By J. P. May
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Extra info for A Concise Course in Algebraic Topology
Therefore E and E ′ are isomorphic if and only if p(π(E , e)) and p′ (π(E ′ , e′ )) are conjugate whenever p(e) = p′ (e′ ). Corollary. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover. That the universal cover does exist will be proved in the next section. It is useful to recast the previous theorem in terms of actions on fibers. Theorem. Let p : E −→ B and p′ : E ′ −→ B be coverings, choose a base object b ∈ B, and let G = π(B, b). If g : E −→ E ′ is a map of coverings, then g restricts to a map Fb −→ Fb′ of G-sets, and restriction to fibers specifies a bijection between Cov(E , E ′ ) and the set of G-maps Fb −→ Fb′ .
Let G act transitively on a set S, choose s ∈ S, and let H = Gs . Then W H is isomorphic to the group AutG (S) of automorphisms of the G-set S. Proof. For n ∈ N H with image n ¯ ∈ W H, define an automorphism φ(¯ n) of S by φ(¯ n)(gs) = gns. For an automorphism φ of S, we have φ(s) = ns for some n ∈ G. For h ∈ H, hns = φ(hs) = φ(s) = ns, hence n−1 hn ∈ Gs = H and n ∈ N H. Clearly φ = φ(¯ n), and it is easy to check that this bijection between W H and AutG (S) is an isomorphism of groups. We shall also need to consider G-maps between different G-sets G/H.
This leads to the following adjointness homeomorphism, which holds without restriction when we work in the category of compactly generated spaces. Proposition. For spaces X, Y , and Z in U , the canonical bijection Z (X×Y ) ∼ = (Z Y )X is a homeomorphism. Observe in particular that a homotopy X × I −→ Y can equally well be viewed as a map X −→ Y I . These adjoint, or “dual,” points of view will play an important role in the next two chapters. PROBLEMS (1) (a) Any subspace of a weak Hausdorff space is weak Hausdorff.
A Concise Course in Algebraic Topology by J. P. May