  
  [1X3 [33X[0;0YCovering spaces[133X[101X
  
  [33X[0;0YLet  [22XY[122X  denote  a  finite  regular  CW-complex.  Let  [22Xwidetilde Y[122X denote its
  universal covering space. The covering space inherits a regular CW-structure
  which  can  be  computed and stored using the datatype of a [22Xπ_1Y[122X-equivariant
  CW-complex.  The cellular chain complex [22XC_∗widetilde Y[122X of [22Xwidetilde Y[122X can be
  computed  and  stored  as  an equivariant chain complex. Given an admissible
  discrete  vector  field  on  [22XY,[122X  we  can  endow [22XY[122X 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 [22XC_∗
  widetilde  Y[122X  involving  one  free  generator  for each critical cell in the
  vector field.[133X
  
  
  [1X3.1 [33X[0;0YCellular chains on the universal cover[133X[101X
  
  [33X[0;0YThe following commands construct a [22X6[122X-dimensional regular CW-complex [22XY≃ S^1 ×
  S^1× S^1[122X homotopy equivalent to a product of three circles.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=[[1,1,1],[1,0,1],[1,1,1]];;[127X[104X
    [4X[25Xgap>[125X [27XS:=PureCubicalComplex(A);;[127X[104X
    [4X[25Xgap>[125X [27XT:=DirectProduct(S,S,S);;[127X[104X
    [4X[25Xgap>[125X [27XY:=RegularCWComplex(T);;[127X[104X
    [4X[28XRegular CW-complex of dimension 6[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(Y);[127X[104X
    [4X[28X110592[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  CW-somplex  [22XY[122X  has  [22X110592[122X  cells.  The  next commands construct a free
  [22Xπ_1Y[122X-equivariant  chain  complex  [22XC_∗widetilde  Y[122X homotopy equivalent to the
  chain  complex of the universal cover of [22XY[122X. The chain complex [22XC_∗widetilde Y[122X
  has just [22X8[122X free generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XY:=ContractedComplex(Y);;[127X[104X
    [4X[25Xgap>[125X [27XCU:=ChainComplexOfUniversalCover(Y);;[127X[104X
    [4X[25Xgap>[125X [27XList([0..Dimension(Y)],n->CU!.dimension(n));[127X[104X
    [4X[28X[ 1, 3, 3, 1 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  commands  construct a subgroup [22XH < π_1Y[122X of index [22X50[122X and the chain
  complex  [22XC_∗widetilde  Y⊗_ ZH Z[122X which is homotopy equivalent to the cellular
  chain  complex  [22XC_∗widetilde  Y_H[122X  of  the  [22X50[122X-fold cover [22Xwidetilde Y_H[122X of [22XY[122X
  corresponding to [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=LowIndexSubgroupsFpGroup(CU!.group,50);;[127X[104X
    [4X[25Xgap>[125X [27XH:=L[Length(L)-1];;[127X[104X
    [4X[25Xgap>[125X [27XIndex(CU!.group,H);[127X[104X
    [4X[28X50[128X[104X
    [4X[25Xgap>[125X [27XD:=TensorWithIntegersOverSubgroup(CU,H);[127X[104X
    [4X[28XChain complex of length 3 in characteristic 0 .[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList([0..3],D!.dimension);[127X[104X
    [4X[28X[ 50, 150, 150, 50 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YGeneral  theory implies that the [22X50[122X-fold covering space [22Xwidetilde Y_H[122X should
  again  be homotopy equivalent to a product of three circles. In keeping with
  this, the following commands verify that [22Xwidetilde Y_H[122X has the same integral
  homology as [22XS^1× S^1× S^1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHomology(D,0);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(D,1);[127X[104X
    [4X[28X[ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(D,2);[127X[104X
    [4X[28X[ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(D,3);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YSpun knots and the Satoh tube map[133X[101X
  
  [33X[0;0YWe'll  contruct  two  spaces  [22XY,W[122X  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.[133X
  
  [33X[0;0YBy  [13Xspinning[113X  a  link  [22XK ⊂ R^3[122X about a plane [22XP⊂ R^3[122X with [22XP∩ K=∅[122X, we obtain a
  collection  [22XSp(K)⊂  R^4[122X  of knotted tori. The following commands produce the
  two  tori  obtained  by  spinning the Hopf link [22XK[122X and show that the space [22XY=
  R^4∖ Sp(K) = Sp( R^3∖ K)[122X is connected with fundamental group [22Xπ_1Y = Z× Z[122X and
  homology  groups  [22XH_0(Y)=  Z[122X,  [22XH_1(Y)= Z^2[122X, [22XH_2(Y)= Z^4[122X, [22XH_3(Y, Z)= Z^2[122X. The
  space  [22XY[122X  is  only  constructed  up  to  homotopy,  and  for  this reason is
  [22X3[122X-dimensional.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHopf:=PureCubicalLink("Hopf");[127X[104X
    [4X[28XPure cubical link.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XY:=SpunAboutInitialHyperplane(PureComplexComplement(Hopf));[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,0);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,1);[127X[104X
    [4X[28X[ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,2);[127X[104X
    [4X[28X[ 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,3);[127X[104X
    [4X[28X[ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,4);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XGY:=FundamentalGroup(Y);;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(GY);[127X[104X
    [4X[28X[ f2, f3 ][128X[104X
    [4X[25Xgap>[125X [27XRelatorsOfFpGroup(GY);[127X[104X
    [4X[28X[ f3^-1*f2^-1*f3*f2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn  alternative embedding of two tori [22XL⊂ R^4[122X can be obtained by applying the
  'tube  map'  of  Shin  Satoh  to  a  welded Hopf link [Sat00]. The following
  commands  construct  the  complement [22XW= R^4∖ L[122X of this alternative embedding
  and  show  that  [22XW[122X has the same fundamental group and integral homology as [22XY[122X
  above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=HopfSatohSurface();[127X[104X
    [4X[28XPure cubical complex of dimension 4.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XW:=ContractedComplex(RegularCWComplex(PureComplexComplement(L)));[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(W,0);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,1);[127X[104X
    [4X[28X[ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,2);[127X[104X
    [4X[28X[ 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,3);[127X[104X
    [4X[28X[ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(W,4);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XGW:=FundamentalGroup(W);;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(GW);[127X[104X
    [4X[28X[ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XRelatorsOfFpGroup(GW);[127X[104X
    [4X[28X[ f1^-1*f2^-1*f1*f2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YDespite  having the same fundamental group and integral homology groups, the
  above  two  spaces  [22XY[122X and [22XW[122X 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 [22X5[122X-fold covers of the spaces.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCY:=ChainComplexOfUniversalCover(Y);[127X[104X
    [4X[28XEquivariant chain complex of dimension 3[128X[104X
    [4X[25Xgap>[125X [27XLY:=LowIndexSubgroups(CY!.group,5);;[127X[104X
    [4X[25Xgap>[125X [27XinvY:=List(LY,g->Homology(TensorWithIntegersOverSubgroup(CY,g),2));;[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCW:=ChainComplexOfUniversalCover(W);[127X[104X
    [4X[28XEquivariant chain complex of dimension 3[128X[104X
    [4X[25Xgap>[125X [27XLW:=LowIndexSubgroups(CW!.group,5);;[127X[104X
    [4X[25Xgap>[125X [27XinvW:=List(LW,g->Homology(TensorWithIntegersOverSubgroup(CW,g),2));;[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSSortedList(invY)=SSortedList(invW);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YCohomology with local coefficients[133X[101X
  
  [33X[0;0YThe  [22Xπ_1Y[122X-equivariant cellular chain complex [22XC_∗widetilde Y[122X of the universal
  cover  [22Xwidetilde  Y[122X  of  a  regular  CW-complex [22XY[122X can be used to compute the
  homology  [22XH_n(Y,A)[122X and cohomology [22XH^n(Y,A)[122X of [22XY[122X with local coefficients in a
  [22XZπ_1Y[122X-module  [22XA[122X.  To illustrate this we consister the space [22XY[122X arising as the
  complement  of  the  trefoil  knot,  with  fundamental  group [22Xπ_1Y = ⟨ x,y :
  xyx=yxy  ⟩[122X.  We  take  [22XA=  Z[122X to be the integers with non-trivial [22Xπ_1Y[122X-action
  given by [22Xx.1=-1, y.1=-1[122X. We then compute[133X
  
  [33X[0;0Y[22Xbeginarraylcl  H_0(Y,A)  &=  &  Z_2  ,  H_1(Y,A)  &= & Z_3 , H_2(Y,A) &= & Z
  .endarray[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK:=PureCubicalKnot(3,1);;[127X[104X
    [4X[25Xgap>[125X [27XY:=PureComplexComplement(K);;[127X[104X
    [4X[25Xgap>[125X [27XY:=ContractedComplex(Y);;[127X[104X
    [4X[25Xgap>[125X [27XY:=RegularCWComplex(Y);;[127X[104X
    [4X[25Xgap>[125X [27XY:=SimplifiedComplex(Y);;[127X[104X
    [4X[25Xgap>[125X [27XC:=ChainComplexOfUniversalCover(Y);;[127X[104X
    [4X[25Xgap>[125X [27XG:=C!.group;;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(G);[127X[104X
    [4X[28X[ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XRelatorsOfFpGroup(G);[127X[104X
    [4X[28X[ f2^-1*f1^-1*f2^-1*f1*f2*f1, f1^-1*f2^-1*f1^-1*f2*f1*f2 ][128X[104X
    [4X[25Xgap>[125X [27Xhom:=GroupHomomorphismByImages(G,Group([[-1]]),[G.1,G.2],[[[-1]],[[-1]]]);;[127X[104X
    [4X[25Xgap>[125X [27XA:=function(x); return Determinant(Image(hom,x)); end;;[127X[104X
    [4X[25Xgap>[125X [27XD:=TensorWithTwistedIntegers(C,A); #Here the function A represents [127X[104X
    [4X[25Xgap>[125X [27X#the integers with twisted action of G.[127X[104X
    [4X[28XChain complex of length 3 in characteristic 0 .[128X[104X
    [4X[25Xgap>[125X [27XHomology(D,0);[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(D,1);[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(D,2);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YDistinguishing between two non-homeomorphic homotopy equivalent spaces[133X[101X
  
  [33X[0;0YThe  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 [22X1[122X homology homomorphisms[133X
  
  [33X[0;0Y[22XH_1(p^-1(B), Z) → H_1(widetilde X_H, Z)[122X[133X
  
  [33X[0;0Ydistinguish between the homeomorphism types of the complements [22XX⊂ R^3[122X of the
  granny knot and the reef knot, where [22XB⊂ X[122X is the knot boundary, and where [22Xp:
  widetilde  X_H  →  X[122X  is  the covering map corresponding to the finite index
  subgroup  [22XH  <  π_1X[122X.  More precisely, [22Xp^-1(B)[122X is in general a union of path
  components[133X
  
  [33X[0;0Y[22Xp^-1(B) = B_1 ∪ B_2 ∪ ⋯ ∪ B_t[122X .[133X
  
  [33X[0;0YThe function [10XFirstHomologyCoveringCokernels(f,c)[110X inputs an integer [22Xc[122X and the
  inclusion  [22Xf:  B↪  X[122X  of  a  knot boundary [22XB[122X into the knot complement [22XX[122X. The
  function  returns  the  ordered  list  of the lists of abelian invariants of
  cokernels[133X
  
  [33X[0;0Y[22Xcoker( H_1(p^-1(B_i), Z) → H_1(widetilde X_H, Z) )[122X[133X
  
  [33X[0;0Yarising  from  subgroups  [22XH  <  π_1X[122X  of index [22Xc[122X. To distinguish between the
  granny and reef knots we use index [22Xc=6[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK:=PureCubicalKnot(3,1);;[127X[104X
    [4X[25Xgap>[125X [27XL:=ReflectedCubicalKnot(K);;[127X[104X
    [4X[25Xgap>[125X [27Xgranny:=KnotSum(K,L);;[127X[104X
    [4X[25Xgap>[125X [27Xreef:=KnotSum(K,K);;[127X[104X
    [4X[25Xgap>[125X [27Xfg:=KnotComplementWithBoundary(ArcPresentation(granny));;[127X[104X
    [4X[25Xgap>[125X [27Xfr:=KnotComplementWithBoundary(ArcPresentation(reef));;[127X[104X
    [4X[25Xgap>[125X [27Xa:=FirstHomologyCoveringCokernels(fg,6);;[127X[104X
    [4X[25Xgap>[125X [27Xb:=FirstHomologyCoveringCokernels(fr,6);;[127X[104X
    [4X[25Xgap>[125X [27Xa=b;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YSecond homotopy groups of spaces with finite fundamental group[133X[101X
  
  [33X[0;0YIf  [22Xp:widetilde  Y  →  Y[122X is the universal covering map, then the fundamental
  group  of  [22Xwidetilde Y[122X is trivial and the Hurewicz homomorphism [22Xπ_2widetilde
  Y→  H_2(widetilde Y, Z)[122X from the second homotopy group of [22Xwidetilde Y[122X to the
  second  integral homology of [22Xwidetilde Y[122X is an isomorphism. Furthermore, the
  map [22Xp[122X induces an isomorphism [22Xπ_2widetilde Y → π_2Y[122X. Thus [22XH_2(widetilde Y, Z)[122X
  is isomorphic to the second homotopy group [22Xπ_2Y[122X.[133X
  
  [33X[0;0YIf the fundamental group of [22XY[122X happens to be finite, then in principle we can
  calculate  [22XH_2(widetilde  Y, Z) ≅ π_2Y[122X. We illustrate this computation for [22XY[122X
  equal  to  the real projective plane. The above computation shows that [22XY[122X has
  second homotopy group [22Xπ_2Y ≅ Z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK:=[ [1,2,3], [1,3,4], [1,2,6], [1,5,6], [1,4,5], [127X[104X
    [4X[25X>[125X [27X        [2,3,5], [2,4,5], [2,4,6], [3,4,6], [3,5,6]];;[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK:=MaximalSimplicesToSimplicialComplex(K);[127X[104X
    [4X[28XSimplicial complex of dimension 2.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XY:=RegularCWComplex(K);  [127X[104X
    [4X[28XRegular CW-complex of dimension 2[128X[104X
    [4X[25Xgap>[125X [27X# Y is a regular CW-complex corresponding to the projective plane.[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XU:=UniversalCover(Y);[127X[104X
    [4X[28XEquivariant CW-complex of dimension 2[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=U!.group;; [127X[104X
    [4X[25Xgap>[125X [27X# G is the fundamental group of Y, which by the next command [127X[104X
    [4X[25Xgap>[125X [27X# is finite of order 2.[127X[104X
    [4X[25Xgap>[125X [27XOrder(G);[127X[104X
    [4X[28X2[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XU:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G))); [127X[104X
    [4X[28XRegular CW-complex of dimension 2[128X[104X
    [4X[25Xgap>[125X [27X#U is the universal cover of Y[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(U,0);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(U,1);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XHomology(U,2);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.6 [33X[0;0YThird homotopy groups of simply connected spaces[133X[101X
  
  
  [1X3.6-1 [33X[0;0YFirst example: Whitehead's certain exact sequence[133X[101X
  
  [33X[0;0YFor  any path connected space [22XY[122X with universal cover [22Xwidetilde Y[122X there is an
  exact sequence[133X
  
  [33X[0;0Y[22X→  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[122X[133X
  
  [33X[0;0Ydue  to  J.H.C.Whitehead. Here [22XK(π_2(widetilde Y),2)[122X is an Eilenberg-MacLane
  space with second homotopy group equal to [22Xπ_2widetilde Y[122X.[133X
  
  [33X[0;0YContinuing  with  the above example where [22XY[122X is the real projective plane, we
  see  that [22XH_4(widetilde Y, Z) = H_3(widetilde Y, Z) = 0[122X since [22Xwidetilde Y[122X is
  a  [22X2[122X-dimensional  CW-space.  The  exact  sequence  implies  [22Xπ_3widetilde Y ≅
  H_4(K(π_2widetilde  Y,2),  Z  )[122X.  Furthermore,  [22Xπ_3widetilde  Y = π_3 Y[122X. The
  following commands establish that [22Xπ_3Y ≅ Z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=AbelianPcpGroup([0]);[127X[104X
    [4X[28XPcp-group with orders [ 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK:=EilenbergMacLaneSimplicialGroup(A,2,5);;[127X[104X
    [4X[25Xgap>[125X [27XC:=ChainComplexOfSimplicialGroup(K);[127X[104X
    [4X[28XChain complex of length 5 in characteristic 0 .[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(C,4);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.6-2 [33X[0;0YSecond example: the Hopf invariant[133X[101X
  
  [33X[0;0YThe following commands construct a [22X4[122X-dimensional simplicial complex [22XY[122X with [22X9[122X
  vertices and [22X36[122X [22X4[122X-dimensional simplices, and establish that[133X
  
  [33X[0;0Y[22Xπ_1Y=0 , π_2Y= Z , H_3(Y, Z)=0, H_4(Y, Z)= Z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsmp:=[ [ 1, 2, 4, 5, 6 ], [ 1, 2, 4, 5, 9 ], [ 1, 2, 5, 6, 8 ], [127X[104X
    [4X[25X>[125X [27X        [ 1, 2, 6, 4, 7 ], [ 2, 3, 4, 5, 8 ], [ 2, 3, 5, 6, 4 ], [127X[104X
    [4X[25X>[125X [27X        [ 2, 3, 5, 6, 7 ], [ 2, 3, 6, 4, 9 ], [ 3, 1, 4, 5, 7 ],[127X[104X
    [4X[25X>[125X [27X        [ 3, 1, 5, 6, 9 ], [ 3, 1, 6, 4, 5 ], [ 3, 1, 6, 4, 8 ], [127X[104X
    [4X[25X>[125X [27X        [ 4, 5, 7, 8, 3 ], [ 4, 5, 7, 8, 9 ], [ 4, 5, 8, 9, 2 ], [127X[104X
    [4X[25X>[125X [27X        [ 4, 5, 9, 7, 1 ], [ 5, 6, 7, 8, 2 ], [ 5, 6, 8, 9, 1 ],[127X[104X
    [4X[25X>[125X [27X        [ 5, 6, 8, 9, 7 ], [ 5, 6, 9, 7, 3 ], [ 6, 4, 7, 8, 1 ], [127X[104X
    [4X[25X>[125X [27X        [ 6, 4, 8, 9, 3 ], [ 6, 4, 9, 7, 2 ], [ 6, 4, 9, 7, 8 ], [127X[104X
    [4X[25X>[125X [27X        [ 7, 8, 1, 2, 3 ], [ 7, 8, 1, 2, 6 ], [ 7, 8, 2, 3, 5 ],[127X[104X
    [4X[25X>[125X [27X        [ 7, 8, 3, 1, 4 ], [ 8, 9, 1, 2, 5 ], [ 8, 9, 2, 3, 1 ], [127X[104X
    [4X[25X>[125X [27X        [ 8, 9, 2, 3, 4 ], [ 8, 9, 3, 1, 6 ], [ 9, 7, 1, 2, 4 ], [127X[104X
    [4X[25X>[125X [27X        [ 9, 7, 2, 3, 6 ], [ 9, 7, 3, 1, 2 ], [ 9, 7, 3, 1, 5 ] ];;[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK:=MaximalSimplicesToSimplicialComplex(smp);[127X[104X
    [4X[28XSimplicial complex of dimension 4.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XY:=RegularCWComplex(Y);[127X[104X
    [4X[28XRegular CW-complex of dimension 4[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XOrder(FundamentalGroup(Y));[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,2);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,3);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XHomology(Y,4);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YPrevious  commands  have  established  [22XH_4(K(  Z,2),  Z)=  Z[122X. So Whitehead's
  sequence reduces to an exact sequence[133X
  
  [33X[0;0Y[22XZ → Z → π_3Y → 0[122X[133X
  
  [33X[0;0Yin  which  the first map is [22XH_4(Y, Z)= Z → H_4(K(π_2Y,2), Z )= Z[122X. Hence [22Xπ_3Y[122X
  is cyclic.[133X
  
  [33X[0;0YHAP  is  currently  unable  to  compute  the  order  of  [22Xπ_3Y[122X  directly from
  Whitehead's sequence. Instead, we can use the [13XHopf invariant[113X. For any map [22Xϕ:
  S^3  →  S^2[122X we consider the space [22XC(ϕ) = S^2 ∪_ϕ e^4[122X obtained by attaching a
  [22X4[122X-cell  [22Xe^4[122X  to [22XS^2[122X via the attaching map [22Xϕ[122X. The cohomology groups [22XH^2(C(ϕ),
  Z)=  Z[122X,  [22XH^4(C(ϕ),  Z)=  Z[122X  are  generated by elements [22Xα, β[122X say, and the cup
  product has the form[133X
  
  [33X[0;0Y[22X- ∪ -: H^2(C(ϕ), Z)× H^2(C(ϕ), Z) → H^4(C(ϕ), Z), (α,α) ↦ h_ϕ β[122X[133X
  
  [33X[0;0Yfor some integer [22Xh_ϕ[122X. The integer [22Xh_ϕ[122X is the [12XHopf invariant[112X. The function [22Xh:
  π_3(S^3)→ Z[122X is a homomorphism and there is an isomorphism[133X
  
  [33X[0;0Y[22Xπ_3(S^2∪ e^4) ≅ Z/⟨ h_ϕ⟩[122X.[133X
  
  [33X[0;0YThe  following commands begin by simplifying the cell structure on the above
  CW-complex [22XY≅ K[122X to obtain a homeomorphic CW-complex [22XW[122X with fewer cells. They
  then  create a space [22XS[122X by removing one [22X4[122X-cell from [22XW[122X. The space [22XS[122X is seen to
  be  homotopy  equivalent  to  a CW-complex [22Xe^2∪ e^0[122X with a single [22X0[122X-cell and
  single  [22X2[122X-cell.  Hence  [22XS≃  S^2[122X  is  homotopy  equivalent  to  the [22X2[122X-sphere.
  Consequently [22XY ≃ C(ϕ ) = S^2∪_ϕ e^4[122X for some map [22Xϕ: S^3 → S^2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XW:=SimplifiedComplex(Y);[127X[104X
    [4X[28XRegular CW-complex of dimension 4[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XS:=RegularCWComplexWithRemovedCell(W,4,6);[127X[104X
    [4X[28XRegular CW-complex of dimension 4[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCriticalCells(S);[127X[104X
    [4X[28X[ [ 2, 6 ], [ 0, 5 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next commands show that the map [22Xϕ[122X in the construction [22XY ≃ C(ϕ)[122X has Hopf
  invariant -1. Hence [22Xh: π_3(S^3)→ Z[122X is an isomorphism. Therefore [22Xπ_3Y=0[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIntersectionForm(K);[127X[104X
    [4X[28X[ [ -1 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[The simplicial complex [22XK[122X in this second example is due to W. Kuehnel and T.
  F. Banchoff and is homeomorphic to the complex projective plane. ][133X
  
  
  [1X3.7 [33X[0;0YComputing the second homotopy group of a space with infinite fundamental[101X
  [1Xgroup[133X[101X
  
  [33X[0;0YThe following commands compute the second integral homology[133X
  
  [33X[0;0Y[22XH_2(π_1W, Z) = Z[122X[133X
  
  [33X[0;0Yof the fundamental group [22Xπ_1W[122X of the complement [22XW[122X of the Hopf-Satoh surface.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=HopfSatohSurface();[127X[104X
    [4X[28XPure cubical complex of dimension 4.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XW:=ContractedComplex(RegularCWComplex(PureComplexComplement(L)));[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XGW:=FundamentalGroup(W);;[127X[104X
    [4X[25Xgap>[125X [27XIsAspherical(GW);[127X[104X
    [4X[28XPresentation is aspherical.[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionAsphericalPresentation(GW);;[127X[104X
    [4X[25Xgap>[125X [27XHomology(TensorWithIntegers(R),2);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFrom Hopf's exact sequence[133X
  
  [33X[0;0Y[22Xπ_2W stackrelh⟶ H_2(W, Z) ↠ H_2(π_1W, Z) → 0[122X[133X
  
  [33X[0;0Yand  the  computation  [22XH_2(W,  Z)= Z^4[122X we see that the image of the Hurewicz
  homomorphism  is  [22Xim(h)= Z^3[122X . The image of [22Xh[122X is referred to as the subgroup
  of [13Xspherical homology classes[113X and often denoted by [22XΣ^2W[122X.[133X
  
  [33X[0;0YThe following command computes the presentation of [22Xπ_1W[122X corresponding to the
  [22X2[122X-skeleton  [22XW^2[122X  and establishes that [22XW^2 = S^2∨ S^2 ∨ S^2 ∨ (S^1× S^1)[122X is a
  wedge of three spheres and a torus.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalGroupOfRegularCWComplex(W,"no simplification");[127X[104X
    [4X[28X< fp group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27XRelatorsOfFpGroup(F);[127X[104X
    [4X[28X[ < identity ...>, f1^-1*f2^-1*f1*f2, < identity ...>, <identity ...> ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  command shows that the [22X3[122X-dimensional space [22XW[122X has two [22X3[122X-cells each
  of  which  is  attached  to the base-point of [22XW[122X with trivial boundary (up to
  homotopy in [22XW^2[122X). Hence [22XW = S^3∨ S^3∨ S^2 ∨ S^2 ∨ S^2 ∨ (S^1× S^1)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCriticalCells(W);[127X[104X
    [4X[28X[ [ 3, 1 ], [ 3, 3148 ], [ 2, 6746 ], [ 2, 20510 ], [ 2, 33060 ], [128X[104X
    [4X[28X  [ 2, 50919 ], [ 1, 29368 ], [ 1, 50822 ], [ 0, 21131 ] ][128X[104X
    [4X[25Xgap>[125X [27XCriticalBoundaryCells(W,3,1);[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XCriticalBoundaryCells(W,3,3148);[127X[104X
    [4X[28X[ -50919, 50919 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTherefore  [22Xπ_1W[122X is the free abelian group on two generators, and [22Xπ_2W[122X is the
  free [22XZπ_1W[122X-module on three free generators.[133X
  
