8 1. Introduction is locally Euclidean, thus every point in M has a neighbourhood which is homeomorphic to an open subset of Rd. A smooth atlas on a d-dimensional topological manifold M is a family (φα)α∈A of homeomorphisms φα : Uα → Vα from open subsets Uα of M to open subsets Vα of Rd, such that the Uα form an open cover of M, and for any α, β ∈ A, the map φβ ◦ φα −1 is smooth (i.e., infinitely differentiable) on the domain of definition φα(Uα ∩ Uβ). Two smooth atlases are equivalent if their union is also a smooth atlas this is easily seen to be an equivalence relation. An equivalence class of smooth atlases is a smooth structure. A smooth manifold is a topological manifold equipped with a smooth structure. A map ψ : M → M from one smooth manifold to another is said to be smooth if φ α ◦ ψ ◦ φ−1 β is a smooth function on the domain of definition Vβ ∩ φ−1(Uβ β ∩ ψ−1(Uα)) for any smooth charts φβ,φα in any the smooth atlases of M, M respectively (one easily verifies that this definition is independent of the choice of smooth atlas in the smooth structure). Note that we do not require manifolds to be connected, nor do we re- quire them to be embeddable inside an ambient Euclidean space such as Rn, although certainly many key examples of manifolds are of this form. The requirement that the manifold be Hausdorff is a technical one, in order to exclude pathological examples such as the line with a doubled point (for- mally, consider the double line R × {0, 1} after identifying (x, 0) with (x, 1) for all x ∈ R\{0}), which is locally Euclidean but not Hausdorff1. Remark 1.1.2. It is a plausible, but nontrivial, fact that a (nonempty) topological manifold can have at most one dimension d associated to it thus a manifold M cannot both be locally homeomorphic to Rd and locally homeomorphic to Rd unless d = d . This fact is a consequence of Brouwer’s invariance of domain theorem see Exercise 6.0.4. On the other hand, it is an easy consequence of the rank-nullity theorem that a smooth manifold can have at most one dimension, without the need to invoke invariance of domain we leave this as an exercise. Definition 1.1.3 (Lie group). A Lie group is a group G = (G, ·) which is also a smooth manifold, such that the group operations · : G × G → G and ()−1 : G → G are smooth maps. (Note that the Cartesian product of two smooth manifolds can be given the structure of a smooth manifold in the obvious manner.) We will also use additive notation G = (G, +) to describe some Lie groups, but only in the case when the Lie group is abelian. 1In some literature, additional technical assumptions such as paracompactness, second count- ability, or metrisability are imposed to remove pathological examples of topological manifolds such as the long line, but it will not be necessary to do so in this text, because (as we shall see later) we can essentially get such properties “for free” for locally Euclidean groups.

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