Covering Space - More On The Group Structure

More On The Group Structure

Let p : C → X be a covering map where both X and C are path-connected. Let x ∈ X be a basepoint of X and let c ∈ C be one of its pre-images in C, that is p(c) = x. There is an induced homomorphism of fundamental groups p_# : π₁(C, c) → π₁(X,x) which is injective by the lifting property of coverings. Specifically if γ is a closed loop at c such that p_# = 1, that is p o γ is null-homotopic in X, then consider a null-homotopy of p o γ as a map f : D2 → X from the 2-disc D2 to X such that the restriction of f to the boundary S1 of D2 is equal to p o γ. By the lifting property the map f lifts to a continuous map g : D2 → C such that the restriction of f to the boundary S1 of D2 is equal to γ. Therefore γ is null-homotopic in C, so that the kernel of p_# : π₁(C,c) → π₁(X,x) is trivial and thus p_# : π₁(C, c) → π₁(X, x) is an injective homomorphism.

Therefore π₁(C,c) is isomorphic to the subgroup p_# (π₁(C, c)) of π₁(X, x). If c₁ ∈ C is another pre-image of x in C then the subgroups p_# (π₁(C, c)) and p_# (π₁(C,c₁)) are conjugate in π₁(X,x) by p-image of a curve in C connecting c to c₁. Thus a covering map p : C → X defines a conjugacy class of subgroups of π₁(X,x) and one can show that equivalent covers of X define the same conjugacy class of subgroups of π₁(X,x).

For a covering p : C → X the group p_#(π₁(C,c)) can also be seen to be equal to

,

the set of homotopy classes of those closed curves γ based at x whose lifts γ_C in C, starting at c, are closed curves at c. If X and C are path-connected, the degree of the cover p (that is, the cardinality of any fiber of p) is equal to the index of the subgroup p_# (π₁(C,c)) in π₁(X, x).

A key result of the covering space theory says that for a "sufficiently good" space X (namely, if X is path-connected, locally path-connected and semi-locally simply connected) there is in fact a bijection between equivalence classes of path-connected covers of X and the conjugacy classes of subgroups of the fundamental group π₁(X, x). The main step in proving this result is establishing the existence of a universal cover, that is a cover corresponding to the trivial subgroup of π₁(X, x). Once the existence of a universal cover C of X is established, if H ≤ π₁(X, x) is an arbitrary subgroup, the space C/H is the covering of X corresponding to H. One also needs to check that two covers of C corresponding to the same (conjugacy class of) subgroup of π₁(X, x) are equivalent. Connected cell complexes and connected manifolds are examples of "sufficiently good" spaces.

Let N(Γ_p) be the normalizer of Γ_p in π₁(X,x). The deck transformation group Aut(p) is isomorphic to the quotient group N(Γ_p)/Γ_p. If p is a universal covering, then Γ_p is the trivial group, and Aut(p) is isomorphic to π₁(X).

Let us reverse this argument. Let N be a normal subgroup of π₁(X, x). By the above arguments, this defines a (regular) covering p : C → X . Let c₁ in C be in the fiber of x. Then for every other c₂ in the fiber of x, there is precisely one deck transformation that takes c₁ to c₂. This deck transformation corresponds to a curve g in C connecting c₁ to c₂.