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2. the Poincaré Dual of a Submanifold

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2. the Poincaré Dual of a Submanifold INTERSECTION THEORY LIVIU I. NICOLAESCU ABSTRACT. I provide more details to the intersection theoretic results in [1]. CONTENTS 1. Transversality and tubular neighborhoods1 2. The Poincare´ dual of a submanifold4 3. Smooth cycles and their intersections8 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∗ 2 V ∗; hv∗; ui = 0; 8u 2 U : Note that dim U + dim U # = dim V = dim V ∗: 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) = U0 \ U1 ; (U0 \ U1) = U0 + U1 : (1.1) If T : V0 ! V1 is a linear map between finite dimensional vector spaces, then it has a dual ∗ ∗ ∗ T : U1 ! U0 : It satisfies the equalities (ker T )# = 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() ker T0 \ ker T1 = 0; (1.3) ∗ ∗ ∗ ∗ ∗ ∗ i.e., T0 t T1 if and only if the systems of linear equations T0 v = 0, T1 v = 0, v 2 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 T0 to U1 is injective. (1.4) Proposition 1.1. Supposet that T0 t U1. Then −1 # ∗ # T0 (U1) = T0 (U1 ): −1 In particular, the codimension of T0 (U1) in U0 is equal to the codimension of U1 in V . Proof. Fix surjection T : V ! W such that ker T = U1. Then −1 T0 (U1) = ker TT0: Using (1.2) we deduce −1 # ∗ ∗ ∗ ∗ T0 (U1) = R(T0 T ) = T0 (R(T )): On the other hand ∗ # # R(T ) = (ker T ) = U1 : ut 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)jS: The quotient bundle (TM)jS=T S 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)jS ! TSM ! 0: (1.5) Dualizing this sequence we obtain a short exact sequence ∗ ∗ ∗ 0 ! (TSM) ! (T M)jS ! T S ! 0: ∗ ∗ The bundle (TSM) is called the conormal bundle of S in M and it is denoted by TS M. 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 xs+1 = ··· = xm = 0; i.e., in these coordinates O \ S can be identified with the coordinate plane (x1; : : : ; xs; 0;:::; 0). We s+1 m ∗ can regard the differentials dx ; : : : ; dx as defining a local trivialization of TS M on O \ S. Note that for any x 2 S we have ∗ # (TS )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)jS TSM are ori- entable , then so is the third, and orientations on two induce orientations on the third via the base-first convention or(TM)jS = 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. ut 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 8x 2 S, t 2 [0; 1]. (iv) H1(N0) = N1. (v) The diagram below is commutative TSM[ Ψ0 [[]Ψ1 N0 w N1 H1 ut If (N; Ψ) is a tubular neighborhood of S in M, then we have a natural projection Ψ π πN : N −! TSM −! M: We will denote by Nx the fiber of πN over x 2 S. Observe that x 2 Nx. Remark 1.4. When S is compact, we can construct a tubular neighborhood of S as follows. • Fix a Riemann metric g on M. • Then for any " > 0 sufficiently small, the set " Ng := x 2 M; distg(x; S) < " determines a tubular neighborhood of S. " " • The natural projection πN : Ng ! S has the following description: for x 2 Ng, the point πN(x) is the point on S closest to x. " " If Ng is as above, then the diffeomorphism Ψ: TSM ! Ng can be constructed as follows. The ? normal bundle TSM can be identified with the orthogonal complement (TS) of TS in (TM)S and ? if x 2 S, X 2 (TxS) ⊂ TxM, then Ψ(X) = expg;x β( jXjg )X ; 1A map H with the properties (i), (ii), (iii) above is called an isotopy (or diffeotopy) of M rel S. 4 LIVIU I. NICOLAESCU where expg;x : TxM ! M is the exponential map defined by the metric g, jXjg is the g-length of X, " and β : [0; 1) ! [0;") is a diffeomorphism such that β(t) = t if t < 4 . ut Suppose M (respectively R; S) is a smooth manifold of dimension m (respectively r; s) and f : R ! M, g : S ! M are smooth maps. We say that f is transversal to g, and we write this f t g, if for any x 2 R and y 2 S such that f(x) = g(y) = z 2 M the differentials f_x : TxR ! TzM and g_ : TyS ! TzM are transversal. When S is a closed submanifold of M and g is the natural inclusion, we say that f is transversal to S and we write this f t S. Proposition 1.5. Suppose M is a manifold of dimension m, S is a closed submanifold of M of codimension c, R is a smooth manifold of dimension r and f : R ! M is a smooth map transversal to S. Then f −1(S) is a closed submanifold of R of codimension c and there exists natural bundle isomorphism _∗ ∗ ∗ ∗ _ ∗ f : f TS M ! Tf −1(S)R; f : Tf −1(S) ! f TSM: Proof. The first claim is local so it suffices to prove it in the special case when M coincides with a vector space of dimension V and S is a subspace U of V of codimension c. Consider a surjection c T : V ! R such that ker T = U = S. Fix x 2 R, set y = f(x) and denote by f_x the differential of f at x, f_x : TxR ! TyM = V . Observe that f −1(S) = fz 2 R; T (f(z)) = 0g The differential of T ◦ f at x 2 R is T ◦ f_x. Since f_x(TxR) + TyS = f_x(TxR) + U0 = TyM = V and the linear map T is surjective with kernel U we deduce that T ◦ f_x is surjective. We can now invoke the implict function theorem to conclude that that f −1(S) is a submanifold. Using Proposition 1.1 and (1.6) we deduce that the dual of f_x defines a linear isomorphism _∗ ∗ ∗ −1 fx :(TS M)f(x) ! (TS−1(S)R)x; 8x 2 f (S): ∗ −1 ∗ ∗ Observing that (TS M)f(x) is the fiber at x 2 f (S) of the pullback f (TS M) we deduce that these isomorphisms define a bundle isomorphism _∗ ∗ ∗ ∗ f : f (TS M) ! Tf −1(S)R: _ ∗ The isomorphism f : Tf −1(S)R ! f TSM is the dual of the above isomorphism. ut 2. THE POINCARE´ DUAL OF A SUBMANIFOLD Let M be a finite type, oriented smooth manifold of dimension m.A(Borel-Moore) cycle of dimension s in M is defined to be a linear map map s s C : Hc (M) ! R;Hc (M) 3 ! 7! hC; !i: m−s The Poincare´ dual of the cycle C is the cohomology class ηC 2 H (M) uniquely determined by the equality Z s ! ^ ηC = hC; !i; 8! 2 Hc (M): M INTERSECTION THEORY 5 Suppose that S is a closed oriented submanifold of dimension s of the oriented manifold M of di- mension m.
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