fundamental theorem of covering spaces



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In topological homotopy theory, the fundamental theorem of covering spaces says that for a sufficiently well-behaved topological space XX, then the functor which sends a covering space of XX to the Set-action (permutation representation) of the fundamental groupoid of XX on the fibers of EE is an equivalence of categories.

This is a basic instance of the general principle of Galois theory.

It follows in particular that for connected XX then the automorphism group of the universal covering space of XX coincides with the fundamental group π 1(X,x)\pi_1(X,x) itself (for any basepojtn xx). This often yields a convient means to determine the fundamental group of XX in the first place.



(fundamental theorem of covering spaces)

Let XX be a locally path-connected and semi-locally simply-connected topological space. Then the operations on

  1. extracting the monodromy Fib EFib_{E} of a covering space EE over XX

  2. reconstructing a covering space from monodromy Rec(ρ)Rec(\rho)

constitute an equivalence of categories

Cov(X)FibRecSet Π 1(X) Cov(X) \underoverset {\underset{Fib}{\longrightarrow}} {\overset{Rec}{\longleftarrow}} {\simeq} Set^{\Pi_1(X)}

between the category of covering spaces, and the category of permutation groupoid representations of the fundamental groupoid of XX.


With the standard definitions of the two functors, both are infact inverse isomorphisms of categories instead of just equivalences of categories (meaning that the required natural isomorphisms from the composites of the two functors to the identity functor are componentwise equalities), which establishes the claim right away. For definiteness, we make this explicit:

Given ρSet Π 1(X)\rho \in Set^{\Pi_1(X)} a permutation representation, we need to exhibit a natural isomorphism of permutation representations.

η ρ:ρFib(Rec(ρ)) \eta_{\rho} \;\colon\; \rho \longrightarrow Fib(Rec(\rho))

First consider what the right hand side is like: By this def. of RecRec and this def. of FibFib we have for every xXx \in X an actual equality

Fib(Rec(ρ))(x)=ρ(x). Fib(Rec(\rho))(x) = \rho(x) \,.

To similarly understand the value of Fib(Rec(ρ))Fib(Rec(\rho)) on morphisms [γ]Π 1(X)[\gamma] \in \Pi_1(X), let γ:[0,1]X\gamma \colon [0,1] \to X be a representing path in XX. As in the proof of the path lifting lemma for covering spaces (this lemma) we find a finite number of paths {γ i} i{1,n}\{\gamma_i\}_{i \in \{1,n\}} such that

  1. regarded as morphisms [γ i][\gamma_i] in Π 1(X)\Pi_1(X) they compose to [γ][\gamma]:

    [γ]=[γ n][γ 2][γ 1] [\gamma] = [\gamma_n] \circ \cdots \circ [\gamma_2] \circ [\gamma_1]
  2. each γ i\gamma_i factors through an open subset U iXU_i \subset X over which Rec(ρ)Rec(\rho) trivializes.

Hence by functoriality of Fib(Rec(ρ))Fib(Rec(\rho)) it is sufficient to understand its value on these paths γ i\gamma_i. But on these we have again by direct unwinding of the definitions that

Fib(Rec(ρ))([γ i])=ρ([γ i]). Fib(Rec(\rho))([\gamma_i]) = \rho([\gamma_i]) \,.

This means that if we take

η ρ(x):ρ(x)=Fib(Rec(ρ)) \eta_\rho(x) \colon \rho(x) \overset{=}{\longrightarrow} Fib(Rec(\rho))

to be the above identification, then this is a natural transformation and hence in a particular a natural isomorphism, as required.

It remains to see that these morphisms η ρ\eta_\rho are themselves natural in ρ\rho, hence that for each morphism ϕ:ρρ\phi \colon \rho \to \rho' the diagram

ρ ϕ ρ eta ρ η ρ Fib(Rec(ρ)) Fib(Rec(ϕ)) Fib(Rec(ρ)) \array{ \rho &\overset{\phi}{\longrightarrow}& \rho' \\ {}^{\mathllap{eta_\rho}}\downarrow && \downarrow^{\mathrlap{\eta_{\rho'}}} \\ Fib(Rec(\rho)) &\underset{Fib(Rec(\phi))}{\longrightarrow}& Fib(Rec(\rho')) }

commutes as a diagram in Rep(Π 1(X),Set)Rep(\Pi_1(X), Set). Since these morphisms are themselves groupoid homotopies (natural isomorphisms) this is the case precisely if for all xXx \in X the corresponding component diagram commutes. But by the above this is

ρ(x) ϕ(x) ρ(x) = = Fib(Rec(ρ))(x) Fib(Rec(ϕ))(x) Fib(Rec(ρ))(x) \array{ \rho(x) &\overset{\phi(x)}{\longrightarrow}& \rho'(x) \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{=}} \\ Fib(Rec(\rho))(x) &\underset{Fib(Rec(\phi))(x) }{\longrightarrow}& Fib(Rec(\rho'))(x) }

and hence this means that the top and bottom horizontal morphism are in fact equal. Directz unwiinding of the definitions shows that this is indeed the case.

Conversely, given ECov(X)E \in Cov(X) a covering space, we need to exhibit a natural isomorphism of covering spaces of the form

ϵ E:Rec(Fib(E))E. \epsilon_E \;\colon\; Rec(Fib(E)) \longrightarrow E \,.

Again by this def. of RecRec and this def. of FibFib the underlying set of Rec(Fib(E))Rec(Fib(E)) is actually equal to that of EE, hence it is sufficient to check that this identity function on underlying sets is a homeomorphism of topological spaces.

By the assumption that XX is locally path-connected and semi-locally simply connected, it is sufficient to check for UXU\subset X an open path-connected subset and xXx \in X a point with the property that π 1(U,x)π 1(X,x)\pi_1(U,x) \to \pi_1(X,x) lands is constant on the trivial element, that the open subsets of EE of the form U×{x^}p 1(U)U \times \{\hat x\} \subset p^{-1}(U) form a basis for the topology of Rec(Fib(E))Rec(Fib(E)). But this is the case by definition of RecRec.

It remains to see that ϵ E\epsilon_E is itself natural in EE. But as for the converse direction, since the components of ϵ E\epsilon_E are in fact equalities, this follows by direct unwinding of the definitions.

This establishes an equivalence as required In fact this is an adjoint equivalence.



Lecture notes include

Revised on July 18, 2017 04:04:03 by Urs Schreiber (