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, Z)= Z, H_1(W, Z)= Z_31, H_2(W, Z)=0, H_3(W, Z)= Z.
gap> Homology(W,0);Homology(W,1);Homology(W,2);Homology(W,3); [ 0 ] [ 31 ] [ ] [ 0 ]
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).
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≥ 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'≅ ± 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∈ {1,2,4}. So L(7,1)≃ L(7,2) ≃ L(7,4), L(7,3)≃ L(7,5)≃(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 ]
The following commands construct the multiplication table (with respect to some basis) for the cohomology rings H^∗(L(7,1), Z_7) and H^∗(L(7,3), 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 ] ]
Given a closed connected orientable 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
L_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)≅ Z. 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 L_W can be defined as the composite
L_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(W) = {L_W(g,g) : g∈ τ H_1(W, 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≤ q≤ 6. This invariant distinguishes between the homotopy type L(7,1)≃ L(7,2) ≃ L(7,4) and the homotopy type L(7,3)≃ L(7,5)≃(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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