MATH 552 C. Homeomorphisms, Manifolds and Diffeomorphisms 2021-09 Bijections Let X and Y be sets, and let U ⊆ X be a subset. A map (or function) f with domain U takes each element x 2 U to a unique element f(x) 2 Y . We write y = f(x); or x 7! f(x): Note that in the textbook and lecture notes, one writes f : X ! Y; even if the domain is a proper subset of X, and often the domain is not specified explicitly. The range (or image) of f is the subset f(U) = f y 2 Y : there exists x 2 U such that y = f(x) g: A map f : X ! Y is one-to-one (or is invertible, or is an injection) if for all x1, x2 in the domain, f(x1) = f(x2) implies x1 = x2 (equivalently, x1 6= x2 implies f(x1) 6= f(x2)). If f is one-to-one, then there is an inverse map (or inverse function) f −1 : Y ! X whose domain is the range f(U) of f. A map f : X ! Y is onto (or is a surjection) if f(U) = Y . We will say a map f : X ! Y is a bijection (or one-to-one correspondence) from X onto Y if its domain is all of X, and it is one-to-one and onto. If f is a bijection, then its inverse map f −1 : Y ! X is defined on all of Y . Homeomorphisms Let X and Y be topological spaces. Examples of topological spaces are: i) Rn; ii) a metric space; iii) a smooth manifold (see below); iv) an open subset of Rn or of a smooth manifold. A map h : X ! Y is a global homeomorphism from X onto Y if its domain is all of X, its range is all of Y , it is a bijection from X onto Y , and both h and its inverse map h−1 are continuous maps. Two topological spaces are equivalent (as topological spaces) if there is a global homeomor- phism from one topological space onto the other. If the domain of h : X ! Y is an open subset U ⊆ X, the range V = h(U) is an open subset V ⊆ Y , h is a bijection from U onto V , and both h and h−1 are continuous on their respective domains, then we call h a local homeomorphism from X into Y . For brevity we call either a global homeomorphism or a local homeomorphism simply a \homeomorphism" and write h : X ! Y . Manifolds and their tangent spaces In this course, we only consider manifolds that are smooth. The definition of a manifold given here is sufficient for us, but there are other, more general, definitions. Let F : Rn ! Rm be differentiable, with m < n. Then the subset of Rn given by n X = f x 2 R : F (x) = 0 g (the solution set of m equations in n unknowns) is a (smooth or differentiable) manifold in Rn if the m × n derivative (matrix) Fx(x) has rank m, the maximum possible, at each point x 2 X. In this case the dimension of X is d = n − m. If the map F is Cp, then we say X is a Cp manifold. Examples of manifolds are: i) Rn; ii) an open subset U ⊆ Rn; iii) the graph in Rn+m f(x; y): y = V (x)g of a smooth function V : Rn ! Rm that is defined on an open subset U ⊆ Rn (as an exercise, show that this satisfies the above definition for a manifold in Rn+m of dimension n ); iv) the 1 1 2 2 2 (1-dimensional) circle S = R =(2πZ) = f(x1; x2) 2 R : x1 + x2 − 1 = 0g; v) the (2-dimensional) cylinder S1 × R1; vi) the 2-dimensional torus or 2-torus T2 = S1 × S1; vii) an open subset of a manifold. A one-dimensional manifold in R2 or R3 is usually called a curve, a two-dimensional manifold in R3 is usually called a surface. If X is a manifold in Rn and x 2 X, then the tangent space to X at x is the vector space n TxX, in R , of all tangent vectors v =γ _ (0) for (smooth) curves γ : R ! X that lie in X with γ(0) = x (v is the tangent vector pointing in the directionγ _ (0) at the point x 2 X; we can think of it as a vector whose \tail" is at the point x and whose \tip" is at the point x + v). Alternatively, TxX is the vector space which is the orthogonal complement of frF1(x); ··· ; rFm(x) g, where Fj is the jth component of F , i.e. ? TxX = frF1(x); ··· ; rFm(x) g : The dimension of the tangent space TxX is equal to the dimension of the manifold X (for any x 2 X). Diffeomorphisms Let X and Y be manifolds. A map h : X ! Y is a global (Cp) diffeomorphism from X onto Y (p ≥ 1) if it is a global homeomorphism from X onto Y , and both h and its inverse map h−1 are (Cp) differentiable (thus, the manifolds X and Y must have the same dimension). If the domain U of h : X ! Y is an open subset of a manifold X, the range V = h(U) is an open subset of a manifold Y , h is a local homeomorphism, and both h and h−1 are (Cp) differentiable on their respective domains, then we call h a local (Cp) diffeomorphism from X into Y . As for homeomorphisms, for brevity we call either a global diffeomorphism or a local diffeomorphism simply a “diffeomorphism", and write h : X ! Y . Exercise. Show that f : R1 ! R1, given by the formula f(x) = x3, is a global homeomorphism, but not a global diffeomorphism. What prevents it from being a global diffeomorphism? 2.