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Covering Spaces Joint work with Kelvin Killeen
Let Y denote a regular CW-Complex, and let U denote the universal covering space of Y. The space U can be constructed and stored as a G-equivariant CW-complex. For instance, the following commands construct a 4-dimensional CW-complex Y which is homotopy equivalent to a 2-dimensional torus. The CW-complex 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 CW-complex 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 Z3+Z3.
gap> G:=U!.group;; gap> L:=LowIndexSubgroupsFpGroup(G,9);; gap> H:=L[58];; gap> AbelianInvariants(G/H); [ 3, 3 ]
The next command constructs the 9-fold 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 CW-complex 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 9-fold covering W. The map p can be constructed as folows.
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
The covering map p induces homomorphisms Hn(p):Hn(W,Z)--->Hn(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 π2(U)-->H2(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 π2(U)-->π2(Y). Thus H2(U) is isomorphic to the second homotopy group π2(Y). If the fundamental group of Y happens to be finite, then we can calculate H2(U) = π2(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 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 ]
The above computation shows that the space Y has infinite cyclic second homotopy group π2(Y) = Z .
Third homotopy groups of simply connected spaces
For any simply connected space U there is an exact sequence ---> π4(U) ---> H4(U) ---> H4( K(π2(U), 2) ) ---> π3(U) ---> H4(U) ---> 0 due to J.H.C.Whitehead. Here K(π2(U), 2) is an Eilenberg-MacLane space with second homotopy group equal to π2(U). FIRST EXAMPLE: Continuing with the above example with Y the real projective plane and U its universal cover, we see that H4(U) = H4(U) = 0 since U is a 2-dimensional CW-space. The exact sequence implies π3(U) = H4(K(π2(U), 2) ). Furthermore, π3(U) = π3(Y) since U is the universal cover. The following commands establish that π3(Y) = Z .
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 ]
SECOND EXAMPLE: The following commands construct a 4-dimensional simplicial complex Y with 9 vertices and 36 4-dimensional simplices, and establish that π1(Y)=0 , π2(Y)=Z , H3(Y)=0, H4(Y)=Z, H4(K(π2(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 CW-complex 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 ----> π3(Y) ---> 0 . The first map H4(Y)=Z ---> H4(K(π2(Y), 2)=Z In order to determine π3(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 π3(Y)=0.]
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