Notes on Differential Topology George Torres Last updated January 4, 2019 Contents 1 Smooth manifolds and Topological manifolds 2 1.1 Smooth Structures . .2 1.2 Product Manifolds . .3 2 Calculus on smooth manifolds 3 2.1 Derivatives . .3 2.1.1 Tangent spaces . .3 2.1.2 Defining df ............................................4 3 The Inverse Function Theorem 5 4 Immersions and Submersions 6 4.1 Immersions . .6 4.2 Submersions . .7 5 Regular Values and Implicit Manifolds 8 5.1 Transversality . .8 5.2 Stability . .9 6 Sard's Theorem 10 6.1 Proof of Sard's Theorem . 10 7 Whitney's Embedding Theorem 12 7.1 Partitions of Unity . 13 7.2 Proof of Whitney . 14 8 Manifolds with Boundary 15 8.1 Facts about Manifolds with Boundary . 15 8.2 Transversality . 16 8.3 Sard's Theorem for Manifolds with boundary . 17 8.4 Classification of 1 dimensional Manifolds . 17 9 Transversality (Revisited) 18 9.1 Transversality and Stability . 19 9.2 Extensions . 20 10 Intersection Theory (Mod 2) 20 1 CONTENTS CONTENTS 11 Intersection Theory 22 11.1 Orientation of Manifolds . 22 11.2 Oriented Intersection Theory . 24 11.3 More General Perspective . 24 12 Lefschetz Fixed Point Theory 26 12.1 Two dimensional maps and the Euler Characteristic . 27 12.2 Non-Lefshetz fixed points . 27 13 Vector Fields and Poincar´e-Hopf 28 13.1 The Euler Characteristic and Triangulations . 29 14 Integration Theory 29 14.1 Differential forms . 29 14.2 Integrating Differential Forms . 31 14.3 Familiar examples . 32 14.4 The Exterior Derivative . 33 14.5 Stokes's Theorem . 34 ||||||||||{ Note to the reader: These are lecture from Harvard's 2014 Differential Topology course Math 132 taught by Dan Gardiner and closely follow Guillemin and Pollack's Differential Topology. 2 1 SMOOTH MANIFOLDS AND TOPOLOGICAL MANIFOLDS 1 Smooth manifolds and Topological manifolds Definition: A topological manifold X is a locally Euclidean space that is Hausdorff and second countable. B To discuss calculus on topological manifolds, they must be equipped with a smooth structure. To define this, we must define the concept of smoothness and smooth manifold. Definition: A smooth map between Euclidean spaces must first be defined on open sets, then on arbitrary sets: • A smooth map between open subsets of Euclidean spaces U ⊂ Rn and V ⊂ Rm is a function f : U ! V that is C1, or infinitely differentiable. • A smooth map from an arbitrary subset A ⊂ Rn is a function f : A ! Rn that extends locally to a smooth function on open sets. Namely, around every a 2 A there is an open set U ⊂ Rn and a smooth map m F : U ! R such that: F = f U\A Definition: Let X ⊂ Rn and Y ⊂ Rm. A function is a diffeomorphism (or smooth equivalence) if f is smooth, bijective and f −1 is smooth. The sets X and Y are said to be diffeomorphic if such a function exists between them. Definition: A smooth k manifold is a subset X ⊂ Rn such that for every point p 2 X, there is an open1 subset V ⊂ X containing p together with a diffeomorphism φ : U ! V , where U ⊂ Rk is some open subset. The maps φ : U ! V are called parameterizations, and the maps φ−1 : V ! U are called coordinate systems. Together, the collection of all such triples f(U; V; φ)g is called an atlas of charts. Example (S1): The 1 sphere S1 = f(x; y) j x2 + y2 = 1g ⊂ R2 is a smooth 1 manifold. To show this, we cover it with four open sets: the top half, the bottom half, the left half and the right half. Each one can be shown to be diffeomorphic to (−1; 1) ⊂ R. For example, let U1 be the top half and define: p 2 φ1 :(−1; 1) ! U1 : x 7! x; 1 − x This is smooth because it is a composition of continuous and C1 functions. The inverse is simply the projection (x; y) 7! x, which is also smooth (as the projection map on all of R2 is smooth). Thus φ is a diffeomorphism. The same process for all other halves of the circle covers all points in S1, so it is a 1 manifold. Definition: A smooth structure on a topological k-manifold X is a smooth k-manifold Y ⊂ Rn together with a homeomorphism X ! Y . 1.1 Smooth Structures Several questions arise when discussing smooth structures: Q: Does every topological manifold admit a smooth structure? A: No. For example, there are 4 manifolds with no smooth structure. These aren't immediately intuitive. Q: If a topological manifold admits a smooth structure, is it unique (up to diffeomorphism)? A: No. For example, the 7 sphere has 28 smooth structures, which form a group under connect sums. 1Open in the subspace topology 3 1.2 Product Manifolds 2 CALCULUS ON SMOOTH MANIFOLDS 1.2 Product Manifolds Proposition: Let X ⊂ Rn and Y ⊂ Rm be smooth manifolds. Then X × Y ⊂ Rn+m is also a smooth manifold. Proof: Let (x; y) 2 X × Y . Since X is a manifold, there is a local parameterization φ : U ! X around X. There is similarly a local parameterization around Y , say : V ! Y . If we take the map φ × : U × V ! X × Y given by: φ × (u; v) = (φ(u); (v)) It is easy to check that this is a parameterization for X × Y around (x; y). 2 Calculus on smooth manifolds The primary use of smooth manifolds is calculus, and the fundamental component of calculus is the derivative as a linear approximation. Namely, if f : Rn ! Rm is a C1 function, then the derivative of f at ~c 2 Rn, denoted by dfc is a linear map satisfying: f(~x) = f(~c) + dfc(~x − ~c) + err(~x) where err(~x) is an error term sufficiently small near ~c. 2.1 Derivatives The goal of this section is to define df, the best linear approximation to a smooth map f : X ! Y between smooth manifolds (possibly of different dimension). This definition will come out of the chain rule between X; Y and their tangent spaces. 2.1.1 Tangent spaces Definition: The tangent space of X at a point x, denoted Tx(X), is the image of dφ~0, where φ is a local paramaterization φ : U ! X with φ(~0) = x. N k B Note that Tx(X) ⊂ R is a k subspace. It is useful to think of it as a copy of R centered at x on the manifold. 1 2 1 Example: Take S ⊂ R . What is T(0;1)(S )? p Take the parameterization φ :(−1; 1) ! S1 that sends x 7! (x; 1 − x2). Then: 1 dφ = p−x 1−x2 At ~0, dφ = (1; 0), so the tangent space is the subspace spanned by (1; 0). 4 2.1 Derivatives 2 CALCULUS ON SMOOTH MANIFOLDS B We now will show that Tx(M) is well defined: Proposition: The tangent space Tx(M) doesn't depend on the choice of parameterization. Proof: Suppose we have distinct parameterizations of X: φ : U ! X φ(0) = x : V ! X (0) = x Since U and V have a nontrivial intersection and they are open, we may shrink them to assume U = V . Now define h = −1 ◦ φ. This is a map U ! U. Since φ and are diffeomorphisms, so is h. The chain rule now says: dφ0 = d 0 ◦ dh0 This implies that the image of dφ0 is restricted to that of d 0, or Im(dφ0) ⊂ Im(d 0). The same argument 0 −1 can be made for h = φ ◦ φ, so Im(d 0) ⊂ Im(dφ0), and we have equality 2.1.2 Defining df Now we return to attempting to define the derivative of a smooth map f : X ! Y between manifolds. Intuitively, the derivative at a point x should be a map between the tangent spaces Tx(X) ! Tx(Y ). We can parameterize around x and f(x) in X and Y respectively using: φ : U ! X φ(0) = x : V ! Y (0) = f(x) If we define h = −1 ◦ f ◦ φ, we require the following diagram to commute, so long as U is chosen small enough: f X Y φ U h V Similarly, we require the derivatives to commute. The derivatives of φ and also map between tangent spaces k ` T0(U) 2 R and T0(V ) 2 R : dfx Tx(X) Tf(x)(Y ) dφ0 d 0 dh0 T0(U) T0(V ) Since dφ0 is an isomorphism (because φ is), dfx is uniquely determined because require the derivatives to −1 commute. Thus dfx = d 0 ◦ dh0 ◦ (dφ0) . 5 3 THE INVERSE FUNCTION THEOREM B The derivative also satisfies the chain rule; that is, given: f g X Y Z we have d(f ◦ g)x = dfx ◦ dg(f(x). The proof of this is straightforward. ||||||| Example: Let S1 ⊂ C be the set fz 2 C j jzj = 1g. We can define F : S1 ! S1 by sending z 7! z2. What is df at i? We can associate S1 with the unit circle in R2 via a + bi ∼ (a; b); in which case the map F is: (x; y) 7! (x2 − y2; 2xy) Now we seek df(0;1). To do this, we need local parameterizations around (0; 1) and F (0; 1) = (−1; 0). One possibility is: p φ :(−1; 1) ! S1 x 7! (x; 1 − x2) p :(−1; 1) ! S1 x 7! (− 1 − x2; x) Now we need h = −1 ◦ f ◦ φ. We note that −1 is the projection onto the y axis, so: p h(x) = 2x 1 − x2 We can easily compute dh0 to be 2.