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<a id="X7BFA4D1587D8DF49" name="X7BFA4D1587D8DF49"></a>
<div class="ChapSects"><a href="chap4.html#X7BFA4D1587D8DF49">4 Three Manifolds</a>
<div class="ContSect"> <a href="chap4.html#X82D1348C79238C2D">4.1 Dehn Surgery</a>

</div>
<div class="ContSect"> <a href="chap4.html#X848EDEE882B36F6C">4.2 Connected Sums</a>

</div>
<div class="ContSect"> <a href="chap4.html#X78AE684C7DBD7C70">4.3 Dijkgraaf-Witten Invariant</a>

</div>
<div class="ContSect"> <a href="chap4.html#X80B6849C835B7F19">4.4 Cohomology rings</a>

</div>
<div class="ContSect"> <a href="chap4.html#X7F56BB4C801AB894">4.5 Linking Form</a>

</div>
<div class="ContSect"> <a href="chap4.html#X850C76697A6A1654">4.6 Determining the homeomorphism type of a lens space</a>

</div>
<div class="ContSect"> <a href="chap4.html#X7EC6B008878CC77E">4.7 Surgeries on distinct knots can yield homeomorphic manifolds</a>

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<div class="ContSect"> <a href="chap4.html#X7B425A3280A2AF07">4.8 Finite fundamental groups of 3-manifolds</a>

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<div class="ContSect"> <a href="chap4.html#X78912D227D753167">4.9 Poincare's cube manifolds

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<div class="ContSect"> <a href="chap4.html#X8761051F84C6CEC2">4.10 There are at least 25 distinct cube manifolds</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7D50795883E534A3">4.10-1 Face pairings for 25 distinct cube manifolds</a>

 <a href="chap4.html#X837811BB8181666E">4.10-2 Platonic cube manifolds</a>

</div></div>
<div class="ContSect"> <a href="chap4.html#X8084A36082B26D86">4.11 There are at most 41 distinct cube manifolds</a>

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<div class="ContSect"> <a href="chap4.html#X7B63C22C80E53758">4.12 There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean</a>

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<div class="ContSect"> <a href="chap4.html#X796BF3817BD7F57D">4.13 Cube manifolds with boundary</a>

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<div class="ContSect"> <a href="chap4.html#X7EC4359B7DF208B0">4.14 Octahedral manifolds</a>

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<div class="ContSect"> <a href="chap4.html#X85FFF9B97B7AD818">4.15 Dodecahedral manifolds</a>

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<div class="ContSect"> <a href="chap4.html#X78B75E2E79FBCC54">4.16 Prism manifolds</a>

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<div class="ContSect"> <a href="chap4.html#X7F31DFDA846E8E75">4.17 Bipyramid manifolds</a>

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</div>

<h3>4 Three Manifolds</h3>

<a id="X82D1348C79238C2D" name="X82D1348C79238C2D"></a>

<h4>4.1 Dehn Surgery</h4>

The following example constructs, as a regular CW-complex, a closed orientable 3-manifold W 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 p/q=17/16.

<div class="example"><pre>
gap> ap:=ArcPresentation(PureCubicalKnot(3,1));;
gap> p:=17;;q:=16;;
gap> W:=ThreeManifoldViaDehnSurgery(ap,p,q);
Regular CW-complex of dimension 3

</pre></div>

The next commands show that this 3-manifold W has integral homology

H_0(W, Z)= Z, H_1(W, Z)= Z_16, H_2(W, Z)=0, H_3(W, Z)= Z

and that the fundamental group π_1(W) is non-abelian.

<div class="example"><pre>
gap> Homology(W,0);Homology(W,1);Homology(W,2);Homology(W,3);
[ 0 ]
[ 16 ]
[ ]
[ 0 ]

gap> F:=FundamentalGroup(W);;
gap> L:=LowIndexSubgroupsFpGroup(F,10);;
gap> List(L,AbelianInvariants);
[ [ 16 ], [ 3, 8 ], [ 3, 4 ], [ 2, 3 ], [ 16, 43 ], [ 8, 43, 43 ] ]

</pre></div>

The following famous result of Lickorish and (independently) Wallace shows that Dehn surgery on knots leads to an interesting range of spaces.

Theorem: Every closed, orientable, connected 3-manifold can be obtained by surgery on a link in S^3. (Moreover, one can always perform the surgery with surgery coefficients ± 1 and with each individual component of the link unknotted.) 

<a id="X848EDEE882B36F6C" name="X848EDEE882B36F6C"></a>

<h4>4.2 Connected Sums</h4>

The following example constructs the connected sum W=A#B of two 3-manifolds, where A is obtained from a 5/1 Dehn surgery on the complement of the first prime knot on 11 crossings and B is obtained by a 5/1 Dehn surgery on the complement of the second prime knot on 11 crossings. The homology groups

H_1(W, Z) = Z_2⊕ Z_594, H_2(W, Z) = 0, H_3(W, Z) = Z

are computed.

<div class="example"><pre>
gap> ap1:=ArcPresentation(PureCubicalKnot(11,1));;
gap> A:=ThreeManifoldViaDehnSurgery(ap1,5,1);;
gap> ap2:=ArcPresentation(PureCubicalKnot(11,2));;
gap> B:=ThreeManifoldViaDehnSurgery(ap2,5,1);;
gap> W:=ConnectedSum(A,B); #W:=ConnectedSum(A,B,-1) would yield A#-B where -B has the opposite orientation
Regular CW-complex of dimension 3

gap> Homology(W,1);
[ 2, 594 ]
gap> Homology(W,2);
[ ]
gap> Homology(W,3);
[ 0 ]

</pre></div>

<a id="X78AE684C7DBD7C70" name="X78AE684C7DBD7C70"></a>

<h4>4.3 Dijkgraaf-Witten Invariant</h4>

Given a closed connected orientable 3-manifold W, a finite group G and a 3-cocycle α∈ H^3(BG,U(1)) Dijkgraaf and Witten define the complex number

$$ Z^{G,\alpha}(W) = \frac{1}{|G|}\sum_{\gamma\in {\rm Hom}(\pi_1W, G)} \langle \gamma^\ast[\alpha], [M]\rangle \ \in\ \mathbb C\ $$ where γ ranges over all group homomorphisms γ: π_1W → G. This complex number is an invariant of the homotopy type of W and is useful for distinguishing between certain homotopically distinct 3-manifolds.

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 H_3(W, Z)≅ Z we represent D_G(W) by the set D_G(W)={ γ_∗(1) }_γ∈ Hom(π_1W, G) where 1 denotes one of the two possible generators of H_3(W, Z).

For any coprime integers p,q≥ 1 the lens space L(p,q) 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 p/q. Lens spaces have cyclic fundamental group π_1(L(p,q))=C_p and homology H_1(L(p,q), Z)≅ Z_p, H_2(L(p,q), Z)≅ 0, H_3(L(p,q), Z)≅ Z. It was proved by J.H.C. Whitehead that two lens spaces L(p,q) and L(p',q') are homotopy equivalent if and only if p=p' and qq'≡ ± n^2 mod p for some integer n.

The following session constructs the two lens spaces L(5,1) and L(5,2). The homology version of the Dijkgraaf-Witten invariant is used with G=C_5 to demonstrate that the two lens spaces are not homotopy equivalent.

<div class="example"><pre>
gap> ap:=[[2,1],[2,1]];; #Arc presentation for the trivial knot

gap> L51:=ThreeManifoldViaDehnSurgery(ap,5,1);;
gap> D:=DijkgraafWittenInvariant(L51,CyclicGroup(5));
[ g1^4, g1^4, g1, g1, id ]

gap> L52:=ThreeManifoldViaDehnSurgery(ap,5,2);;
gap> D:=DijkgraafWittenInvariant(L52,CyclicGroup(5));
[ g1^3, g1^3, g1^2, g1^2, id ]

</pre></div>

A theorem of Fermat and Euler states that if a prime p is congruent to 3 modulo 4, then for any q exactly one of ± q is a quadratic residue mod p. For all other primes p either both or neither of ± q is a quadratic residue mod p. Thus for fixed p ≡ 3 mod 4 the lens spaces L(p,q) form a single homotopy class. There are precisely two homotopy classes of lens spaces for other p.

The following commands confirm that L(13,1) ≄ L(13,2).

<div class="example"><pre>
gap> L13_1:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,1);;
gap> DijkgraafWittenInvariant(L13_1,CyclicGroup(13));
[ g1^12, g1^12, g1^10, g1^10, g1^9, g1^9, g1^4, g1^4, g1^3, g1^3, g1, g1, id ]
gap> L13_2:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,2);;
gap> DijkgraafWittenInvariant(L13_2,CyclicGroup(13));
[ 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>

<a id="X80B6849C835B7F19" name="X80B6849C835B7F19"></a>

<h4>4.4 Cohomology rings</h4>

The following commands construct the multiplication table (with respect to some basis) for the cohomology rings H^∗(L(13,1), Z_13) and H^∗(L(13,2), Z_13). These rings are isomorphic and so fail to distinguish between the homotopy types of the lens spaces L(13,1) and L(13,2).

<div class="example"><pre>
gap> L13_1:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,1);;
gap> L13_2:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,2);;
gap> L13_1:=BarycentricSubdivision(L13_1);;
gap> L13_2:=BarycentricSubdivision(L13_2);;
gap> A13_1:=CohomologyRing(L13_1,13);;
gap> A13_2:=CohomologyRing(L13_2,13);;
gap> M13_1:=List([1..4],i->[]);;
gap> B13_1:=CanonicalBasis(A13_1);;
gap> M13_2:=List([1..4],i->[]);;
gap> B13_2:=CanonicalBasis(A13_2);;
gap> for i in [1..4] do
> for j in [1..4] do
> M13_1[i][j]:=B13_1[i]*B13_1[j];
> od;od;
gap> for i in [1..4] do
> for j in [1..4] do
> M13_2[i][j]:=B13_2[i]*B13_2[j];
> od;od;
gap> Display(M13_1);
[ [ 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 ] ]
gap> Display(M13_2);
[ [ 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>

<a id="X7F56BB4C801AB894" name="X7F56BB4C801AB894"></a>

<h4>4.5 Linking Form</h4>

Given a closed connected oriented 3-manifold W let τ H_1(W, Z) denote the torsion subgroup of the first integral homology. The linking form is a bilinear mapping

Lk_W: τ H_1(W, Z) × τ H_1(W, Z) ⟶ Q/ Z.

To construct this form note that we have a Poincare duality isomorphism

ρ: H^2(W, Z) stackrel≅⟶ H_1(W, Z), z ↦ z∩ [W]

involving the cap product with the fundamental class [W]∈ H^3(W, Z). That is, [M] is the generator of H^3(W, Z)≅ Z determining the orientation. The short exact sequence Z ↣ Q ↠ Q/ Z gives rise to a cohomology exact sequence

→ H^1(W, Q) → H^1(W, Q/ Z) stackrelβ⟶ H^2(W, Z) → H^2(W, Q) →

from which we obtain the isomorphism β : τ H^1(W, Q/ Z) stackrel≅⟶ τ H^2(W, Z). The linking form Lk_W can be defined as the composite

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

where ev(x,α) evaluates a 1-cocycle α on a 1-cycle x.

The linking form can be used to define the set

I^O(W) = {Lk_W(g,g) : g∈ τ H_1(W, Z)}

which is an oriented-homotopy invariant of W. Letting W^+ and W^- denote the two possible orientations on the manifold, the set

I(W) ={I^O(W^+), I^O(W^-)}

is a homotopy invariant of W which in this manual we refer to as the linking form homotopy invariant.

The following commands compute the linking form homotopy invariant for the lens spaces L(13,q) with 1≤ q≤ 12. This invariant distinguishes between the two homotopy types that arise.

<div class="example"><pre>
gap> LensSpaces:=[];;
gap> for q in [1..12] do
> Add(LensSpaces,ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,q));
> od;
gap> Display(List(LensSpaces,LinkingFormHomotopyInvariant));;
[ [ [ 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>

<a id="X850C76697A6A1654" name="X850C76697A6A1654"></a>

<h4>4.6 Determining the homeomorphism type of a lens space</h4>

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 3-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 Reidermeister torsion, and Brody's method, using tubular neighbourhoods of 1-cycles, both require identifying a suitable "preferred" generator of H_1(L(p,q), Z). 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 H_1(L(p,q), Z).

Przytycki and Yasukhara proved the following.

Theorem. Let ρ: S^ 3 → L(p, q) be the p-fold cyclic cover and K a knot in L(p, q) that represents a generator of H_1 (L(p, q), Z). If ρ ^-1 (K) is the trivial knot, then Lk_ L(p,q) ([K], [K]) = q/p or = overline q/p ∈ Q/ Z where qoverline q ≡ 1 mod 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.

The following commands construct a small lens space L=L(p,q) with unknown values of p,q. Subsequent commands will determine the homeomorphism type of L.

<div class="example"><pre>
gap> p:=Random([2,3,5,7,11,13,17]);;
gap> q:=Random([1..p-1]);;
gap> L:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],p,q);
Regular CW-complex of dimension 3

</pre></div>

We can readily determine the value of p=11 by calculating the order of π_1(L).

<div class="example"><pre>
gap> F:=FundamentalGroupWithPathReps(L);;
gap> StructureDescription(F);
"C11"

</pre></div>

The next commands take the default edge path γ: S^1→ L representing a generator of the cyclic group π_1(L) and lift it to an edge path tildeγ: S^1→ tilde L.

<div class="example"><pre>
gap> U:=UniversalCover(L);;
gap> G:=U!.group;;
gap> p:=EquivariantCWComplexToRegularCWMap(U,Group(One(G)));;
gap> U:=Source(p);;
gap> gamma:=[];;
gap> gamma[2]:=F!.loops[1];;
gap> gamma[2]:=List(gamma[2],AbsInt);;
gap> gamma[1]:=List(gamma[2],k->L!.boundaries[2][k]);;
gap> gamma[1]:=SSortedList(Flat(gamma[1]));;
gap> 
gap> gammatilde:=[[],[],[],[]];;
gap> for k in [1..U!.nrCells(0)] do
> if p!.mapping(0,k) in gamma[1] then Add(gammatilde[1],k); fi;
> od;
gap> for k in [1..U!.nrCells(1)] do
> if p!.mapping(1,k) in gamma[2] then Add(gammatilde[2],k); fi;
> od;
gap> gammatilde:=CWSubcomplexToRegularCWMap([U,gammatilde]);
Map of regular CW-complexes

</pre></div>

The next commands check that the path tildeγ is unknotted in tilde L≅ S^3 by checking that π_1(tilde L∖ image(tildeγ)) is infinite cyclic.

<div class="example"><pre>
gap> C:=RegularCWComplexComplement(gammatilde);
Regular CW-complex of dimension 3

gap> G:=FundamentalGroup(C);
<fp group of size infinity on the generators [ f2 ]>

</pre></div>

Since tildeγ is unkotted the cycle γ represents the preferred generator [γ]∈ H_1(L, Z). The next commands compute Lk_L([γ],[γ])= 7/11.

<div class="example"><pre>
gap> LinkingFormHomeomorphismInvariant(L);
[ 7/11 ]

</pre></div>

The classification of Moise/Brody states that L(p,q)≅ L(p,q') if and only if qq'≡ ± 1 mod p. Hence the lens space L has the homeomorphism type

L≅ L(11,7) ≅ L(11,8) ≅ L(11,4) ≅ L(11,3).

<a id="X7EC6B008878CC77E" name="X7EC6B008878CC77E"></a>

<h4>4.7 Surgeries on distinct knots can yield homeomorphic manifolds</h4>

The lens space L(5,1) is a quotient of the 3-sphere S^3 by a certain action of the cyclic group C_5. It can be realized by a p/q=5/1 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 W obtained from a p/q=1/5 Dehn filling of the complement of the trefoil and show that W at least has the same integral homology and same fundamental group as L(5,1).

<div class="example"><pre>
gap> ap:=ArcPresentation(PureCubicalKnot(3,1));;
gap> W:=ThreeManifoldViaDehnSurgery(ap,1,5);;
gap> Homology(W,1);
[ 5 ]
gap> Homology(W,2);
[ ]
gap> Homology(W,3);
[ 0 ]

gap> F:=FundamentalGroup(W);;
gap> StructureDescription(F);
"C5"

</pre></div>

The next commands construct the universal cover widetilde W and show that it has the same homology as S^3 and trivivial fundamental group π_1(widetilde W)=0.

<div class="example"><pre>
gap> U:=UniversalCover(W);;
gap> G:=U!.group;;
gap> Wtilde:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G)));
Regular CW-complex of dimension 3

gap> Homology(Wtilde,1);
[ ]
gap> Homology(Wtilde,2);
[ ]
gap> Homology(Wtilde,3);
[ 0 ]

gap> F:=FundamentalGroup(Wtilde);
<fp group on the generators [ ]>

</pre></div>

By construction the space widetilde W is a manifold. Had we not known how the regular CW-complex widetilde W had been constructed then we could prove that it is a closed 3-manifold by creating its barycentric subdivision K=sdwidetilde W, which is homeomorphic to widetilde W, and verifying that the link of each vertex in the simplicial complex sdwidetilde W is a 2-sphere. The following command carries out this proof.

<div class="example"><pre>
gap> IsClosedManifold(Wtilde);

true

</pre></div>

The Poincare conjecture (now proven) implies that widetilde W is homeomorphic to S^3. Hence W=S^3/C_5 is a quotient of the 3-sphere by an action of C_5 and is hence a lens space L(5,q) for some q.

The next commands determine that W is homeomorphic to L(5,4)≅ L(5,1).

<div class="example"><pre>
gap> Lk:=LinkingFormHomeomorphismInvariant(W);
[ 0, 4/5 ]

</pre></div>

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

<a id="X7B425A3280A2AF07" name="X7B425A3280A2AF07"></a>

<h4>4.8 Finite fundamental groups of 3-manifolds</h4>

Lens spaces are examples of 3-manifolds with finite fundamental groups. The complete list of finite groups G arising as fundamental groups of closed connected 3-manifolds is recalled in <a href="chap7.html#X79B1406C803FF178">7.12</a> where one method for computing their cohomology rings is presented. Their cohomology could also be computed from explicit 3-manifolds W with π_1W=G. For instance, the following commands realize a closed connected 3-manifold W with π_1W = C_11× SL_2( Z_5).

<div class="example"><pre>
gap> ap:=ArcPresentation(PureCubicalKnot(3,1));;
gap> W:=ThreeManifoldViaDehnSurgery(ap,1,11);;
gap> F:=FundamentalGroup(W);;
gap> Order(F);
1320
gap> AbelianInvariants(F);
[ 11 ]
gap> StructureDescription(F);
"C11 x SL(2,5)"

</pre></div>

Hence the group G=C_11× SL_2( Z_5) of order 1320 acts freely on the 3-sphere widetilde W. It thus has periodic cohomology with


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.


for n > 0.

<a id="X78912D227D753167" name="X78912D227D753167"></a>

<h4>4.9 Poincare's cube manifolds

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 <a href="http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_cube_manifolds">Manifold Atlas Project (MAP)</a>. 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

<img src="images/pcube.png" align="center" height="200" alt="cube"/>

and barycentric subdivision is used to ensure that the quotient is represented as a regular CW-complex.

Examples 3 and 4 from Poincare's paper, described in the following figures taken from MAP,

<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"/>

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.)

<div class="example"><pre>
gap> A:=1;;C:=2;;D:=3;;B:=4;;
gap> Ap:=5;;Cp:=6;;Dp:=7;;Bp:=8;;

gap> L:=[[A,B,D,C],[Bp,Dp,Cp,Ap]];;
gap> M:=[[A,B,Bp,Ap],[Cp,C,D,Dp]];;
gap> N:=[[A,C,Cp,Ap],[D,Dp,Bp,B]];;
gap> Ex3:=PoincareCubeCWComplex(L,M,N);
Regular CW-complex of dimension 3

gap> IsClosedManifold(Ex3);

true

gap> L:=[[A,B,D,C],[Bp,Dp,Cp,Ap]];;
gap> M:=[[A,B,Bp,Ap],[C,D,Dp,Cp]];;
gap> N:=[[A,C,Cp,Ap],[B,D,Dp,Bp]];;
gap> Ex4:=PoincareCubeCWComplex(L,M,N);
Regular CW-complex of dimension 3

gap> IsClosedManifold(Ex4);
true

gap> List([0..3],k->Homology(Ex3,k));
[ [ 0 ], [ 2, 2 ], [ ], [ 0 ] ]
gap> List([0..3],k->Homology(Ex4,k));
[ [ 0 ], [ 2, 0 ], [ 0 ], [ 0 ] ]

</pre></div>

<a id="X8761051F84C6CEC2" name="X8761051F84C6CEC2"></a>

<h4>4.10 There are at least 25 distinct cube manifolds</h4>

The function <code class="code">PoincareCubeCWComplex(A,G)</code> can also be applied to two inputs where A is a pairing of the six faces such as A=[[1,2],[3,4],[5,6]] and G is a list of three elements of the dihedral group of order 8 such as G=[(2,4),(2,4),(2,4)*(1,3)]. 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 A and G 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 cube manifold. Distinct values of A and G 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

A=[ [1,2], [3,4], [5,6] ], A=[ [1,2], [3,5], [4,6] ], A=[ [1,4], [2,6], [3,5] ]

and all G in D_8× D_8× D_8.

The following commands iterate through these 3×8^3 = 1536 pairs (A,G) and show that in precisely 163 cases (just over 10% of cases) the quotient CW-complex is a closed 3-manifold.

<div class="example"><pre>
gap> A1:= [ [1,2], [3,4], [5,6] ];;
gap> A2:=[ [1,2], [3,5], [4,6] ];;
gap> A3:=[ [1,4], [2,6], [3,5] ];;
gap> D8:=DihedralGroup(IsPermGroup,8);;

gap> Manifolds:=[];;
gap> for A in [A1,A2,A3] do
> for x in D8 do
> for y in D8 do
> for z in D8 do
> G:=[x,y,z];
> F:=PoincareCubeCWComplex(A,G);
> b:=IsClosedManifold(F);
> if b=true then Add(Manifolds,F); fi;
> od;od;od;od;

gap> Size(Manifolds);
163

</pre></div>

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..

<div class="example"><pre>
gap> invariant1:=function(m);
> return List([1..3],k->Homology(m,k));
> end;;

gap> C:=Classify(Manifolds,invariant1);;

gap> invariant2:=function(m)
> local L;
> L:=FundamentalGroup(m);
> if GeneratorsOfGroup(L)= [] then return [];fi;
> L:=LowIndexSubgroupsFpGroup(L,5);
> L:=List(L,AbelianInvariants);
> L:=SortedList(L);
> return L;
> end;;

gap> D:=RefineClassification(C,invariant2);;
gap> List(D,Size);
[ 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>

The next commands construct a list of 18 orientable cube manifolds and a list of 7 non-orientable cube manifolds.

<div class="example"><pre>
gap> Manifolds:=List(D,x->x[1]);;
gap> OrientableManifolds:=Filtered(Manifolds,m->Homology(m,3)=[0]);;
gap> NonOrientableManifolds:=Filtered(Manifolds,m->Homology(m,3)=[]);;
gap> Length(OrientableManifolds);
18
gap> Length(NonOrientableManifolds);
7

</pre></div>

The next commands show that the 7 non-orientable cube manifolds all have infinite fundamental groups.

<div class="example"><pre>
gap> List(NonOrientableManifolds,m->IsFinite(FundamentalGroup(m)));
[ false, false, false, false, false, false, false ]

</pre></div>

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.

<div class="example"><pre>
gap> List(OrientableManifolds{[4,8,10,11,12,13,15,16,18]},m->
 IsFinite(FundamentalGroup(m)));
[ true, true, true, true, true, true, true, true, true ]

gap> List(OrientableManifolds{[4,8,10,11,12,13,15,16,18]},m->
 StructureDescription(FundamentalGroup(m)));
[ "Q8", "C2", "C4", "C3 : C4", "C12", "C8", "C14", "C6", "1" ]

</pre></div>

<a id="X7D50795883E534A3" name="X7D50795883E534A3"></a>

<h5>4.10-1 Face pairings for 25 distinct cube manifolds</h5>

The following are the face pairings of 25 non-homeomorphic cube manifolds, with vertices of the cube numbered as describe above.

<div class="example"><pre>
gap> for i in [1..25] do 
> p:=Manifolds[i]!.cubeFacePairings;
> Print("Manifold ",i," has face pairings:\n");
> Print(p[1],"\n",p[2],"\n",p[3],"\n");
> Print("Fundamental group is: ");
> if i in [ 1, 9, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 25 ] then
> Print(StructureDescription(FundamentalGroup(Manifolds[i])),"\n");
> else Print("infinite non-cyclic\n"); fi;
> if Homology(Manifolds[i],3)=[0] then Print("Orientable, ");
> else Print("Non orientable, "); fi;
> Print(ManifoldType(Manifolds[i]),"\n");
> for x in Manifolds[i]!.edgeDegrees do
> Print(x[2]," edges of \"degree\" ",x[1],", ");
> od;
> Print("\n\n");
> od;

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>

<a id="X837811BB8181666E" name="X837811BB8181666E"></a>

<h5>4.10-2 Platonic cube manifolds</h5>

A platonic solid is a convex, regular polyhedron in 3-dimensional euclidean E^3 or spherical S^3 or hyperbolic space H^3. Being regular 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 π/2. A platonic cube in spherical space has dihedral angles 2π/3. A platonic cube in hyperbolic space has dihedral angles 2π/5. This can alternatively be expressed by saying that in a tessellation of E^3 by platonic cubes each edge is adjacent to 4 square faces. In a tessellation of S^3 by platonic cubes each edge is adjacent to 3 square faces. In a tessellation of H^3 by platonic cubes each edge is adjacent to 5 five square faces.

Any cube manifold M induces a cubical CW-decomposition of its universal cover widetilde M. We say that M is a platonic cube manifold if every edge in widetilde M is adjacent to 4 faces in the euclidean case widetilde M= E^3, is adjacent to 3 faces in the spherical case widetilde M= S^3, is adjacent to 5 faces in the hyperbolic case widetilde M= H^3.

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.

<a id="X8084A36082B26D86" name="X8084A36082B26D86"></a>

<h4>4.11 There are at most 41 distinct cube manifolds</h4>

Using the <a href="https://simpcomp-team.github.io/simpcomp/README.html">Simpcomp</a> 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.

The following commands establish homeomorphisms (simplicial complex isomorphisms) between manifolds in each equivalence class D[i] above for 1 ≤ i≤ 25, 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.

<div class="example"><pre>
gap> LoadPackage("Simpcomp");;

gap> inv3:=function(m)
> local K;
> K:=BarycentricSubdivision(m);
> K:=MaximalSimplicesOfSimplicialComplex(K);
> K:=SC(K);
> if not SCIsStronglyConnected(K) then Print("WARNING!\n"); fi;
> return SCExportIsoSig( K );
> end;
function( m ) ... end

gap> DD:=[];;
gap> for x in D do
> y:=Classify(x,inv3);
> Add(DD,List(y,z->z[1]));
> od;

gap> List(DD,Size);
[ 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>

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.

<div class="example"><pre>
gap> DD:=List(DD,x->List(x,y->PoincareCubeCWComplexNS(
> y!.cubeFacePairings[1],y!.cubeFacePairings[2],y!.cubeFacePairings[3])));;

gap> D:=[];;
gap> for x in DD do
> y:=Classify(x,inv3);
> Add(D,List(y,z->z[1]));
>od;;

gap> List(D,Size);
[ 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>

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.

<div class="example"><pre>
gap> DD:=[];;
gap> for x in D do
> if Length(x)>1 then
> Add(DD, List(x,y->BarycentricallySimplifiedComplex(y)));
> else Add(DD,x);
> fi;
> od;
gap> D:=[];;

gap> for x in DD do
> y:=Classify(x,inv3);
> Add(D,List(y,z->z[1]));
> od;

gap> List(D,Size);
[ 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 ]

gap> DD:=List(D,x->List(x,y->PoincareCubeCWComplexNS(
> y!.cubeFacePairings[1],y!.cubeFacePairings[2],y!.cubeFacePairings[3])));;

gap> D1:=[];;
gap> for x in DD do
> if Length(x)>1 then
> Add(D1, List(x,y->BarycentricallySimplifiedComplex(RegularCWComplex(BarycentricSubdivision(y)))));
> else Add(D1,x);
> fi;
> od;

gap> DD:=[];;
gap> for x in D1 do
> y:=Classify(x,inv3);
> Add(DD,List(y,z->z[1]));
> od;;

gap> Print(List(DD,Size),"\n");
[ 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>

The two manifolds in DD[14] have fundamental group C_2 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 C_4. 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.

These observations can be incorporated into our list DD of equivalence classes of manifolds as follows.

<div class="example"><pre>
gap> DD[14]:=DD[14]{[1]};;
gap> DD[17]:=DD[17]{[1]};;
gap> DD[25]:=DD[25]{[1]};;
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--> maximum size reached

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