3 Covering spaces
  
  Let  Y  denote  a  finite  regular  CW-complex.  Let  widetilde Y denote its
  universal covering space. The covering space inherits a regular CW-structure
  which  can  be  computed and stored using the datatype of a π_1Y-equivariant
  CW-complex.  The cellular chain complex C_∗widetilde Y of widetilde Y can be
  computed  and  stored  as  an equivariant chain complex. Given an admissible
  discrete  vector  field  on  Y,  we  can  endow Y with a smaller non-regular
  CW-structre  whose  cells  correspond  to  the  critical cells in the vector
  field. This smaller CW-structure leads to a more efficient chain complex C_∗
  widetilde  Y  involving  one  free  generator  for each critical cell in the
  vector field.
  
  
  3.1 Cellular chains on the universal cover
  
  The following commands construct a 6-dimensional regular CW-complex Y≃ S^1 ×
  S^1× S^1 homotopy equivalent to a product of three circles.
  
    Example  
    gap> A:=[[1,1,1],[1,0,1],[1,1,1]];;
    gap> S:=PureCubicalComplex(A);;
    gap> T:=DirectProduct(S,S,S);;
    gap> Y:=RegularCWComplex(T);;
    Regular CW-complex of dimension 6
    
    gap> Size(Y);
    110592
    
  
  
  The  CW-somplex  Y  has  110592  cells.  The  next commands construct a free
  π_1Y-equivariant  chain  complex  C_∗widetilde  Y homotopy equivalent to the
  chain  complex of the universal cover of Y. The chain complex C_∗widetilde Y
  has just 8 free generators.
  
    Example  
    gap> Y:=ContractedComplex(Y);;
    gap> CU:=ChainComplexOfUniversalCover(Y);;
    gap> List([0..Dimension(Y)],n->CU!.dimension(n));
    [ 1, 3, 3, 1 ]
    
  
  
  The  next  commands  construct a subgroup H < π_1Y of index 50 and the chain
  complex  C_∗widetilde  Y⊗_ ZH Z which is homotopy equivalent to the cellular
  chain  complex  C_∗widetilde  Y_H  of  the  50-fold cover widetilde Y_H of Y
  corresponding to H.
  
    Example  
    gap> L:=LowIndexSubgroupsFpGroup(CU!.group,50);;
    gap> H:=L[Length(L)-1];;
    gap> Index(CU!.group,H);
    50
    gap> D:=TensorWithIntegersOverSubgroup(CU,H);
    Chain complex of length 3 in characteristic 0 .
    
    gap> List([0..3],D!.dimension);
    [ 50, 150, 150, 50 ]
    
  
  
  General  theory implies that the 50-fold covering space widetilde Y_H should
  again  be homotopy equivalent to a product of three circles. In keeping with
  this, the following commands verify that widetilde Y_H has the same integral
  homology as S^1× S^1× S^1.
  
    Example  
    gap> Homology(D,0);
    [ 0 ]
    gap> Homology(D,1);
    [ 0, 0, 0 ]
    gap> Homology(D,2);
    [ 0, 0, 0 ]
    gap> Homology(D,3);
    [ 0 ]
    
  
  
  
  3.2 Spun knots and the Satoh tube map
  
  We'll  contruct  two  spaces  Y,W  with  isomorphic  fundamental  groups and
  isomorphic  intergal  homology,  and  use  the  integral  homology of finite
  covering  spaces  to  establsh  that  the  two spaces have distinct homotopy
  types.
  
  By  spinning  a  link  K ⊂ R^3 about a plane P⊂ R^3 with P∩ K=∅, we obtain a
  collection  Sp(K)⊂  R^4  of knotted tori. The following commands produce the
  two  tori  obtained  by  spinning the Hopf link K and show that the space Y=
  R^4∖ Sp(K) = Sp( R^3∖ K) is connected with fundamental group π_1Y = Z× Z and
  homology  groups  H_0(Y)=  Z,  H_1(Y)= Z^2, H_2(Y)= Z^4, H_3(Y, Z)= Z^2. The
  space  Y  is  only  constructed  up  to  homotopy,  and  for  this reason is
  3-dimensional.
  
    Example  
    gap> Hopf:=PureCubicalLink("Hopf");
    Pure cubical link.
    
    gap> Y:=SpunAboutInitialHyperplane(PureComplexComplement(Hopf));
    Regular CW-complex of dimension 3
    
    gap> Homology(Y,0);
    [ 0 ]
    gap> Homology(Y,1);
    [ 0, 0 ]
    gap> Homology(Y,2);
    [ 0, 0, 0, 0 ]
    gap> Homology(Y,3);
    [ 0, 0 ]
    gap> Homology(Y,4);
    [  ]
    gap> GY:=FundamentalGroup(Y);;
    gap> GeneratorsOfGroup(GY);
    [ f2, f3 ]
    gap> RelatorsOfFpGroup(GY);
    [ f3^-1*f2^-1*f3*f2 ]
    
  
  
  An  alternative embedding of two tori L⊂ R^4 can be obtained by applying the
  'tube  map'  of  Shin  Satoh  to  a  welded Hopf link [Sat00]. The following
  commands  construct  the  complement W= R^4∖ L of this alternative embedding
  and  show  that  W has the same fundamental group and integral homology as Y
  above.
  
    Example  
    gap> L:=HopfSatohSurface();
    Pure cubical complex of dimension 4.
    
    gap> W:=ContractedComplex(RegularCWComplex(PureComplexComplement(L)));
    Regular CW-complex of dimension 3
    
    gap> Homology(W,0);
    [ 0 ]
    gap> Homology(W,1);
    [ 0, 0 ]
    gap> Homology(W,2);
    [ 0, 0, 0, 0 ]
    gap> Homology(W,3);
    [ 0, 0 ]
    gap> Homology(W,4);
    [  ]
    
    gap> GW:=FundamentalGroup(W);;
    gap> GeneratorsOfGroup(GW);
    [ f1, f2 ]
    gap> RelatorsOfFpGroup(GW);
    [ f1^-1*f2^-1*f1*f2 ]
    
  
  
  Despite  having the same fundamental group and integral homology groups, the
  above  two  spaces  Y and W were shown by Kauffman and Martins [KFM08] to be
  not  homotopy  equivalent.  Their technique involves the fundamental crossed
  module  derived  from the first three dimensions of the universal cover of a
  space,  and  counts  the  representations of this fundamental crossed module
  into a given finite crossed module. This homotopy inequivalence is recovered
  by the following commands which involves the 5-fold covers of the spaces.
  
    Example  
    gap> CY:=ChainComplexOfUniversalCover(Y);
    Equivariant chain complex of dimension 3
    gap> LY:=LowIndexSubgroups(CY!.group,5);;
    gap> invY:=List(LY,g->Homology(TensorWithIntegersOverSubgroup(CY,g),2));;
    
    gap> CW:=ChainComplexOfUniversalCover(W);
    Equivariant chain complex of dimension 3
    gap> LW:=LowIndexSubgroups(CW!.group,5);;
    gap> invW:=List(LW,g->Homology(TensorWithIntegersOverSubgroup(CW,g),2));;
    
    gap> SSortedList(invY)=SSortedList(invW);
    false
    
  
  
  
  3.3 Cohomology with local coefficients
  
  The  π_1Y-equivariant cellular chain complex C_∗widetilde Y of the universal
  cover  widetilde  Y  of  a  regular  CW-complex Y can be used to compute the
  homology  H_n(Y,A) and cohomology H^n(Y,A) of Y with local coefficients in a
  Zπ_1Y-module  A.  To illustrate this we consister the space Y arising as the
  complement  of  the  trefoil  knot,  with  fundamental  group π_1Y = ⟨ x,y :
  xyx=yxy  ⟩.  We  take  A=  Z to be the integers with non-trivial π_1Y-action
  given by x.1=-1, y.1=-1. We then compute
  
  beginarraylcl  H_0(Y,A)  &=  &  Z_2  ,  H_1(Y,A)  &= & Z_3 , H_2(Y,A) &= & Z
  .endarray
  
    Example  
    gap> K:=PureCubicalKnot(3,1);;
    gap> Y:=PureComplexComplement(K);;
    gap> Y:=ContractedComplex(Y);;
    gap> Y:=RegularCWComplex(Y);;
    gap> Y:=SimplifiedComplex(Y);;
    gap> C:=ChainComplexOfUniversalCover(Y);;
    gap> G:=C!.group;;
    gap> GeneratorsOfGroup(G);
    [ f1, f2 ]
    gap> RelatorsOfFpGroup(G);
    [ f2^-1*f1^-1*f2^-1*f1*f2*f1, f1^-1*f2^-1*f1^-1*f2*f1*f2 ]
    gap> hom:=GroupHomomorphismByImages(G,Group([[-1]]),[G.1,G.2],[[[-1]],[[-1]]]);;
    gap> A:=function(x); return Determinant(Image(hom,x)); end;;
    gap> D:=TensorWithTwistedIntegers(C,A); #Here the function A represents 
    gap> #the integers with twisted action of G.
    Chain complex of length 3 in characteristic 0 .
    gap> Homology(D,0);
    [ 2 ]
    gap> Homology(D,1);
    [ 3 ]
    gap> Homology(D,2);
    [ 0 ]
    
  
  
  
  3.4 Distinguishing between two non-homeomorphic homotopy equivalent spaces
  
  The  granny  knot  is  the sum of the trefoil knot and its mirror image. The
  reef  knot  is  the  sum  of  two  identical copies of the trefoil knot. The
  following commands show that the degree 1 homology homomorphisms
  
  H_1(p^-1(B), Z) → H_1(widetilde X_H, Z)
  
  distinguish between the homeomorphism types of the complements X⊂ R^3 of the
  granny knot and the reef knot, where B⊂ X is the knot boundary, and where p:
  widetilde  X_H  →  X  is  the covering map corresponding to the finite index
  subgroup  H  <  π_1X.  More precisely, p^-1(B) is in general a union of path
  components
  
  p^-1(B) = B_1 ∪ B_2 ∪ ⋯ ∪ B_t .
  
  The function FirstHomologyCoveringCokernels(f,c) inputs an integer c and the
  inclusion  f:  B↪  X  of  a  knot boundary B into the knot complement X. The
  function  returns  the  ordered  list  of the lists of abelian invariants of
  cokernels
  
  coker( H_1(p^-1(B_i), Z) → H_1(widetilde X_H, Z) )
  
  arising  from  subgroups  H  <  π_1X  of index c. To distinguish between the
  granny and reef knots we use index c=6.
  
    Example  
    gap> K:=PureCubicalKnot(3,1);;
    gap> L:=ReflectedCubicalKnot(K);;
    gap> granny:=KnotSum(K,L);;
    gap> reef:=KnotSum(K,K);;
    gap> fg:=KnotComplementWithBoundary(ArcPresentation(granny));;
    gap> fr:=KnotComplementWithBoundary(ArcPresentation(reef));;
    gap> a:=FirstHomologyCoveringCokernels(fg,6);;
    gap> b:=FirstHomologyCoveringCokernels(fr,6);;
    gap> a=b;
    false
    
  
  
  
  3.5 Second homotopy groups of spaces with finite fundamental group
  
  If  p:widetilde  Y  →  Y is the universal covering map, then the fundamental
  group  of  widetilde Y is trivial and the Hurewicz homomorphism π_2widetilde
  Y→  H_2(widetilde Y, Z) from the second homotopy group of widetilde Y to the
  second  integral homology of widetilde Y is an isomorphism. Furthermore, the
  map p induces an isomorphism π_2widetilde Y → π_2Y. Thus H_2(widetilde Y, Z)
  is isomorphic to the second homotopy group π_2Y.
  
  If the fundamental group of Y happens to be finite, then in principle we can
  calculate  H_2(widetilde  Y, Z) ≅ π_2Y. We illustrate this computation for Y
  equal  to  the real projective plane. The above computation shows that Y has
  second homotopy group π_2Y ≅ Z.
  
    Example  
    gap> K:=[ [1,2,3], [1,3,4], [1,2,6], [1,5,6], [1,4,5], 
    >         [2,3,5], [2,4,5], [2,4,6], [3,4,6], [3,5,6]];;
    
    gap> K:=MaximalSimplicesToSimplicialComplex(K);
    Simplicial complex of dimension 2.
    
    gap> Y:=RegularCWComplex(K);  
    Regular CW-complex of dimension 2
    gap> # Y is a regular CW-complex corresponding to the projective plane.
    
    gap> U:=UniversalCover(Y);
    Equivariant CW-complex of dimension 2
    
    gap> G:=U!.group;; 
    gap> # G is the fundamental group of Y, which by the next command 
    gap> # is finite of order 2.
    gap> Order(G);
    2
    
    gap> U:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G))); 
    Regular CW-complex of dimension 2
    gap> #U is the universal cover of Y
    
    gap> Homology(U,0);
    [ 0 ]
    gap> Homology(U,1);
    [  ]
    gap> Homology(U,2);
    [ 0 ]
    
  
  
  
  3.6 Third homotopy groups of simply connected spaces
  
  
  3.6-1 First example: Whitehead's certain exact sequence
  
  For  any path connected space Y with universal cover widetilde Y there is an
  exact sequence
  
  →  π_4widetilde  Y  →  H_4(widetilde Y, Z) → H_4( K(π_2widetilde Y,2), Z ) →
  π_3widetilde Y → H_3(widetilde Y, Z) → 0
  
  due  to  J.H.C.Whitehead. Here K(π_2(widetilde Y),2) is an Eilenberg-MacLane
  space with second homotopy group equal to π_2widetilde Y.
  
  Continuing  with  the above example where Y is the real projective plane, we
  see  that H_4(widetilde Y, Z) = H_3(widetilde Y, Z) = 0 since widetilde Y is
  a  2-dimensional  CW-space.  The  exact  sequence  implies  π_3widetilde Y ≅
  H_4(K(π_2widetilde  Y,2),  Z  ).  Furthermore,  π_3widetilde  Y = π_3 Y. The
  following commands establish that π_3Y ≅ Z.
  
    Example  
    gap> A:=AbelianPcpGroup([0]);
    Pcp-group with orders [ 0 ]
    
    gap> K:=EilenbergMacLaneSimplicialGroup(A,2,5);;
    gap> C:=ChainComplexOfSimplicialGroup(K);
    Chain complex of length 5 in characteristic 0 .
    
    gap> Homology(C,4);
    [ 0 ]
    
  
  
  
  3.6-2 Second example: the Hopf invariant
  
  The following commands construct a 4-dimensional simplicial complex Y with 9
  vertices and 36 4-dimensional simplices, and establish that
  
  π_1Y=0 , π_2Y= Z , H_3(Y, Z)=0, H_4(Y, Z)= Z.
  
    Example  
    gap> smp:=[ [ 1, 2, 4, 5, 6 ], [ 1, 2, 4, 5, 9 ], [ 1, 2, 5, 6, 8 ], 
    >         [ 1, 2, 6, 4, 7 ], [ 2, 3, 4, 5, 8 ], [ 2, 3, 5, 6, 4 ], 
    >         [ 2, 3, 5, 6, 7 ], [ 2, 3, 6, 4, 9 ], [ 3, 1, 4, 5, 7 ],
    >         [ 3, 1, 5, 6, 9 ], [ 3, 1, 6, 4, 5 ], [ 3, 1, 6, 4, 8 ], 
    >         [ 4, 5, 7, 8, 3 ], [ 4, 5, 7, 8, 9 ], [ 4, 5, 8, 9, 2 ], 
    >         [ 4, 5, 9, 7, 1 ], [ 5, 6, 7, 8, 2 ], [ 5, 6, 8, 9, 1 ],
    >         [ 5, 6, 8, 9, 7 ], [ 5, 6, 9, 7, 3 ], [ 6, 4, 7, 8, 1 ], 
    >         [ 6, 4, 8, 9, 3 ], [ 6, 4, 9, 7, 2 ], [ 6, 4, 9, 7, 8 ], 
    >         [ 7, 8, 1, 2, 3 ], [ 7, 8, 1, 2, 6 ], [ 7, 8, 2, 3, 5 ],
    >         [ 7, 8, 3, 1, 4 ], [ 8, 9, 1, 2, 5 ], [ 8, 9, 2, 3, 1 ], 
    >         [ 8, 9, 2, 3, 4 ], [ 8, 9, 3, 1, 6 ], [ 9, 7, 1, 2, 4 ], 
    >         [ 9, 7, 2, 3, 6 ], [ 9, 7, 3, 1, 2 ], [ 9, 7, 3, 1, 5 ] ];;
    
    gap> K:=MaximalSimplicesToSimplicialComplex(smp);
    Simplicial complex of dimension 4.
    
    gap> Y:=RegularCWComplex(Y);
    Regular CW-complex of dimension 4
    
    gap> Order(FundamentalGroup(Y));
    1
    gap> Homology(Y,2);
    [ 0 ]
    gap> Homology(Y,3);
    [  ]
    gap> Homology(Y,4);
    [ 0 ]
    
  
  
  Previous  commands  have  established  H_4(K(  Z,2),  Z)=  Z. So Whitehead's
  sequence reduces to an exact sequence
  
  Z → Z → π_3Y → 0
  
  in  which  the first map is H_4(Y, Z)= Z → H_4(K(π_2Y,2), Z )= Z. Hence π_3Y
  is cyclic.
  
  HAP  is  currently  unable  to  compute  the  order  of  π_3Y  directly from
  Whitehead's sequence. Instead, we can use the Hopf invariant. For any map ϕ:
  S^3  →  S^2 we consider the space C(ϕ) = S^2 ∪_ϕ e^4 obtained by attaching a
  4-cell  e^4  to S^2 via the attaching map ϕ. The cohomology groups H^2(C(ϕ),
  Z)=  Z,  H^4(C(ϕ),  Z)=  Z  are  generated by elements α, β say, and the cup
  product has the form
  
  - ∪ -: H^2(C(ϕ), Z)× H^2(C(ϕ), Z) → H^4(C(ϕ), Z), (α,α) ↦ h_ϕ β
  
  for some integer h_ϕ. The integer h_ϕ is the Hopf invariant. The function h:
  π_3(S^3)→ Z is a homomorphism and there is an isomorphism
  
  π_3(S^2∪ e^4) ≅ Z/⟨ h_ϕ⟩.
  
  The  following commands begin by simplifying the cell structure on the above
  CW-complex Y≅ K to obtain a homeomorphic CW-complex W with fewer cells. They
  then  create a space S by removing one 4-cell from W. The space S is seen to
  be  homotopy  equivalent  to  a CW-complex e^2∪ e^0 with a single 0-cell and
  single  2-cell.  Hence  S≃  S^2  is  homotopy  equivalent  to  the 2-sphere.
  Consequently Y ≃ C(ϕ ) = S^2∪_ϕ e^4 for some map ϕ: S^3 → S^2.
  
    Example  
    gap> W:=SimplifiedComplex(Y);
    Regular CW-complex of dimension 4
    
    gap> S:=RegularCWComplexWithRemovedCell(W,4,6);
    Regular CW-complex of dimension 4
    
    gap> CriticalCells(S);
    [ [ 2, 6 ], [ 0, 5 ] ]
    
  
  
  The  next commands show that the map ϕ in the construction Y ≃ C(ϕ) has Hopf
  invariant -1. Hence h: π_3(S^3)→ Z is an isomorphism. Therefore π_3Y=0.
  
    Example  
    gap> IntersectionForm(K);
    [ [ -1 ] ]
    
  
  
  [The simplicial complex K in this second example is due to W. Kuehnel and T.
  F. Banchoff and is homeomorphic to the complex projective plane. ]
  
  
  3.7 Computing the second homotopy group of a space with infinite fundamental
  group
  
  The following commands compute the second integral homology
  
  H_2(π_1W, Z) = Z
  
  of the fundamental group π_1W of the complement W of the Hopf-Satoh surface.
  
    Example  
    gap> L:=HopfSatohSurface();
    Pure cubical complex of dimension 4.
    
    gap> W:=ContractedComplex(RegularCWComplex(PureComplexComplement(L)));
    Regular CW-complex of dimension 3
    
    gap> GW:=FundamentalGroup(W);;
    gap> IsAspherical(GW);
    Presentation is aspherical.
    true
    gap> R:=ResolutionAsphericalPresentation(GW);;
    gap> Homology(TensorWithIntegers(R),2);
    [ 0 ]
    
  
  
  From Hopf's exact sequence
  
  π_2W stackrelh⟶ H_2(W, Z) ↠ H_2(π_1W, Z) → 0
  
  and  the  computation  H_2(W,  Z)= Z^4 we see that the image of the Hurewicz
  homomorphism  is  im(h)= Z^3 . The image of h is referred to as the subgroup
  of spherical homology classes and often denoted by Σ^2W.
  
  The following command computes the presentation of π_1W corresponding to the
  2-skeleton  W^2  and establishes that W^2 = S^2∨ S^2 ∨ S^2 ∨ (S^1× S^1) is a
  wedge of three spheres and a torus.
  
    Example  
    gap> F:=FundamentalGroupOfRegularCWComplex(W,"no simplification");
    < fp group on the generators [ f1, f2 ]>
    gap> RelatorsOfFpGroup(F);
    [ < identity ...>, f1^-1*f2^-1*f1*f2, < identity ...>, <identity ...> ]
    
  
  
  The  next  command shows that the 3-dimensional space W has two 3-cells each
  of  which  is  attached  to the base-point of W with trivial boundary (up to
  homotopy in W^2). Hence W = S^3∨ S^3∨ S^2 ∨ S^2 ∨ S^2 ∨ (S^1× S^1).
  
    Example  
    gap> CriticalCells(W);
    [ [ 3, 1 ], [ 3, 3148 ], [ 2, 6746 ], [ 2, 20510 ], [ 2, 33060 ], 
      [ 2, 50919 ], [ 1, 29368 ], [ 1, 50822 ], [ 0, 21131 ] ]
    gap> CriticalBoundaryCells(W,3,1);
    [  ]
    gap> CriticalBoundaryCells(W,3,3148);
    [ -50919, 50919 ]
    
  
  
  Therefore  π_1W is the free abelian group on two generators, and π_2W is the
  free Zπ_1W-module on three free generators.