Lecture 2. 2 Smooth maps and the notion of equivalence Let X and Y be smooth manifolds. A continuous map f : X → Y is called smooth if for all charts (U, φ) for and X and (V , ψ) for Y we have that the composition ψ ◦ f ◦ φ −1 : φ(U ∩ f −1 (V )) → ψ(V ) is smooth. Two manifolds X and Y are called diffeomorphic if there is a homeomorphism h : X → Y so that h and h −1 are smooth. 3 Standard pathologies. The condition that X be Hausdorff and second countable does not follow from the existence of an atlas. The line with two origins. Let X be the quotient space of R × {0, 1} by the equivalence relation (t, 1) ≡ (t, 0) unless t = 0. Then X is not Hausdorff, how ever X admits an atlas with two charts. Let Ui be the image of R ×{i } in X . These maps invert to give coordinates. Remark 1. Actually nonHausdorff spaces which satisfy all the other properties arise in real life for example in the theory of foliations or when taking quotients by non compact group actions. More work is required to come up with a useful notions to replace that of manifolds in this context. The long line. Let S� denote the smallest uncountable totally ordered set. Consider the product X = S� × (0, 1] with dictionary order topology. Then give X charts as follows. For (ω, t ) ∈ X if t � = 1 let U(ω,t) = {ω} × (0, 1) and φ(ω,t ) : U → R be given by φ(ω,t ) (ω, t ) = t . If t = 1 let S(ω) denote the successor of ω. Set U(ω,1) = {ω} × (0, 1] sup{S(ω)} × (0, 1) and � t if η = ω φ(ω,t ) (η, t ) = t + 1 if η = S(ω). Exercise 5. Check that overlaps are smooth. 5 The collection {U(ω,1/2) }ω∈Sω is uncountable and consists of disjoint open sets, so X is not second countable. Different charts Consider R1 denote R with the following charts (R, x ) and R2 with the chart (R, x 3 ). Identity map R1 → R2 is smooth but not R2 → R1 . R1 and R2 are diffeomorphic by the map x � → x 3 thought of as a map from R1 → R2 . These pathologies are simple problems to keep in mind when thinking about the deﬁnitions. There are far more subtle issues that arise. Given a topological manifold we can ask can carry an atlas, and if it carries an atlas how many non diffeomorphic atlases does it carry. The ﬁrst observation of this phenomenon is due to John Milnor who showed that the sevensphere admits an atlas (with two charts!) which is not diffeomorphic to the standard differentiable structure. We’ll examine this example later in the course. 6

18.966 Geometry of Manifolds: Lecture 2

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