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<h2>About HAP: Covering Spaces</h2>

<h3>Joint work with Kelvin Killeen</h3>


Let <math>Y</math> denote a finite regular CW-complex.
Let <math>U</math> denote its universal covering space.
The covering space inherits a regular CW-structure which
can be computed and stored using the data type of a
<math>π1Y</math>-equivariant CW-complex. The
cellular chain complex <math>CU</math> of <math>U</math> can
be computed and stored as an equivariant chain complex.
Given an admissible discrete vector field on
<math>Y</math>, we can endow <math>Y</math> 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
<math>CU</math> involving one free generator for each critical cell in the vector field.


<h3>Cellular chains on the universal cover</h3>

The following commands construct a <math>6</math>-dimensional
regular CW-complex
<math>Y⋍S1×
S1×S1</math>
homotopy equivalent to a product of three circles.


<div><code>
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);; 
gap> Size(Y); 
110592
</code></div>


The CW-somplex <math>Y</math> has <math>110592</math> cells.
The next commands construct a free
<math>π1Y</math>-equivariant chain complex
<math>CU</math> homotopy equivalent to the chain complex of the
universal cover of <math>Y</math>. The chain complex <math>CU</math>
has just <math>8</math> free generators.


<div><code>
gap> Y:=ContractedComplex(Y); 
gap> CU:=ChainComplexOfUniversalCover(Y,"vector field");; 
gap> List([0..Dimension(Y)],n->CU!.dimension(n)); 
[ 1, 3, 3, 1 ]
</code></div>

The next commands construct a subgroup <math>H < π1Y</math>
of index <math>50</math> and the chain complex
<math>CU⊗ℤHℤ</math> which is
homotopy equivalent to the cellular chain complex
<math>CUH</math> of the <math>50</math>-fold cover
<math>UH</math> of
<math>Y</math> corresponding to <math>H</math>.


<div><code>
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 . 
</code></div>

 General theory implies that the <math>50</math>-fold covering space
<math>UH</math> should again be homotopy equivalent to a
product of three circles. As a check for this, the following commands
establish that <math>UH</math> has the same integral homology
as <math>S1× S1×S1</math>

<div><code>
gap> Homology(D,0); 
[ 0 ] 
gap> Homology(D,1); 
[ 0, 0, 0 ] 
gap> Homology(D,2); 
[ 0, 0, 0 ] 
gap> Homology(D,3); 
[ 0 ]
</code></div>

<h3>Cohomology with local coefficients</h3>


The <math>π1Y</math>-equivariant cellular chain complex
<math>CU</math> of the universal cover <math>U</math> of a regular
CW-complex <math>Y</math> can be used to compute the homology
<math>Hn(Y,A)</math> and cohomology <math>Hn(Y,A)</math>
of <math>Y</math> with local coefficients in a
<math>ℤπ1Y</math>-module <math>A</math>.
To illustrate this we consister the space <math>Y</math> arising as the
complement of the trefoil knot, with fundamental group
<math>π1Y = <x,y : xyx=yxy ></math>.
We take <math>A= ℤ</math> to be the integers with non-trivial
<math>π1Y</math>-action given by <math>x.1=-1, y.1=-1</math>.
We then compute

<center>
<math>H0(Y,A) = ℤ2</math> , 
<math>H1(Y,A) = ℤ3</math> , 
<math>H2(Y,A) = ℤ</math> .
</center>

<div><code>
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 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 ] 
</code></div>

<h3>Finite covers as regular CW-complexes</h3>

We next construct a 4-dimensional CW-complex <math>Y⋍S1×S1</maths>
homotopy equivalent to a 2-dimensional torus, involving
2304 cells. We choose a subgroup <math>H < π1Y</math>
of index <math>50</math> and construct the covering space <math>UH</math>
corresponding to <math>H</math> as a finite regular CW-complex. The fundamental group
<math>π1(UH)</math> is shown to be free abelian on two generators.
This is in keeping with the fact that <math>UH</math> is homotopy equivalent to
<math>Y</math>.


<div><code>
gap> G:=U!.group;; 
gap> L:=LowIndexSubgroupsFpGroup(G,50);; 
gap> H:=L[Length(L)-3];;Index(G,H); 
50 
gap> W:=EquivariantCWComplexToRegularCWComplex(U,H); 
Regular CW-complex of dimension 4 
gap> Size(W); 
115200 
 
gap> F:=FundamentalGroup(W); 
<fp group of size infinity on the generators [ f1, f2 ]> 
gap> RelatorsOfFpGroup(F); 
[ f2^-1*f1*f2*f1^-1 ]
</code></div>

<h3>Covering maps</h3>

It may be that we are interested in the covering map
<math>p:UH → Y</math> and not just the covering <math>UH</math> itself. As an illustration we construct the map <math>p</math> for
<math>Y</math> homotopy equivalent to a torus, for <math>H<π1Y</math> a subgroup with

<center>
<math>π1Y / H ≅ ℤ3⊕ℤ3</math>,
</center>
and for <math>p</math> the corresponding covering map.


<div><code>
gap> A:=[[1,1,1],[1,0,1],[1,1,1]];; 
gap> S:=PureCubicalComplex(A);; 
gap> T:=DirectProduct(S,S);; 
gap> Y:=RegularCWComplex(T);; 
gap> U:=UniversalCover(Y); 
Equivariant CW-complex of dimension 4 
gap> G:=U!.group;; 
gap> L:=LowIndexSubgroupsFpGroup(G,9);; 
gap> H:=L[58];; 
gap> AbelianInvariants(G/H); 
[ 3, 3 ] 
gap> p:=EquivariantCWComplexToRegularCWMap(U,H); 
Map of regular CW-complexes 

gap> Source(p); 
Regular CW-complex of dimension 4 
gap> Size(Source(p)); 
20736 

gap> Target(p); 
Regular CW-complex of dimension 4 
gap> Size(Target(p)); 
2304 
</code></div>


The covering map <math>p</math> induces homomorphisms
<math>Hn(p):Hn(W,Z) → Hn(Y,Z)</math>
on integral homology. These homomorphisms, together with their cokernels,
can be computed as follows.


<div><code>
gap> P:=ChainMap(p); 
Chain Map between complexes of length 4 . 

gap> h0:=Homology(P,0);; 
gap> AbelianInvariants(Target(h0)/Image(h0)); 
[ ] 

gap> h1:=Homology(P,1);; 
gap> AbelianInvariants(Target(h1)/Image(h1)); 
[ 3, 3 ] 

gap> h2:=Homology(P,2);; 
gap> AbelianInvariants(Target(h2)/Image(h2)); 
[ 9 ] 
</code></div>

<h3>Second homotopy groups of spaces</h3>

If <math>p:U → Y</math> is the map from the universal cover
<math>U</math> of <math>Y</math>, then the fundamental group of
<math>U</math> is trivial and the Hurewicz homomorphism
<math>π2U-->H2(U,ℤ)</math> from the second
homotopy group of <math>U</math> to the second integral homology of
<math>U</math> is an isomorphism. Furthermore, the map <math>p</math>
induces an isomorphism <math>π2U →
π2Y</math>. Thus <math>H2(U,ℤ)</math>
is isomorphic to
the second homotopy group <math>π2Y</math>.


If the fundamental group of <math>Y</math> happens to be finite, then
in principle we
can calculate <math>H2(U.ℤ) ≅ π2Y</math>.
We illustrate this computation for <math>Y</math> equal to the
real projective plane.


<div><code>
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); # Y is a regular CW-complex corresponding to the projective plane. 
Regular CW-complex of dimension 2

gap> U:=UniversalCover(Y); 
Equivariant CW-complex of dimension 2 

gap> G:=U!.group;; #G is the fundamental group of Y, which by the next command is finite of order 2.
gap> Order(G); 
2 

gap> U:=EquivariantCWComplexToRegularCWComplex(U,Group(One(G))); #U is the universal cover of Y
Regular CW-complex of dimension 2

gap> Homology(U,0); 
[ 0 ] 
gap> Homology(U,1); 
[ ] 
gap> Homology(U,2); 
[ 0 ] 
</code></div>


The above computation shows that the space <math>Y</math> has
infinite cyclic second homotopy group <math>π2Y = ℤ</math> .


<h3>Third homotopy groups of simply connected spaces</h3>


For any simply connected space <math>U</math> there is an exact sequence

<center>
→ π4U → H4(U,ℤ) →
H4( K(π2U,2), ℤ ) → π3U
→ H4(U,ℤ) → 0
</center>

due to J.H.C.Whitehead. Here
<math>K(π2U,2)</math> is an Eilenberg-MacLane space with
second homotopy group equal to <math>π2U</math>.

<center>First Example</center>

Continuing with the above example where <math>Y</math> is the real
projective plane and <math>U</math> its universal cover, we see that
<math>H4(U,ℤ) = H4(U,ℤ) = 0</math>
since <math>U</math> is a 2-dimensional CW-space. The exact sequence implies
<math>π3U ≅ H4(K(π2U,2), ℤ )</math>. Furthermore, <math>π3U = π3Y</math> since
<math>U</math> is the universal cover. The following commands establish that

<center>
<math>
π3Y ≅ ℤ .
</math></center>
<div><code>
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 ] 
</code></div>

<center>Second Example</center>

The following commands construct a 4-dimensional simplicial complex
<math>Y</math> with 9 vertices and 36 4-dimensional simplices,
and establish that

<center>
<math>π1Y=0 , π2Y=ℤ , H3(Y,ℤ)=0, H4(Y,ℤ)=ℤ, H4(K(π2U,2), ℤ) =ℤ</math> .
</center>

<div><code>
gap> Y:=[ [ 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> Y:=MaximalSimplicesToSimplicialComplex(Y); 
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 ] 
</code></div>


Whitehead's sequence reduces to an exact sequence

<center>
<math>ℤ → ℤ → π3Y → 0</math>
</center>

in which the first map is
<math>
H4(Y,ℤ)=ℤ → H4(K(π2Y,2), ℤ)=ℤ
</math>.
In order to determine <math>π3Y</math> it remains compute this first map. This computation is currently not available in HAP.


[The simplicial complex in this second example is due to W. Kiihnel and
T. F. Banchoff and is of the homotopy type of the complex projective plane.
So, assuming this extra knowledge, we have <math>π3Y=0</math>.]

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