 4 from Poincare's paper, described in the following figures taken from <span class="URL"><a href="http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_cube_manifolds">MAP</a></span>,</p>
<p><img src="images/Poincares_cube_manifolds3.png" align="center" width="250" alt="cube manifold"/> <img src="images/Poincares_cube_manifolds5.png" align="center" width="250" alt="cube manifold"/></p>
<p>are constructed in the following example. Both are checked to be orientable manifolds, and are shown to have different homology. (Note that the second example in Poincare's paper is not a manifold -- the links of some of its vertices are not homeomorphic to a 2-sphere.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=1;;C:=2;;D:=3;;B:=4;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ap:=5;;Cp:=6;;Dp:=7;;Bp:=8;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=[[A,B,D,C],[Bp,Dp,Cp,Ap]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=[[A,B,Bp,Ap],[Cp,C,D,Dp]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=[[A,C,Cp,Ap],[D,Dp,Bp,B]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ex3:=PoincareCubeCWComplex(L,M,N);</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Ex3);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=[[A,B,D,C],[Bp,Dp,Cp,Ap]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=[[A,B,Bp,Ap],[C,D,Dp,Cp]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=[[A,C,Cp,Ap],[B,D,Dp,Bp]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ex4:=PoincareCubeCWComplex(L,M,N);</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Ex4);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..3],k->Homology(Ex3,k));</span>
[ [ 0 ], [ 2, 2 ], [ ], [ 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..3],k->Homology(Ex4,k));</span>
[ [ 0 ], [ 2, 0 ], [ 0 ], [ 0 ] ]
</pre></div>
<p><a id="X8761051F84C6CEC2" name="X8761051F84C6CEC2"></a></p>
<h4>4.10 <span class="Heading">There are at least 25 distinct cube manifolds</span></h4>
<p>The function <code class="code">PoincareCubeCWComplex(A,G)</code> can also be applied to two inputs where <span class="SimpleMath">A</span> is a pairing of the six faces such as <span class="SimpleMath">A=[[1,2],[3,4],[5,6]]</span> and <span class="SimpleMath">G</span> is a list of three elements of the dihedral group of order <span class="SimpleMath">8</span> such as <span class="SimpleMath">G=[(2,4),(2,4),(2,4)*(1,3)]</span>. The dihedral elements specify how each pair of faces are glued together. With these inputs it is easy to iterate over all possible values of <span class="SimpleMath">A</span> and <span class="SimpleMath">G</span> in order to construct all possible closed 3-manifolds arising from the pairwise identification of faces of a cube. We call such a manifold a <em><strong class="button">cube manifold</strong></em>. Distinct values of <span class="SimpleMath">A</span> and <span class="SimpleMath">G</span> can of course yield homeomorphic spaces. To ensure that each possible cube manifold is constructed, at least once, up to homeomorphism it suffices to consider</p>
<p><span class="SimpleMath">A=[ [1,2], [3,4], [5,6] ]</span>, <span class="SimpleMath">A=[ [1,2], [3,5], [4,6] ]</span>, <span class="SimpleMath">A=[ [1,4], [2,6], [3,5] ]</span></p>
<p>and all <span class="SimpleMath">G</span> in <span class="SimpleMath">D_8× D_8× D_8</span>.</p>
<p>The following commands iterate through these <span class="SimpleMath">3×8^3 = 1536</span> pairs <span class="SimpleMath">(A,G)</span> and show that in precisely 163 cases (just over 10% of cases) the quotient CW-complex is a closed 3-manifold.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A1:= [ [1,2], [3,4], [5,6] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A2:=[ [1,2], [3,5], [4,6] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A3:=[ [1,4], [2,6], [3,5] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D8:=DihedralGroup(IsPermGroup,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Manifolds:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for A in [A1,A2,A3] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for x in D8 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for y in D8 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for z in D8 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">G:=[x,y,z];</span>
<span class="GAPprompt">></span> <span class="GAPinput">F:=PoincareCubeCWComplex(A,G);</span>
<span class="GAPprompt">></span> <span class="GAPinput">b:=IsClosedManifold(F);</span>
<span class="GAPprompt">></span> <span class="GAPinput">if b=true then Add(Manifolds,F); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;od;od;od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Manifolds);</span>
163
</pre></div>
<p>The following additional commands use integral homology and low index subgroups of fundamental groups to establish that the 163 cube manifolds represent at least 25 distinct homotopy equivalence classes of manifolds. One homotopy class is represented by up to 40 of the manifolds, and at least four of the homotopy classes are each represented by a single manifold..</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">invariant1:=function(m);</span>
<span class="GAPprompt">></span> <span class="GAPinput">return List([1..3],k->Homology(m,k));</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=Classify(Manifolds,invariant1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">invariant2:=function(m)</span>
<span class="GAPprompt">></span> <span class="GAPinput">local L;</span>
<span class="GAPprompt">></span> <span class="GAPinput">L:=FundamentalGroup(m);</span>
<span class="GAPprompt">></span> <span class="GAPinput">if GeneratorsOfGroup(L)= [] then return [];fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">L:=LowIndexSubgroupsFpGroup(L,5);</span>
<span class="GAPprompt">></span> <span class="GAPinput">L:=List(L,AbelianInvariants);</span>
<span class="GAPprompt">></span> <span class="GAPinput">L:=SortedList(L);</span>
<span class="GAPprompt">></span> <span class="GAPinput">return L;</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=RefineClassification(C,invariant2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D,Size);</span>
[ 40, 2, 10, 15, 8, 6, 2, 6, 2, 5, 7, 1, 4, 11, 7, 7, 10, 4, 4, 2, 1, 3, 1,
1, 4 ]
</pre></div>
<p>The next commands construct a list of 18 orientable cube manifolds and a list of 7 non-orientable cube manifolds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Manifolds:=List(D,x->x[1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrientableManifolds:=Filtered(Manifolds,m->Homology(m,3)=[0]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NonOrientableManifolds:=Filtered(Manifolds,m->Homology(m,3)=[]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(OrientableManifolds);</span>
18
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NonOrientableManifolds);</span>
7
</pre></div>
<p>The next commands show that the 7 non-orientable cube manifolds all have infinite fundamental groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(NonOrientableManifolds,m->IsFinite(FundamentalGroup(m)));</span>
[ false, false, false, false, false, false, false ]
</pre></div>
<p>The final commands show that (at least) 9 of the orientable manifolds have finite fundamental groups and list the isomorphism types of these finite groups. Note that it is now known that any closed 3-manifold with finite fundamental group is spherical (i.e. is a quotient of the 3-sphere). Spherical manifolds with cyclic fundamental group are, by definition, lens spaces.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(OrientableManifolds{[4,8,10,11,12,13,15,16,18]},m-></span>
IsFinite(FundamentalGroup(m)));
[ true, true, true, true, true, true, true, true, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(OrientableManifolds{[4,8,10,11,12,13,15,16,18]},m-></span>
StructureDescription(FundamentalGroup(m)));
[ "Q8", "C2", "C4", "C3 : C4", "C12", "C8", "C14", "C6", "1" ]
</pre></div>
<p><a id="X7D50795883E534A3" name="X7D50795883E534A3"></a></p>
<h5>4.10-1 <span class="Heading">Face pairings for 25 distinct cube manifolds</span></h5>
<p>The following are the face pairings of 25 non-homeomorphic cube manifolds, with vertices of the cube numbered as describe above.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..25] do </span>
<span class="GAPprompt">></span> <span class="GAPinput">p:=Manifolds[i]!.cubeFacePairings;</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print("Manifold ",i," has face pairings:\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print(p[1],"\n",p[2],"\n",p[3],"\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print("Fundamental group is: ");</span>
<span class="GAPprompt">></span> <span class="GAPinput">if i in [ 1, 9, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25 ] then</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print(StructureDescription(FundamentalGroup(Manifolds[i])),"\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">else Print("infinite non-cyclic\n"); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">if Homology(Manifolds[i],3)=[0] then Print("Orientable, ");</span>
<span class="GAPprompt">></span> <span class="GAPinput">else Print("Non orientable, "); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print(ManifoldType(Manifolds[i]),"\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">for x in Manifolds[i]!.edgeDegrees do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print(x[2]," edges of \"degree\" ",x[1],", ");</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print("\n\n");</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
Manifold 1 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 7, 8, 4 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 8, 7, 6 ] ]
[ [ 1, 4, 8, 5 ], [ 3, 2, 6, 7 ] ]
Fundamental group is: Z x C2
Non orientable, other
4 edges of "degree" 2, 4 edges of "degree" 4,
Manifold 2 has face pairings:
[ [ 1, 5, 6, 2 ], [ 7, 8, 4, 3 ] ]
[ [ 1, 2, 3, 4 ], [ 1, 5, 8, 4 ] ]
[ [ 5, 8, 7, 6 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: infinite non-cyclic
Non orientable, other
2 edges of "degree" 1, 2 edges of "degree" 3, 2 edges of "degree" 8,
Manifold 3 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 7, 8, 4 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
[ [ 1, 4, 8, 5 ], [ 2, 3, 7, 6 ] ]
Fundamental group is: infinite non-cyclic
Non orientable, euclidean
6 edges of "degree" 4,
Manifold 4 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 7, 8, 4 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
[ [ 1, 4, 8, 5 ], [ 6, 7, 3, 2 ] ]
Fundamental group is: infinite non-cyclic
Non orientable, euclidean
6 edges of "degree" 4,
Manifold 5 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 7, 8, 4 ] ]
[ [ 1, 2, 3, 4 ], [ 6, 5, 8, 7 ] ]
[ [ 1, 4, 8, 5 ], [ 2, 6, 7, 3 ] ]
Fundamental group is: infinite non-cyclic
Non orientable, euclidean
6 edges of "degree" 4,
Manifold 6 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
[ [ 1, 4, 8, 5 ], [ 2, 3, 7, 6 ] ]
Fundamental group is: infinite non-cyclic
Orientable, euclidean
6 edges of "degree" 4,
Manifold 7 has face pairings:
[ [ 1, 5, 6, 2 ], [ 7, 3, 4, 8 ] ]
[ [ 1, 2, 3, 4 ], [ 1, 5, 8, 4 ] ]
[ [ 5, 8, 7, 6 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: infinite non-cyclic
Orientable, other
2 edges of "degree" 1, 2 edges of "degree" 3, 2 edges of "degree" 8,
Manifold 8 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 7, 8, 5, 6 ] ]
[ [ 1, 4, 8, 5 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: infinite non-cyclic
Orientable, other
4 edges of "degree" 2, 2 edges of "degree" 8,
Manifold 9 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 8, 5, 6, 7 ] ]
[ [ 1, 4, 8, 5 ], [ 6, 2, 3, 7 ] ]
Fundamental group is: Q8
Orientable, spherical
8 edges of "degree" 3,
Manifold 10 has face pairings:
[ [ 1, 5, 6, 2 ], [ 4, 8, 7, 3 ] ]
[ [ 1, 2, 3, 4 ], [ 7, 8, 5, 6 ] ]
[ [ 1, 4, 8, 5 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: infinite non-cyclic
Orientable, other
4 edges of "degree" 2, 4 edges of "degree" 4,
Manifold 11 has face pairings:
[ [ 1, 5, 6, 2 ], [ 4, 3, 7, 8 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
[ [ 1, 4, 8, 5 ], [ 2, 3, 7, 6 ] ]
Fundamental group is: infinite non-cyclic
Non orientable, euclidean
6 edges of "degree" 4,
Manifold 12 has face pairings:
[ [ 1, 5, 6, 2 ], [ 4, 8, 7, 3 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
[ [ 1, 4, 8, 5 ], [ 2, 3, 7, 6 ] ]
Fundamental group is: Z x Z x Z
Orientable, euclidean
6 edges of "degree" 4,
Manifold 13 has face pairings:
[ [ 1, 5, 6, 2 ], [ 4, 8, 7, 3 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
[ [ 1, 4, 8, 5 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: infinite non-cyclic
Orientable, euclidean
6 edges of "degree" 4,
Manifold 14 has face pairings:
[ [ 1, 5, 6, 2 ], [ 7, 3, 4, 8 ] ]
[ [ 1, 2, 3, 4 ], [ 7, 8, 5, 6 ] ]
[ [ 1, 4, 8, 5 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: C2
Orientable, spherical
12 edges of "degree" 2,
Manifold 15 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 7, 8, 4 ] ]
[ [ 1, 2, 3, 4 ], [ 1, 5, 8, 4 ] ]
[ [ 5, 8, 7, 6 ], [ 2, 3, 7, 6 ] ]
Fundamental group is: Z
Non orientable, other
4 edges of "degree" 1, 2 edges of "degree" 2, 2 edges of "degree" 8,
Manifold 16 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 1, 5, 8, 4 ] ]
[ [ 5, 8, 7, 6 ], [ 2, 3, 7, 6 ] ]
Fundamental group is: Z
Orientable, other
4 edges of "degree" 1, 2 edges of "degree" 2, 2 edges of "degree" 8,
Manifold 17 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 1, 5, 8, 4 ] ]
[ [ 5, 8, 7, 6 ], [ 3, 7, 6, 2 ] ]
Fundamental group is: C4
Orientable, spherical
2 edges of "degree" 1, 2 edges of "degree" 3, 2 edges of "degree" 8,
Manifold 18 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 8, 4, 1, 5 ] ]
[ [ 5, 8, 7, 6 ], [ 6, 2, 3, 7 ] ]
Fundamental group is: C3 : C4
Orientable, spherical
2 edges of "degree" 2, 4 edges of "degree" 5,
Manifold 19 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 8, 4, 1, 5 ] ]
[ [ 5, 8, 7, 6 ], [ 3, 7, 6, 2 ] ]
Fundamental group is: C12
Orientable, spherical
2 edges of "degree" 2, 2 edges of "degree" 3, 2 edges of "degree" 7,
Manifold 20 has face pairings:
[ [ 1, 5, 6, 2 ], [ 3, 4, 8, 7 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 8, 4, 1 ] ]
[ [ 5, 8, 7, 6 ], [ 3, 7, 6, 2 ] ]
Fundamental group is: C8
Orientable, spherical
8 edges of "degree" 3,
Manifold 21 has face pairings:
[ [ 1, 5, 6, 2 ], [ 7, 3, 4, 8 ] ]
[ [ 1, 2, 3, 4 ], [ 8, 4, 1, 5 ] ]
[ [ 5, 8, 7, 6 ], [ 7, 6, 2, 3 ] ]
Fundamental group is: infinite non-cyclic
Orientable, euclidean
6 edges of "degree" 4,
Manifold 22 has face pairings:
[ [ 1, 5, 6, 2 ], [ 5, 6, 7, 8 ] ]
[ [ 3, 7, 8, 4 ], [ 7, 6, 2, 3 ] ]
[ [ 1, 2, 3, 4 ], [ 8, 4, 1, 5 ] ]
Fundamental group is: C14
Orientable, spherical
2 edges of "degree" 2, 4 edges of "degree" 5,
Manifold 23 has face pairings:
[ [ 1, 5, 6, 2 ], [ 5, 6, 7, 8 ] ]
[ [ 3, 7, 8, 4 ], [ 7, 6, 2, 3 ] ]
[ [ 1, 2, 3, 4 ], [ 5, 8, 4, 1 ] ]
Fundamental group is: C6
Orientable, spherical
6 edges of "degree" 2, 2 edges of "degree" 6,
Manifold 24 has face pairings:
[ [ 1, 5, 6, 2 ], [ 7, 8, 5, 6 ] ]
[ [ 3, 7, 8, 4 ], [ 2, 3, 7, 6 ] ]
[ [ 1, 2, 3, 4 ], [ 4, 1, 5, 8 ] ]
Fundamental group is: infinite non-cyclic
Orientable, euclidean
6 edges of "degree" 4,
Manifold 25 has face pairings:
[ [ 1, 5, 6, 2 ], [ 6, 7, 8, 5 ] ]
[ [ 3, 7, 8, 4 ], [ 3, 7, 6, 2 ] ]
[ [ 1, 2, 3, 4 ], [ 1, 5, 8, 4 ] ]
Fundamental group is: 1
Orientable, spherical
4 edges of "degree" 1, 4 edges of "degree" 5,
</pre></div>
<p><a id="X837811BB8181666E" name="X837811BB8181666E"></a></p>
<h5>4.10-2 <span class="Heading">Platonic cube manifolds</span></h5>
<p>A <em>platonic solid</em> is a convex, regular polyhedron in <span class="SimpleMath">3</span>-dimensional euclidean <span class="SimpleMath">E^3</span> or spherical <span class="SimpleMath">S^3</span> or hyperbolic space <span class="SimpleMath">H^3</span>. Being <em>regular</em> means that all edges are congruent, all faces are congruent, all angles between adjacent edges in a face are congruent, all dihedral angles between adjacent faces are congruent. A platonic cube in euclidean space has six congruent square faces with diherdral angle <span class="SimpleMath">π/2</span>. A platonic cube in spherical space has dihedral angles <span class="SimpleMath">2π/3</span>. A platonic cube in hyperbolic space has dihedral angles <span class="SimpleMath">2π/5</span>. This can alternatively be expressed by saying that in a tessellation of <span class="SimpleMath">E^3</span> by platonic cubes each edge is adjacent to 4 square faces. In a tessellation of <span class="SimpleMath">S^3</span> by platonic cubes each edge is adjacent to 3 square faces. In a tessellation of <span class="SimpleMath">H^3</span> by platonic cubes each edge is adjacent to 5 five square faces.</p>
<p>Any cube manifold <span class="SimpleMath">M</span> induces a cubical CW-decomposition of its universal cover <span class="SimpleMath">widetilde M</span>. We say that <span class="SimpleMath">M</span> is a <em>platonic cube manifold</em> if every edge in <span class="SimpleMath">widetilde M</span> is adjacent to 4 faces in the euclidean case <span class="SimpleMath">widetilde M= E^3</span>, is adjacent to 3 faces in the spherical case <span class="SimpleMath">widetilde M= S^3</span>, is adjacent to 5 faces in the hyperbolic case <span class="SimpleMath">widetilde M= H^3</span>.</p>
<p>In the above list of 25 cube manifolds we see that the euclidean manifolds 3, 4, 5, 6, 11 are platonic and that the spherical manifolds 9, 20 are platonic.</p>
<p><a id="X8084A36082B26D86" name="X8084A36082B26D86"></a></p>
<h4>4.11 <span class="Heading">There are at most 41 distinct cube manifolds</span></h4>
<p>Using the <span class="URL"><a href="https://simpcomp-team.github.io/simpcomp/README.html">Simpcomp</a></span> package for GAP we can show that many of the 163 cube manifolds constructed above are homeomorphic. We do this by showing that barycentric subdivisions of many of the manifolds are combinatorially the same.</p>
<p>The following commands establish homeomorphisms (simplicial complex isomorphisms) between manifolds in each equivalence class D[i] above for <span class="SimpleMath">1 ≤ i≤ 25</span>, and then discard all but one manifold in each homeomorphism class. We are left with 59 cube manifolds, some of which may be homeomorphic, representing at least 25 distinct homeomorphism classes. The 59 manifolds are stored in the list DD of length 25 each of whose terms is a list of cube manifolds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("Simpcomp");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inv3:=function(m)</span>
<span class="GAPprompt">></span> <span class="GAPinput">local K;</span>
<span class="GAPprompt">></span> <span class="GAPinput">K:=BarycentricSubdivision(m);</span>
<span class="GAPprompt">></span> <span class="GAPinput">K:=MaximalSimplicesOfSimplicialComplex(K);</span>
<span class="GAPprompt">></span> <span class="GAPinput">K:=SC(K);</span>
<span class="GAPprompt">></span> <span class="GAPinput">if not SCIsStronglyConnected(K) then Print("WARNING!\n"); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">return SCExportIsoSig( K );</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;</span>
function( m ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">DD:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for x in D do</span>
<span class="GAPprompt">></span> <span class="GAPinput">y:=Classify(x,inv3);</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(DD,List(y,z->z[1]));</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(DD,Size);</span>
[ 9, 1, 3, 3, 3, 1, 1, 1, 1, 1, 2, 1, 2, 7, 4, 4, 3, 1, 1, 1, 1, 3, 1, 1, 3 ]
</pre></div>
<p>The function <code class="code">PoincareCubeCWCompex()</code> applies cell simplifications in its construction of the quotient of a CW-complex. A variant <code class="code">PoincareCubeCWCompexNS()</code> performs no cell simplifications and thus returns a bigger cell complex which we can attempt to use to establish further homeomorphisms. This is done in the following session and succeeds in showing that there are at most 51 distinct homeomorphism types of cube manifolds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DD:=List(DD,x->List(x,y->PoincareCubeCWComplexNS(</span>
<span class="GAPprompt">></span> <span class="GAPinput">y!.cubeFacePairings[1],y!.cubeFacePairings[2],y!.cubeFacePairings[3])));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for x in DD do</span>
<span class="GAPprompt">></span> <span class="GAPinput">y:=Classify(x,inv3);</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(D,List(y,z->z[1]));</span>
>od;;
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D,Size);</span>
[ 8, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 4, 3, 1, 1, 1, 1, 1, 1, 1, 2 ]
</pre></div>
<p>Making further modifications to the cell structures of the manifolds that leave their homeomorphism types unchanged can help to identify further simplicial isomorphisms between barycentric subdivisions. For instance, the following commands succeed in establishing that there are at most 45 distinct homeomorphism types of cube manifolds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DD:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for x in D do</span>
<span class="GAPprompt">></span> <span class="GAPinput">if Length(x)>1 then</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(DD, List(x,y->BarycentricallySimplifiedComplex(y)));</span>
<span class="GAPprompt">></span> <span class="GAPinput">else Add(DD,x);</span>
<span class="GAPprompt">></span> <span class="GAPinput">fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for x in DD do</span>
<span class="GAPprompt">></span> <span class="GAPinput">y:=Classify(x,inv3);</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(D,List(y,z->z[1]));</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D,Size);</span>
[ 7, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DD:=List(D,x->List(x,y->PoincareCubeCWComplexNS(</span>
<span class="GAPprompt">></span> <span class="GAPinput">y!.cubeFacePairings[1],y!.cubeFacePairings[2],y!.cubeFacePairings[3])));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D1:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for x in DD do</span>
<span class="GAPprompt">></span> <span class="GAPinput">if Length(x)>1 then</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(D1, List(x,y->BarycentricallySimplifiedComplex(RegularCWComplex(BarycentricSubdivision(y)))));</span>
<span class="GAPprompt">></span> <span class="GAPinput">else Add(D1,x);</span>
<span class="GAPprompt">></span> <span class="GAPinput">fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DD:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for x in D1 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">y:=Classify(x,inv3);</span>
<span class="GAPprompt">></span> <span class="GAPinput">Add(DD,List(y,z->z[1]));</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print(List(DD,Size),"\n");</span>
[ 6, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 2 ]
</pre></div>
<p>The two manifolds in DD[14] have fundamental group <span class="SimpleMath">C_2</span> and are thus lens spaces. There is only one homeomorphism class of such lens spaces and so these two manifolds are homeomorphic. The three manifolds in DD[17] are lens spaces with fundamental group <span class="SimpleMath">C_4</span>. Again, there is only one homeomorphism class of such lens spaces and so these three manifolds are homeomorphic. The two manifolds in DD[25] have trivial fundamental group and are hence both homeomorphic to the 3-sphere. These observations mean that there are at most 41 closed manifolds arising from a cube by identifying the cube's faces pairwise.</p>
<p>These observations can be incorporated into our list DD of equivalence classes of manifolds as follows.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DD[14]:=DD[14]{[1]};;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DD[17]:=DD[17]{[1]};;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DD[25]:=DD[25]{[1]};;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(DD,Size);</span>
[ 6, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
</pre></div>
<p><a id="X7B63C22C80E53758" name="X7B63C22C80E53758"></a></p>
<h4>4.12 <span class="Heading">There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean</span></h4>
<p>The following commands show that there are at least 18 and at most 21 orientable cube manifolds.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DDorient:=Filtered(DD,x->Homology(x[1],3)=[0]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(DDorient,Size);</span>
[ 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
</pre></div>
<p>The next commands show that the fundamental groups of the two manifolds in DDorient[7] are isomorphic to <span class="SimpleMath">Z × Z : Z</span>, and that the fundamental groups of the three manifolds in DDorient[9] are isomorphic to <span class="SimpleMath">Z</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1:=FundamentalGroup(DDorient[7][1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g2:=FundamentalGroup(DDorient[7][2]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(g1);</span>
[ f1^-1*f2*f1*f2^-1, f3^-1*f1*f3*f1, f3^-1*f2^-1*f3*f2^-1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(g2);</span>
[ f1*f2*f1^-1*f2^-1, f1^-1*f3*f1^-1*f3^-1, f3*f2*f3^-1*f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">h1:=FundamentalGroup(DDorient[9][1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h2:=FundamentalGroup(DDorient[9][2]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h3:=FundamentalGroup(DDorient[9][3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(h1);</span>
"Z"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(h2);</span>
"Z"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(h3);</span>
"Z"
</pre></div>
<p>Since neither <span class="SimpleMath">Z× Z : Z</span> nor <span class="SimpleMath">Z</span> is a free product of two non-trivial groups we conclude that the manifolds in DDorient[7] and DDorient[9] are prime. Since oriented prime 3-manifolds are determined up to homeomorphism by their fundamental groups we can conclude that there are precisely 18 orientable closed manifolds arising from a cube by identifying the cube's faces pairwise.</p>
<p>A compact 3-manifold <span class="SimpleMath">M</span> is <em>spherical</em> if it is of the form <span class="SimpleMath">M=S^3/Γ</span> where <span class="SimpleMath">Γ</span> is a finite group acting freely as rotations on <span class="SimpleMath">S^3</span>. The fundamental group of <span class="SimpleMath">M</span> is then the finite group <span class="SimpleMath">Γ</span>. Perelmen showed that a compact 3-manifold is spherical if and only if its fundamental group is finite.</p>
<p>A compact 3-manifold is <em>euclidean</em> if it is of the form <span class="SimpleMath">M= R^3/Γ</span> where <span class="SimpleMath">Γ</span> is a group of affine transformations acting freely on <span class="SimpleMath">R^3</span>. The fundamental group is then <span class="SimpleMath">Γ</span> and is called a <em>Bieberbach group</em> of dimension 3. It can be shown that a group <span class="SimpleMath">Γ</span> is isomorphic to a Bieberbach group of dimension <span class="SimpleMath">n</span> if and only if there is a short exact sequence <span class="SimpleMath">Z^n ↣ Γ ↠ P</span> with <span class="SimpleMath">P</span> a finite group.</p>
<p>The following command establishes that there are precisely 9 orientable spherical manifolds and 5 closed orientable euclidean manifolds arising from pairwise identifications of the faces of the cube.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(OrientableManifolds,ManifoldType);</span>
[ "euclidean", "other", "other", "spherical", "other", "euclidean",
"euclidean", "spherical", "other", "spherical", "spherical", "spherical",
"spherical", "euclidean", "spherical", "spherical", "euclidean", "spherical" ]
</pre></div>
<p><a id="X796BF3817BD7F57D" name="X796BF3817BD7F57D"></a></p>
<h4>4.13 <span class="Heading">Cube manifolds with boundary</span></h4>
<p>If a space <span class="SimpleMath">Y</span> obtained from identifying faces of the cube fails to be a manifold then it fails because one or more vertices of <span class="SimpleMath">Y</span> fail to have a spherical link. By using barycentric subdivision if necessary, we can ensure that the stars of any two non-manifold vertices of <span class="SimpleMath">Y</span> have trivial intersection. Removing the stars of the non-manifold vertices from <span class="SimpleMath">Y</span> yields a 3-manifold with boundary <span class="SimpleMath">hat Y</span>.</p>
<p>The following commands show that there are 367 combinatorially different regular CW-complexes <span class="SimpleMath">Y</span> that arise by identifying faces of a cube in pairs and which fail to be manifolds. The commands also show that these spaces give rise to at least 180 non-homeomorphic manifolds <span class="SimpleMath">hat Y</span> with boundary.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A1:= [ [1,2], [3,4], [5,6] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A2:=[ [1,2], [3,5], [4,6] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A3:=[ [1,4], [2,6], [3,5] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D8:=DihedralGroup(IsPermGroup,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NonManifolds:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for A in [A1,A2,A3] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for x in D8 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for y in D8 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for z in D8 do</span>
<span class="GAPprompt">></span> <span class="GAPinput">G:=[x,y,z];</span>
<span class="GAPprompt">></span> <span class="GAPinput">F:=PoincareCubeCWComplex(A,G);</span>
<span class="GAPprompt">></span> <span class="GAPinput">b:=IsClosedManifold(F);</span>
<span class="GAPprompt">></span> <span class="GAPinput">if b=false then Add(NonManifolds,F); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;od;od;od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=Classify(NonManifolds,inv3); #See above for inv3</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=List(D,x->x[1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(D);</span>
367
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=List(D,ThreeManifoldWithBoundary);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=Classify(M,invariant1);; #See above for invariant1 </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(C,Size);</span>
[ 33, 13, 3, 18, 21, 7, 6, 13, 51, 2, 1, 15, 11, 11, 1, 35, 2, 2, 6, 15,
17, 2, 3, 2, 14, 17, 3, 1, 25, 8, 4, 1, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">inv5:=function(m) </span>
<span class="GAPprompt">></span> <span class="GAPinput">local B;</span>
<span class="GAPprompt">></span> <span class="GAPinput">B:=BoundaryOfPureRegularCWComplex(m);;</span>
<span class="GAPprompt">></span> <span class="GAPinput">return invariant1(B);</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CC:=RefineClassification(C,inv5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(CC,Size);</span>
[ 25, 5, 3, 5, 4, 4, 2, 1, 11, 3, 4, 7, 3, 6, 4, 1, 5, 1, 1, 5, 1, 13, 4,
6, 40, 1, 2, 1, 11, 4, 5, 3, 1, 2, 7, 4, 1, 14, 11, 10, 2, 2, 6, 9, 3, 3,
2, 15, 2, 3, 2, 14, 17, 2, 1, 1, 4, 7, 14, 8, 3, 1, 1, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CC:=RefineClassification(CC,invariant2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(CC,Size); </span>
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 1, 2, 1, 4, 2, 3, 2, 3,
4, 3, 2, 1, 1, 3, 2, 4, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 13, 3, 1, 4, 2, 1,
2, 2, 3, 3, 3, 4, 4, 2, 4, 4, 4, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 3, 2, 2, 1, 1, 2, 1, 1, 2,
1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 5, 2, 3, 2, 14, 17, 1, 1, 1,
1, 4, 5, 2, 9, 1, 4, 7, 1, 3, 1, 1, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(CC);</span>
180
</pre></div>
<p><a id="X7EC4359B7DF208B0" name="X7EC4359B7DF208B0"></a></p>
<h4>4.14 <span class="Heading">Octahedral manifolds</span></h4>
<p>The above construction of 3-manifolds as quotients of a cube can be extended to other polytopes. A polytope of particular interest, and one that appears several times in the classic book on Three-Manifolds by William Thurston <a href="chapBib.html#biBthurston">[Thu02]</a>, is the octahedron. The function <code class="code">PoincareOctahahedronCWComplex()</code> can be used to construct any 3-dimensional CW-complex arising from an octahedron by identifying the eight faces pairwise; the vertices and faces of the octahedron are numbered as follows.</p>
<p><img src="images/octahedron.png" align="center" height="350" alt="octahedron"/></p>
<p>The following commands construct a spherical 3-manifold Y with fundamental group equal to the binary tetrahedral group <span class="SimpleMath">G</span>. The commands then use the universal cover of this manifold to construct the first four terms of a free periodic <span class="SimpleMath">ZG</span>-resolution of <span class="SimpleMath">Z</span> of period <span class="SimpleMath">4</span>. The resolution has one free generator in dimensions <span class="SimpleMath">4n</span> and <span class="SimpleMath">4n+3</span> for <span class="SimpleMath">n≥ 0</span>. It has two free generators in dimensions <span class="SimpleMath">4n+1</span> and <span class="SimpleMath">4n+2</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=[ [ 1, 4, 5 ], [ 2, 6, 3 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=[ [ 3, 4, 5 ], [ 6, 1, 2 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=[ [ 2, 3, 5 ], [ 6, 4, 1 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:=[ [ 1, 2, 5 ], [ 6, 3, 4 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=PoincareOctahedronCWComplex(L,M,N,P);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Y);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=FundamentalGroup(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(G);</span>
"SL(2,3)"
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ChainComplexOfUniversalCover(Y);</span>
Equivariant chain complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..3],R!.dimension);</span>
[ 1, 2, 2, 1 ]
</pre></div>
<p><a id="X85FFF9B97B7AD818" name="X85FFF9B97B7AD818"></a></p>
<h4>4.15 <span class="Heading">Dodecahedral manifolds</span></h4>
<p>Another polytope of interest, and one that can be used to construct the Poincare homology sphere, is the dodecahedron. The function <code class="code">PoincareDodecahedronCWComplex()</code> can be used to construct any 3-dimensional CW-complex arising from a dodecahedron by identifying the <span class="SimpleMath">12</span> pentagonal faces pairwise; the vertices of the prism are numbered as follows.</p>
<p><img src="images/dodecahedron.png" align="center" height="250" alt="dodecahedron"/></p>
<p>The following commands construct the Poincare homology <span class="SimpleMath">3</span>-sphere (with fundamental group equal to the binary icosahedral group of order 120).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=PoincareDodecahedronCWComplex(</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[1,2,3,4,5],[6,7,8,9,10]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[1,11,16,12,2],[19,9,8,18,14]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[2,12,17,13,3],[20,10,9,19,15]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[3,13,18,14,4],[16,6,10,20,11]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[4,14,19,15,5],[17,7,6,16,12]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[5,15,20,11,1],[18,8,7,17,13]]);</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Y);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..3],n->Homology(Y,n));</span>
[ [ 0 ], [ ], [ ], [ 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(FundamentalGroup(Y));</span>
"SL(2,5)"
</pre></div>
<p>The following commands construct Seifert-Weber space, a rational homology sphere.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=PoincareDodecahedronCWComplex(</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[1,2,3,4,5],[7,8,9,10,6]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[1,11,16,12,2],[9,8,18,14,19]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[2,12,17,13,3],[10,9,19,15,20]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[3,13,18,14,4],[6,10,20,11,16]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[4,14,19,15,5],[7,6,16,12,17]],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[[5,15,20,11,1],[8,7,17,13,18]]);</span>
Regular CW-complex of dimension 3
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(W);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..3],n->Homology(W,n));</span>
[ [ 0 ], [ 5, 5, 5 ], [ ], [ 0 ] ]
</pre></div>
<p><a id="X78B75E2E79FBCC54" name="X78B75E2E79FBCC54"></a></p>
<h4>4.16 <span class="Heading">Prism manifolds</span></h4>
<p>Another polytope of interest is the prism constructed as the direct product <span class="SimpleMath">D_n× [0,1]</span> of an n-gonal disk <span class="SimpleMath">D_n</span> with the unit interval. The function <code class="code">PoincarePrismCWComplex()</code> can be used to construct any 3-dimensional CW-complex arising from a prism with even <span class="SimpleMath">n≥ 4</span> by identifying the <span class="SimpleMath">n+2</span> faces pairwise; the vertices of the prism are numbered as follows.</p>
<p><img src="images/prismnam.png" align="center" height="150" alt="prism"/></p>
<p>The case <span class="SimpleMath">n=4</span> is that of a cube. The following commands construct a manifold <span class="SimpleMath">Y</span> arising from a hexagonal prism (<span class="SimpleMath">n=6</span>) with fundamental group <span class="SimpleMath">π_1Y=C_5× Q_32</span> equal to the direct product of the cyclic group of order <span class="SimpleMath">5</span> and the quaternion group of order <span class="SimpleMath">32</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=[[1,2,3,4,5,6],[11,12,7,8,9,10]];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=[[1,7,8,2],[4,5,11,10]];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=[[2,8,9,3],[6,1,7,12]];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:=[[3,9,10,4],[6,12,11,5]];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=PoincarePrismCWComplex(L,M,N,P);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Y);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=FundamentalGroup(Y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(G);</span>
"C5 x Q32"
</pre></div>
<p>An exhaustive search through all manifolds constructed from a hexagonal prism by identify faces pairwise shows that the finite groups arising as fundamental groups are precisely: <span class="SimpleMath">Q_8</span>, <span class="SimpleMath">Q_16</span>, <span class="SimpleMath">C_4</span>, <span class="SimpleMath">C_3 : C_4</span>, <span class="SimpleMath">C_5 : C_4</span>, <span class="SimpleMath">C_8</span>, <span class="SimpleMath">C_16</span>, <span class="SimpleMath">C_12</span>, <span class="SimpleMath">C_20</span>, <span class="SimpleMath">C_2</span>, <span class="SimpleMath">C_6</span>, <span class="SimpleMath">C_3 × Q_8</span>, <span class="SimpleMath">C_3 × Q_16</span>, <span class="SimpleMath">C_5 × Q_32</span>. Each of these finite groups <span class="SimpleMath">G=π_1Y</span> is either cyclic (in which case the corresponding manifold is a lens space) or else has the propert that <span class="SimpleMath">G/Z(G)</span> is dihedral (in which case the corresponding manifold is called a <em>prism manifold</em>). The majority of the manifolds arising from a hexagonal prism have infinite fundamental group.</p>
<p>Infinite families of spherical <span class="SimpleMath">3</span>-maniolds can be constructed from the infinite family of prisms. For instance, a prism manifold which we denote by <span class="SimpleMath">P_r</span> can be obtained from a prism <span class="SimpleMath">D_2r× [0,1]</span> by identifying the left and right side under a twist of <span class="SimpleMath">π/r</span>, and identifying opposite square faces under a twist of <span class="SimpleMath">π/2</span>. Its fundamental group <span class="SimpleMath">π_1P_r</span> is the binary dihedral group of order <span class="SimpleMath">4r</span>. The following commands construct <span class="SimpleMath">P_r</span> for <span class="SimpleMath">r=3</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=[[1,2,3,4,5,6],[8,9,10,11,12,7]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=[[1,7,8,2],[11,10,4,5]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=[[2,8,9,3],[12,11,5,6]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:=[[3,9,10,4],[7,12,6,1]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y:=PoincarePrismCWComplex(L,M,N,P);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClosedManifold(Y);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(FundamentalGroup(Y));</span>
"C3 : C4"
</pre></div>
<p><a id="X7F31DFDA846E8E75" name="X7F31DFDA846E8E75"></a></p>
<h4>4.17 <span class="Heading">Bipyramid manifolds</span></h4>
<p>Yet another polytope of interest is the bipyramid constructed as the suspension of an n-gonal disk <span class="SimpleMath">D_n</span>. The function <code class="code">PoincareBipyramidCWComplex()</code> can be used to construct any 3-dimensional CW-complex arising from a bipyramid with <span class="SimpleMath">n≥ 3</span> by identifying the <span class="SimpleMath">2n</span> faces pairwise; the vertices of the prism are numbered as follows.</p>
<p><img src="images/bipyramid.png" align="center" height="150" alt="bipyramid"/></p>
<p>For <span class="SimpleMath">n=4</span> the bipyramid is the octahedron.</p>
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