Cobordism and Hopf's Theorem E.J. Sanchez 1 Introduction In this paper we will provide an exposition of some of Milnor's Topology from the Differentiable Viewpoint, introducing the theory of cobordism and using it to prove the following theorem due to Heinz Hopf: Theorem 1 (Hopf's Theorem). Let M be a connected, oriented manifold without boundary of dimension p, and let f,g : M −! Sp be smooth maps. Then f and g are homotopic , deg(f) = deg(g). Before proving this theorem, we will lay some groundwork with a brief overview of cobordism, framed cobordism, and the Pontryagin manifold. We will then prove Theorem 2, which connects the Pontryagin manifold (a geometric construction) with the homotopy class of the map (a topolog- ical notion) and will be the most important ingredient in proving Hopf's Theorem: Theorem 2. With assumptions as in Hopf's Theorem, f,g : M −! Sp are homotopic , the associated Pontryagin manifolds are framed cobordant. 2 Cobordism and Framed Cobordism In this section we define the notions of cobordism and framed cobordism, and give a few basic examples. Definition 1. Let N 0 and N be manifolds without boundary, of dimension p, embedded in some ambient manifold M. Then we say a submanifold X of M × I is a cobordism of N 0 and N if @X = N 0 × f0g [ N × f1g. 1 Thus, cobordism forms an equivalence relation on the set of smooth mani- folds without boundary. To see this, note that reflexivity is clear if we simply choose X = N × I. Symmetry also follows immediately from the definition. Finally, transitivity holds since if X1 is a cobordism of M and N, and X2 is a cobordism of N and P , then we can rescale each by one half and then glue X1 and X2 along the boundary manifold N to get a manifold with boundary equal to M × f0g [ P × f1g. To utilize the theory of cobordism, we place additional structure on our manifolds, namely the framing of a submanifold. Note that in the following definition and in the rest of the paper, we will denote the dimension of manifolds M and N by m and n, respectively. Definition 2. A framing of the submanifold N ⊂ M is a smooth function v which takes each y 2 N and assigns to it a basis of the normal vectors to N in M at y. That is, v(y) = fv1(y); : : : vm−n(y)g where the vi(y) form a basis of ? TyN ⊂ TyM To see a straightforward example of a framing, take the n-sphere Sn em- bedded in Rn+1. Then there is the natural framing which assigns to each point x 2 Sn the unit normal vector to x, i.e. thought of as one unit outward along the line from y to the origin. The normal tangent space has dimension equal to one, so each unit vector forms a basis. Given framed submanifolds N and N 0 in M, we can naturally expand the definition of cobordism by introducing the notion of framed cobordism: N and N 0 are framed cobordant if there is a cobordism X ⊂ M, where X itself is a framed submanifold of M that on N and N 0 restrics to the given framings on those submanifolds. Now, given a map f : M −! N, observe that this notion of a fram- ing comes up somewhat naturally. If we choose a regular value y 2 N in the image of f, then we have that −1 Txf (y) is defined by the vanishing of the differential dxf, for any x 2 M in 2 −1 the fiber f (y). Thus, since dxf is a linear map between vector spaces, the subspace spanned by the tangent vectors normal to f −1(y) at x is isomorphic to the tangent space of Imf at y. We then consider the particular case of f : M −! Sp, and select a regular p value y 2 S . Let υ = (v1; : : : vp) be a positively oriented basis for the tangent p space TyS , and observe by what was stated above that we have the natural p ∼ ? −1 isomorphism TyS = Tx f (y). Thus υ induces a framing of the submanifold f −1(y). This submanifold f −1(y) ⊂ M with the induced framing (denoted f ∗υ) is called the Pontryagin manifold associated to f at y. The following theorem justifies the use of the article \the" Pontryagin manifold associated to f; in fact the Pontryagin manifold is unique up to framed cobordism: Theorem 3. The Pontryagin manifolds associated to f at any two regular values are framed cobordant. Proving this statement requires a few preparatory results however, the first of which shows that any two positively oriented framings of f −1(y) yield framed cobordant submanifolds. 0 ? −1 Lemma 2.1. Let υ and υ be positively oriented bases of Tx f (y). Then (f −1(y); υ) and (f −1(y); υ0) are framed cobordant. Proof. Note that we have the obvious cobordism f −1(y) × I, so we simply need to find compatible framing. But the space of all positively oriented ? −1 bases of Tx f (y) can be identified with the subspace of matrices in GLn(R) that have positive determinant, and since this is a connected space there is a path connecting υ and υ0. Each point of this path represents a positively oriented basis which varies continuously, so this gives the compatible framing as desired. The next result shows that if regular values are sufficiently close together, then their inverse images are frame cobordant. Lemma 2.2. Given a regular value y of f in Sp, if we choose any z suffi- ciently close to y, then f −1(y) is framed cobordant to f −1(z). Proof. First observe that the set of critical values, denoted f(C), in Sp is compact, so we may choose some small > 0 so that the closed -ball around 3 y contains only regular values. Now, given z in this -ball, choose a smooth p p family of rotations rt : S −! S so that r1(y) = z and the following are satisfied: • r0 = id, • rt = r1 for 0 < t ≤ 1, −1 • 8t, rt (z) lies on the great circle from y to z. p We may now define a homotopy F : M × I −! S by F (x; t) = rt · f(x). p For each t, z is a regular value of the composition rt ◦ f : M −! S , and so z is also a regular value of the homotopy F . Therefore, F −1(z) is a framed submanifold of M × I and gives a framed −1 −1 −1 −1 −1 cobordism between f (z) and (r1 ◦ f) (z) = f ◦ r1 (z) = f (y), thus proving the lemma. Lemma 2.3. If f and g are smooth homotopic and y is a regular value for both, then f −1(y) is framed cobordant to g−1(y). Proof. Let F be a homotopy from f to g, and choose a regular value z for F which is sufficiently close to y so that both f −1(y) and g−1(y) are framed cobordant to f −1(z) and g−1(z) respectively. F −1(z) then serves as a framed cobordism for f −1(z) and g−1(z), and the claim follows from Lemma 2.2. Now, given a map f : M −! Sp, we would like to finish showing that the Pontryagin manifold f −1(y) of f is independent (up to framed cobordism) of the regular value y chosen. So we suppose two regular values y and z of f, and note that we may choose rotations rt, just as in Lemma 2.2 of the sphere so that r0 = id and r1(y) = z. Then F (x; t) = rt ◦ f is a homotopy from f −1 to r1 ◦ f and hence by Lemma 2.3 we have that f (y) is framed cobordant −1 −1 −1 −1 to (r1 ◦ f) (y) = f ◦ r1 (y) = f (z), thus proving Theorem 3. So we are justified in referring to \the" Pontryagin manifold of a map f : M −! Sp, up to framed cobordism class. But in fact much more is true: Any compact framed submanifold N of codimension p in a manifold M is the Pontryagin manifold of some mapping f : M −! Sp. We will not prove this here, but the idea is to find an appropriate \product" neighborhood of N (in a sense which will be defined later) and compose maps from this neighborhood to Rp with an embedding 4 Rp −! Sp. This is an interesting fact which nicely rounds out the basic theory of Pontryagin manifolds, and will be needed to prove the final statement. Returning to our immediate goal of proving Theorem 2, we observe that Lemma 2.3 finishes showing the forward direction; we now proceed to argue for the reverse direction, which will suffice to show the more difficult impli- cation in Hopf's Theorem. To begin this, we will first argue for a slightly weaker statement, where the Pontryagin manifolds are assumed to be strictly equal rather than merely framed cobordant. But in fact most of the work consists of proving this weaker statement. Lemma 2.4. If the framed manifold (f −1(y); f ∗υ) is equal to (g−1(y); g∗υ), then f is smoothly homotopic to g. The proof of this statement will require an auxiliary result, for which we merely sketch a brief argument. Lemma 2.5 (Product Neighborhood Theorem).