INTRODUCTION TO DIFFERENTIAL TOPOLOGY

Transcription

1 INTRODUCTION TO DIFFERENTIAL TOPOLOGY Joel W. Robbin UW Madison Dietmar A. Salamon ETH Zürich 29 April 2018

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3 Preface These are notes for the lecture course Differential Geometry II held by the second author at ETH Zürich in the spring semester of A prerequisite is the foundational chapter about smooth manifolds in [16] as well as some basic results about geodesics and the exponential map. For the benefit of the reader we summarize some of the relevant background material in the introduction. The first half of this book deals with degree theory and the Pointaré Hopf theorem, the Pontryagin construction, intersection theory, and Lefschetz numbers. In this part we follow closely the beautiful exposition of Milnor in [10]. For the additional material on intersection theory and Lefschetz numbers a useful reference is the book by Guillemin and Pollack [5]. The second half of this book is devoted to differential forms and de Rham cohomology. It begins with an elemtary introduction into the subject and continues with some deeper results such as Poincaré duality, the Čech de Rham complex, and the Thom isomorphism theorem. Many of our proofs in this part are taken from the classical textbook of Bott and Tu [2] which is also a highly recommended reference for a deeper study of the subject (including sheaf theory, homotopy theory, and characteristic classes). 29 April 2018 Joel W. Robbin and Dietmar A. Salamon iii

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5 Contents Introduction 1 1 Degree Theory Smooth Manifolds and Smooth Maps Manifolds with Boundary Proof of Sard s Theorem The Degree Modulo Two of a Smooth Map The Borsuk Ulam Theorem The Brouwer Degree Oriented Manifolds and the Brouwer Degree Zeros of a Vector Field Isolated Zeros Nondegenerate Zeros The Poincaré Hopf Theorem Homotopy and Framed Cobordisms The Pontryagin Construction The Product Neighborhood Theorem The Hopf Degree Theorem Intersection Theory Transversality Intersection Numbers Intersection Numbers Modulo Two Orientation and Intersection Numbers Isolated Intersections Self-Intersection Numbers The Lefschetz Number of a Smooth Map v

6 vi CONTENTS 5 Differential Forms Exterior Algebra Alternating Forms Exterior Product and Pullback Differential Forms on Manifolds The Exterior Differential and Integration The Exterior Differential on Euclidean Space The Exterior Differential on Manifolds Integration The Theorem of Stokes The Lie Derivative Cartan s Formula Integration and Exactness Volume Forms Integration and Degree The Gauß Bonnet Formula Moser Isotopy De Rham Cohomology The Poincaré Lemma The Mayer Vietoris Sequence Long Exact Sequences Finite Good Covers The Künneth Formula Compactly Supported Differential Forms Definition and Basic Properties The Mayer Vietoris Sequence for Hc Poincaré Duality The Poincaré Pairing Proof of Poincaré Duality Poincaré Duality and Intersection Numbers Euler Characteristic and Betti Numbers Examples and Exercises The Čech de Rham Complex The Čech Complex The Isomorphism The Čech de Rham Complex Product Structures Remarks on De Rham s Theorem

7 CONTENTS vii 7 Vector Bundles and the Euler Class Vector Bundles The Thom Class Integration over the Fiber Thom Forms The Thom Isomorphism Theorem Intersection Theory Revisited The Euler Class The Euler Number The Euler Class The Product Structure on H (CP n ) Connections and Curvature Connections Vector Valued Differential Forms Connections Parallel Transport Structure Groups Pullback Connections Curvature Definition and basic properties The Bianchi Identity Gauge Transformations Flat Connections Chern Weil Theory Invariant Polynomials Characteristic Classes The Euler Class of an Oriented Rank-2 Bundle Two Examples Chern Classes Definition and Properties Construction of the Chern Classes Proof of Existence and Uniqueness Tensor Products of Complex Line Bundles Chern Classes in Geometry Complex Manifolds The Adjunction Formula Complex Surfaces Almost Complex Structures on Four-Manifolds Low-Dimensional Manifolds

8 viii CONTENTS A Notes 247 A.1 Paracompactness A.2 Partitions of Unity A.3 Embedding a Manifold into Euclidean Space A.4 The Exponential Map A.5 Classifying Smooth One-Manifolds References 263 Index 264

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11 Chapter 1 Degree Theory 1.1 Smooth Manifolds and Smooth Maps 1.2 Manifolds with Boundary This section introduces the concept of a manifold with boundary. positive integer m and introduce the notations H m := { x = (x 1,..., x m ) R m xm 0 }, H m := { x = (x 1,..., x m ) R m xm = 0 }, Fix a (1.2.1) for the m-dimensional upper half space and its boundary. Definition A smooth m-manifold with boundary consists of a (second countable Haudorff) topological space M, and open cover {U α } α A of M, and a collection of homeomorphisms φ α : U α Ω α onto open subsets Ω α H m, one for every α A, such that, for every pair α, β A, the transition map φ βα := φ β φ 1 α : φ α (U α U β ) φ β (U α U β ) is a diffeomorphism (see Figure 1.1). The homeomorphisms φ α : U α Ω α are called coordinate charts, the collection {φ α, U α } α A is called an atlas of M, and the subset M = { p M φα (p) H m for every α A with p U α }. (1.2.2) is called the boundary of M. 3

12 4 CHAPTER 1. DEGREE THEORY M Uα U β φ α φ β φ βα Figure 1.1: A manifold with boundary. Remark Let (M, {φ α, U α } α A ) be a manifold with boundary. (i) The domain Ω αβ := φ α (U α U β ) H m of the transition map φ βα in Definition need not be an open subset of R m. If x Ω αβ H m is a boundary point of Ω αβ, then the map φ βα is called smooth near x iff there exists an open neighborhood U R m of x and a smooth map Φ : U R m such that Φ(x) = φ βα (x) for all x Ω αβ U. (ii) If p M and let α, β A such that p U α U β. Then φ α (p) H m φ β (p) H m (1.2.3) To see this, assume that x := φ α (p) Ω αβ \ H m and φ β (p) H m. Then the mth coordinate φ βα,m : Ω αβ R has a local minimum at x and hence the Jacobi matrix dφ βα ( x) is not invertible, a contradiction. (iii) The boundary M admits the natural structure of an (m 1)-manifold without boundary. (Exercise: Prove this.) (iv) The tangent space of M at p M is defined as the quotient T p M := p U α {α} R m/ (1.2.4) under the equivalence relation [α, ξ] [β, η] iff η = dφ βα (φ α (p))ξ. Thus the tangent space at each boundary point p M is a vector space (and not a half space). For p M and α A such that p U α, define the linear map dφ α (p) : T p M R m by dφ α (p)v := ξ for v = [α, ξ] T p M. Here [α, ξ] denotes the equivalence class of the pair (α, ξ) with ξ R m. (v) Let p M. A tangent vector v T p M is called outward pointing if dφ α (p)v R m \ H m for some, and hence every, α A such that p U α. (Exercise: Prove that this condition is independent of the choice of α.)

13 1.2. MANIFOLDS WITH BOUNDARY 5 Lemma Let M be a smooth m-manifold without boundary and suppose that g : M R is a smooth function such that 0 is a regular value of g. Then the set M 0 := { p M g(x) 0 } is an m-manifold with boundary M 0 := { p M g(x) = 0 }. Proof. Fix an element p 0 M such that g(p 0 ) = 0. By [16, Theorem ] the set g 1 (0) M is a smooth (m 1)-dimensional submanifold of M. Hence there exists an open neighborhood U M of p 0 and a coordinate chart φ : U Ω with values in an open set Ω R m such that φ(u g 1 (0)) = Ω (R m 1 {0}). Adding a constant vector in R m 1 {0} to φ and shrinking U, if necessary, we may assume without loss of generality that φ(p 0 ) = 0, Ω = { x R m x < r } for some constant r > 0. Thus, for every p U, we have g(p) = 0 φ m (p) = 0. Thus (g φ 1 )(x) = 0 for all x Ω with x m = 0. Since zero is a regular value of g, this implies that x m (g φ 1 )(x) 0 for all x = (x 1,..., x m 1, 0) Ω. This set is connected and so the sign is independent of x. Replacing φ by its composition with the reflection (x 1,..., x m ) (x 1,..., x m 1, x m ), if necessary, we may assume that x m (g φ 1 )(x) > 0 for all x = (x 1,..., x m 1, 0) Ω. Since Ω = {x R m x < r}, this implies p U M 0 φ m (p) 0 for all p U. Thus U 0 := U M 0 = {p U g(p) 0} is an open neoghborhood of p 0 with respect to the relative topology of M 0 and φ 0 : U 0 Ω 0 := { x Ω xm 0 } H m is a homeomorphism. Cover M 0 by such open sets to obtain an atlas with smooth transition maps. This proves Lemma

14 6 CHAPTER 1. DEGREE THEORY Example The closed unit disc is a smooth manifold with boundary D m := {x R m x 1} D m = S m 1 = { x R m x = 1 }. This follows from Lemma with M = R m and g(x) = 1 m i=1 x2 i. In Lemma the manifold M has empty boundary, the submanifold M 0 M has codimension zero. and near each boundary point of M 0 there exists a coordinate chart of M on an open set U M that sends the intersection U M 0 to an open subset of the upper half space H m. The next definition introduces the notion of a submanifold with boundary of any codimension such that the boundary of the submanifold is contained in the boundary of the ambient manifold M R m n X x U 1 F (0) H m φ Ω 0 n R Figure 1.2: A submanifold with boundary. Definition Let M be a smooth m-manifold with boundary. A subset X M is called a d-dimensional submanifold with boundary X = X M, if, for every p X, there exists an open neighborhood U M of p and a coordinate chart φ : U Ω with values in an open set Ω H m such that φ(u X) = Ω ({0} H d ). Lemma Let M be a smooth m-manifold with boundary, let N be a smooth n-manifold without boundary, let f : M N be a smooth map, and let q N be a regular value of f and a regular value of f M. Then the set X := f 1 (q) = { p M f(p) = q } M is an (m n)-dimensional submanifold with boundary X = X M.

15 1.2. MANIFOLDS WITH BOUNDARY 7 Proof. This is a local statement. Hence it suffices to assume that M = H m, N = R n, q = 0 R n. Let f : H m R n be a smooth map such that zero is a regular value of f and of f H m. If f 1 (0) H m = the result follows from [16, Theorem ]. Thus assume f 1 (0) H m and let x H m with f( x) = 0. Choose an open neighborhood U R m of x and a smooth map F : U R n such that F (x) = f(x) for all x U H m. Since zero is a regular value of f the derivative df ( x) = df( x) : R m R n is surjective. Now denote by e 1,..., e m the standard basis of R m. We prove the following. Claim. There exist integers 1 i 1 < < i n m 1 such that span{e i1,..., e in } ker df ( x) = {0} (1.2.5) Denote by v 1,..., v m R n the columns of the Jacobi matrix df ( x) R n m. Then the linear map d(f H m)( x) : T x H m = R m 1 {0} R n is given by d(f H m)( x)ξ = m 1 i=1 ξ iv i for ξ = (ξ 1,..., ξ m 1, 0) R m 1 {0}. Since this linear map is surjective, there exist integers 1 i 1 < < i n m 1 such that det(v i1,..., v in ) 0. These indices satisfy (1.2.5) and this proves the claim. Reordering the coordinates x 1,..., x m 1, if necessary, we may assume without loss of generality that i ν = ν for ν = 1,..., n. Now define the map Φ : U R m = R n R m n by Φ(x) := (F (x), x n+1,..., x m ) for x = (x 1,..., x m ) U. Then dφ( x)ξ = (df ( x)ξ, ξ n+1,..., ξ m ) for ξ = (ξ 1,..., ξ m ) R m. By the claim with i ν = ν for ν = 1,..., n the linear map dφ( x) : R m R m is injective and hence bijective. Thus the inverse function theorem asserts that the restriction of Φ to a sufficiently small neighborhood of x is a diffeomorphism onto its image. Shrink U, if necessary, to obtain that Φ(U) is an open subset of R m and Φ : U Φ(U) is a diffeomorphism. Then U H m is an open neighborhood of x in M = H m, the set Ω := Φ(U H m ) = Φ(U) H m is an open subset of H m, the restriction φ := Φ U H m : U H m Ω is a diffeomorphism and hence a coordinate chart of M, and φ(u X) = Ω ({0} H m n ) (see Figure 1.2). This proves Lemma

16 8 CHAPTER 1. DEGREE THEORY 1.3 Proof of Sard s Theorem 1.4 The Degree Modulo Two of a Smooth Map 1.5 The Borsuk Ulam Theorem

17 Chapter 2 The Brouwer Degree 2.1 Oriented Manifolds and the Brouwer Degree 2.2 Zeros of a Vector Field Isolated Zeros Let M be a smooth manifold without boundary and let X Vect(M). Definition (Isolated Zero). A point p 0 M is called an isolated zero of X if X(p 0 ) = 0 and there exists an open set U M such that p 0 U and X(p) 0 for all p U \ {p 0 }. The goal of this section is to assign an index ι(p 0, X) Z to each isolated zero of X. As a first step we consider the special case of a smooth vector field ξ : Ω R m on an open set Ω R m. Definition (Index). Let Ω R m be an open set, let ξ : Ω R m be a smooth vector field, and let x 0 Ω be an isolated zero of ξ. Choose ε > 0 such that, for all x R m, 0 < x ε = ξ(x) 0. Then the integer ( ι(x 0, ξ) := deg S m 1 S m 1 : x ξ(x ) 0 + εx) ξ(x 0 + εx) (2.2.1) is independent of the choice of ε and is called the index of ξ at x 0. 9

18 10 CHAPTER 2. THE BROUWER DEGREE Nondegenerate Zeros Lemma Let X Vect(M) and let p M be a nondgenerate zero of X. Then p is an isolated zero of X and ι(p, X) = sign ( det(dx(p)) ) { +1, if DX(p) is orientation preserving, (2.2.2) = 1, if DX(p) is orientation reversing. Proof.

19 2.3. THE POINCARÉ HOPF THEOREM The Poincaré Hopf Theorem Theorem (Poinaré Hopf). Let M be a compact smooth m-dimensional manifold with boundary and let X Vect(M) be a smooth vector field on M that points out on the boundary. Assume that X has only isolated zeros. Then m ι(p, X) = ( 1) k dim H k (M), (2.3.1) p M, X(p)=0 k=0 where H (M) denotes the de Rham cohomology of M. In particular, the left hand side is independent of the choice of the vector field X. It is called the Euler characteristic of M and is denoted by χ(m) := ι(p, X). (2.3.2) Proof. See page 12. p M, X(p)=0 Theorem was proved in 1885 by Poincaré in the case dim(m) = 2. After partial results by Brouwer and Hadamard, the theorem was established in full generality in 1926 by Hopf. In this section we will only prove that the sum of the indices of the zeros of a vector field with with only isolated zeros that points out on the boundary is independent of the choice of the vector field. The formula (2.3.1) for the de Rham cohomology groups will be established in Theorem Lemma (Hopf). Let N R n be a compact smooth n-dimensional submanifold with boundary, i.e. N is compact and its boundary agrees with the boundary of its interior and is a smooth (n 1)-dimensional submanifold of R n. Let Y : N R n be a smooth vector field with only isolated zeros such that Y (x) 0 for all x N. Then ( ) Y ι(x, Y ) = deg : N Sn 1. (2.3.3) Y x N, Y (x)=0 If, in addition, the vector field Y points out of N on the boundary, then ( ) Y deg : N Sn 1 = deg(g), (2.3.4) Y where g : N S n 1 denotes the Gauß map, i.e. for every x N the unit vector g(x) S n 1 is orthogonal to T x N and points out of N, so that x + tg(x) R n \ N for every sufficiently small real number t > 0. Proof.

20 12 CHAPTER 2. THE BROUWER DEGREE Lemma Let M be a smooth manifold with boundary, let X be a smooth vector field on M, and let p 0 M \ M be an isolated zero of X. Choose an open neighborhood U M \ M of p 0 such that p 0 is the only zero of X in U. Then there exists a smooth vector field X on M such that X (p) = X(p) for all p M \ U, the zeros of X in U are all nondegenerate, and ι(p, X ) = ι(p 0, X). (2.3.5) Proof. Proof of Theorem p U, X (p)=0

21 Chapter 3 Homotopy and Framed Cobordisms 3.1 The Pontryagin Construction 3.2 The Product Neighborhood Theorem 3.3 The Hopf Degree Theorem 13

22 14 CHAPTER 3. HOMOTOPY AND FRAMED COBORDISMS

23 Chapter 4 Intersection Theory The purpose of the present chapter is to extend the degree theory developed in Chapters 1 and 2 to smooth maps between manifolds of different dimensions. The relevant transversality theory is the subject of Section 4.1, orientation and intersection numbers are introduced in Section 4.2, selfintersection numbers are discussed in Section 4.3, and Section 4.4 examines the Lefschetz number of a smooth map from a compact manifold to itself and establishes the Lefschetz Hopf theorem and the Lefschetz fixed point theorem. 4.1 Transversality This section introduces the notion of transversality of a smooth map to a submanifold of the target space. Definition (Transversality). Let m, n, k be nonnegative integers such that k n, let M be a smooth m-manifold, let N be a smooth n- manifold, and let Q N be a smooth submanifold of dimension n k. The number k is called the codimension of Q and is denoted by codim(q) := dim(n) dim(q). Let f : M N be a smooth map and let p f 1 (Q). The map f is said to be transverse to Q at p if T f(p) N = im (df(p)) + T f(p) Q (4.1.1) It is called transverse to Q if it is transverse to Q at every p f 1 (Q). The notation f Q signifies that the map f is transverse to the submanifold Q. 15

24 16 CHAPTER 4. INTERSECTION THEORY M f Q Figure 4.1: Transverse and nontransverse intersections. Example (i) If Q = N, then every smooth map f : M N is transverse to Q. (ii) If Q = {q} is a single point in N, then a smooth map f : M N is transverse to Q if and only if q is a regular value of f. (iii) If f : M N is an embedding, then its image P := f(m) is a smooth submanifold of N (see [16, Theorem 2.3.4]). In this situation f is transverse to Q if and only if T q N = T q P + T q Q for all q P Q. (4.1.2) If (4.1.2) holds we say that P is transverse to Q and write P Q. (iv) Assume M =, let T M = {(p, v) p M, v T p M} be the tangent bundle, and let Z = {(p, v) T M v = 0} be the zero section in T M. Identify a vector field X Vect(M) with the map M T M : p (p, X(p)). This map is transverse to the zero section if and only if the vector field X has only nondegenerate zeros. (Exercise: Prove this). (v) Assume M =. Then the graph of a smooth map f : M M is transverse to the diagonal = {(p, p) p M} M M if and only if every fixed point p = f(p) M is nondegenerate, i.e. det(1l df(p)) 0. (Exercise: Prove this). The next lemma generalizes the observation that the preimage of a regular value is a smooth submanifold (see [16, Theorem ]). Lemma Let M be an m-manifold with boundary, let N be an n- manifold without boundary, and let Q N be a codimension-k submanifold without boundary. Assume f and f M are transverse to Q. Then the set P := f 1 (Q) = { p M f(p) Q } is a codimension-k submanifold of M with boundary P = P M and its tangent space at p P is the linear subspace T p P = { v T p M df(p)v Tf(p) Q }.

25 4.1. TRANSVERSALITY 17 Proof. Let p 0 P = f 1 (Q) and define q 0 := f(p 0 ) Q. Then it follows from [16, Theorem 2.3.4] that there exists an open neighborhood V N of q 0 and a smooth map g : V R k such that the origin 0 R k is a regular value of g and V Q = g 1 (0). We prove the following. Claim: Zero is a regular value of the map g f : U := f 1 (V ) R k and also of the map g f U M : U M R k. To see this, fix an element p U such that g(f(p)) = 0 and let η R k. Then q := f(p) V Q, g(q) = 0. Since zero is a regular value of g, there exists a vector w T q N such that dg(q)w = η. Since f is transverse to Q, there exists a vector v T p M such that Since T q Q = ker dg(q), this implies w df(p)v T q Q. d(g f)(p)v = dg(q)df(p)v = dg(q)w = η. Thus zero is a regular value of g f : U R k, and the same argument shows that zero is also a regular value of the restriction of g f to U M. By Lemma it follows from the claim that the set P U = f 1 (Q) U = (g f) 1 (0) is a smooth (m k)-dimensional submanifold of M with boundary and the tangent spaces (P U) = P U M T p P = ker d(g f)(p) = ker dg(q)df(p) = { v T p M df(p) ker dg(q) = T q Q } for p U with q := f(p) Q. This proves Lemma The next goal is to show that, given a compact submanifold Q N without boundary, every smooth map f : M N is smoothly homotopic to a map that is transverse to Q. This is in contrast to Sard s theorem in Chapter 1 which asserts, in the case where Q = {q} is a singleton, that almost every element q N is a regular value of f. Instead, the results of the present section imply that, given an element q N, every smooth map f : M N is homotopic to one that has q as a regular value.

26 18 CHAPTER 4. INTERSECTION THEORY Thom Smale Transversality Assume throughout that M is a smooth m-manifold with boundary, that N is smooth n-manifold without boundary, and that Q N is a codimension-k submanifold without boundary that is closed as a subset of N. Definition (Relative Homotopy). Let A M be any subset and let f, g : M N be smooth maps such that f(p) = g(p) for all p A. (4.1.3) A smooth map F : [0, 1] M N is called a homotopy from f to g relative to A if F (0, p) = f(p) and F (1, p) = g(p) for all p M and F (t, p) = f(p) = g(p) for all t [0, 1] and all p A. (4.1.4) The maps f and g are called homotopic relative to A if there exists a smooth homotopy from f to g relative to A. We write f A g to mean that f is homotopic to g relative to A. That relative homotopy is an equivalence relation is shown as in Section 1.4. Theorem (Local Transversality). Let f : M N be a smooth map and let U M be an open set with compact closure such that Then the following holds. f ( U \ U ) Q =. (i) There exists a smooth map g : M N such that g is homotopic to f relative to M \ U and both g U and g U M are transverse to Q. (ii) If f U M is transverse to Q, then there exists a smooth map g : M N such that g is homotopic to f relative to M (M \ U) and g U M is transverse to Q. Proof. See page 22. Corollary (Global Transversality). Assume M is compact. Then every smooth map f : M N is homotopic to a smooth map g : M N such that both g and g M are transverse to Q, and the homotopy can be chosen relative to the boundary whenever the restriction of f to the boundary is transverse to Q. Proof. Theorem with U = M. The proof of Theorem relies on the following lemma.

27 4.1. TRANSVERSALITY 19 Lemma Let N be an n-manifold without boundary, let K N be a compact set, and let V N be an open neighborhood of K with compact closure. Then there exists an integer l 0 and a smooth map G : R l N N such that G(0, q) = q, (4.1.5) G(λ, q) K = q V, (4.1.6) { } G q V = T G(λ,q) N = span (λ, q) i = 1,..., l (4.1.7) λ i for all λ R l and all q N. Proof. The proof has three steps. Step 1. Let W N be a neighborhood of V with compact closure. Then there exists vector fields X 1,..., X l Vect(N) such that supp(x i ) W for all i and T q N = span {X 1 (q),..., X l (q)} for all q V. Assume without loss of generality that N R l is a smooth submanifold of the Euclidean space R l for some integer l and that N is a closed subset of R l (see Theorem A.3.1). By Theorem A.2.2 there exists a partition of unity subordinate to the open cover M = W (M \ V ) and hence there exists a smooth function ρ : M [0, 1] such that supp(ρ) W, ρ V 1. Define the vector fields X 1,..., X l Vect(N) by X i (q) := ρ(q)π(q)e i for i = 1,..., l and q N, where Π(q) R k k denotes the orthogonal projection onto T q N and e 1,..., e l denote the standard basis of R l. These vector fields have support in U and the vectors X 1 (q),..., X l (q) span the tangent space T q N for every q V. This proves Step 1. Step 2. Let W and X 1,..., X l be as in Step 1, for each i let φ t i Diff(M) be the flow of X i, and define the map ψ : R l N N by ψ(t 1,..., t l, q) := φ t 1 1 φ t 2 2 φ t l l (q) for t i R and q N. Then ψ(0, q) = q for all q N and there exists a constant ε > 0 such that the following holds. (I) If q V and t R l satisfies max i t i < ε, then { } ψ T ψ(t,q) N = span (t, q) t i i = 1,..., l. (4.1.8) (II) If q N \ V and t R l satisfies max i t i < ε, then ψ(t, q) / K.

28 20 CHAPTER 4. INTERSECTION THEORY The vector fields X i have compact support and hence are complete. Thus the map ψ : R l N N is well defined. It satisfies ψ(0, q) = q, ψ t i (0, q) = X i (q) for all q N and all i {1,..., l}. Hence (4.1.8) holds for t = 0 by Step 1 and so assertion (I) follows from the fact that V is compact and the set of all pairs (t, q) R l N that satisfy (4.1.8) is open. To prove (II) we argue by contradition, and assume (I) is wrong for every ε > 0. Then there exist sequences t ν R l and q ν N \ V such that lim ν t ν = 0 and ψ(t ν, q ν ) K for all ν. Then ψ(t ν, q ν ) q ν and so q ν W for all ν. Since W has compact closure, there exists a subsequence (still denoted by q ν ) that converges to an element q N. Since V is an open neighborhood of K and q ν N \ V for all ν, it follows that q N \ V. On the other hand, since G is continuous and K is a closed subset of N, we have q = ψ(0, q) = lim ν ψ(t ν, q ν ) K. This is a contradiction and so (II) must hold for some ε > 0. This proves Step 2. Step 3. We prove Lemma Let ψ be as in Step 2 and define the map G : R l N N by G(λ 1,..., λ l, q) := ψ ελ 1 ελ,..., l, q (4.1.9) ε 2 + λ 2 1 ε 2 + λ 2 l for λ i R and q N. Then G(0, q) = q for all q N and so G satisfies (4.1.5). Moreover, G satisfies (4.1.6) by (II) and satisfies (4.1.7) by (I). This proves Lemma Remark The assertion of Lemma holds with l 2n. To see this, suppose that the vector fields X 1,..., X l satisfy the requirements of Step 1 in the proof of Lemma with l > 2n. Choose a Riemannian metric on N and define the map f : T N R l by f(q, w) := ( w, X 1 (q),..., w, X l (q) ) for q N and w T q N. This map has a regular value ξ = (ξ 1,..., ξ l ) R l by Sard s theorem. Since l > 2n = dim(t N), we have ξ / f(t N) and, in particular, ξ 0. Assume without loss of generality that ξ l 0 and define Y i Vect(N) by Y i (q) := X i (q) ξ i ξ l X l (q) for q N and i = 1,... l 1. Then, since ξ / f(t N), it follows that T q N = span {Y 1 (q),..., Y l 1 (q)} for all q K. (Exercise: Prove this.)

29 4.1. TRANSVERSALITY 21 We also need the following lemma. Assume that Q N is a compact codimension-k submanifold without boundary. Let F : R l M N be a smooth map such that both F and F R l M are transverse to Q. Then Lemma asserts that the set { M := F 1 (Q) = (λ, p) R l M } F (λ, p) Q is a smooth submanifold of R l M with boundary M = M (R l M). Denote by π : M R l the obvious projection. Lemma Fix an element λ R l and define the map F λ : M N by F λ (p) := F (λ, p) for p M. Then the following holds. (i) λ is a regular value of π if and only if F λ is transverse to Q. (ii) λ is a regular value of π M if and only if F λ M is transverse to Q. Proof. Choose an element p M such that q := F λ (p) = F (λ, p) Q. Then (λ, p) M, the tangent space of M at (λ, p) is given by { T (λ,p) M = ( λ, v) R l M } df (λ, p)( λ, v) T q Q, and dπ(λ, p)( λ, v) = λ for ( λ, v) T (λ,p) M. The following are equivalent. (A) The differential dπ(λ, p) : T (λ,p) M R l is surjective. (B) T q N = im (df λ (p)) + T q Q. Assume first that (B) holds and fix an element λ R l. Define w := l i=1 λ i F λ i (λ, p) T q N. By (B) there exists a vector v T p M such that w df λ (p)v T q Q. Hence df (λ, p)( λ, v) = df λ (p)v + l i=1 λ i F λ i (λ, p) = df λ (p)v w T q Q. Hence ( λ, v) T (λ,p) M and dπ(λ, p)( λ, v) = λ, and so (A) holds. Conversely, assume (A) and fix an element w T q N. Then, since F is transverse to Q, there exists a pair ( λ, v) R l T p M such that w df (λ, p)( λ, v) T q Q. Now it follows from (A) that there exists a tangent vector v 0 T p M such that ( λ, v 0 ) T (λ,p) M and so df (λ, p)( λ, v 0 ) T q Q. This implies w df λ (p)(v v 0 ) = w df (λ, p)( λ, v) df (λ, p)( λ, v 0 ) T q Q and so (B) holds. This shows that (A) is equivalent to (B) and this proves (i). The proof of (ii) is analogous and this proves Lemma

30 22 CHAPTER 4. INTERSECTION THEORY Proof of Theorem We prove part (i). We claim that the set K := f(u) Q N is compact. To see this, note that f(u \ U) Q =, hence K = f(u) Q, and so the set K is compact because U M is compact and Q N is closed. Moreover, K N \ f(u \ U), and hence Lemma A.1.2 asserts that there exists an open neighborhood V N of K with compact closure such that f(u \ U) V =. This implies that the set B := U f 1 (V ) is compact. Hence there exists a smooth function β : M [0, 1] such that supp(β) U, β B = 1. (4.1.10) (See Theorem A.2.2.) Choose a map G : R l N N as in Lemma and define F : R l M N by F (λ, p) := F λ (p) := G(β(p)λ, f(p)) for (λ, p) R l M. (4.1.11) Then F 0 = f, F λ M\U = f M\U for all λ by (4.1.5) in Lemma We prove that F R l U and F R l (U M) are transverse to Q. If (λ, p) R l U with F (λ, p) Q, then f(p) V by (4.1.6), thus p B, and so the vectors F λ i (λ, p) = β(p) G λ i (β(p)λ, f(p)) span the tangent space T F (λ,p) N by (4.1.7) in Lemma Lemma 4.1.3, the set Hence, by M := (R l U) F 1 (Q) is a smooth submanifold of R l U with boundary M = R l (U M). By Sard s theorem there exists a common regular value λ R l of the projection π : M R l and of π M : M R l. Hence, by Lemma 4.1.9, the homotopy f t (p) := F (tλ, p) satisfies the requirements of part (i).

31 4.1. TRANSVERSALITY 23 We prove part (ii). Thus assume that f U M is transverse to Q. As in the proof of (i), define the compact set K := f(u) Q N, choose an open neighborhood V N of K with compact closure such that f(u \ U) V =, and define the compact set B M by B := U f 1 (V ). We prove that there exists a smooth function β : M R such that supp(β) U, β U M = 0, β B\ M > 0. (4.1.12) To see this choose a smooth function β 1 : M [0, 1] with supp(β 1 ) U, β 1 B = 1 as in (4.1.10). Choose an atlas {U α, φ α } α A on M and let ρ α : M [0, 1] be a partition of unity subordinate to the cover, i.e. each point in M has an open neighborhood on which only finitely many of the ρ α do not vanish and supp(ρ α ) U α, ρ α = 1. (See Theorem A.2.2.) For α A define β α : U α R by α β α φ 1 α (x) := x m for x φ α (U α ) H m. Then the function ρ α β α : U α R extends uniquely to a smooth function on M that vanishes on M \ U α, the function β 0 := α ρ α β α : M R vanishes on the boundary and is positive in the interior, and so the product function β := β 0 β 1 satisfies (4.1.12). With this understood, the proof of part (ii) proceeds exactly as the proof of (i). The key observation is that the function F : R l M N in (4.1.11) still has the property that F R l U and F R l (U M) are transverse to Q, because F (λ, ) M = f M for all λ R l and f U M is transverse to Q by assumption. This proves Theorem

32 24 CHAPTER 4. INTERSECTION THEORY 4.2 Intersection Numbers Intersection Numbers Modulo Two Let N be a n-manifold without boundary, let Q N be a codimension-m submanifold without boundary that is closed as a subset of N, and let M be a compact m-manifold with boundary. If f : M N is a smooth map that is transverse to Q and satisfies f( M) Q =, (4.2.1) then the set f 1 (Q) M \ M is a compact zero-dimensional submanifold by Lemma and hence is a finite set. Theorem (Intersection Number Modulo Two). Let f : M N be a smooth map satisfying (4.2.1). Then the following holds. (i) There exists a smooth map g : M N that is transverse to Q and homotopic to f relative to the boundary. (ii) Let g be as in (i). Then the number #g 1 (Q) is finite and its residue class modulo two is independent of the choice of g. It is called the intersection number of f and Q modulo two and is denoted by { 0, if #g I 2 (f, Q) := 1 (Q) is even, 1, if #g 1 for g M f with g Q. (4.2.2) (Q) is odd, (iii) Let f 0, f 1 : M N be smooth maps satisfying the condition (4.2.1) and let F : [0, 1] M N be a smooth homotopy from f 0 to f 1 such that Then F ([0, 1] M) Q =. (4.2.3) I 2 (f 0, Q) = I 2 (f 1, Q). (iv) Let W be a compact (m+1)-manifold with boundary and let F : W N be a smooth map. Then I 2 (F W, Q) = 0. Proof. See page 25. Lemma Let f 0, f 1 : M N be smooth maps that satisfy (4.2.1) and are transverse to Q. Let F : [0, 1] M N be a smooth homotopy from f 0 to f 1 that satisfies (4.2.3). Then there exists a smooth homotopy G : [0, 1] M N from f 0 to f 1 such that G is transverse to Q and Moreover, #f 1 0 G(t, p) = F (t, p) 1 (Q) #f1 (Q) (modulo 2). for all t [0, 1] and all p M.

33 4.2. INTERSECTION NUMBERS 25 Proof. Since A := F 1 (Q) is a compact subset of W := [0, 1] (M \ M), there exists an open subset U [0, 1] M such that A U U W. Now W is a noncompact manifold with boundary W = {0, 1} (M \ M) and the homotopy F restricts to a smooth map F : W N such that F W is transverse to Q. Hence it follows from part (ii) of Theorem that there exists a smooth map G : W N such that G is transverse to Q and G W (W \U) = F W (W \U). Since W U is an open neighborhood of [0, 1] M, the map G extends to a smooth homotopy on all of [0, 1] M that satisfies G(t, p) := F (t, p) for all (t, p) [0, 1] M. Since G is continuous, the set X := G 1 (Q) [0, 1] M is compact. Since G([0, 1] M) Q =, we have X = G 1 (Q) [0, 1] (M \ M) = W. Since G W and G W are transverse to Q, it follows from Lemma that X is a 1-dimensional submanifold of W with boundary X = X W = ( {0} f 1 0 (Q)) ( {1} f 1 0 (Q)). Hence #f0 1 1 (Q) + #f1 (Q) = # X 2Z by Theorem A.5.1 and this proves Lemma Proof of Theorem Part (i) follows directly from Corollary We prove part (ii). Assume that g, h : M N are both transverse to Q and homotopic to f relative to the boundary. Then g is homotopic to h relative to the boundary and hence #g 1 (Q) #h 1 (Q) (modulo 2) by Lemma This proves (ii). We prove part (iii). For i = 0, 1 it follows from (i) that there exists a smooth map g i : M N such that g i is transverse to Q and homotopic to f i relative to the boundary. Compose the homotopies to obtain a smooth homotopy G : [0, 1] M N from g 0 to g 1 with G([0, 1] M) Q =. Then #g0 1 (Q) #g 1 1 (Q) (modulo 2) by Lemma and this proves (iii). We prove part (iv). Corollary asserts that there exists a smooth map G : W N such that G is homotopic to F and both G and G W are transverse to Q. By Lemma the set X := G 1 (Q) W is a compact 1-dimensional submanifold with boundary X = X W = (G W ) 1 (Q). Hence #(G W ) 1 (Q) is an even number by Theorem A.5.1. Since F W is smoothly homotopic to G W it follows that I 2 (F W, Q) = 0. This proves Theorem

34 26 CHAPTER 4. INTERSECTION THEORY Example Let N = RP n be the real projective space and fix an integer 0 < m < n. Define the inclusion f : RP m RP n by f([x 0 : : x m ]) := ([x 0 : : x m : 0 : : 0]) for [x 0 : : x m ] RP m and consider the submanifold Q := { [x 0 : x 1 : : x n ] RP n x 0 = = x m 1 = 0 }. Then f is transverse to Q and I 2 (f, Q) = 1. Hence f is not homotopic to a constant map. With m = 1 this shows that RP n is not simply connected. Exercise Let M R n be a compact connected smooth codimension-1 submanifold without boundary. Then R n \ M has two connected components and M is orientable. Step 1. There exists a constant ε > 0 such that p + v / M for all p M and all v T p M with 0 < v ε, and the set U ε := {p + v p M, v T p M, v < ε} is an open neighborhood of M. Hint: Let V R n be an open set and let f : V R be a smooth function such that zero is a regular value of f and f 1 (0) = V M =: W. Define the normal vector field X : W R n by X := f 1 f and consider the map W R R n : (p, t) p + tx(p). Step 2. Let p M, let v T p M S n 1, and let ε > 0 be as in Step 1. Define the curve γ : [ ε, ε] R n by γ(t) = p + tv. Then I 2 (γ, M) = 1 and hence p + εv and p εv cannot be joined by a curve in R n \ M. Step 3. Let p 0, p 1 M. Then there exist smooth curves γ : [0, 1] M, v : [0, 1] S n 1 such that γ(0) = p 0, γ(1) = p 1, and v(t) T γ(t) M for 0 t 1. Step 4. Let U ε be as in Step 1. Then U ε \ M has precisely two connected components. Hint: By Step 2 the set U ε \ M has at least two connected components and by Step 3 it has at most two connected components. Step 5. The set R n \ M has precisely two connected components. Hint: Every element of R n \ M can be joined to U ε \ M by a curve in R n \ M. Step 6. There exists a smooth map X : M S n 1 such that X(p) T p M for all p M. Hence M is orientable. Exercise Let N be a connected manifold without boundary and let M N be a compact connected codimension-1 submanifold without boundary. Find an example where N \ M is connected. If N is simply connected, show that N \ M has two connected components.

35 4.2. INTERSECTION NUMBERS Orientation and Intersection Numbers Let M and N be oriented smooth manifolds and let Q N be an oriented submanifold with dim(m) = m, dim(n) = n, and dim(q) = n k. The next definition shows how the orientations of M, Q, N induce an orientation of the manifold f 1 (Q) whenever f : M N is tranverse to Q. Definition (Orientation). Let f : M N be a smooth map that is transverse to Q. The manifold P := f 1 (Q) M is oriented by a map which assigns to every basis of every tangent space of P a sign ν {±1}. Let p P and fix a basis v 1,..., v m k of T p P. The sign ν(p; v 1,..., v m k ) {±1} is defined as follows. Choose tangent vectors v m k+1,..., v m T p M such that the vectors v 1,..., v m form a positive basis of T p M and choose a positive basis w k+1,..., w n of T f(p) Q. Then define ν(p; v 1,..., v m k ) := +1, if the vectors w 1,..., w n, with w i := df(p)v m k+i for 1 i k, form a positive basis of T f(p) N, 1, otherwise. (4.2.4) If k = 0 then Q N and P M are open sets and the sign is determined by the orientation of T p M. If k {m, n} the sign is understood as follows. Case 1: k = m < n. In this case P is a zero-dimensional submanifold of M, there is only the empty basis of T p P = {0}, and the sign is denoted by ν(p). Thus ν(p) = +1 if and only if signs match in T f(p) N = im (df(p)) T f(p) Q. Case 2: k = m = n. In this case Q N and P M are zero-dimensional submanifolds, the orientation of Q is a function ε : Q {±1}, the derivative df(p) : T p M T f(p) N is a vector space isomorphism, and +ε(f(p)), if df(p) : T p M T f(p) N ν(p) := is orientation preserving, (4.2.5) ε(f(p)), otherwise. Note that this formula is consitent with Case 1 and equation (4.2.4). Case 3: k = n < m. In this case Q has dimension zero and the orientation is a map ε : Q {±1}. Now choose v m n+1,..., v m T p M such that v 1,..., v m form a positive basis of T p M. Then +ε(f(p)), if df(p)v m n+1,..., df(p)v m ν(p, v 1,..., v m k ) := is a positive basis of T f(p) N, (4.2.6) ε(f(p)), otherwise.

36 28 CHAPTER 4. INTERSECTION THEORY Intersection Indices The next definition introduces the intersection index of a transverse intersection in the case of complementary dimensions. Definition (Intersection Index). Let M be a compact oriented m-manifold with boundary, let N be an oriented n-manifold without boundary, and let Q N be oriented (n m)-dimensional submanifold without boundary that is closed as a subset of N. Let f : M N be a smooth map that satisfies f( M) Q = and is transverse to Q. Fix an element p f 1 (Q) M \ M. Then T f(p) N = im (df(p)) T f(p) Q. and the intersection index of f and Q at p is defined as the sign ν(p; f, Q) obtained by comparing orientations in this decomposition. Thus +1, if df(p)v 1,..., df(p)v m, w m+1,... w n is a positive basis of T f(p) N ν(p; f, Q) := for every positive basis v 1,..., v m of T p M and every positive basis w m+1,..., v n of T f(p) Q, 1, otherwise. This corresponds to Case 1 in Definition Theorem (Intersection Number). Let M and Q N be as in Definition and let f : M N be a smooth map with f( M) Q =. Then the following holds. (i) There exists a smooth map g : M N that is transverse to Q and homotopic to f relative to the boundary. (ii) Let g be as in (i). Then the integer I(g, Q) := p g 1 (Q) ν(p; g, Q) is independent of the choice of g. It is called the intersection number of f and Q and is denoted by I(f, Q) := f Q := ν(p; g, Q) for g M f with g Q. (4.2.7) p g 1 (Q) (iii) Let f 0, f 1 : M N be smooth maps satisfying f i ( M) Q = and let F : [0, 1] M N be a smooth homotopy from f 0 to f 1 such that F ([0, 1] M) Q =. Then I(f 0, Q) = I(f 1, Q). (iv) Let W be a compact oriented (m + 1)-manifold with boundary and let F : W N be a smooth map. Then I(F W, Q) = 0. Proof. See page 30.

37 4.2. INTERSECTION NUMBERS 29 Lemma (Vanishing). Let W be an oriented smooth (m+1)-manifold with boundary and let F : W N be a smooth map such that F and F W is transverse to Q. Assume that the set F 1 (Q) W is compact. Then the intersection F 1 (Q) W is a finite set and ν(p; F W, Q) = 0. p F 1 (Q) W Proof. By Lemma 4.1.3, the set X := F 1 (Q) W is a compact oriented smooth 1-manifold with boundary X = X W = (F W ) 1 (Q). Thus X is a finite union of circles and arcs by Theorem A.5.1. Let A X be an arc and choose an orientation preserving diffeomorphism γ : [0, 1] A. Then γ(0), γ(1) W, the vector γ(0) points into W, and γ(1) points out of W. Let v 1,..., v m be a positive basis of T γ(1) W and let w m+1,..., w n be a positive basis of T F (γ(1)) Q. Since γ(1) is outward pointing, it follows from the definition of the boundary orientation that γ(1), v 1,..., v m is a positive basis of T γ(1) W. Since γ(1) is a positive tangent vector in T γ(t) X it follows from the sign convention in Definition that the vectors df (γ(1))v 1,..., df (γ(1))v m, w m+1,..., w n form a positive basis of T F (γ(1)) N. Hence it follows from the definition of the intersection index in Definition that ν(γ(1); F W, Q) = +1. Since γ(0) points in to W, the same argument shows that ν(γ(0); F W, Q) = 1. Thus ν(γ(0); F W, Q) + ν(γ(1); F W, Q) = 0. Since this holds for the endpoints of every arc A X, we obtain ν(p; F W, Q) = 0. This proves Lemma p F 1 (Q) W

38 30 CHAPTER 4. INTERSECTION THEORY Lemma (Homotopy). Let f 0, f 1 : M N be smooth maps that satisfy (4.2.1), are transverse to Q, and are smoothly homotopic by a homotopy that satisfies (4.2.3). Then p f 1 0 (Q) ν(p; f 0, Q) = p f 1 1 (Q) ν(p; f 1, Q). Proof. By Lemma there exists a smooth homotopy F : [0, 1] M N from f 0 to f 1 that satisfies (4.2.3) and is transverse to Q. Thus F 1 (Q) is compact and contained in the set W := [0, 1] (M \ M). This set is an oriented (m + 1)-manifold with boundary W = {0, 1} M. The boundary orientation of W agrees with the orientation of M at t = 1 and is opposite to the orientation of M at t = 0. Moreover, F W is transverse to Q by assumption. Hence it follows from Lemma that 0 = ν((t, p); F W, Q) (t,p) F 1 (Q) W = p f 1 This proves Lemma (Q) ν(p; f 1, Q) p f 1 0 (Q) ν(p; f 0, Q). Proof of Theorems Part (i) follows directly from Corollary We prove part (ii). Assume that g, h : M N are both transverse to Q and homotopic to f relative to the boundary. Then g is homotopic to h relative to the boundary and hence ν(p; g, Q) = ν(p; h, Q). p g 1 (Q) p h 1 (Q) by Lemma This proves (ii). We prove part (iii). For i = 0, 1 it follows from (i) that there exists a smooth map g i : M N such that g i is transverse to Q and homotopic to f i relative to the boundary. Compose the homotopies to obtain a smooth homotopy G : [0, 1] M N from g 0 to g 1 with G([0, 1] M) Q =. Then I(f 0, Q) = I(g 0, Q) = I(g 1, Q) = I(f 1, Q) by Lemma and this proves (iii). We prove part (iv). Corollary asserts that there exists a smooth map G : W N such that G and G W are transverse to Q and G is homotopic to F. Then F W is homotopic to G W and G 1 (Q) is compact because W is compact. Hence I(F W, Q) = I(G W, Q) = 0 by Lemma This proves Theorem

39 4.2. INTERSECTION NUMBERS 31 Exercise Let P, Q, N be compact oriented smooth manifolds without boundary such that dim(p ) + dim(q) = dim(n) and let f : P N and g : Q N be smooth maps. The map f is called transverse to g if every pair (p, q) P Q with f(p) = g(q) satisfies T f(p) N = im (df(p)) im (dg(q)). (4.2.8) In the transverse case the intersection index ν(p, q; f, g) {±1} is defined to be ±1 according to whether or not the orientations match in the direct sum (4.2.8), and the intersection number of f and g is defined by I(f, g) := f g := ν(p, q; f, g). (4.2.9) f(p)=g(q) (i) Prove that every smooth map f : P N is smoothly homotopic to a map f : P N that is transverse to g. (ii) If f 0, f 1 : P N are transverse to g, prove that I(f 0, g) = I(f 1, g). Deduce that the intersection number I(f, g) is well defined for every pair of smooth maps f : P N and g : Q N, transverse or not. (iii) Prove that I(g, f) = ( 1) dim(p ) dim(q) I(f, g). (4.2.10) (iv) Define the map f g : P Q N N by (f g)(p, q) := (f(p), g(q)) for p P and q Q and let N N be the diagonal. Prove that I(f, g) = ( 1) dim(q) I(f g, ). (4.2.11) Exercise Let N := CP 2. A smooth map f : CP 1 CP 2 is called a polynomial map of degree deg(f) = d if it has the form f([z 0 : z 1 ]) = [f 0 (z 0, z 1 ) : f 1 (z 0, z 2 ) : f 2 (z 0, z 1 )], d f i (z 0, z 1 ) = a ij z j 0 zd j 1, j=0 with a ij C and the homogeneous polynomials f i : C 2 \ {0} C have no common zeros. Let f, g : CP 1 CP 2 be polynomial maps. Prove that f g = deg(f) deg(g). Hint: Show that any two polynomial maps from CP 1 to CP 2 of degree d are smoothly homotopic. Consider the examples f([z 0 : z 1 ]) = [z0 d zd 1 : 0 : zd 1 ] and g([z 0 : z 1 ]) = [0 : z0 d zd 1 : zd 1 ] and show that f is transverse to g.

40 32 CHAPTER 4. INTERSECTION THEORY Isolated Intersections In this subsection we assign an intersection index to each isolated intersection which agrees with the index in Definition in the transverse case. Definition (The Index of an Isolated Intersection). Let M be a compact oriented m-manifold with boundary, let N be an oriented n-manifold without boundary, and let Q N be an oriented codimension-m submanifold without boundary that is closed as a subset of N, and let f : M N be a smooth map such that f( M) Q =. An element p 0 M \ M is called an isolated intersection of f and Q if f(p 0 ) Q and there is an open neighborhood U of p 0 such that f(p) / Q for all p U \ {p 0 }. Let p 0 M be an isolated intersection. Choose an orientation preserving diffeomorphism ψ : V R n, defined on an open neighborhood V N of f(p 0 ) such that ψ(v Q) = {0} R n m and the map V Q R n m : q (ψ m+1 (q),..., ψ n (q)) is an orientation preserving diffeomorphism. Choose an orientation preserving diffeomorphism φ : U R m, defined on an open neighborhood U M of p 0 such that f(u) V. Let x 0 := φ(p 0 ) and ε > 0. Then the integer ν(p 0 ; f, Q) := deg ( S m 1 S m 1 : x ξ(x 0 + εx) ξ(x 0 + εx) ξ := (ψ 1,..., ψ m ) f φ 1 : R m R m, is called the intersection index of f and Q at p 0 (see Figure 4.2). ), (4.2.12) Theorem Let M, Q, N, and f : M N be as in Definition Then the following holds. (i) The intersection index of f and Q at an isolated intersection p 0 is independent of the choice of the coordinate charts φ and ψ used to define it. (ii) If f and Q intersect transversally at p 0, then the intersection index in Definition agrees with the intersection index in Definition (iii) If f and Q have only isolated intersections, then ν(p; f, Q) = I(f, Q). (4.2.13) Proof. See page 34. p f 1 (Q)

41 4.2. INTERSECTION NUMBERS 33 Lemma (Perturbation). Let M, Q, N, f be as in Definition Let p 0 M \ M be an isolated intersection of f and Q and let U M be an open neighborhood of p 0 such that U f 1 (Q) = {p 0 } and U M =. Then there exists a smooth map g : M N that is homotopic to f relative to M \ U such that g U is transverse to Q and ν(p 0 ; f, Q) = ν(p; g, Q). (4.2.14) p U g 1 (Q) Proof. Shrinking U, if necessary, we may assume that there exist coordinate charts φ : U R m and ψ : V R n as in Definition The resulting map ξ := (ψ 1,..., ψ m ) f φ 1 : R m R m is a smooth vector field on R m with an isolated zero at x 0 = φ(p 0 ) and no other zeros. Moreover, the index in (4.2.12) agrees with the index of the isolated zero x 0 of the vector field ξ in Definition 2.2.2, i.e. ν(p 0 ; f, Q) = ι(x 0, ξ). (4.2.15) We prove the following. Claim 1: p 0 is a transverse intersection of f and Q if and only if the Jacobi matrix dξ(x 0 ) R m m is nonsingular. Claim 2: If p 0 is a transverse intersection of f and Q then the intersection index in Definition is given by ν(p 0 ; f, Q) = sign(det(dξ(x 0 ))) and agrees with the intersection index in Definition To see this, observe that the transversality condition in local coordinates takes the form im (df(p 0 )) T f(p0 )Q = T f(p0 )N (4.2.16) im ( d(ψ f φ 1 )(x 0 ) ) ( {0} R n m) = R n. This holds if and only if the linear map dξ(x 0 ) : R m R m is bijective, which proves Claim 1. To prove Claim 2, assume (4.2.16). Then it follows from (4.2.15) and Lemma that ν(p 0 ; f, Q) = ι(x 0, ξ) = sign ( det(dξ(x 0 )) ) { +1, if dξ(x0 ) is orientation preserving, = 1, if dξ(x 0 ) is orientation reversing. This sign is +1 if and only if the orientations match in the direct sum decomposition (4.2.16) and this proves Claim 2.