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Differentiable Topology and Geometry

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Differentiable Topology and Geometry Differentiable Topology and Geometry April 8, 2004 These notes were taken from the lecture course ”Differentiable Manifolds” given by Mladen Bestvina, at University of Utah, U.S., Fall 2002, and LATEX- ed by Adam Keenan. Contents 1 Introduction 4 2 Tangent Spaces 16 2.1 Differentiation........................... 18 2.2 CurveDefinition ......................... 19 2.3 Derivations ............................ 20 2.4 Immersions ............................ 24 2.5 Submersions............................ 30 2.6 Grassmannians .......................... 36 3 Transversality Part I 37 4 Stability 41 5 Imbeddings 45 5.1 Sard’sTheorem .......................... 45 6 Transversality Part II 47 7 Partitions of Unity 49 8 Vector Bundles 53 8.1 TangentBundle.......................... 53 8.2 CotangentBundle. .. .. 54 8.3 NormalBundle .......................... 54 1 8.4 VectorBundles .......................... 60 9 Vector Fields 69 9.1 LieAlgebras............................ 72 9.2 IntegralCurves .......................... 73 9.3 LieDerivatives .......................... 79 10 Morse Theory 83 11 Manifolds with Boundary 89 12 Riemannian Geometry 97 12.1Connections ............................100 12.2Collars...............................101 13 Intersection Theory 105 13.1 IntersectionTheorymod2 . .105 13.2 DegreeTheorymod2. .110 13.3Orientation ............................114 13.4 OrientedIntersectionNumber . 121 13.5 MappingClassGroups . .123 13.6 Degreetheory . .. .. .126 14 Vector Fields and the Poincar´e–Hopf Theorem 129 14.1 AlgebraicCurves . .137 15 Group Actions and Lie Groups 139 15.1 ComplexProjectiveSpace . .140 15.2LieGroups.............................143 15.3 FibreBundles . .. .. .146 15.4 ActionsofLieGroups . .147 15.5 DiscreteGroupActions. .147 15.6 LieGroupsasRiemannianManifolds . 149 16 The Pontryagin Construction 150 17 Surgery 155 18 Integration on Manifolds 157 18.1 Multilinear Algebra and Tensor products . 157 18.2Forms ...............................160 18.3 Integration.............................163 2 18.4 Exteriorderivative . .165 18.5 Stoke’stheorem . .167 19 de Rham Cohomology 171 19.1 Mayer-VietorisSequence . .175 19.2ThomClass ............................189 19.3Curvature .............................190 20 Duality 192 21 Subbundles and Integral Manifolds 195 3 1 Introduction n We all know from multivariable calculus what it means for a function f : R m −→ R to be smooth. We would like to extend this definition to spaces other than Euclidean ones. Is there a reasonable definition of smoothness for a function f : X Y ? Since smoothness is a local condition these spaces X and Y should look−→ like Euclidean spaces locally. This motivates the definition of a smooth manifold. Definition 1.0.1. A topological space M is a topological manifold if: (1) M is a Hausdorff space. (2) For all x M there exists an open neighbourhood Ux of x and a homeo- ∈ n(x) morphism Ux ∼= R for some n(x). n n R Example 1.0.1. R (as well as any open set in ) with the standard topology is a topological manifold. 1 2 Example 1.0.2. If we give S the subspace topology of R , then it is cer- tainly a Hausdorff space. Now consider the following four subspaces of S1: 1 U1 = (x, y) S : x> 0 , { ∈ 1 } U2 = (x, y) S : x< 0 , U = {(x, y) ∈ S1 : y > 0}, 3 { ∈ } U = (x, y) S1 : y < 0 . 4 { ∈ } U1 All of these subspaces are homeomorphic to ( 1, 1), for example, the home- omorphism φ from U to ( 1, 1) is given by−φ (x, y) = y. Now ( 1, 1) is 1 1 − 1 − R homeomorphic to R , hence each Ui is homeomorphic to . Since each point in S1 lies in one of these subspaces we have that S1 is a topological manifold. 4 Lemma 1.0.1. Any open set of a topological manifold is a topological man- ifold. n Theorem 1.0.1 (Invariance of Domain). Let U R be an open set and n ⊆ n R f : U R be a continuous injective map. Then f(U) is open . → We shall give a proof of this later on in the notes. n Corollary 1.0.1. If n = m, then an open set in R cannot be homeomorphic m 6 to an open set in R . In the definition n(x) is well-defined by invariance of domain. Also, n(x) is locally constant. Hence if the manifold is connected, then n(x) is constant on the whole manifold. In which case the dimension of the manifold is defined to be n(x). Lemma 1.0.2. Let M and N be topological manifolds. Then M N is also × a topological manifold. Proof. From basic topology we know that M N is a Hausdorff space. × Let (p1,p2) M N. Then there exists open neighbourhoods U1 and U2, ∈ × ni containing p1 and p2 respectively, and a homeomorphism φi from Ui to R . n1 +n2 Therefore φ φ is a homeomorphism from U U to R . 1 × 2 1 × 2 Example 1.0.3. The torus T 2 = S1 S1 is a topological manifold. × Why do we want Hausdorff in the definition? Example 1.0.4 (A non-Hausdorff manifold). Let M =( , 0) (0, ) −∞ ∐ ∞ ∐ 0+, 0− , with the standard topology on ( , 0) (0, ). A basis of neigh- bourhoods{ } of 0+ is ( ǫ, 0) (0, ǫ) 0+ −∞, where∪ ǫ >∞0. A basis of neigh- bourhoods of 0− is {(−ǫ, 0) ∪ (0, ǫ) ∪{0− }}, where ǫ> 0. { − ∪ ∪{ }} 0+ 0 - 5 Notice that in this space there is a sequence with a nonunique limit. Thus, we need the Hausdorff condition to guarantee unique limits. 2 Example 1.0.5. Let M = (x, y) R : xy = 0 . If we give M the in- 2 { ∈ } duced topology from R , then M is a Hausdorff topological space. But it is not a topological manifold because there is no neighbourhood of the origin n homeomorphic to an open set in R . Proposition 1.0.1. For a topological manifold the following are equivalent: (1) each component is σ-compact; (2) each component is second countable; (3) M is metrisable; (4) M is paracompact. Consider the Natural numbers with the order topology. Now take N × [0, 1), this space is given the lexicographical ordering. What you end up with is the half line with the standard topology. The half long line occurs when you replace the natural numbers by a well-ordered uncountable set. Example 1.0.6 (The Long Line). An example of a topological manifold which fails these conditions is the long line. Let Ω = set of all countable ordinals. This is well-ordered and uncountable. Let M = Ω [0, 1) with the lexicographical order. The order topology on M is the half× long line. The space M M/(0, 0) (0, 0) is called the long line. ∐ ∼ n Example 1.0.7 (Real Projective Space). We define R P as a set to be Sn/x x (This is equivalent to the space of lines through the origin in ∼ − 6 n+1 n n R R R ). If we give P the quotient topology, then P becomes a topo- logical space. So how do we make this into a manifold? One can check n that with this topology R P is a Hausdorff space. Now for each x we have n + to find a neighbourhood homeomorphic to an open set in R . Let Ui = (x , x ,...,x ) Sn : x > 0 and U − = (x , x ,...,x ) Sn : x < 0 . { 0 1 n ∈ i } i { 0 1 n ∈ i } Then U ± = (x , x ,...,x , x˜ , x ,...,x ): x2 + x2 + . x˜2 + + x2 < 1 , i ∼ { 0 1 i−1 i i+1 n 0 1 i ··· n } ± wherex ˜i just means that the term is omitted. It is clear that Ui is home- n n R omorphic to an open set in R . Now let Ui P be the image under the n n + − ⊂ −1 + − quotient map π : S R P of Ui (or Ui ). Then π (Ui)= Ui Ui . In order to show that π−→: U + U is a homeomorphism we need to show∪ that 0 −→ 0 the inverse is continuous. Let [x, x] U0 with x =(x0, x1,...,xn). We may − ∈ + assume x0 > 0 by replacing x by x if necessary. Now define ρ: U0 U0 − + − + −→ by ρ([x, x]) = x. The composition ρ π : U0 U0 U0 is continuous, hence ρ is− continuous. ◦ ∪ −→ 1 1 Remark: R P is homeomorphic to S and the homeomorphism is given by [z, z] z2. This map is continuous because the map S1 S1 given by z z2−is continuous.→ → → S1 BB BB BB BB z→z2 π BB BB BB BB B! 1 1 R / P [z,−z]→z2 S The map is a homeomorphism since S1 is compact. 1 1 The fact that R P is homeomorphic to S should be clear geometrically. From now on a topological manifold will be a topological manifold which is metrisable. For spaces that are Hausdorff and metrisable one can construct partitions of unity (which we shall define later). This is an indespensable tool in differentiable topology. 7 Definition 1.0.2. Let M be a topological manifold. A chart on M is a n homeomorphism φ: U φ(U) R , where U M is open, φ(U) is open n −→ ⊆ ⊆ in R . Definition 1.0.3. Two charts φ: U φ(U) and ψ : V ψ(V ) are C∞ related if −→ −→ ψ φ−1 : φ(U V ) ψ(U V ) ◦ ∩ −→ ∩ and φ ψ−1 : ψ(U V ) φ(U V ) ◦ ∩ −→ ∩ are both C∞. V U ψ φ φψ −1 ψ (V) φ(U) ψφ−1 3 ∞ R Remark: The map R given by x x is a C homeomorphism but the inverse given by x−→ x1/3 is not C∞→(it is not even C1). → Definition 1.0.4. An atlas on M is a family (Ui,φi) of charts, such that U is an open cover of M, and any two charts{ are C∞} related. { i} n n R Example 1.0.8.
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