INTERSECTION THEORY

I provide more details to the intersection theoretic results in [1]. CONTENTS 1. Transversality and tubular neighborhoods 1 2. The Poincaré dual of a submanifold 4 3. Smooth cycles and their intersections 8 4. Applications 14 5. The Euler class of an oriented rank two real vector bundle 18 References 20 1. TRANSVERSALITY AND TUBULAR NEIGHBORHOODS We need to introduce a bit of microlocal terminology. We begin by reviewing a few facts of linear algebra. If V is a finite dimensional real vector space and U is a subspace of V , then its annihilator is the subspace U := { v∗ ∈ V ∗; 〈v∗, u〉 = 0, ∀u ∈ U } . Note that dimU + dimU = dimV = dimV ∗. If we identify the bidual V ∗∗ with V in a canonical fashion, then we deduce (U) = U. If U0, U1 are subspaces in V then (U0 + U1) = U # 0 ∩ U # 1 , (U0 ∩ U1) # = U 0 + U # 1 . (1.1) If T : V0 → V1 is a linear map between finite dimensional vector spaces, then it has a dual T ∗ : U∗ 1 → U∗ 0 . It satisfies the equalities (kerT ) = R(T ∗), R(T ) = ker(T ∗) (1.2) where R denotes the range of a linear operator. Two linear maps Ti : U0 → V , i = 0, 1 are said to be transversal, and we write this T0 t T1, if R(T0) + R(T1) = V. Date: Started Oct. 6, 2011. Completed on Oct. 14, 2011. Last revised October 31, 2011. Notes for Topics class, Fall 2011. 1 2 LIVIU I. NICOLAESCU Using (1.1) and (1.2) we deduce that T0 t T1⇐⇒ kerT ∗ 0 ∩ kerT ∗ 1 = 0, (1.3) i.e., T0 t T1 if and only if the systems of linear equations T ∗ 0 v ∗ = 0, T ∗ 1 v ∗ = 0, v∗ ∈ V ∗, has only the trivial solution v∗ = 0. Suppose now that U1 is a subspace in V and T1 is the natural inclusion. In this case we say that T0 is transversal to U1 and we write this T0 t U1. From (1.3) we deduce T0 t U1 ⇐⇒ the restriction of T ∗ 0 to U # 1 is injective. (1.4) Proposition 1.1. Supposet that T0 t U1. Then T−1 0 (U1) # = T ∗ 0 (U # 1 ). In particular, the codimension of T−1 0 (U1) in U0 is equal to the codimension of U1 in V . Proof. Fix surjection T : V →W such that kerT = U1. Then T−1 0 (U1) = kerTT0. Using (1.2) we deduce T−1 0 (U1) # = R(T ∗ 0 T ∗) = T ∗ 0 (R(T ∗)). On the other hand R(T ∗) = (kerT ) = U 1 . u t Suppose now that M is a smooth manifold of dimension m and S is a closed submanifold of M . The tangent bundle of S is naturally a subbundle of the restriction to S of the tangent bundle of M , TS ↪→ (TM)|S . The quotient bundle (TM)|S/TS is called the normal bundle of S in M and it is denoted by TSM . In particular, we have a short exact sequence of vector bundles over S 0→ TS → (TM)|S → TSM → 0. (1.5) Dualizing this sequence we obtain a short exact sequence 0→ (TSM) → (T M)|S → T ∗S → 0. The bundle (TSM) is called the conormal bundle of S in M and it is denoted by T ∗ SM . It can be given the following intuitive description. Suppose that (x1, . . . , xm) are local coordinates on an open set O ⊂M such that in these coordinates the intersection O ∩ S has the simple description x = · · · = x = 0, i.e., in these coordinates O∩ S can be identified with the coordinate plane (x1, . . . , xs, 0, . . . , 0). We can regard the differentials dxs+1, . , dxm as defining a local trivialization of T ∗ SM on O ∩ S. Note that for any x ∈ S we have (T ∗ S)x = (TxS) . (1.6) Any Riemann metric on M defines an isomorphism between the normal and conormal bundles of S. From the exact sequence (1.5) we deduce that if two of the bundles TS, (TM)|S TSM are orientable , then so is the third, and orientations on two induce orientations on the third via the base-first convention or(TM)|S = or TS ∧ or TSM. (1.7) INTERSECTION THEORY 3 A co-orientation on a closed submanifold S ⊂M is by definition an orientation on the normal bundle TSM . Thus, an orientation on the ambient space M together with a co-orientation on S induce an orientation on S. Definition 1.2. Suppose that S is a closed submanifold of the smooth manifold M . A tubular neighborhood of S in M is a pair (N,Ψ), where N is an open neighborhood of S in M , and Ψ : TSM → N is a diffeomorphism such that its restriction to S ↪→ TSN coincides with the identity map 1S : S → S. u t We have the following fundamental result, [2, 3]. Theorem 1.3. (a) Let S and M be as in the above definition. Then for any open neighborhood U of S in M there exists a tubular neighborhood (N,Ψ) such that N ⊂ U. (b) If (N0,Ψ0) and (N1,Ψ1) are two tubular neighborhoods of S, then there exists a smooth map H : [0, 1]×M →M, x 7→ Ht(x), with the following properties.1 (i) H0 = 1M . (ii) Ht is a diffeomorphism of M . (iii) Ht(x) = x ∀x ∈ S, t ∈ [0, 1]. (iv) H1(N0) = N1. (v) The diagram below is commutative TSM