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4 Three Manifolds
 4.1 Dehn Surgery
 4.2 Dijkgraaf-Witten Invariant
 4.3 Cohomology rings
 4.4 Linking Form

4 Three Manifolds

4.1 Dehn Surgery

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

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

The next commands show that this closed compact \(3\)-manifold \(W\) has integral homology

\( H_0(W,\mathbb Z)=\mathbb Z\), \( H_1(W,\mathbb Z)=\mathbb Z_{31}\), \( H_2(W,\mathbb Z)=0\), \( H_3(W,\mathbb Z)=\mathbb Z\).

gap> Homology(W,0);Homology(W,1);Homology(W,2);Homology(W,3);
[ 0 ]
[ 31 ]
[  ]
[ 0 ]

4.2 Dijkgraaf-Witten Invariant

Given a closed connected orientable \(3\)-manifold \(W\), a finite group \(G\) and a 3-cocycle \(\alpha\in H^3(BG,U(1))\) 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 \(\gamma\) ranges over all group homomorphisms \(\gamma\colon \pi_1W \rightarrow G\). 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 \(H_3(W,\mathbb Z)\cong \mathbb Z\) we represent \(D_G(W)\) by the set \(D_G(W)=\{ \gamma_\ast(1) \}_{\gamma\in {\rm Hom}(\pi_1W, G)}\) where \(1\) denotes one of the two possible generators of \(H_3(W,\mathbb Z)\).

The following session constructs the two Lens spaces \(L(5,1)\) and \(L(5,2)\). These are \(3\)-dimensional manifolds with cyclic fundamental group of order \(5\) and with identical integral homology groups. 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.

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 ]

A lens space \(L(p,q)\) exists for any coprime integers \(p,q\ge 1\). It was proved by Paul Olum that two lens spaces \(L(p,q)\) and \(L(p',q')\) are homotopy equivalent if and only if \(p=p'\) and \(qq'\cong \pm n^2 \mod p\) for some integer \(n\). For instance, \(L(7,q)\) is homotopy equivalent to \(L(7,q')\) if and only if \(qq' \mod 7\in \{1,2,4\}\). So \(L(7,1)\simeq L(7,2) \simeq L(7,4)\), \(L(7,3)\simeq L(7,5)\simeq(7,6)\). The following session shows that the Dijkgraaf-Witten invariant, with \(G=C_7\), fails to distinguish between \(L(7,1)\) and \(L(7,3)\).

gap> L71:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],7,1);;
gap> L73:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],7,3);;
gap> DijkgraafWittenInvariant(L71,CyclicGroup(7));
[ g1^6, g1^6, g1^5, g1^5, g1^3, g1^3, id ]
gap> DijkgraafWittenInvariant(L73,CyclicGroup(7));
[ g1^6, g1^6, g1^5, g1^5, g1^3, g1^3, id ]

4.3 Cohomology rings

The following commands construct the multiplication table (with respect to some basis) for the cohomology rings \(H^\ast(L(7,1),\mathbb Z_7)\) and \(H^\ast(L(7,3),\mathbb Z_7)\). These rings are isomorphic and so fail to distinguish between the homotopy types of the lens spaces \(L(7,1)\) and \(L(7,3)\).

gap> L71:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],7,1);;
gap> L73:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],7,3);;
gap> L71:=BarycentricSubdivision(L71);;
gap> L73:=BarycentricSubdivision(L73);;
gap> A71:=CohomologyRing(L71,7);;
gap> A73:=CohomologyRing(L73,7);;
gap> M71:=List([1..4],i->[]);;
gap> B71:=CanonicalBasis(A71);;
gap> M73:=List([1..4],i->[]);;
gap> B73:=CanonicalBasis(A73);;
gap> for i in [1..4] do
> for j in [1..4] do
> M71[i][j]:=B71[i]*B71[j];
> od;od;
gap> for i in [1..4] do
> for j in [1..4] do
> M73[i][j]:=B73[i]*B73[j];
> od;od;
gap> Display(M71);
[ [           v.1,           v.2,           v.3,           v.4 ],
  [           v.2,         0*v.1,  (Z(7)^3)*v.4,         0*v.1 ],
  [           v.3,  (Z(7)^3)*v.4,         0*v.1,         0*v.1 ],
  [           v.4,         0*v.1,         0*v.1,         0*v.1 ] ]
gap> Display(M73);
[ [           v.1,           v.2,           v.3,           v.4 ],
  [           v.2,         0*v.1,  (Z(7)^5)*v.4,         0*v.1 ],
  [           v.3,  (Z(7)^5)*v.4,         0*v.1,         0*v.1 ],
  [           v.4,         0*v.1,         0*v.1,         0*v.1 ] ]

4.4 Linking Form

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

\(L_W\colon \tau H_1(W,\mathbb Z) \times \tau H_1(W,\mathbb Z) \longrightarrow \mathbb Q/\mathbb Z\).

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

\(\rho\colon H^2(W,\mathbb Z) \stackrel{\cong}{\longrightarrow} H_1(W,\mathbb Z), z \mapsto z\cap [W]\)

involving the cap product with the fundamental class \([W]\in H^3(W,\mathbb Z)\cong \mathbb Z\). The short exact sequence \(\mathbb Z \rightarrowtail \mathbb Q \twoheadrightarrow \mathbb Q/\mathbb Z\) gives rise to a cohomology exact sequence

\( \rightarrow H^1(W,\mathbb Q) \rightarrow H^1(W,\mathbb Q/\mathbb Z) \stackrel{\beta}{\longrightarrow} H^2(W,\mathbb Z) \rightarrow H^2(W,\mathbb Q) \rightarrow \)

from which we obtain the isomorphism \(\beta \colon \tau H^1(W,\mathbb Q/\mathbb Z) \stackrel{\cong}{\longrightarrow} \tau H^2(W,\mathbb Z)\). The linking form \(L_W\) can be defined as the composite

\(L_W\colon \tau H_1(W,\mathbb Z) \times \tau H_1(W,\mathbb Z) \stackrel{1\times \rho^{-1}}{\longrightarrow} \tau H_1(W,\mathbb Z) \times \tau H^2(W,\mathbb Z) \stackrel{1\times \beta^{-1}}{\longrightarrow} \tau H_1(W,\mathbb Z) \times \tau H^1(W,\mathbb Q/\mathbb Z) \stackrel{ev}{\longrightarrow } \mathbb Q/\mathbb Z \)

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

The linking form can be used to define the set

\(I(W) = \{L_W(g,g) \ \colon \ g\in \tau H_1(W,\mathbb Z)\}\)

which is a homotopy invariant of \(W\) and which we refer to as the linking form invariant.

The following commands compute the linking form invariant for the lens spaces \(L(7,q)\) with \(1\le q\le 6\). This invariant distinguishes between the homotopy type \(L(7,1)\simeq L(7,2) \simeq L(7,4)\) and the homotopy type \(L(7,3)\simeq L(7,5)\simeq(7,6)\).

gap> LensSpaces:=[];;
gap> for q in [1..6] do
> Add(LensSpaces,ThreeManifoldViaDehnSurgery([[1,2],[1,2]],7,q));
> od;
gap> Display(List(LensSpaces,LinkingFormInvariant));;
[ [    0,  3/7,  3/7,  5/7,  5/7,  6/7,  6/7 ],
  [    0,  3/7,  3/7,  5/7,  5/7,  6/7,  6/7 ],
  [    0,  1/7,  1/7,  2/7,  2/7,  4/7,  4/7 ],
  [    0,  3/7,  3/7,  5/7,  5/7,  6/7,  6/7 ],
  [    0,  1/7,  1/7,  2/7,  2/7,  4/7,  4/7 ],
  [    0,  1/7,  1/7,  2/7,  2/7,  4/7,  4/7 ] ]

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