


Covering Spaces Joint work with Kelvin Killeen 

Let
Y denote a regular CWComplex, and let U denote the universal covering
space of Y. The space U
can be constructed and stored as a Gequivariant CWcomplex. For instance, the following commands construct a 4dimensional CWcomplex Y which is homotopy equivalent to a 2dimensional torus. The CWcomplex Y involves 2304 cells. 

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> Size(Y); 2304 

The
next commands then construct the universal covering space U of Y. 

gap>
U:=UniversalCover(Y); Equivariant CWcomplex of dimension 4 

The
next commands set G equal to the fundamental group of Y, and then
construct a subgroup H<G of index 9. The group G is free abelian on
two generators, and the quotient G/H is shown to be isomorphic to Z_{3}+Z_{3}. 

gap>
G:=U!.group;; gap> L:=LowIndexSubgroupsFpGroup(G,9);; gap> H:=L[58];; gap> AbelianInvariants(G/H); [ 3, 3 ] 

The
next command constructs the 9fold covering space W of Y for which the
covering map p:W>Y sends the fundamental group of W injectively
onto the subgroup H<G. The space W has 20736 cells. 

gap>
W:=EquivariantCWComplexToRegularCWComplex(U,H); Regular CWcomplex of dimension 4 gap> Size(W); 20736 

General
theory
implies
that
the
covering
space
W
should also be homotopy
equivalent to a torus. As a check for this, the following commands
establish that W has the same integral homology as a torus, as well as
the same fundamental group. 

gap>
Homology(W,0); [ 0 ] gap> Homology(W,1); [ 0, 0 ] gap> Homology(W,2); [ 0 ] gap> Homology(W,3); [ ] gap> Homology(W,4); [ ] gap> F:=FundamentalGroup(W); #I there are 2 generators and 1 relator of total length 4 <fp group of size infinity on the generators [ f1, f2 ]> gap> RelatorsOfFpGroup(F); [ f2*f1*f2^1*f1^1 ] 

It
may be that we are interested in the covering map p:W>Y and not
just the 9fold covering W. The map p can be constructed as folows. 

gap>
p:=EquivariantCWComplexToRegularCWMap(U,H); Map of regular CWcomplexes gap> Source(p); Regular CWcomplex of dimension 4 gap> Size(Source(p)); 20736 gap> Target(p); Regular CWcomplex of dimension 4 gap> Size(Target(p)); 2304 

The
covering map p induces homomorphisms H_{n}(p):H_{n}(W,Z)>H_{n}(Y,Z)
on
integral
homology.
These
homomorphisms, together with their
cokernels, can be computed as follows. 

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 ] 

Second homotopy groups of spaces with
finite fundamental groups 

If
p:U>Y is the map from the universal cover U of Y, then the
fundamental group of U is trivial and the Hurewicz homomorphism π_{2}(U)>H_{2}(U)
from
the
second
homotopy
group
of U to the second integral homology of
U is an isomorphism. Furthermore, the map p induces an
isomorphism
π_{2}(U)>π_{2}(Y). Thus H_{2}(U) is
isomorphic to the second homotopy group π_{2}(Y). If the fundamental group of Y happens to be finite, then we can calculate H_{2}(U) = π_{2}(Y) as follows. We illustrate the computation for Y equal to the real projective plane. 

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 CWcomplex corresponding to the projective plane. Regular CWcomplex of dimension 2 gap> U:=UniversalCover(Y); Equivariant CWcomplex 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 CWcomplex of dimension 2 gap> Homology(U,0); [ 0 ] gap> Homology(U,1); [ ] gap> Homology(U,2); [ 0 ] 

The
above computation shows that the space Y has infinite cyclic second
homotopy group π_{2}(Y) = Z .


Third homotopy groups of simply connected
spaces 

For
any simply connected space U there is an exact sequence > π_{4}(U) >
H_{4}(U) > H_{4}( K(π_{2}(U), 2) )
> π_{3}(U) > H_{4}(U) > 0
due to J.H.C.Whitehead. Here K(π_{2}(U), 2) is an EilenbergMacLane space with second homotopy group equal to π_{2}(U). FIRST EXAMPLE: Continuing with the above example with Y the real projective plane and U its universal cover, we see that H_{4}(U) = H_{4}(U) = 0 since U is a 2dimensional CWspace. The exact sequence implies π_{3}(U) = H_{4}(K(π_{2}(U), 2) ). Furthermore, π_{3}(U) = π_{3}(Y) since U is the universal cover. The following commands establish that π_{3}(Y) = Z .


gap>
A:=AbelianPcpGroup([0]); Pcpgroup 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 ] 

SECOND EXAMPLE: The following commands construct a 4dimensional simplicial complex Y with 9 vertices and 36 4dimensional simplices, and establish that π_{1}(Y)=0 , π_{2}(Y)=Z , H_{3}(Y)=0, H_{4}(Y)=Z, H_{4}(K(π_{2}(U), 2) =Z . 

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 ] ];; Y:=MaximalSimplicesToSimplicialComplex(Y); Simplicial complex of dimension 4. Y:=RegularCWComplex(K); Regular CWcomplex of dimension 4 gap> Order(FundamentalGroup(Y)); 1 gap> Homology(Y,2); [ 0 ] gap> Homology(Y,3); [ ] gap> Homology(Y,4); [ 0 ] 

Whitehead's
sequence yields the exact sequence Z > Z > π_{3}(Y)
> 0 .
The first map
H_{4}(Y)=Z > H_{4}(K(π_{2}(Y),
2)=Z
In order to determine π_{3}(Y)
it remains compute this first map.
[The simplicial complex 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 π_{3}(Y)=0.] 

