Differential Topology Andrew Kobin Fall 2015 Contents Contents Contents 0 Introduction 1 0.1 Review of General Topology . .1 1 Smooth Manifolds 4 1.1 Smooth Maps and the Tangent Space . .5 1.2 Submanifolds . .7 1.3 Abstract Manifolds . .8 1.4 Manifolds With Boundary . 13 2 Regular Values 15 2.1 Regular and Critical Values . 15 2.2 Sard's Theorem . 22 2.3 Homotopy and Degree mod 2 . 24 2.4 Orientation . 27 2.5 Brouwer Degree . 28 3 Transversality and Embeddings 34 3.1 Transversality, Immersion and Submersion . 34 3.2 Embedding Theorems . 38 4 Vector Bundles 41 4.1 Vector and Tangent Bundles . 41 4.2 Sections . 44 4.3 Constructing New Vector Bundles . 47 i 0 Introduction 0 Introduction These notes cover a course in differential topology taught by Dr. Slava Krushkal in the fall of 2015 at the University of Virginia. The main references for the course are Wallace's Differ- ential Topology: First Steps and Milnor's Princeton lectures, Topology from the differentiable viewpoint. The topics covered in the course are: Smooth maps and manifolds Tangent spaces Submanifolds of Rn and the embedding theorem Transversality Regular and critical points Sard's theorem and Brouwer's fixed point theorem Degree of maps Vector fields and the Euler characteristic An introduction to vector bundles and differential forms. 0.1 Review of General Topology The formal objects in general topology were only laid out about a century ago, but the concepts they represent go back much further in time. Definition. A topology T on a set X is a collection of subsets of X which satisfy (1) ?;X 2 T . S (2) For every arbitrary collection Uα 2 T , α Uα 2 T . Tk (3) For every finite collection U1;:::;Uk 2 T , i=1 Ui 2 T . The pair (X; T ) is called a topological space, although this is often abbreviated by X when the topology is understood. One of the most important topologies is the one generated by a metric. Definition. A metric on a space X is a function d : X × X ! R such that (1) (Positive semidefinite) d(x; y) ≥ 0 for all x; y 2 X and d(x; y) = 0 if and only if x = y. (2) (Symmetric) d(x; y) = d(y; x) for all x; y 2 X. (3) (Triangle inequality) d(x; y) + d(y; z) ≤ d(x; z) for all x; y; z 2 X. 1 0.1 Review of General Topology 0 Introduction The sets that make up a topology are by convention called open sets or neighborhoods, while their complements are called closed sets. Definition. A point p 2 X is a limit point of a set A ⊂ X if every neighborhood U of p intersects A in a point other than p. Definition. A point p 2 X lies in the closure of A, denoted A, if every neighborhood U of p intersects A nontrivially. It is easy to see from these two definitions that a set is closed if and only if it contains all of its limit points. For this reason it's common to write the closure as A = A [ L, where L is the set of limit points of A. One of the most important conceits in mathematics is the idea of comparison. We generally accomplish this by constructing maps between objects (categorically, maps are called morphisms) that preserve a desired structure. In topology, the objects are spaces and the natural maps between them are continuous functions. Definition. A function f : X ! Y between topological spaces is continuous at a point p 2 X if for every open set U containing f(p), f −1(U) ⊂ X is also open. The function is continuous (on X) if it is continuous at every point in X. Definition. A continuous map f : X ! Y is called a homeomorphism if it is one-to-one and onto (a bijection) and has a continuous inverse. In this case, X and Y are said to be homeomorphic, denoted X ∼= Y . One of the principal themes in topology is the study of topological properties, or properties preserved under homeomorphism. We recall several key topological properties next. Definition. Given a topological space X, a subset A ⊂ X is compact if every open cover of A has a finite subcover. Proposition 0.1.1. Closed subspaces of compact spaces are compact. The Separation Axioms are additional assumptions we make on a topological space to ensure there are \enough" open sets in X to do certain things. The first two are listed below. Definition. A space X is T1 if every point set is closed. Definition. A space X is Hausdorff (also called T2) if for every pair of distinct x; y 2 X, there exist disjoint open sets U and V such that x 2 U and y 2 V . Proposition 0.1.2. If X is Hausdorff and A ⊂ X is compact, then A is closed. The converse of this statement is not true in general, and even in nice spaces (metric spaces) there are additional conditions (e.g. boundedness) to imply compactness. Definition. A topological space X is said to be disconnected provided there exist nonempty, disjoint open sets U and V whose union is X. If no such sets exist, we say X is connected. 2 0.1 Review of General Topology 0 Introduction In this course, we will implicitly assume that every map is smooth, i.e. infinitely differ- entiable on an open subset U ⊂ Rn. However, we need to define the notion of a smooth map on any subset of Rn, not just the open ones. Definition. For an arbitrary subset S ⊂ Rn, a function f : S ! Rm is smooth on S if for every x 2 S, there exists a neighborhood W ⊂ n containing x and an extension R f~ : W ! Rm which is smooth on W and satisfies f~ = f. W \S In differential topology, the fundamental notion of equivalence is even stronger than homeomorphism. Definition. A diffeomorphism is a map f : X ! Y that is a homeomorphism (continuous and invertible, with continuous inverse) that is smooth and whose inverse is also smooth. In the case that f is a diffeomorphism, we say X and Y are diffeomorphic. An important theorem going forward is the Inverse Function Theorem from analysis, which we recall here. Theorem 0.1.3 (Inverse Function Theorem). Suppose f : U ! Rn is a smooth map on an open set U ⊂ n, and for a point x 2 U, assume the differential D f is non-singular, or R x equivalently, the Jacobian @fi (x) is invertible. Then there exists a neighborhood V ⊂ U @xj containing x such that fjV : V ! f(V ) is a diffeomorphism. 3 1 Smooth Manifolds 1 Smooth Manifolds The fundamental object in differential topology is the smooth manifold. Imprecisely, a manifold is a topological space which is locally Euclidean. In this section we will present two different definitions of a smooth manifold, but ultimately we will see that they are equivalent. Definition. A smooth n-dimensional manifold is a subspace M ⊂ Rk, k ≥ n, such that for any point x 2 M, there is a neighborhood U ⊂ M containing x which is diffeomorphic to an open subset of Rn. We will often write M n to convey that M is an n-dimensional manifold. Example 1.0.4. The unit circle S1 ⊂ R2 is a smooth one-dimensional manifold. R2 ( x ) Given a point x 2 S1, there are many good choices of a diffeomorphism, including Coordinate projection: f(x1; x2) = x1. Stereographic projection from N = (0; 1). Polar coordinates: (cos θ; sin θ) 7! θ for θ 2 (0; 2π). Each of these will be a diffeomorphism on a different neighborhood of x. However notice that there's no global diffeomorphism S1 ! R since these spaces aren't homeomorphic to begin with. n Definition. A collection of diffeomorphisms gx : Ux ! Vx ⊂ R , one for each point x 2 M, is called an atlas of M. Each Ux is called a coordinate chart (or coordinate system) −1 around x, and the inverse map gx is called a parametrization about x. Note that when dim M ≥ 1, around any point x 2 M, there are uncountably many choices of parametrizations and coordinate systems. 4 1.1 Smooth Maps and the Tangent Space 1 Smooth Manifolds 1.1 Smooth Maps and the Tangent Space Loosely, the tangent space to a manifold M at a point x 2 M is the set of all tangent vectors to M at x. This generalizes the concepts of tangent lines and tangent planes from multivariable calculus: tangent plane M tangent line f(x) Definition. The tangent space to a smooth manifold M at a point x 2 M is the set of tangent vectors to all parametrized differentiable curves lying in M and passing through x: 0 d TxM := α (0) = dt α(t)jt=0 α :(−"; ") ! M is smooth and α(0) = x : Remarks. α 1 A curve α :(−"; ") ! M is smooth if the composition (−"; ") −! M,! Rk is smooth, which in turn means the composition with every coordinate projection is smooth: α mi (−"; ") −! M −! R: 2 It is not immediately clear that TxM is closed under addition { however, scalar multi- ples are easily obtained by reparametrization of the curve. Suppose M m and N n are smooth manifolds, possibly of different dimensions. What does it mean for a map f : M ! N to be smooth? Definition. If M ⊂ Rk and N ⊂ R`, then f is smooth if for every x 2 M there is an open k ~ ` ~ neighborhood W ⊂ R of x and a smooth map f : W ! R such that fjW \M = fjW \M .