  
  [1X4 [33X[0;0YThree Manifolds[133X[101X
  
  
  [1X4.1 [33X[0;0YDehn Surgery[133X[101X
  
  [33X[0;0YThe  following  example  constructs,  as  a  regular  CW-complex,  a  closed
  orientable 3-manifold [22XW[122X 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 [22Xp/q=17/16[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xap:=ArcPresentation(PureCubicalKnot(3,1));;[127X[104X
    [4X[25Xgap>[125X [27Xp:=17;;q:=16;;[127X[104X
    [4X[25Xgap>[125X [27XW:=ThreeManifoldViaDehnSurgery(ap,p,q);[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe next commands show that this [22X3[122X-manifold [22XW[122X has integral homology[133X
  
  [33X[0;0Y[22XH_0(W, Z)= Z[122X, [22XH_1(W, Z)= Z_16[122X, [22XH_2(W, Z)=0[122X, [22XH_3(W, Z)= Z[122X[133X
  
  [33X[0;0Yand that the fundamental group [22Xπ_1(W)[122X is non-abelian.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHomology(W,0);Homology(W,1);Homology(W,2);Homology(W,3);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[ 16 ][128X[104X
    [4X[28X[  ][128X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalGroup(W);;[127X[104X
    [4X[25Xgap>[125X [27XL:=LowIndexSubgroupsFpGroup(F,10);;[127X[104X
    [4X[25Xgap>[125X [27XList(L,AbelianInvariants);[127X[104X
    [4X[28X[ [ 16 ], [ 3, 8 ], [ 3, 4 ], [ 2, 3 ], [ 16, 43 ], [ 8, 43, 43 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  famous result of Lickorish and (independently) Wallace shows
  that Dehn surgery on knots leads to an interesting range of spaces.[133X
  
  [33X[0;0Y[12XTheorem:[112X  [13X Every closed, orientable, connected [22X3[122X-manifold can be obtained by
  surgery on a link in [22XS^3[122X. (Moreover, one can always perform the surgery with
  surgery  coefficients  [22X±  1[122X  and  with each individual component of the link
  unknotted.) [113X[133X
  
  
  [1X4.2 [33X[0;0YConnected Sums[133X[101X
  
  [33X[0;0YThe following example constructs the connected sum [22XW=A#B[122X of two [22X3[122X-manifolds,
  where  [22XA[122X  is obtained from a [22X5/1[122X Dehn surgery on the complement of the first
  prime  knot  on  11 crossings and [22XB[122X is obtained by a [22X5/1[122X Dehn surgery on the
  complement of the second prime knot on 11 crossings. The homology groups[133X
  
  [33X[0;0Y[22XH_1(W, Z) = Z_2⊕ Z_594[122X, [22XH_2(W, Z) = 0[122X, [22XH_3(W, Z) = Z[122X[133X
  
  [33X[0;0Yare computed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xap1:=ArcPresentation(PureCubicalKnot(11,1));;[127X[104X
    [4X[25Xgap>[125X [27XA:=ThreeManifoldViaDehnSurgery(ap1,5,1);;[127X[104X
    [4X[25Xgap>[125X [27Xap2:=ArcPresentation(PureCubicalKnot(11,2));;[127X[104X
    [4X[25Xgap>[125X [27XB:=ThreeManifoldViaDehnSurgery(ap2,5,1);;[127X[104X
    [4X[25Xgap>[125X [27XW:=ConnectedSum(A,B);[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(W,1);[127X[104X
    [4X[28X[ 2, 594 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,2);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,3);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YDijkgraaf-Witten Invariant[133X[101X
  
  [33X[0;0YGiven  a  closed  connected  orientable [22X3[122X-manifold [22XW[122X, a finite group [22XG[122X and a
  3-cocycle [22Xα∈ H^3(BG,U(1))[122X Dijkgraaf and Witten define the complex number[133X
  
  [33X[0;0Y$$  Z^{G,\alpha}(W)  =  \frac{1}{|G|}\sum_{\gamma\in  {\rm  Hom}(\pi_1W, G)}
  \langle  \gamma^\ast[\alpha], [M]\rangle \ \in\ \mathbb C\ $$ where [22Xγ[122X ranges
  over  all  group  homomorphisms  [22Xγ:  π_1W  →  G[122X.  This  complex number is an
  invariant of the homotopy type of [22XW[122X and is useful for distinguishing between
  certain homotopically distinct [22X3[122X-manifolds.[133X
  
  [33X[0;0YA  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  [22XH_3(W,  Z)≅  Z[122X  we  represent  [22XD_G(W)[122X by the set [22XD_G(W)={ γ_∗(1) }_γ∈
  Hom(π_1W,  G)[122X  where  [22X1[122X denotes one of the two possible generators of [22XH_3(W,
  Z)[122X.[133X
  
  [33X[0;0YFor  any coprime integers [22Xp,qge 1[122X the [13Xlens space[113X [22XL(p,q)[122X 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 [22Xp/q[122X. Lens spaces have cyclic
  fundamental   group   [22Xπ_1(L(p,q))=C_p[122X  and  homology  [22XH_1(L(p,q),  Z)≅  Z_p[122X,
  [22XH_2(L(p,q), Z)≅ 0[122X, [22XH_3(L(p,q), Z)≅ Z[122X. It was proved by J.H.C. Whitehead that
  two  lens  spaces [22XL(p,q)[122X and [22XL(p',q')[122X are homotopy equivalent if and only if
  [22Xp=p'[122X and [22Xqq'≡ ± n^2 mod p[122X for some integer [22Xn[122X.[133X
  
  [33X[0;0YThe  following session constructs the two lens spaces [22XL(5,1)[122X and [22XL(5,2)[122X. The
  homology  version  of  the  Dijkgraaf-Witten invariant is used with [22XG=C_5[122X to
  demonstrate that the two lens spaces are not homotopy equivalent.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xap:=[[2,1],[2,1]];; #Arc presentation for the trivial knot[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XL51:=ThreeManifoldViaDehnSurgery(ap,5,1);;[127X[104X
    [4X[25Xgap>[125X [27XD:=DijkgraafWittenInvariant(L51,CyclicGroup(5));[127X[104X
    [4X[28X[ g1^4, g1^4, g1, g1, id ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XL52:=ThreeManifoldViaDehnSurgery(ap,5,2);;[127X[104X
    [4X[25Xgap>[125X [27XD:=DijkgraafWittenInvariant(L52,CyclicGroup(5));[127X[104X
    [4X[28X[ g1^3, g1^3, g1^2, g1^2, id ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  theorem  of  Fermat  and Euler states that if a prime [22Xp[122X is congruent to 3
  modulo  4,  then  for any [22Xq[122X exactly one of [22X± q[122X is a quadratic residue mod p.
  For  all other primes [22Xp[122X either both or neither of [22X± q[122X is a quadratic residue
  mod  [22Xp[122X.  Thus  for  fixed  [22Xp  ≡ 3 mod 4[122X the lens spaces [22XL(p,q)[122X form a single
  homotopy  class. There are precisely two homotopy classes of lens spaces for
  other [22Xp[122X.[133X
  
  [33X[0;0YThe following commands confirm that [22XL(13,1) ≄ L(13,2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL13_1:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,1);;[127X[104X
    [4X[25Xgap>[125X [27XDijkgraafWittenInvariant(L13_1,CyclicGroup(13));[127X[104X
    [4X[28X[ g1^12, g1^12, g1^10, g1^10, g1^9, g1^9, g1^4, g1^4, g1^3, g1^3, g1, g1, id ][128X[104X
    [4X[25Xgap>[125X [27XL13_2:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,2);;[127X[104X
    [4X[25Xgap>[125X [27XDijkgraafWittenInvariant(L13_2,CyclicGroup(13));[127X[104X
    [4X[28X[ g1^11, g1^11, g1^8, g1^8, g1^7, g1^7, g1^6, g1^6, g1^5, g1^5, g1^2, g1^2, [128X[104X
    [4X[28X  id ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.4 [33X[0;0YCohomology rings[133X[101X
  
  [33X[0;0YThe  following  commands construct the multiplication table (with respect to
  some  basis)  for  the  cohomology rings [22XH^∗(L(13,1), Z_13)[122X and [22XH^∗(L(13,2),
  Z_13)[122X.  These  rings  are  isomorphic and so fail to distinguish between the
  homotopy types of the lens spaces [22XL(13,1)[122X and [22XL(13,2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL13_1:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,1);;[127X[104X
    [4X[25Xgap>[125X [27XL13_2:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,2);;[127X[104X
    [4X[25Xgap>[125X [27XL13_1:=BarycentricSubdivision(L13_1);;[127X[104X
    [4X[25Xgap>[125X [27XL13_2:=BarycentricSubdivision(L13_2);;[127X[104X
    [4X[25Xgap>[125X [27XA13_1:=CohomologyRing(L13_1,13);;[127X[104X
    [4X[25Xgap>[125X [27XA13_2:=CohomologyRing(L13_2,13);;[127X[104X
    [4X[25Xgap>[125X [27XM13_1:=List([1..4],i->[]);;[127X[104X
    [4X[25Xgap>[125X [27XB13_1:=CanonicalBasis(A13_1);;[127X[104X
    [4X[25Xgap>[125X [27XM13_2:=List([1..4],i->[]);;[127X[104X
    [4X[25Xgap>[125X [27XB13_2:=CanonicalBasis(A13_2);;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..4] do[127X[104X
    [4X[25X>[125X [27Xfor j in [1..4] do[127X[104X
    [4X[25X>[125X [27XM13_1[i][j]:=B13_1[i]*B13_1[j];[127X[104X
    [4X[25X>[125X [27Xod;od;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..4] do[127X[104X
    [4X[25X>[125X [27Xfor j in [1..4] do[127X[104X
    [4X[25X>[125X [27XM13_2[i][j]:=B13_2[i]*B13_2[j];[127X[104X
    [4X[25X>[125X [27Xod;od;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(M13_1);[127X[104X
    [4X[28X[ [            v.1,            v.2,            v.3,            v.4 ],[128X[104X
    [4X[28X  [            v.2,          0*v.1,  (Z(13)^6)*v.4,          0*v.1 ],[128X[104X
    [4X[28X  [            v.3,  (Z(13)^6)*v.4,          0*v.1,          0*v.1 ],[128X[104X
    [4X[28X  [            v.4,          0*v.1,          0*v.1,          0*v.1 ] ][128X[104X
    [4X[25Xgap>[125X [27XDisplay(M13_2);[127X[104X
    [4X[28X[ [          v.1,          v.2,          v.3,          v.4 ],[128X[104X
    [4X[28X  [          v.2,        0*v.1,  (Z(13))*v.4,        0*v.1 ],[128X[104X
    [4X[28X  [          v.3,  (Z(13))*v.4,        0*v.1,        0*v.1 ],[128X[104X
    [4X[28X  [          v.4,        0*v.1,        0*v.1,        0*v.1 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.5 [33X[0;0YLinking Form[133X[101X
  
  [33X[0;0YGiven  a  closed  connected [12Xoriented[112X [22X3[122X-manifold [22XW[122X let [22Xτ H_1(W, Z)[122X denote the
  torsion  subgroup  of  the  first  integral  homology. The [13Xlinking form[113X is a
  bilinear mapping[133X
  
  [33X[0;0Y[22XLk_W: τ H_1(W, Z) × τ H_1(W, Z) ⟶ Q/ Z[122X.[133X
  
  [33X[0;0YTo construct this form note that we have a Poincare duality isomorphism[133X
  
  [33X[0;0Y[22Xρ: H^2(W, Z) stackrel≅⟶ H_1(W, Z), z ↦ z∩ [W][122X[133X
  
  [33X[0;0Yinvolving  the  cap  product with the fundamental class [22X[W]∈ H^3(W, Z)[122X. That
  is,  [22X[M][122X  is  the generator of [22XH^3(W, Z)≅ Z[122X determining the orientation. The
  short exact sequence [22XZ ↣ Q ↠ Q/ Z[122X gives rise to a cohomology exact sequence[133X
  
  [33X[0;0Y[22X→ H^1(W, Q) → H^1(W, Q/ Z) stackrelβ⟶ H^2(W, Z) → H^2(W, Q) →[122X[133X
  
  [33X[0;0Yfrom  which we obtain the isomorphism [22Xβ : τ H^1(W, Q/ Z) stackrel≅⟶ τ H^2(W,
  Z)[122X. The linking form [22XLk_W[122X can be defined as the composite[133X
  
  [33X[0;0Y[22XLk_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[122X[133X
  
  [33X[0;0Ywhere [22Xev(x,α)[122X evaluates a [22X1[122X-cocycle [22Xα[122X on a [22X1[122X-cycle [22Xx[122X.[133X
  
  [33X[0;0YThe linking form can be used to define the set[133X
  
  [33X[0;0Y[22XI^O(W) = {Lk_W(g,g) : g∈ τ H_1(W, Z)}[122X[133X
  
  [33X[0;0Ywhich is an oriented-homotopy invariant of [22XW[122X. Letting [22XW^+[122X and [22XW^-[122X denote the
  two possible orientations on the manifold, the set[133X
  
  [33X[0;0Y[22XI(W) ={I^O(W^+), I^O(W^-)}[122X[133X
  
  [33X[0;0Yis a homotopy invariant of [22XW[122X which in this manual we refer to as the [13Xlinking
  form homotopy invariant[113X.[133X
  
  [33X[0;0YThe  following  commands compute the linking form homotopy invariant for the
  lens  spaces  [22XL(13,q)[122X  with [22X1le qle 12[122X. This invariant distinguishes between
  the two homotopy types that arise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLensSpaces:=[];;[127X[104X
    [4X[25Xgap>[125X [27Xfor q in [1..12] do[127X[104X
    [4X[25X>[125X [27XAdd(LensSpaces,ThreeManifoldViaDehnSurgery([[1,2],[1,2]],13,q));[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(List(LensSpaces,LinkingFormHomotopyInvariant));;[127X[104X
    [4X[28X[ [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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, [128X[104X
    [4X[28X          2/13, 5/13, 5/13, 6/13, 6/13, 7/13, 7/13, 8/13, 8/13, 11/13, 11/13 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ], [128X[104X
    [4X[28X[128X[104X
    [4X[28X  [ [ 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 ], [128X[104X
    [4X[28X      [ 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 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.6 [33X[0;0YDetermining the homeomorphism type of a lens space[133X[101X
  
  [33X[0;0YIn  1935  K.  Reidemeister  [Rei35] classified lens spaces up to orientation
  preserving  PL-homeomorphism.  This  was  generalized by E. Moise [Moi52] in
  1952 to a classification up to homeomorphism -- his method requred the proof
  of  the  Hauptvermutung  for  [22X3[122X-dimensional  manifolds. In 1960, following a
  suggestion  of  R. Fox, a proof was given by E.J. Brody [Bro60] that avoided
  the  need  for the Hauptvermutung. Reidemeister's method, using what is know
  termed   [13XReidermeister   torsion[113X,   and   Brody's   method,   using  tubular
  neighbourhoods  of [22X1[122X-cycles, both require identifying a suitable "preferred"
  generator  of  [22XH_1(L(p,q),  Z)[122X. In 2003 J. Przytycki and A. Yasukhara [PY03]
  provided  an  alternative method for classifying lens spaces, which uses the
  linking  form  and  again  requires  the  identification  of  a  "preferred"
  generator of [22XH_1(L(p,q), Z)[122X.[133X
  
  [33X[0;0YPrzytycki and Yasukhara proved the following.[133X
  
  [33X[0;0Y[12XTheorem.[112X  [13XLet  [22Xρ:  S^ 3 → L(p, q)[122X be the [22Xp[122X-fold cyclic cover and [22XK[122X a knot in
  [22XL(p, q)[122X that represents a generator of [22XH_1 (L(p, q), Z)[122X. If [22Xρ ^-1 (K)[122X is the
  trivial  knot,  then  [22XLk_  L(p,q)  ([K], [K]) = q/p[122X or [22X= overline q/p ∈ Q/ Z[122X
  where [22Xqoverline q ≡ 1 mod p[122X. [113X[133X
  
  [33X[0;0YThe  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.[133X
  
  [33X[0;0YThe  following  commands  construct a small lens space [22XL=L(p,q)[122X with unknown
  values  of [22Xp,q[122X. Subsequent commands will determine the homeomorphism type of
  [22XL[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:=Random([2,3,5,7,11,13,17]);;[127X[104X
    [4X[25Xgap>[125X [27Xq:=Random([1..p-1]);;[127X[104X
    [4X[25Xgap>[125X [27XL:=ThreeManifoldViaDehnSurgery([[1,2],[1,2]],p,q);[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  can  readily  determine  the  value  of [22Xp=11[122X by calculating the order of
  [22Xπ_1(L)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalGroupWithPathReps(L);;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(F);[127X[104X
    [4X[28X"C11"[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  commands  take  the  default  edge  path [22Xγ: S^1→ L[122X representing a
  generator  of  the  cyclic  group [22Xπ_1(L)[122X and lift it to an edge path [22Xtildeγ:
  S^1→ tilde L[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XU:=UniversalCover(L);;[127X[104X
    [4X[25Xgap>[125X [27XG:=U!.group;;[127X[104X
    [4X[25Xgap>[125X [27Xp:=EquivariantCWComplexToRegularCWMap(U,Group(One(G)));;[127X[104X
    [4X[25Xgap>[125X [27XU:=Source(p);;[127X[104X
    [4X[25Xgap>[125X [27Xgamma:=[];;[127X[104X
    [4X[25Xgap>[125X [27Xgamma[2]:=F!.loops[1];;[127X[104X
    [4X[25Xgap>[125X [27Xgamma[2]:=List(gamma[2],AbsInt);;[127X[104X
    [4X[25Xgap>[125X [27Xgamma[1]:=List(gamma[2],k->L!.boundaries[2][k]);;[127X[104X
    [4X[25Xgap>[125X [27Xgamma[1]:=SSortedList(Flat(gamma[1]));;[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27Xgammatilde:=[[],[],[],[]];;[127X[104X
    [4X[25Xgap>[125X [27Xfor k in [1..U!.nrCells(0)] do[127X[104X
    [4X[25X>[125X [27Xif p!.mapping(0,k) in gamma[1] then Add(gammatilde[1],k); fi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27Xfor k in [1..U!.nrCells(1)] do[127X[104X
    [4X[25X>[125X [27Xif p!.mapping(1,k) in gamma[2] then Add(gammatilde[2],k); fi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27Xgammatilde:=CWSubcomplexToRegularCWMap([U,gammatilde]);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe next commands check that the path [22Xtildeγ[122X is unknotted in [22Xtilde L≅ S^3[122X by
  checking that [22Xπ_1(tilde L∖ image(tildeγ))[122X is infinite cyclic.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC:=RegularCWComplexComplement(gammatilde);[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=FundamentalGroup(C);[127X[104X
    [4X[28X<fp group of size infinity on the generators [ f2 ]>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince [22Xtildeγ[122X is unkotted the cycle [22Xγ[122X represents the preferred generator [22X[γ]∈
  H_1(L, Z)[122X. The next commands compute [22XLk_L([γ],[γ])= 7/11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLinkingFormHomeomorphismInvariant(L);[127X[104X
    [4X[28X[ 7/11 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe classification of Moise/Brody states that [22XL(p,q)≅ L(p,q')[122X if and only if
  [22Xqq'≡ ± 1 mod p[122X. Hence the lens space [22XL[122X has the homeomorphism type[133X
  
  [33X[0;0Y[22XL≅ L(11,7) ≅ L(11,8) ≅ L(11,4) ≅ L(11,3)[122X.[133X
  
  
  [1X4.7 [33X[0;0YSurgeries on distinct knots can yield homeomorphic manifolds[133X[101X
  
  [33X[0;0YThe  lens space [22XL(5,1)[122X is a quotient of the [22X3[122X-sphere [22XS^3[122X by a certain action
  of the cyclic group [22XC_5[122X. It can be realized by a [22Xp/q=5/1[122X 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 [22XW[122X
  obtained  from  a  [22Xp/q=1/5[122X Dehn filling of the complement of the trefoil and
  show  that  [22XW[122X  at  least has the same integral homology and same fundamental
  group as [22XL(5,1)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xap:=ArcPresentation(PureCubicalKnot(3,1));;[127X[104X
    [4X[25Xgap>[125X [27XW:=ThreeManifoldViaDehnSurgery(ap,1,5);;[127X[104X
    [4X[25Xgap>[125X [27XHomology(W,1);[127X[104X
    [4X[28X[ 5 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,2);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,3);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalGroup(W);;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(F);[127X[104X
    [4X[28X"C5"[128X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe next commands construct the universal cover [22Xwidetilde W[122X and show that it
  has  the  same homology as [22XS^3[122X and trivivial fundamental group [22Xπ_1(widetilde
  W)=0[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XU:=UniversalCover(W);;[127X[104X
    [4X[25Xgap>[125X [27XG:=U!.group;;[127X[104X
    [4X[25Xgap>[125X [27XWtilde:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G)));[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(Wtilde,1);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Wtilde,2);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Wtilde,3);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalGroup(Wtilde);[127X[104X
    [4X[28X<fp group on the generators [  ]>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YBy  construction  the  space [22Xwidetilde W[122X is a manifold. Had we not known how
  the  regular CW-complex [22Xwidetilde W[122X had been constructed then we could prove
  that  it  is  a  closed  [22X3[122X-manifold  by creating its barycentric subdivision
  [22XK=sdwidetilde  W[122X,  which  is homeomorphic to [22Xwidetilde W[122X, and verifying that
  the  link  of  each  vertex  in  the  simplicial  complex [22Xsdwidetilde W[122X is a
  [22X2[122X-sphere. The following commands carry out this proof: each link is shown to
  admit  a  discrete  vector  field  with  precisely two critical cells -- one
  [22X0[122X-cell and one [22X2[122X-cell.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK:=BarycentricSubdivision(Wtilde);[127X[104X
    [4X[28XSimplicial complex of dimension 3.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XL:=[];;[127X[104X
    [4X[25Xgap>[125X [27Xfor v in K!.vertices do[127X[104X
    [4X[25X>[125X [27XAdd(L,CriticalCells(RegularCWComplex(VertexLink(K,v))));[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27XSSortedList(List(L,Size));[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XC:=Concatenation(L);;[127X[104X
    [4X[25Xgap>[125X [27XSSortedList(List(C,x->x[1]));[127X[104X
    [4X[28X[ 0, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   Poincare   conjecture   (now  proven)  implies  that  [22Xwidetilde  W[122X  is
  homeomorphic  to  [22XS^3[122X.  Hence  [22XW=S^3/C_5[122X is a quotient of the [22X3[122X-sphere by an
  action of [22XC_5[122X and is hence a lens space [22XL(5,q)[122X for some [22Xq[122X.[133X
  
  [33X[0;0YThe next commands determine that [22XW[122X is homeomorphic to [22XL(5,4)≅ L(5,1)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLk:=LinkingFormHomeomorphismInvariant(W);[127X[104X
    [4X[28X[ 0, 4/5 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YMoser  [Mos71]  gives  a  precise decription of the lens spaces arising from
  surgery  on the trefoil knot and more generally from surgery on torus knots.
  Greene [Gre13] determines the lens spaces that arise by integer Dehn surgery
  along a knot in the three-sphere[133X
  
  
  [1X4.8 [33X[0;0YFinite fundamental groups of [22X3[122X[101X[1X-manifolds[133X[101X
  
  [33X[0;0YLens  spaces are examples of [22X3[122X-manifolds with finite fundamental groups. The
  complete  list  of  finite  groups [22XG[122X arising as fundamental groups of closed
  connected  [22X3[122X-manifolds  is  recalled  in [14X7.11[114X where one method for computing
  their cohomology rings is presented. Their cohomology could also be computed
  from  explicit  [22X3[122X-manifolds  [22XW[122X  with  [22Xπ_1W=G[122X.  For  instance,  the following
  commands  realize  a  closed  connected [22X3[122X-manifold [22XW[122X with [22Xπ_1W = C_11× SL_2(
  Z_5)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xap:=ArcPresentation(PureCubicalKnot(3,1));;[127X[104X
    [4X[25Xgap>[125X [27XW:=ThreeManifoldViaDehnSurgery(ap,1,11);;[127X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalGroup(W);;[127X[104X
    [4X[25Xgap>[125X [27XOrder(F);[127X[104X
    [4X[28X1320[128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariants(F);[127X[104X
    [4X[28X[ 11 ][128X[104X
    [4X[25Xgap>[125X [27XStructureDescription(F);[127X[104X
    [4X[28X"C11 x SL(2,5)"[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHence the group [22XG=C_11× SL_2( Z_5)[122X of order [22X1320[122X acts freely on the [22X3[122X-sphere
  [22Xwidetilde W[122X. It thus has periodic cohomology with[133X
  
  
  [24X[33X[0;6YH_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.[133X
  
  [124X
  
  [33X[0;0Yfor [22Xn > 0[122X.[133X
  
