Geometry and Topology

Contents

14 Vector bundle basics 14–1 14.1 Introduction...... 14–1 14.2 Definitions and Examples...... 14–2 14.3 Examples...... 14–2 14.4 The tangent bundle...... 14–4 14.4.1 Submanifolds and the normal bundle...... 14–5 14.5 Operations on vector bundles...... 14–5 14.5.1 Morphisms, sub-bundles, quotient bundles, pull-back...... 14–5 14.5.2 Homotopy property of vector bundles...... 14–6 14.6 Riemannian and Hermitian structures on vector bundles...... 14–6

15 Connections on vector bundles 15–1 15.1 Introduction...... 15–1 15.2 Definitions and examples...... 15–2 15.2.1 Description in terms of local data...... 15–3 15.2.2 Existence of connections...... 15–5 15.2.3 Pull-back...... 15–5 15.3 Connections and inner products...... 15–5

16 Curvature 16–1 16.1 Parallel transport...... 16–1 16.2 Curvature of a connection...... 16–2 16.3 Curvature in terms of parallel transport...... 16–5

17 Characteristic classes 17–1 17.1 Introduction...... 17–1 17.2 The first Chern class...... 17–2 17.2.1 First Chern class of a line bundle...... 17–2 17.2.2 First Chern class of higher rank bundles...... 17–2 17.3 Higher Chern classes...... 17–4 17.3.1 Traces of powers of the curvature...... 17–4 17.3.2 Digression on invariant functions and symmetric functions...... 17–5 17.3.3 The Chern class and the Chern character...... 17–7 17.4 Pontryagin Classes...... 17–11 17.5 A serious calculation...... 17–12 17.6 The Euler class and the Pfaffian...... 17–14 17.7 Gauss-Bonnet and Poincar´e-Hopf...... 17–15 17.8 A theorem of Bott on foliations and characteristic classes...... 17–21

Part C of the Geometry/Topology stream will be organized as follows. These notes cover the essential matter of this part of the course, and contain the important definitions, examples and theorems. The lectures themselves will not be an exercise in reading out these notes, but will back them up with a more detailed treatment of examples and discussion of the proofs.

0–1 (i)

A large part of this material is covered in the excellent book of Bott and Tu. Unfortunately they do not treat connections and curvature, but for almost everything else it is still the best available source on this subject. These notes owe a great deal to that book. Geometry and Topology Lecture 14: Vector bundle basics

after Michael Singera

Contents 14.1 Introduction...... 14–1 14.2 Definitions and Examples...... 14–2 14.3 Examples...... 14–2 14.4 The tangent bundle...... 14–4 14.4.1 Submanifolds and the normal bundle...... 14–5 14.5 Operations on vector bundles...... 14–5 14.5.1 Morphisms, sub-bundles, quotient bundles, pull-back...... 14–5 14.5.2 Homotopy property of vector bundles...... 14–6 14.6 Riemannian and Hermitian structures on vector bundles...... 14–6

14.1 Introduction

In essence a vector bundle is a continuously varying family of vector spaces, parameterized by some additional variables. This apparently abstract notion is of great importance in geometry and topology, and is the correct setting for gauge theory of mathematical physics. Here is a typical example. Suppose X is a topological space and we have a given continuous map Φ: X → Hom(Rm, Rn). In other words, for each x ∈ X, Φ(x) is an m×n matrix, depending continuously m upon x. Then for each x ∈ X we have Kx ⊂ R , the kernel of Φx. As x varies in X, Kx should vary continuously, and indeed this is the case provided that the rank of Φx does not change. If we suppose that the rank of Φ(x) does not depend upon x ∈ X, then the vector spaces Kx fit together to form a vector bundle over X. We shall explain precisely what is meant by ‘fit together to form a vector bundle’ later in this lecture. There may not exist any obvious vector bundles over a general topological space. However, if M is a smooth manifold of dimension n, there is a natural vector bundle called the tangent bundle, denoted TM. For each point x, this can be pictured as the vector space of dimension n which ‘approximates M at x’. (This makes sense since x has a neighbourhood diffeomorphic to Rn.) TM was introduced in lecture 3 of this course, though not referred to there as a vector bundle.

Given a vector bundle (which we continue to think of informally as a family of vector spaces Vx parame- terized by a space X) it is natural to ask whether one can choose, for each x ∈ X, a basis

(e1(x), . . . , em(x)) of Vx whose elements depend continuously on x. The answer in general is no. If, however, the answer is ‘yes’ then we say that the vector bundle is trivial. We shall see in subsequent lectures how to define topological invariants (characteristic classes) living in the (deRham) cohomology of X, with the property that if the vector bundle is trivial, then all characteristic classes must vanish. For simplicity in what follows, I shall work in the category of smooth manifolds (or smooth manifolds with boundary) unless otherwise stated. Most of the results remain true, but with perhaps different proofs, in the topological category as well.

a [email protected]

14–1 14–2

14.2 Definitions and Examples

How do we formalize the notion of a ‘smoothly varying family of vector spaces’? It is convenient to introduce a more general notion and then specialize.

Definition 14.2.1 Let π : Z → M be a smooth map from the manifold Z to the manifold M. We say that the data (Z, π) define a smooth fibre bundle, with typical fibre F , if there is a covering of M by open sets Uα with the following property: for each α, there is a diffeomorphism

−1 φα : Uα × F → π (Uα) (14.1) such that the first projection pr1 : Uα × F → Uα agrees with the composite π ◦ φα. The space Z is referred to as the total space of the fibre bundle, M is called the base.

This definition is supposed to capture the idea of a smoothly varying family of manifolds, each diffeo- morphic to F . For it follows from the definition that for each x ∈ M, the subset π−1(x) of Z is a smooth submanifold of Z diffeomorphic to F , though not in a canonical fashion. We now specialize to define vector bundles.

Definition 14.2.2 A smooth fibre bundle (E, π : E → M) with typical fiber Rk is called a k-dimensional real vector bundle if (i) for each x ∈ M, the set π−1(x) is equipped with the structure of a k-dimensional real vector space and (ii) the diffeomorphisms φα can be chosen in such a way that for each point x ∈ Uα, −1 the bijective map z 7→ φα(x, z) ∈ π (x) is linear. Variations: complex k-dimensional vector bundle E ... typical fiber Ck ... Instead of k-dimensional vector bundle, we also say vector bundle of rank k. If the rank is equal to 1, then E is also called a (real or complex) line bundle.

A (smooth) section of a fibre bundle π : Z → M is a (smooth) map s : M → Z with the property that −1 π ◦ s = idM . In other words, s(x) lies in the fibre of π (x) for every x ∈ M. Strictly speaking we defined a vector bundle as a map π : E → M, and the source of that map is the total space of the vector bundle. However it is extremely common to say and write the vector bundle E.

Definition 14.2.3 Let π : E → M and π0 : E0 → M be smooth vector bundles. A smooth map f : E → E0 is called a morphism of vector bundles if it commutes with the projections (π0 = π ◦ f) and 0 is linear on the fibres (fx : Ex → Ex is linear for every x ∈ M). The morphism is said to be injective (surjective) if each of the maps fx is injective (surjective). The morphism is called an isomorphism if it is both injective and surjective.

Definition 14.2.4 The data E = M × V and π = pr1 define a vector bundle over M called the product bundle (with fibre V ). A vector bundle isomorphic to a product bundle is called a trivial vector bundle.

One of the main goals of the (topological) study of vector bundles is to identify topological invariants which detect the non-triviality of vector bundles. Now, however, it is high time for some examples.

14.3 Examples

This section is deliberately short on detail. The detail will be supplied in the first lecture.

Example 14.3.1 (M¨obiusbundle) This is a real line-bundle over the circle S1 = R/Z. It can be defined as the quotient (orbit space) of R×R by the action of Z on R×R given by n·(x, y) = (x+n, −y). It can also be defined in terms of local data as follows. Let U0 = {−π < θ < π}, U1 = {0 < θ < 2π}. 1 Then U0 ∩ U1 = {0 < θ < π} ∪ {π < θ < 2π}. To specify a line-bundle over S = U0 ∪ U1, it is enough ∗ to write down a smooth map from U0 ∪ U1 into GL(1, R) = R . We take this map to be identically equal to 1 on {0 < θ < π} and identically equal to −1 on {π < θ < 2π}. 14–3

The M¨obius bundle is non-trivial. To trivialize it one would have to find a nowhere vanishing section. One can see that such a section does not exist by the intermediate value theorem.

Example 14.3.2 (The tautological line-bundle over the projective line) Let V be a 2-dimensional real (complex) vector space. Then P(V ) is the space of real (complex) 1-dimensional subspaces of V .(In previous lectures we assumed V = R2 and wrote RP 1 instead of P(V ).) Recall that P(V ) is a manifold. To introduce charts, first identify V with k ⊕ k (k = R or C). Every 1-dimensional subspace apart from 0 ⊕ k contains a unique element of the form (1, x), and similarly every 1-dimensional subspace apart from k ⊕ 0 contains a unique element of the form (y, 1). Thus P(V ) is covered by two sets, each diffeomorphic to k, and with transition function x = 1/y. This follows because if y 6= 0, then (y, 1) spans the same 1-dimensional subspace as (1, 1/y). Consider the product P(V ) × V . Each point x of P(V ) is a subspace of V , so we have a subset

L = {(x, v) ∈ P(V ) × V : v is in the subspace x}. (14.2)

There is a projection π : L → P(V ) given by restriction of the second projection P(V ) × V → P(V ). I claim that L is a line-bundle over P(V ), real if V is real, complex if V is complex. This is called the tautological line bundle over P(V ). To see this, use the two subsets we used to cover P(V ),

U0 = {subspaces spanned by (1, x): x ∈ k},U1 = {subspaces spanned by (y, 1) : y ∈ k}, (14.3)

Form U0 × k and U1 × k and map them into L by

φ0(x, λ) = λ(1, x), φ1(y, µ) = µ(y, 1) (14.4)

If y = 1/x and φ0(x, λ) = φ1(1/x, µ), then we have λ = yµ. (14.5)

Since multiplication by y is an invertible linear map, this shows that L is a line-bundle over P(V ).

As a substantial generalization of the previous example, we may consider the Grassmannians and their universal bundles.

Example 14.3.3 (Universal bundle of a Grassmannian) Let V be an N-dimensional real or com- plex vector space. Denote by

Grn(V ) = {U ⊂ V : U is a linear subspace of dimension n} (14.6)

We claim first that Grn(V ) is a manifold of real dimension N(N − n) if V is real and 2N(N − n) if V is complex. We will prove this in a moment. Denote by

E = {(x, v) ∈ Grn(V ) × V : v lies on the subspace x}. (14.7)

The first projection of Grn(V ) × V restricts to E to give a smooth map π : E → Grn(V ). We shall see that E is a vector bundle (real if V is real, complex if V is complex), of rank n over Grn(V ).

Let L be an n-dimensional subspace of V , representing a point of Grn(V ). Let W ⊂ V be a complementary subspace, so V = L ⊕ W (14.8) For every linear map T ∈ Hom(L, W ), we get a subspace of V , the graph of T :

G(T ) = {(λ, T λ) ∈ L ⊕ W : λ ∈ L}. (14.9)

In this way we have identified a certain subset U(W ) of Grn(V ) with the vector space Hom(L, W ), which has the claimed dimension. We note that an n-dimensional linear subspace L0 ⊂ V arises as a graph in this way if and only if L0 ∩ W = 0. To figure out the change of coordinate maps, let us take a new decomposition of V

V = M ⊕ Y (14.10) 14–4 where M is n-dimensional. Then we have another set U(Y ) consisting of subspaces L0 which are transverse to Y , and such subspaces are graphs

G(S) = {(µ, Sµ) ∈ M ⊕ Y : µ ∈ M} (14.11)

Let the block-matrix a b (14.12) c d represent the identity as a map from L ⊕ W to M ⊕ Y . Then we can compare the two descriptions of a subspace, namely the decomposition of (λ, T λ) with respect to M ⊕ Y is just

(aλ + bT λ, cλ + dT λ) (14.13)

If a + bT ∈ Hom(M,M) is invertible we can set

µ = (a + bT )λ, λ = (a + bT )−1µ (14.14) and we see that the graph of T relative to L ⊕ W is equal to the graph of (c + dT )(a + bT )−1 relative to M ⊕ Y . The invertibility of a + bT is precisely the condition that our graph is transverse to Y , and so we see that we have a smooth change of coordinates map from an open subset of Hom(L, W ) to an open subset of Hom(M,Y ).

Let us now prove that E is a vector bundle over Grn(V ). We have done most of the work already. With the notation as above, consider

U(W ) × L identified with Hom(L, W ) × L (14.15) via the graph construction. Given (T, λ) on the RHS we get the pair

(G(T ), λ + T λ) ∈ E (14.16) and this is clearly a diffeomorphism. To do the transition business, we see that the relation µ = (a+bT )λ gives the linear transition function

Hom(L, W ) × L → Hom(M,Y ) × M. (14.17)

Example 14.3.4 (Patching construction) If E → M is a vector bundle, then by definition, there is a −1 cover (Uα) of M by open sets over each of which E is trivial. That is we have maps φα : Uα×V → π (Uα) −1 and whenever Uα ∩ Uβ 6= ∅, the map gαβ = φα φβ : Uαβ × V → Uαβ × V is linear for each point x ∈ Uαβ. (Here Uαβ = Uα ∩ Uβ.)

Clearly we have gαβgβγ gγα = id whenever Uαβγ 6= ∅. Here gαβgβγ gγα denotes the composition of the three transition maps, wherever defined; composition symbol ◦ suppressed.

Conversely, given an open cover (Uα)α∈I of M where the index set I is ordered and for each pairwise intersection Uαβ a smooth map gαβ : Uαβ → GL(V ) (14.18) (where V is some vector space) satisfying the cocycle conditions

−1 gβα = gαβ (14.19) and gαβgβγ gγα = 1 (14.20) whenever Uαβγ 6= ∅ then these data define a vector bundle E with typical fibre V .

14.4 The tangent bundle

Let M be a smooth manifold of dimension n with atlas A. We have the tangent bundle TM from chapter 3, a smooth manifold in its own right. There is a projection π : TM → M. For each x ∈ M, the preimage 14–5

−1 π (x) = TxM has a preferred structure of vector space. This suggests, of course, that π : TM → M is a vector bundle. Indeed, for each chart ψ : U → M in the atlas (where U is some open set in Rn), we have T ψ : TU → TM and we know that there is an identification TU =∼ U × Rn. Therefore we can write n T ψ : U × R → TM. n n n If we restrict this to {x} × R for some x ∈ R , then we get the linear map R → TxM which is the differential of ψ at x ∈ U. In particular T ψ satisfies the linearity condition which makes it into a vector bundle chart. Therefore π : TM → M is indeed a vector bundle.

14.4.1 Submanifolds and the normal bundle

Let S ⊂ M be a k-dimensional smooth submanifold of M. At each point s of S, the tangent space TsS sits inside TsM as a k-dimensional linear subspace. Thus TS is a sub-vector-bundle of TM|S (see below). There is a quotient bundle, the normal bundle of S in M, denoted by NS/M . This is a vector bundle on S whose sections may be thought of vector fields pointing ‘out of S’. The tubular neighbourhood theorem states that the submanifold S possesses an open neighbourhood diffeomorphic to NS/M . Just as TxM gives a ‘first-order approximation’ to M at X, so the bundle NS/M gives a first-order approximation to M along S. Later on we shall use this fact to have a closer look at the Poincar´edual of S, assuming everything is oriented.

14.5 Operations on vector bundles

Given any ‘natural operation’ on vector spaces, there is a corresponding ‘natural operation’ on vector bundles (over a fixed smooth manifold M). For example

(i) Direct Sum If E → M and F → M are two real vector bundles, then there is a vector bundle E ⊕ F → M, called the direct sum. If gαβ is a system of transition functions for E and hαβ is a system of transition functions for F , then gαβ ⊕ hαβ is a system of transition functions for E ⊕ F . The fibre of E ⊕ F at x ∈ M is Ex ⊕ Fx (where Ex and Fx are the respective fibers of E and F at x, that is, the preimages of x under E → M and F → M).

∗ ∗ ∗ (ii) Dual bundle If E → M is a real vector bundle then E is the bundle with (E )x = (Ex) (the dual vector space). (iii) Homomorphism bundles If E and F are vector bundles on M, then Hom(E,F ), the homomorphism bundle, is the bundle whose fibre at x is the vector space of linear maps from Ex to Fx. One can check that Hom(E, R) = E∗, if R is the product bundle M × R. (iv) Tensor product If E and F are bundles then E ⊗ F is the bundle whose fibre at x is the tensor ∗ product Ex ⊗ Fx. We have E ⊗ F = Hom(E ,F ), and this can be taken as defining tensor product. (v) Exterior and symmetric powers If E is a bundle, then ΛpE is the bundle whose fibre at x is the p p ∗ p-th exterior power Λ Ex of the fibre Ex. This is also naturally isomorphic to alt (Ex). Similarly one defines the p-th symmetric power SpE. The top exterior power is sometimes called the determinant bundle, ΛnE = det(E), if n = rankE.

14.5.1 Morphisms, sub-bundles, quotient bundles, pull-back

Recall that we defined the notion of a vector bundle morphism at the beginning of this lecture. A bundle morphism is the same thing as a cross-section of the bundle Hom(E,F ). A subbundle E0 ⊂ E of E is a vector bundle, each of whose fibres is a vector subspace of the corresponding 0 00 0 fibre of E. If E ⊂ E, there is a quotient bundle E whose fibre at x is equal to Ex/Ex. If we have an exact sequence of vector bundles, 0 → E0 → E → E00 → 0 14–6

(so E0 is isomorphic to a subbundle of E, and E00 is isomorphic to E/E0). Let M and N be manifolds (or just topological spaces), let f : M → N be a smooth (or just continuous) map and let E → N be a bundle. The pull-back f ∗E of E by f is a bundle over M defined as f ∗(E) = {(x, e) ∈ M × E : π(e) = f(x)}. ∗ Thus f (E)x = Ef(x) for all x in M. In terms of transition functions, if we have a local description of E ∗ −1 in terms of a cover (Uα) of N and transition functions gαβ, then f (E) has a trivializing cover f (Uα) ∗ with transition functions f (gαβ).

14.5.2 Homotopy property of vector bundles

The following is a very important property of vector bundles: Let h : M × I → N be a smooth homotopy. Let E be any vector bundle over N. Then the two pull-backs ∗ ∗ h0(E) and h1(E) are isomorphic vector bundles over M. The proof (for compact M) can be found in Bott and Tu, p. 58. A very important corollary is: if M is contractible, then any vector bundle over M is trivial. Let f : M → pt and g : pt → M be maps such that gf is homotopic to the identity on M. By the homotopy property, E ' (gf)∗E = f ∗(g∗E), but g∗E must be trivial, as a bundle over the point. The pull-back of a trivial bundle is also trivial, so (gf)∗E is trivial.

14.6 Riemannian and Hermitian structures on vector bundles

Let E → M be a real vector bundle of rank n.A Riemannian structure on E is a choice of inner product (alias nonsingular symmetric bilinear form)

Ex × Ex → R (14.21) for every x ∈ M, depending smoothly on x. As an example, if M is a smooth submanifold of euclidean space Rk for some k, then the tangent bundle TM comes with a canonical Riemannian structure because k each tangent space TxM is identified with a liner subspace of R , and so inherits an inner product from the usual one in Rk. In the case of a complex vector bundle E → M of rank n, a Hermitian structure on E is a choice of nonsingular hermitian form Ex × Ex → C for every x ∈ M, depending smoothly on x. A Riemannian vector bundle E → M of rank n admits Riemannian local trivialisations. To show this we note the following. If U is open in M and s1, . . . , sn : U → E are sections of E|U such that {s1(x), . . . , sn(x)} is a basis of Ex, for every x ∈ M, then we may apply the Gram-Schmidt orthonor- malisation process to each basis {s1(x), . . . , sn(x)}. At the end of the process we have a new basis 0 0 0 0 {s1(x), . . . , sn(x)} of Ex which is orthonormal. The sections s1, . . . , sn of U|E are smooth. They allow us to identify E|U with U × Rn as a Riemannian vector bundle. It can be very useful to have a Riemannian structure on a (real) vector bundle E → M. Suppose for example that E0 ⊂ E is a sub-vector bundle of E. Then, if E comes with a Riemannian structure, we 00 0 00 can form the orthogonal complement E ⊂ E of E , defined so that Ex is the orthogonal complement of 0 ∼ 0 00 Ex in Ex, for every x ∈ M. It follows that E = E ⊕ E . Similar remarks apply, mutatis mutandis, to complex vector bundles and Hermitian structures on them. Lemma 14.6.1 Every real vector bundle E → M admits a Riemannian structure and every complex vector bundle E → M admits a Hermitian structure.

Proof We prove this in the real case. Choose a locally finite open cover of M by open sets Uα such that E|Uα is a trivial vector bundle. Then E|Uα clearly admits a Riemannian structure, say h , iα. Next choose a smooth partition of unity (ϕα) subordinate to the open covering (Uα). Thus each ϕα : M → [0, 1] is a smooth function with support contained in Uα, and the sum of the ϕα is equal to 1 everywhere. Now, for x ∈ M and vectors v, w ∈ Ex , let X hv, wi = ϕα(x) · hv, wiα (14.22) α where the RHS is to be read as 0 if x∈ / Uα. This defines a Riemannian structure on E.  Geometry and Topology Lecture 15: Connections on vector bundles

after Michael Singera

Contents 15.1 Introduction...... 15–1 15.2 Definitions and examples...... 15–2 15.2.1 Description in terms of local data...... 15–3 15.2.2 Existence of connections...... 15–5 15.2.3 Pull-back...... 15–5 15.3 Connections and inner products...... 15–5

15.1 Introduction

Given a vector bundle E → M there is no obvious way of matching up elements in different fibres, or in other words of ‘comparing’ elements in fibres Ex , Ey for points x, y in M which are close to each other. Connections give a differential-geometric way of attempting to do this. There are a number of approaches to the theory of connections. We shall adopt one which emphasises the algebraic and analytic aspects: connections as first order differential operators

∇ : Γ(M,E) = Ω0(E) −→ Ω1(M,E) = Γ(M, hom(TM,E)), (15.1) where on the RHS we have the space of E-valued 1-forms, i.e. sections of the vector bundle

hom(TM,E) → M.

To bring this into focus recall that we have the exterior derivative operator

d : Ω0(M) → Ω1(M); (15.2) a general connection is a ‘souped-up’ version of d. A product bundle E = M × V , where V is a vector space, has a standard connection, given by applying d to a V -valued function on M. Notice that for this connection, a section s satisfies ds = 0 if and only if it is a ‘constant’ element of V . Conversely (recalling that one of our tasks is to decide when a given bundle is trivial), we may attempt to trivialize a bundle equipped with a connection by trying to find a basis of parallel sections, that is to say, a set of sections (s1, . . . , sn) such that

∇sj = 0 (15.3) for all j. Unfortunately it is not in general possible to solve ∇s = 0 for a single nonzero section s, let alone to find an entire parallel basis. The obstruction is given by the curvature F (∇) of the connection, which is a 2-form with values in End(E). If F (∇) = 0, we say that ∇ is flat. We shall see that if a bundle E can be equipped with a flat connection then (at least if the base M is simply connected) then the bundle E is indeed trivial. If X is a vector field on M, then one can form the contraction

∇X s = hX, ∇si, (15.4)

a [email protected]

15–1 15–2 pairing a vector field with an E-valued 1-form to get a section of E. This is the directional derivative of s, in the direction X. While curvature obstructs the solution of ∇s = 0, it turns out that (15.4) can always be solved along a smooth curve γ : [0, 1] → M in M, with an arbitrary initial condition s(γ(0)) = v ∈ Eγ(0) and with X the tangent vector field along the curve. Using this idea we can, for example, reprove the result that a vector bundle over a contractible space is trivial.

15.2 Definitions and examples

Let E → M be a real or complex vector bundle of rank n over a smooth manifold M. We write altp(TM,E) for the smooth vector bundle on M whose fiber at x ∈ M is the linear space of p-linear p alternating maps TxM → Ex. Also Ω (M,E) is the vector space (usually infinite dimensional) of smooth sections of altp(M,E) → M. Important special cases: Ω0(M,E) = Γ(M,E) is the space of smooth sections of the vector bundle E itself, and

Ω1(M,E) = Γ(M, hom(TM,E)) is the space of sections of the vector bundle hom(TM,E) → M. In the real case, if E is the trivial line bundle, E = M × R, then Ωp(M,E) is what we have previously denoted by Ωp(M). If E is a trivial vector bundle, E = M × Rk, then

k Y Ωp(M,E) =∼ Ωp(M) . i=1

∗ p Returning to the general case, we know from earlier chapters that Ω (M) = (Ω (M))p≥0 is a graded ring ∗ p with the wedge product ∧. It turns out that Ω (M,E) = (Ω (M))p≥0 is a graded module. Namely, for p, q ≥ 0 there is a bilinear map

p q p+q Ω (M) × Ω (M,E) −→ Ω (M,E);(ω1, ω2) 7→ ω1 ∧ ω2 (15.5) defined by exactly the same formula as the “internal” wedge product in Ω∗(M):

(1) (2) (k+`) (ω1 ∧ ω2)(x)(u , u , . . . , u )

X (σ(1)) (σ(k)) (σ(k+1)) (σ(k+`)) = sign(σ) · ω1(x)(u , . . . , u ) · ω2(x)(u , . . . , u ) (15.6) σ∈ shuf(k,`) for x ∈ M. For the time being we are mostly interested in the cases where at least one of p, q is zero. If p = 0, then ω1 is a function M → R and ω1 ∧ ω2 is the pointwise product of ω1 and ω2. If q = 0, then ω2 is a section of E → M and again ω1 ∧ ω2 is the pointwise product of ω1 and ω2 ; more precisely,

(1) (p) (1) (p) (ω1 ∧ ω2)(x)(u , . . . , u ) = ω1(x)u , . . . , u ) · ω2(x).

A connection on E is a linear map

∇ :Ω0(M,E) → Ω1(M,E) (15.7) which satisfies the Leibniz rule

∇(f ∧ s) = df ∧ s + f ∧ ∇s for all f ∈ Ω0(M), s ∈ Ω0(M,E). (15.8)

Of course you may also write this equation in the form

∇(f · s) = df · s + f · ∇s . (15.9)

0 You are supposed to think of ∇ as follows. Suppose that s ∈ Ω (M,E) and suppose v ∈ TxM. Choose an open interval J ⊂ R containing 0, and a smooth curve γ : J → M with γ(0) = x and γ0(0) = v. Then (∇s)(x)(v) ∈ R is the rate of change of t 7→ s(γ(t)) at t = 0. We shall prove in §15.2.2 that there exist connections on any bundle E. 15–3

Lemma 15.2.1 Let x ∈ M. If s1 ad s2 are smooth sections of E which agree in a neighbourhood of x, then ∇s1 = ∇s2 in a neighbourhood of x.

Proof Let r = s1 − s2. Then r vanishes in a neighbourhood of x. We can find a smooth function f : M → R such that f vanishes in a smaller neighbourhood of x and f · r = r. Then

∇s1 − ∇s2 = ∇r = ∇(f · r) = df · r + f · ∇r and the RHS clearly vanishes near x.  The argument also shows that, if a section s of E is only defined in a neighbourhood U of x, then ∇s(x) is well defined. (Choose a section s1 which is defined on all of M and agrees with s in a smaller neighbourhood of x. Define ∇s(x) = ∇s1(x). By the lemma, the result is well defined.) Let U be an open set of M and suppose that E|U is trivial. Then we have a basis of local sections e1, . . . , en, say of E|U, and the general element s of Ω0(U, E) can be expanded as a linear combination P j 0 s = fje , where fj ∈ Ω (M). Using only linearity and the Leibniz rule, we compute

X j X j j ∇s = ∇ fj ∧ e = (dfj ∧ e + fj ∧ ∇e ). (15.10) j j

j P j i j Now ∇e is itself a 1-form with values in E, so can be expanded in the form i ωi ∧ e , where the ωi are honest differential 1-forms on U. The conclusion is that we have

X j X j X j i ∇ fje = dfj ∧ e + fj ∧ ωi ∧ e . (15.11) j i,j

j The locally defined matrix of 1-forms (ωi ) is called the connection matrix relative to the given local trivialization of E. It is an n × n matrix (where n is the fiber dimension of E) whose entries are elements 1 j of Ω (M). The term ωi is in row i and column j. (That’s a convention which is not universally accepted.) Equation (15.11) implies that ∇ is a first order linear differential operator. That is, the value ∇s(x) depends only on the value s(x) and the total first derivative of s at x, assuming a choice of local coordinates for M near x and a local trivialization for the vector bundle E over some neighbourhood of x.

15.2.1 Description in terms of local data

Suppose that our bundle E is described in terms of an open cover (Uα) and local trivializations

φα : Uα × V → E|Uα (15.12) and transition functions −1 gαβ = φα φβ : Uα ∩ Uβ → GL(V ). (15.13) 0 −1 If s ∈ Ω (M,E), then we set sα : Uα → V , sα = φα (s|Uα ). The collection of V -valued functions (sα) then satisfy the patching conditions

−1 −1 −1 sα = φα s = (φα φβ)(φβ s) = gαβsβ. (15.14)

∞ 1 Given a connection ∇ on E we may consider the locally defined operators ∇α : C (Uα,V ) → Ω (Uα,V ),

−1 ∇αsα = φα ∇(φαsα), (15.15) or −1 ∇α = φα ∇φα (15.16) in an abbreviated notation. (This must be understood as a composition of operators.) We have seen that ∇α must have the shape 1 ∇α = d + Aα where Aα ∈ Ω (Uα, End(V )); (15.17) this followed from the Leibniz rule.

Now on Uα ∩ Uβ, we have the two representations for ∇:

−1 −1 φα∇αφα = ∇ = φβ∇φβ (15.18) 15–4 so that −1 ∇α = gαβ ∇βgαβ. (15.19)

From this follows the transformation law for the (Aα), −1 −1 Aα = gαβAβgαβ − (dgαβ) gαβ in Uαβ. (15.20) [ Computation filled in by MW, May 2018:

φα(d + Aα)(sα) = ∇s = φβ(d + Aβ)(sβ), on Uα ∩ Uβ, therefore

(d + Aβ)(sβ) = gβα(d + Aβ)(sα) = gβα(d + Aα)gαβ(sβ) = gβα(dgαβ)sβ + dsβ + (gβαAαgαβ)(sβ) so that Aβ = gβαd(gαβ) + gβαAαgαβ. −1 Multiply both sides of this equation from the left with gαβ and from the right with gβα = gαβ to obtain (15.20). ] Thus a connection corresponds precisely to a collection of locally defined matrix-valued 1-forms Aα which satisfy the relations (15.20) for all α, β.

Example 15.2.2 (Line bundles) For real or complex line bundles, the above formulae simplify slightly, because GL1(R) and GL1(C) are abelian. Indeed the patching relation (15.20) simplifies to −1 Aα = Aβ − dgαβ gαβ in Uαβ (15.21) in this case, where Aα is now just a real or complex 1-form in Uα for each α.

Example 15.2.3 (Connection on M¨obiusbundle) Recall that the M¨obiusbundle L → S1 can be 1 defined in terms of local data as in Example 14.3.1: we have a two-set cover U0,U1 of S , with U0 ∩ U1 = ` 0 0 W W and the transition function g01 is equal to 1 on W and to −1 on W . Looking at (15.21) we see that the patching condition is just A0 = A1 in this case, so we can for example take A0 = A1 = 0. More generally, let L → M be any real line-bundle. It is a fact that L has local trivialisations in which all gαβ are equal to ±1 (assuming, as we may, that Uαβ is connected for all α, β). Again, in this case, the local connections ∇α = d (i.e. Aα = 0) patch together to form a global connection in this case.

Example 15.2.4 (Connection on tautological bundle over CP 1) From Example 14.3.2 the tauto- logical complex line bundle L → CP 1 is described locally by

U0 × C 3 (ψ(z), e),U1 × C 3 (ϕ(w), f) (15.22) where ψ(z) ⊂ C2 is the line spanned by (1, z) and ϕ(w) is the line spanned by (w, 1). The pairs (ψ(z), e) and (ϕ(w), f) represent the point-on-line constellations (e, ze) ∈ ψ(z) and (wf, f) ∈ ϕ(w). The transition maps were obtained by assuming these constellations equal. Then we must have zw = 1 and f = ze. −1 Thus g01 : U01 → GL1(C) is the map taking C(1, z) to the invertible 1 × 1-matrix (z ). Then the locally defined complex 1-forms z w A = dz, A = dw, (15.23) 0 1 + zz 1 1 + ww satisfy the patching relation (15.21) so that they give a local description of a connection on this complex line-bundle. These formulae look ad hoc, but in fact have a natural geometric explanation which we will now give.

Example 15.2.5 (Subbundles of trivial bundles) Let E ⊂ M × CN be a subbundle, and equip CN with its standard hermitian inner product. Then for each point x ∈ M, we have the orthogonal projection N Px : C → Ex, that is Pxex = ex if ex ∈ Ex,Pxv = 0 if v ⊥ Ex. (15.24) 2 N In particular, Px = Px for all x. The dependence of Px on x is smooth, that is P : M → End(C ) is a smooth map. We can define a connection on E by projecting the standard product connection on CN , ∇s = P ds (15.25) where on the RHS we have differentiated s as a CN -valued function on M and then projected the result back into E. The Leibniz rule is easily verified, and I claim, moreover, that the previous example is a special case of this construction, viewing L → CP 1 as a subbundle of CP 1 × C2. 15–5

15.2.2 Existence of connections

Let us return to our general vector bundle E with its description in terms of local data as above. Let (ρα) be a partition of unity subordinate to the cover (Uα). Then the definition

X −1 ∇s := φαd(φ ραs) (15.26) α defines a connection on E. To see that it is well-defined, notice that ραs has support in Uα, so the α-th 1 term in the sum is a well defined element of Ω (M,E), with support in Uα. The definition is clearly R- or C-linear and the Leibniz rule can be checked (lecture). Thus we have constructed a connection on E.

Remark 15.2.6 It is an exercise to show that every real vector bundle on a compact smooth manifold is a sub-bundle of a product bundle M × Rk for some k. Granted this, we can also construct a connection by the projection construction of Example 15.2.5.

Proposition 15.2.7 The set of all connections on a bundle E is an affine space modelled on the vector space Ω1(M, End(E)).

Proof What this statement means is that if ∇0 is a connection on E then every other connection ∇ 1 has the form ∇0 + α, where α ∈ Ω (M, End(E)). To prove it, we take two connections ∇0 and ∇1 on E. 0 Let α = ∇1 − ∇0. Let s be a smooth section of E and let f ∈ Ω (M). Then

α(fs) = ∇1(fs) − ∇0(fs) = f∇1s − f∇0s = fα(s) by applying the Leibniz rule to ∇0 and ∇1. By the reasoning which led us to ... it follows that (αs)(x) for x ∈ M depends only on s(x), and is well defined in terms of s(x). Thus α is a smooth ‘family’ of linear maps Ex → hom(TxM,Ex), one for each x ∈ M. Noting that ∼ hom(Ex, hom(TxM,Ex)) = hom(TxM, End(Ex)) ,

1 we see that α is just an element of Ω (End(E)). 

15.2.3 Pull-back

Let f : M → N be a smooth map and E → N a bundle on N, with connection ∇. Recall that there is a pull-back bundle f ∗E → M. I claim that there is an induced pull-back connection f ∗∇ on f ∗E, with the property (f ∗∇)(f ∗s) = f ∗(∇s). (15.27) Since any section of f ∗E is a combination of sections of the form u ∧ f ∗(s) where u ∈ Ω0(M), this, together with the Leibniz rule, serves to determine and define f ∗∇. ∗ −1 Alternatively, recall that the local description of f E is in terms of the open cover f (Uα), gαβ ◦ f. ∗ Then one can verify that if ∇ is described locally in terms of matrix-valued 1-forms Aα, then f Aα patch together correctly to define a connection on f ∗E.

15.3 Connections and inner products

Let E → M be a real vector bundle equipped with a Riemannian structure. Then it is of interest to look for a metric (alias orthogonal) connection on E. A metric connection on E is a differential geometric attempt to match up fibres Ex and Ey for x, y ∈ M “infinitesimally” close to each other, and that by a linear isometry. It is easy to make this precise using the definition of a connection as an operator. We say that a connection

∇ :Ω0(M,E) → Ω1(M,E) is metric if for any two smooth sections s1, s2 of E we have

dhs1, s2i = h∇s1, s2i + hs1, ∇s2i . (15.28) 15–6

0 1 Here hs1, s2i is to be understood as an element of Ω (M), so that the LHS is an element of Ω (M). The two summands in the RHS should also be read as elements of Ω1(M). Let U ⊂ M be open and such that E|U is trivial. We choose a trivialisation, E =∼ U × Rn, and express the connection ∇ in the usual way by an n × n matrix ω whose entries are differential 1-forms on U, as in/after equation(15.11). Then ∇s = ds + ω(s) for a smooth section s of E|U. To decode the RHS, think of ω as an element of Ω1(U, End(E)) and plug in s ∈ Ω0(U, E) to get ω(s) ∈ Ω1(U, E). If the connection ∇ is metric w.r.t. a given Riemannian structure on E, and the trivialisation of E|U which we picked is a Riemannian trivialisation, what does this mean for ω ? For sections s1 and s2 of E|U we calculate

h∇s , s i + hs , ∇s i = hds , s i + hs , ds i + hω(s ), s )i + hs , ω(s )i 1 2 1 2 1 2 1 2 1 2 1 2 (15.29) = dhs1, s2i + hω(s1), s2)i + hs1, ω(s2)i .

1 Hence we must have hω(s1), s2)i + hs1, ω(s2)i = 0 ∈ Ω (U), for all sections s1 and s2 of E|U. This means quite simply that ω is a skew-symmetric matrix of differential 1-forms on U, or in other words, ω is a differential 1-form with values in the skew-symmetric real n × n-matrices. This should not come as a complete surprise. The orthogonal group O(n) is contained in the general linear group GL(n) as a closed topological subgroup, and its tangent space at the identity element is the vector space of skew-symmetric n × n-matrices. (It is contained in the tangent space of GL(n), which is the vector space of all n × n-matrices, since GL(n) is open in the vector space of all n × n-matrices.)

Similar definitions and observations apply to Hermitian structures on complex vector bundles E → M. A connection ∇ on a Hermitian E → M is metric (alias unitary) if it satisfies

dhs1, s2i = h∇s1, s2i + hs1, ∇s2i . (15.30) for all smooth section s1 and s2 of E. Suppose that U ⊂ M is open and E|U admits a Hermitian trivialisation, so that E|U =∼ U × Cn and ∇ can be expressed in the form ∇s = ds + ω(s) for an n × n-matrix with entries in Ω1(U, End(E)). Then the condition which corresponds to ∇ being metric is that ω is a skew-hermitian matrix:

i j − ωj = complex conjugate of ωi . (15.31)

Lemma 15.3.1 Every Riemannian (real) vector bundle admits an orthogonal connection. Every Hermi- tian (complex) vector bundle admits a unitary connection.

Indeed the formula of equation 15.26 defines such a connection, provided that the local trivialisations for the vector bundle which appear in the formula are Riemannian (or Hermitian, as appropriate). Geometry and Topology Lecture 16: Curvature

after Michael Singera

Contents 16.1 Parallel transport...... 16–1 16.2 Curvature of a connection...... 16–2 16.3 Curvature in terms of parallel transport...... 16–5

16.1 Parallel transport

The definition of connection was motivated by the desire to ‘connect’ nearby fibres in a vector bundle. So far, there has been a lot of formalism, and the reader may have lost track of the original motivation! This section is aimed at remedying this situation. Let E be an n-dimensional vector bundle on a smooth manifold M equipped with a connection ∇. If X is a vector field on M we can form the contraction ∇X , the covariant directional derivative in the direction X. Let γ : [0, 1] → M be a regular smooth curve in the base M. In particular the tangent vector field X(t) =γ ˙ (t) ∈ Tγ(t)M never vanishes and we can consider the equation

∇X s = 0 (16.1) for a section s of E defined along γ. A section satisfying this equation is said to be parallel or covariant- constant along γ. By solving this equation (which we shall do shortly) we define a linear map

Pγ : Ex → Ey, x = γ(0), y = γ(1). (16.2)

Indeed we solve the equation with s(0) = e (an arbitrary initial value in Ex) and then set Pγ e = s(1). This is well-defined and linear by the uniqueness of the solution.

Proposition 16.1.1 Let γ : [0, 1] → M be a regular curve with γ(0) = x, γ(1) = y. Then the parallel transport operator Pγ exists.

Proof The interval is contractible, so we may trivialize the vector bundle γ∗E over [0, 1]. Then the equation to be solved becomes ds + A(t)s(t) = 0, s(0) = e. (16.3) dt where A is a smooth function on [0, 1] with n × n matrix values. To solve this equation, integrate from 0 to t to get Z t s(t) + A(τ)s(τ)dτ = e (16.4) 0 For e fixed and T ∈ (0, 1], consider the operator

Z t R : L∞([0,T ],V ) → L∞([0,T ],V ),R[s] = − A(τ)s(τ)dτ (16.5) 0

a [email protected]

16–1 16–2

Then we can write (16.4) in the form (1 − R)s = e (16.6) where e is viewed as a constant function in L∞[0,T ]. ∞ Here L is the space of continuous V -valued functions with the norm supt∈[0,T ] |s(t)| and | · | is the ordinary euclidean norm in V . Similarly |A| is the euclidean norm of the matrix A. Then we have |As| ≤ |A||s|. To show that a solution exists for small t, choose T so that

Z T |A(τ)|dτ ≤ N < 1 (16.7) 0 Then the operator norm of R is < 1 and so (1 − R) has an inverse,

(1 − R)−1 = 1 + R + R2 + ··· .

Hence (16.4) has a unique solution in [0,T ] if T is sufficiently small. At first, we know only that s is continuous, but the RHS of (16.4) is then C1, so s is C1. Hence the RHS is C2, so s is C3. It follows that s is actually smooth. Finally, we show that a solution exists for the whole interval. Since A is continuous, hence bounded on [0, 1], there exists T > 0 such that a solution exists on [a, a+T ] for any a. So we may cut [0, 1] into finitely many intervals [Tj,Tj+1]length ≤ T , solve the ODE separately on each one by the above argument, and match the solutions so that sj+1(Tj+1) = sj(Tj+1). In this way we get a solution over the whole interval [0, 1]. 

Remark 16.1.2 You will have noticed that the construction of the linear isomorphism Pγ : Ex → Ey did not use the assumption that γ : [0, 1] → M is regular. In fact Pγ is well defined without that assumption. To clarify, when we mentioned a section s of E along γ, that should really have been regarded as a section of γ∗E, a vector bundle on [0, 1]. The vector bundle γ∗E comes with a connection, γ∗∇. What our proof shows is simply that a connection on a vector bundle on [0, 1] determines a unique trivialization of the vector bundle on [0, 1]. This distinguished trivialization has the property that sections which are constant according the trivialization have zero covariant derivative w.r.t. that connection.

Remark 16.1.3 We can use parallel transport to ‘spread out’ a trivialization over contractible sets, thereby getting a different proof of the theorem that a bundle over a contractible manifold is trivial. The idea is as follows. Suppose H : M × I → M is a contraction of M, in the sense that H(x, 1) = x for all x but H(x, 0) = p for all x. Then for fixed x, t 7→ H(x, t) is a curve joining p to x. For any connection on

E, THx : Ep → Ex defines an isomorphism of the fixed fibre Ep with Ex, and it can be shown that the dependence of THx on x is smooth. Thus we get a smooth trivialization of E over M.

16.2 Curvature of a connection

Let ∇ be a connection on a vector bundle E → M. Then ∇ defines, in a natural way, connections on other vector bundles on M naturally associated to E, for example the dual bundle E∗ and the tensor powers of E. The definitions are meant to be natural with respect to the Leibniz rule for differentiation. Example: we would like to define ∇ of an endomorphism A of E in such a way that

∇(As) = (∇A)s + A∇s (16.8) for every section s of E. We note that the LHS and the second term of the RHS already make sense, so we know what (∇A)s should be for every section s:

(∇A)s = ∇(As) − A∇s . (16.9)

One checks easily that the RHS is zero at x ∈ M if s(x) = 0. It follows that the value of (∇A)s at x ∈ M depends only on the value s(x), so that ∇A is a section of

hom(E, hom(TM,E)) =∼ hom(TM, hom(E,E)) = Ω1(M, End(E)) 16–3 as required. Furthermore it is not hard to check that A 7→ ∇A satisfies the Leibniz rule, and so defines a connection on End(E).

As a further exercise in the Leibniz rule we can extend ∇ to act on differential p-forms on M with values in E. This is so important here that we formulate the result as a theorem. Before we start, recall (15.5).

Theorem 16.2.1 Let ∇ be a connection on a vector bundle E → M. Then there exist unique R-linear p p+1 maps d∇ :Ω (M,E) −→ Ω (M,E), for p ≥ 0, which satisfy the following conditions:

0 1 • d∇ :Ω (M,E) → Ω (M; E) is just ∇ itself;

p q p • for λ ∈ Ω (M) and µ ∈ Ω (M,E), we have d∇(λ ∧ µ) = dλ ∧ µ + (−1) λ ∧ d∇µ.

p p+1 Proof To start with, we note that if d∇ :Ω (M,E) −→ Ω (M,E) exists and satisfies these conditions, p p+1 then for s ∈ Ω (M,E) and x ∈ M, the alternating form (d∇s)(x) ∈ alt (TxM,Ex) depends only on the restriction s|U for an arbitrarily small open neighbourhood U of x in M. The proof of this is similar p p+1 to the proof of lemma 15.2.1. Therefore d∇ :Ω (M,E) −→ Ω (M,E), if it exists, determines R-linear maps p p+1 d∇ :Ω (U, E) −→ Ω (U, E) (16.10) for every open U ⊂ M, and these will still satisfy the two conditions. Now it will be enough to show that the maps d∇ of (16.10) exist and are unique for every open U in M which is a coordinate patch and over which the vector bundle E admits a trivialization. (In particular, the uniqueness ensures that on an intersection U ∩ V of overlapping coordinate patches U and V , the locally defined d∇ agree.) Or to put it differently, we have reduced the general case of our theorem to the special case where M is an open set in euclidean space Rm and the vector bundle E → M is a trivial vector bundle. In this situation, we have preferred “basic” sections s1, . . . , sn of E such that {s1(x), . . . , sn(x)} is a basis for Ex whenever x ∈ M. Every µ ∈ Ωp(M,E) can be written uniquely in the form

n X µ = µi ∧ si i=1

p where µi ∈ Ω (M). The conditions on d∇ force us to define

n n X p X d∇µ = dµi ∧ si + (−1) µi ∧ ∇si . i=1 i=1

p p+1 Thus we have a definition for d∇ :Ω (M,E) → Ω (M,E), and it only remains to check that this 0 satisfies the two conditions. The first condition, which requires that d∇ agree with ∇ on Ω (M,E), is satisfied by construction. The second is a straightforward consequence of the associativity of the wedge product and the Leibniz rule for the exterior derivative d. 

If you compare the characterization of d∇ in theorem 16.2.1 with the characterization of the exterior derivative d itself, you will note what it does not say. It does not say that d∇d∇ = 0. The reason is that d∇d∇ is not always zero. The composition d∇d∇ is called the curvature (operator) of the connection.

Lemma 16.2.2 For λ ∈ Ωp(M) and µ ∈ Ωq(M,E) we have

d∇d∇(λ ∧ µ) = λ ∧ d∇d∇µ .

Proof By the Leibniz rule, d∇(λ ∧ µ) = dλ ∧ µ ± λ ∧ d∇µ , so that

d∇d∇(λ ∧ µ) = ddλ ∧ µ ± dλ ∧ d∇µ ∓ dλ ∧ d∇µ + λ ∧ d∇d∇µ (16.11) which simplifies to λ ∧ d∇d∇µ. 

It follows from the special case p = 0 of the lemma that d∇d∇ can be described pointwise in M, so amounts to a collection of linear maps

q q+2 Fq(x) : alt (TxM,Ex) → alt (TxM,Ex) 16–4

which of course depend smoothly on x. The general case of the lemma then implies that F∗(x) is ∗ an endomorphism, raising degree by two, of the graded module alt (TxM,Ex) over the graded ring ∗ alt (TxM). As the graded module in question is free, generated by elements in degree 0, it is clear that the endomorphism is determined by what it does in degree 0. That is why we normally focus on the 2 linear maps F0(x) = F (x): Ex → alt (TxM,Ex). As x varies, these determine a vector bundle morphism F : E → alt2(TM,E) or equivalently, an element

F = F (∇) ∈ Ω2(M, End(E)). (16.12)

This is the curvature of the connection ∇ on E.

Let’s now calculate the curvature F (∇) in the case of a trivial vector bundle. More precisely let U be an open set of M and suppose that E|U is trivial. Then we have a basis of local sections e1, . . . , en. We are in the situation described around equation (15.10). So the connection ∇ can be defined by an i 1 n × n-matrix ω = (ωj) with entries in Ω (U). Then ∇s = ds + ωs (16.13)

n P k k for a section s of E|U, alias smooth map U → R . In other words if s = k fke , where the e for k = 1, . . . , n are the basic sections, then

P k P P k j ∇s = k dfk ∧ e + k j fkωj ∧ e .

We find that

k k P k j P k j P k j (d∇d∇)e = d∇(∇e ) = d∇( j ωj ∧ e ) = j dωj ∧ e − j ωj ∧ ∇e P k j P k P j i P k j P P j k i = j dωj ∧ e − j ωj ∧ ( i ωi ∧ e ) = j dωj ∧ e + i j ωi ∧ ωj ∧ e .

As this is true for all k = 1, . . . , n, we have by linearity F (∇)s = (dω + ω ∧ ω)s for all sections s of E|U. So we may simply write F (∇) = dω + ω ∧ ω . (16.14) This is often called the Maurer-Cartan equation. It is one of the most important equations in all of mathematics (many would say, among the top ten). Note that dω + ω ∧ ω is an n × n-matrix whose entries are elements of Ω2(U). The entry in row i and column k is

k X j k dωi + ωi ∧ ωj . (16.15) j

[ Added by MW in June 2018: some of us might prefer to reason as follows.

d∇d∇(s) = (d + ω)(d + ω)(s) = (d + ω)(ds + ω ∧ s) = dds + ω ∧ ds + dω ∧ s − ω ∧ ds + ω ∧ ω ∧ s

... so that F (∇) = d∇d∇ = dω + ω ∧ ω. The notation here is not very consistent with that used in earlier sections, where we had capital roman letters A etc. instead of ω.]

Remark 16.2.3 If we choose local coordinates in U as well, then the connection can be written in the form m X j s 7→ (∂j + A(j))s · dx j=1 where s : U → Rn can be any smooth map. Here A(j) is an n × n matrix with entries in Ω0(U). Namely, q q j k A(j)p = ωp(ξj) where ξj is the j-th basic vector field on U. Then the dx ∧ dx -component of the curvature F is given by the commutator

[∂j + A(j), ∂k + A(k)] = ∂jA(k) − ∂kA(j) + [A(j),A(k)].

Example 16.2.4 (Line-bundles) If L → M is a complex line-bundle, then the ω ∧ ω term in the definition of the curvature is zero and we have

F = dω. (16.16) 16–5

This is now an honest differential 2-form, independent of local trivialisations of L chosen, for End(L) is canonically isomorphic to the trivial bundle. [ It is not claimed that ω is independent of the local trivialisation. ] In the case of a U(1)-connection, this will be a 2-form with purely imaginary values. For the connection on L → CP 1 discussed in example 15.2.4, with the local coordinates on CP 1 used there, we see that dz ∧ dz F = (16.17) (1 + zz)2 which is indeed purely imaginary. In fact this is i/2 times the standard round area-form on the sphere S2 = CP 1. [ Added by MW, June 2018: as I am not used to this notation I find the following helpful. If we write points in R2 as pairs (x, y), and consequently complex numbers in the form x + iy, then dz means dx + idy and d¯z means dx − idy. Note that these are C-valued alternating 1-forms on C = R2.]

We saw above that in the case of a line-bundle the curvature-form is closed by (16.16). It is not in general 1 exact (as the example L → CP illustrates) because the Aα [ also known as ω ] do not patch together to form a globally defined 1-form.

16.3 Curvature in terms of parallel transport

The definition of curvature given above seems unavoidable from the point of view of differential forms, but there is another point of view which is older. It describes curvature in terms of parallel transport.

Let E → M be a real vector bundle of rank n with a connection ∇. Let x ∈ M and v, w ∈ TxM be given. We choose a smooth map g : R2 → M such that dg dg x = g(0, 0) ∈ M , v = (0, 0) , w = (0, 0) . ds dt 2 2 For (s, t) ∈ R , let γs,t : [0, 2s + 2t] → R be the (piecewise smooth) path given by travelling on a straight line segment from (0, 0) to (s, 0), then from there straight to (s, t), then from there to (0, t) and from there straight back to (0, 0), all with unit velocity. We now define

2 K : R → End(TxM) so that K(s, t): TxM → TxM is the parallel transport operator determined by ∇ and the path

g ◦ γs,t : [0, 2s + 2t] → M.

If either s = 0 or t = 0, then K(s, t) is equal to id ∈ End(TxM). This implies that the differential of K at (0, 0) is zero. For the same reason the second partial derivatives ∂2K/∂s2 and ∂2K/∂t2 at (0, 0) must vanish. But ∂2K/∂s∂t at (0, 0) is interesting:

∂2K Proposition 16.3.1 (0, 0) = − F (∇)(x)(v, w). ∂s∂t

Having stated this, we may as well prove something much more useful. Namely for arbitrary s ∈ R there is an equality ∂K Z s (s, 0) = − F (∇)(xr)(vr, wr) dr (16.18) ∂t 0 2 where xr = g(r, 0) ∈ M and vr, wr ∈ Txr M are the images of the standard basis vectors in R under the differential of g at (r, 0). An extra convention is needed to decode the RHS: although F (∇)(xr)(vr, wr) is strictly speaking a linear endomorphism of Exr , we want to think of it as an endomorphism of Ex = Ex0 by conjugating with the parallel transport Ex0 → Exr determined by ∇ and the path r 7→ g(r, 0) in M, with r ∈ [0, s].

A sketch proof of equation (16.18) is as follows. There is no loss of generality in assuming that M = R2 and g is the identity (if not, replace E by g∗E and ∇ by g∗∇, a connection on g∗E). Then K(s, t) is 2 simply parallel transport along the path γs,t. Since M = R is contractible, we know that the vector bundle E can be trivialised. More precisely, we trivialise it by

∼ n – first choosing a linear isomorphism E(0,0) = R ; 16–6

– then identifying each fiber E(s,0) with E(0,0) by means of parallel transport along the straight line segment from (s, 0) to (0, 0) ;

– then identifying each fiber E(s,t) with E(s,0) by means of parallel transport along the straight line segment from (s, t) to (s, 0).

Now we may write 2 n E = R × R . (16.19) The connection ∇ can be described by an n × n matrix of 1-forms on R2, which is much the same thing as a pair of matrices A and B with entries in Ω0(R2). (Compare remark 16.2.3, where these matrices had different names.) As a reward for choosing the trivialization of E carefully, we find that A vanishes on the axis R × 0 ⊂ R2 and B vanishes everywhere. Therefore we have

K(r, t) = Qt(r) (16.20)

2 n where Qt(r) is parallel transport in E = R × R , with connection ∇, along the edge from (r, t) to (0, t). The defining equation for Qt(r) now implies that ∂K (r, t) = K(r, t) · A(r, t). (16.21) ∂r

For t close to 0 and r ∈ [0, s], the difference K(r, t) − In has entries comparable in size to |t| because K(r, 0) = In. Also A(r, t) has entries comparable in size to |t| because A(r, 0) is the zero matrix. Therefore equation (16.21), divided by t and with t → 0, turns into

∂2K ∂A (r, 0) = (r, 0) . ∂r∂t ∂t Integrating over r ∈ [0, s] we obtain ∂K Z s ∂A Z s ∂A ∂B (s, 0) = (r, 0) dr = (r, 0) − (r, 0) dr . ∂t 0 ∂t 0 ∂t ∂r By remark 16.2.3 the integrand in the RHS is equal to −F (∇) evaluated at position (r, 0) and on the standard basis vectors, which we have agreed to call v and w. This completes the sketch proof. It is clear that equation (16.18) must have strong consequences when the curvature F (∇) is zero every- where on M. In that case ∇ is called a flat connection on E. Assume that ∇ is flat, let x, y ∈ M.

Theorem 16.3.2 Let β0 : [0, 1] → M and β1 : [0, 1] → M be smooth paths in M connecting x to y. If β0 and β1 are smoothly homotopic (with fixed start- and endpoints), then the parallel transport operators

Pβ0 ,Pβ1 : Ex −→ Ey determined by the flat connection ∇ on E are the same.

Proof The smooth homotopy whose existence we assume is a smooth map

2 [0, 1] → M ;(s, t) 7→ βt(s) such that βt(0) = x and βt(1) = y for all t ∈ [0, 1]. There are some lemmas in existence saying that such a map can always be extended to a smooth map R2 → M. We choose such an extension. Calling it g, and plugging it into equation (16.18), we deduce immediately that the “rate of change” d/dt of

Pβt : Ex → Ey

is zero at t = 0. Here Pβt for fixed t ∈ [0, 1] is the parallel transport operator associated with the path [0, 1] 3 s 7→ βt(s) ∈ M. As there is nothing special to t = 0 here, we conclude that the rate of change d/dt of Pβt : Ex → Ey is zero for all t. It follows that Pβ0 = Pβ1 .  To illustrate, suppose that M is a simply connected smooth manifold and E → M is a vector bundle on M with a flat connection ∇. For x and y in M, there exists a smooth path β from x to y (as M is path-connected). All such paths are smoothly homotopic with fixed start- and endpoints, (as M is simply connected). Therefore the parallel transport isomorphism Pβ : Ex → Ey is independent of β by theorem 16.3.2. We may fix a random x ∈ M, and obtain canonical isomorphisms Ex → Ey for all y ∈ M, depending smoothly on y. This shows that E is a trivial vector bundle. Geometry and Topology Lecture 17: Characteristic classes

after Michael Singera

Contents 17.1 Introduction...... 17–1 17.2 The first Chern class...... 17–2 17.2.1 First Chern class of a line bundle...... 17–2 17.2.2 First Chern class of higher rank bundles...... 17–2 17.3 Higher Chern classes...... 17–4 17.3.1 Traces of powers of the curvature...... 17–4 17.3.2 Digression on invariant functions and symmetric functions...... 17–5 17.3.3 The Chern class and the Chern character...... 17–7 17.4 Pontryagin Classes...... 17–11 17.5 A serious calculation...... 17–12 17.6 The Euler class and the Pfaffian...... 17–14 17.7 Gauss-Bonnet and Poincar´e-Hopf...... 17–15 17.8 A theorem of Bott on foliations and characteristic classes...... 17–21

17.1 Introduction

In this section we introduce some characteristic classes of real and complex vector bundles. In the setting of smooth manifolds and deRham cohomology, a k-dimensional characteristic class for real/complex vector bundles of rank n is a rule which to every smooth manifold M and real/complex vector bundle E → M of rank n assigns naturally a deRham cohomology class

k v(E) ∈ HdR(M).

The naturality condition has a precise meaning, which is that isomorphic vector bundles E, E0 on M must have v(E) = v(E0), and moreover

∗ ∗ k 0 v(f E) = f v(E) ∈ HdR(M ) (17.1) whenever f : M 0 → M is a smooth map and E is a rank n real/complex vector bundle on M. The manifolds can be with or without boundary, compact or not. k Many characteristic classes v are independent of rank, so that v(E) ∈ HdR(M) is defined whatever the rank of E, and satisfies v(E) = v(E ⊕ E0) (17.2) whenever E0 is a trivial vector bundle on M (real or complex, as appropriate). In such a case we may say that v is a k-dimensional stable characteristic class for vector bundles.

Among the best known characteristic classes are the Chern classes ck for complex vector bundles. These are stable characteristic classes. The Chern class ck(E) of a complex vector bundle E → M lives in 2k HdR(M). We will first discuss c1(L) for a complex line bundle L. The definition is almost obvious and we have already seen it in example 16.2.4.

a [email protected]

17–1 17–2

The Pontryagin characteristic classes pj are defined for real vector bundles. They are also stable and can be defined by much the same procedure as the Chern classes. But the result is zero in half of all cases, 4j so that we only get characteristic classes in HdR: 4j pj(E) ∈ HdR(M) for a real vector bundle E → M. Because pj is defined by (almost) the same formula as c2j, we have

j 4j pj(E) = (−1) c2j(E ⊗ C) ∈ HdR(M) (17.3) where E ⊗ C is the complexification of E, a complex vector bundle on M.

17.2 The first Chern class

17.2.1 First Chern class of a line bundle

In the previous lecture we noted that if L → M is a complex line-bundle and ∇ is a connection on L, then the curvature F = F (∇) is a closed differential 2-form on M, that is, dF = 0. Now F is a complex-valued differential 2-form. Looking at the real and imaginary parts separately, we may say that

2 [F ] ∈ HdR(M) ⊗ C . (17.4)

Proposition 17.2.1 If ∇ and ∇1 are two connections on L then F (∇) − F (∇1) is exact (in the image of d), so the cohomology class represented is independent of the chosen connection.

Proof Recall that if ∇ and ∇1 are two connections on the same line-bundle, then ∇1 = ∇ + a, where a is some 1-form with values in End(L). Now the line bundle End(L) is canonically trivial, so a is an ordinary complex-valued 1-form. It follows that

F (∇1) = F (∇) + da as required.  2 It follows from proposition 17.2.1 that [F ] in equation 17.4 belongs to the imaginary part of HdR(M)⊗C. Namely, we may put a Hermitian structure on the line bundle, and then a unitary connection ∇. For a unitary connection ∇, the curvature F (∇) has purely imaginary values because a skew-hermitian 1 × 1 matrix with complex entries is the same thing as a purely imaginary complex number. See equation 15.31. Therefore we define i c (L) := [F ] ∈ H2 (M) (17.5) 1 2π dR for a complex line bundle L → M with connection ∇ and curvature form F = F (∇). The constant 2π in the denominator ensures that for the line bundle L → CP 1 discussed in example 15.2.4, we get Z c1(L) = − 1 . 1 CP More generally, it ensures that, in the case where L is a complex line bundle over a compact smooth 2-dimensional manifold M, the integral of c1(L) over M is an integer. (The proof will not be given here.)

17.2.2 First Chern class of higher rank bundles

To deal with the higher-rank case we need some results concerning traces. Recall that on End(Cn) there is a trace map n tr : End(C ) → C (17.6) (sum of the diagonal elements) and tr(AB) = tr(BA) (17.7) for all matrices A and B. This implies also that tr(A) = tr(BAB−1). It follows that the trace is not only a number that we extract from a square matrix, but a number that we extract from an endomorphism, i.e., we have tr : End(W ) → C (17.8) 17–3 where W can be any finite dimensional real vector space. Apart from all that, the trace map is obviously a linear map. We need to generalise these simple facts to a multilinear form setting. Let V be a finite dimensional vector space over C. There is a trace map

k n k tr : alt (V, End(C )) → alt (V ) (17.9) which in the case k = 0 reduces to the map (17.6). Elements in the LHS of (17.9) can be thought of as alternating k-forms on V with values in the ring of n × n-matrices, or as n × n-matrices whose entries are alternating k-forms on V . With the second interpretation, the trace map is defined by taking the sum of the diagonal elements in a matrix of alternating k-forms on V . Keeping that interpretation, we have a bilinear “matrix multiplication” map

k n ` n k+` n alt (V, End(C )) × alt (V, End(C )) −→ alt (V, End(C )) (17.10)

q denoted by (A, B) 7→ A ∧ B. In formulae, if A is an n × n matrix with entries Ap which are alternating r k-forms on V , and B is an n × n matrix with entries Bq which are alternating `-forms on V , then A ∧ B P q s is again an n × n matrix whose entry in row p and column r is q Ap ∧ Bq , an alternating (k + `)-form. Therefore tr(A ∧ B) = (−1)k`tr(B ∧ A) (17.11) by the same verification which proves equation (17.7). Meanwhile the map tr in (17.9) can also be defined by taking an alternating k-form on V “with values” in End(Cn) and replacing each of these values by its trace, which is a number. Also the multiplication map in (17.10) can be defined by the same formula as the usual wedge product,

altk(V ) × alt`(V ) −→ altk+`(V ) except that products of “values” in the ring C have to be replaced by products of “values” in the ring End(Cn). These alternative definitions make it clear that Cn in (17.9) and (17.10) can be replaced by an arbitrary finite dimensional complex vector space W , so that we have

tr : altk(V, End(W )) → altk(V ) (17.12) and altk(V, End(W )) × alt`(V, End(W )) −→ altk+`(V, End(W )) (17.13) and equation (17.11) remains valid for the map (17.13).

Now let us imagine that E → M is a vector bundle of rank n on a smooth manifold. Substituting TxM for V and Ex for W in (17.12), and letting x run through M, we obtain

k k tr : Ω (M, End(E)) −→ Ω (M) ⊗R C. (17.14) Similarly (17.13) leads to

Ωk(M, End(E)) × Ω`(M, End(E)) −→ Ωk+`(M, End(E)) (17.15) and we still have tr(κ ∧ λ) = (−1)k`tr(λ ∧ κ) (17.16) for κ ∈ Ωk(M, End(E)) and λ ∈ Ω`(M, End(E)).

Applying the trace map (17.14) to the curvature F = F (∇) ∈ Ω2(M, End(E)) of a connection ∇ on E, we obtain 2 tr(F ) ∈ Ω (M) ⊗ C .

Proposition 17.2.2 The 2-form tr(F ) is closed: d(tr(F )) = 0 ∈ Ω3(M) ⊗ C. 17–4

Proof Let U be open in M and suppose that E|U is trivialised. The connection ∇ is then described by an n × n-matrix ω with entries in Ω1(U) ⊗ C, so that ∇s = ds + ω · s for any smooth section s of E|U. The curvature is given by the Maurer-Cartan formula F (∇) = dω + ω ∧ ω where d is the usual exterior derivative applied entry-wise. Therefore d(tr(F )) = d(tr(dω + ω ∧ ω)) = tr(d(dω + ω ∧ ω)) = tr(dω ∧ ω − ω ∧ dω) = tr(dω ∧ ω) − tr(ω ∧ dω).

By equation (17.16), the RHS is zero.  Proposition 17.2.3 The cohomology class represented by tr(F ) is independent of the chosen connection.

Proof Let ∇0 and ∇1 be two connections on E. Form E × R, viewed as a vector bundle on M × R of the same rank n. It is easy to construct a connection ∇∞ on E × R → M × R which restricts to ∇0 on E × {0} → M × {0} and to ∇1 on E × {0} → M × {0}. Let 2 z = [F (∇∞)] ∈ HdR(M × R) ⊗ C .

Let j0, j1 : M → M × R be the embeddings defined by x 7→ (x, 0) and x 7→ (x, 1). By construction ∗ ∗ 2 j0 (z) = [F (∇0)] and j1 (z) = [F (∇1)] in HdR(M) ⊗ C. As j0 and j1 are homotopic, it follows that [F (∇0)] = [F (∇1)].  Finally we observe that if E → M is a Hermitian complex vector bundle with a unitary connection ∇, then the differential 2-form tr(F ∇) belongs to the purely imaginary part of Ω2(M) ⊗ C. This can be verified using Hermitian local trivialisations of E and the observation that the trace of a skew-hermitian (square) matrix is purely imaginary. As any complex vector bundle E → M admits a Hermitian structure and a unitary connection, this together with proposition 17.2.3 demonstrates that [tr(F (∇))] always belongs to the purely imaginary part of Ω2(M) ⊗ C, for any connection ∇ on E. This leads to the definition i c (E) = [tr(F )] ∈ H2 (M) 1 2π dR where F is the curvature of a connection on E. Obviously this is also in agreement with the earlier definition of c1(E) in the case of complex line bundles.

17.3 Higher Chern classes

17.3.1 Traces of powers of the curvature

Now we can build higher topological invariants of a complex rank-n vector bundle E → M by considering traces of powers of the curvature. So let us look at ik θ (F ) = tr(F k) ∈ Ω2k(M) ⊗ (17.17) k (2π)k C where F = F (∇) is the curvature of some connection ∇ on E. Here we have written F k for the k-th wedge power F ∧ F ∧ · · · ∧ F in the sense of (17.15).

Proposition 17.3.1 d(θk(F )) = 0. Proof This is very similar to the proof of proposition (17.2.2). In the notation there, using a local trivialisation of E over an open set U ⊂ M, the curvature is still given by the Maurer-Cartan formula F (∇) = dω + ω ∧ ω. As before, dF = dω ∧ ω − ω ∧ dω (17.18) where d is the standard exterior derivative applied entry-wise. It is more useful in the following to convert this to dF = F ∧ ω − ω ∧ F (17.19) which follows by substituting F − ω ∧ ω for dω in equation (17.18). Equation (17.19) is sometimes called Bianchi’s identity. It implies that

k−1 k−1 k−1 X X X d(F k) = F r ∧ dF ∧ F k−r−1 = (F r+1 ∧ ω ∧ F k−r−1) − (F r ∧ ω ∧ F k−r) = F k−1 ∧ ω − ω ∧ F k−1 . r=0 r=0 r=0 k k Therefore, using formula (17.16) and the linearity of the trace, d(tr(F )) = tr(d(F )) = 0.  17–5

2k Proposition 17.3.2 The class [θk(F )] ∈ HdR(M) ⊗ C is independent of the choice of connection ∇ on E. It is purely imaginary when k is odd, and real when k is even.

Proof The first part can be proved exactly like proposition (17.2.3). The second part can be proved by assuming that E comes with a Hermitian structure and choosing a unitary connection ∇. Then it only remains to observe that the k-th power of a skew-Hermitian matrix is skew-Hermitian if k is odd, and Hermitian if k is even. The trace of such a matrix power is therefore purely imaginary (if k is odd) or real (if k is even).  We now have a sequence of cohomological invariants 2k [θk(F )] ∈ HdR(M) (17.20) associated to a complex vector bundle E on M. To recapitulate, F is the curvature of some connection ∇ on E. The closed differential form θk(F ) depends on ∇, but its cohomology class does not. The cohomology classes (17.20) are not yet the Chern classes of E, but the Chern classes are easily constructed out of them.

17.3.2 Digression on invariant functions and symmetric functions

A symmetric function f of the variables x1, . . . , xn is a function with the property

f(xσ(1), . . . , xσ(n)) = f(x1, . . . , xn) (17.21) for every permutation σ of {1, . . . , n}. In practice we will be dealing with polynomial functions and formal power series (i.e. power series where you don’t worry about convergence) so it may be preferable to think of the xj as indeterminates. Example 17.3.3 The power sums

π1 = x1 + x2 + ··· + xn, 2 2 2 π2 = x1 + x2 + ··· + xn, 3 3 3 π3 = x1 + x2 + ··· + xn, ··· are symmetric functions.

Example 17.3.4 The elementary symmetric functions

σ1 = x1 + x2 + ··· + xn, X σ2 = xixj, i

It is clear that any polynomial in the πk or the σk gives a polynomial symmetric function. Conversely, it can be proved that any polynomial symmetric function can be written as a polynomial function of the πk or the σk. In particular, the σk can be written in terms of the πk and conversely. For example,   X 1 X 1 X X 1 σ = x x = x x = x x − x2 = (π2 − π ) (17.22) 2 i j 2 i j 2  i j i  2 1 2 i

Example 17.3.5 The function Φ(A) = tr Ak is invariant.

Example 17.3.6 The function Φ(A) = det(I + A) is invariant.

There is a correspondence between invariant matrix polynomial functions and symmetric polynomials which goes as follows. If A = diag(x1, . . . , xn) is a diagonal matrix, then Φ(A) = f(x1, . . . , xn) defines the function f which corresponds to the invariant matrix polynomial Φ. Taking Q to be a permutation matrix, that is, a matrix with precisely one 1 in each row and column, and all other entries zero, the −1 identity Φ(QAQ ) = Φ(A) shows that f must be a symmetric function of the xj.

Example 17.3.7 The symmetric function corresponding to tr Ak is obtained by assuming A to be diag- k k k onal with entries (x1, . . . , xn). Then A = diag(x1 , . . . , xn) and

k X k tr A = xi = πk.

Similarly,   1 + x1 0 ··· 0  0 1 + x2 ··· 0  det(I + A) = det   = (1 + x )(1 + x ) ··· (1 + x )  . .. .  1 2 n  . . .  0 0 ··· 1 + xn

In terms of the elementary symmetric functions,

det(I + A) = 1 + σ1 + σ2 + ··· + σn.

Remark 17.3.8 Added by MW, June 2018. We need more precise statements about the relationship between invariant matrix polynomial functions and symmetric polynomials. Let us assume that Φ is an invariant polynomial for (n × n)-matrices. More precisely, Φ is a polynomial with complex coefficients in variables tjk where j ∈ {1, 2, . . . , n} and k ∈ {1, 2, . . . , n}. We can evaluate Φ on an (n × n)-matrix A with entries in C by substituting the entry ajk ∈ C for the variable tjk. This gives Φ(A) ∈ C. The invariance condition means that Φ(BAB−1) = Φ(A) for all A and invertible B.

As explained above, the matrix polynomial Φ determines a symmetric polynomial f = fΦ of the variables x1, . . . , xn. This is obtained by substituting xj for tjj and 0 for the tjk where j 6= k.

The map taking the n × n invariant matrix polynomial Φ to the symmetric polynomial fΦ is injective. Reason: if fΦ is zero, then Φ(A) is zero for all diagonal matrices A (matrices having zero entries in all the off-diagonal positions). It follows (by the invariance condition) that Φ(A) is zero for all diagonalizable matrices A, Since these form a dense open subset of the vector space of all n × n matrices with entries in C, it follows that Φ(A) = 0 for arbitrary A. But this implies that Φ is zero as a polynomial in the variables tjk .

We should also explain how Φ can be recovered from f = fΦ. If we begin with f, symmetric polynomial in the variables x1, . . . , xn , then we can write f(x1, . . . , xn) = g(σ1, . . . , σn) where g is a polynomial with coefficients in C and σ1, . . . , σn are the elementary symmetric polynomials in the variables x1, . . . , xn. (This is the fundamental theorem about symmetric polynomials. It is usually stated for the ground ring Z, which makes for a stronger statement.) Now we set

Φ = Φ((tjk)) := g(s1, s2, . . . , sn) where 1, s1, s2, . . . , sn are the coefficients of the polynomial

det(I + λ(tjk)) which we view as a polynomial in the single variable λ. The coefficients of this polyomial belong to the P polynomial ring C[..., tjk, ..]. For example, s1 is the trace of (tjk), also known as tjj, and sn is the 17–7

determinant of the matrix (tjk), not to be confused with the product t11t22 ··· tnn. It is straightforward to verify that fΦ for this Φ is the symmetric polynomial f that we started with. That being done, we should remember that Φ is a polynomial with complex coefficients of the variables tjk. Therefore we can make sense of Φ(A) for any (n × n)-matrix A with entries in any commutative C-algebra. For example, this could be the algebra

M 2p Ω (U) ⊗R C p≥0 where U is a smooth manifold (for us typically an open subset of a bigger smooth manifold M).

17.3.3 The Chern class and the Chern character

Now suppose Φ is a polynomial invariant matrix function. If F is the curvature form of a connection in a complex vector bundle E, we define Φ(iF/2π), which will be a differential form on M, according to the following recipe.

(i) Find the corresponding symmetric polynomial f in the variables x1, . . . , xn .

(ii) Express f as a polynomial in the power sums πk,

f(x1, . . . , xn) = P (π1, π2, ··· ).

(iii) Replace πj by θj in this, Φ(iF/2π) := P (θ1, θ2,...).

k For example, if Φ(A) = tr(A ), then Φ(iF/2π) = θk(F ). Permitted alternative: choose local trivializa- tions of E over open subsets Uα of M, so that iF/2π can be written as a matrix with entries in

2 M 2p Ω (Uα) ⊗R C ⊂ Ω (Uα) ⊗R C p≥0

L 2p and apply Φ directly to that matrix to obtain an element in p≥0 Ω (Uα) ⊗R C. By the invariance property of Φ, these elements agree on the intersections Uα ∩ Uβ. With all this algebraic machinery out of the way, we can define the Chern classes of a complex vector bundle.

Definition 17.3.9 Let E → M be a complex vector bundle of rank n and let ∇ be any connection in E, with curvature F . The total Chern class c(E) ∈ Heven(M) is the class of Φ(i/2πF ) where Φ(A) = dR det(I + A). Explicitly,

c(E) = 1 + c1(E) + c2(E) + ··· + cn(E) = [det(1 + iF/2π)]. (17.25)

2j Also cj(E) = [σj(iF/2π)] ∈ HdR(M) is called the j-th Chern class of E.

We shall develop the properties of the Chern classes shortly. Here is another important definition:

Definition 17.3.10 Let E → M be a complex vector bundle of rank n and let ∇ be any connection in E, with curvature F . The Chern character Ch(E) ∈ Heven(M) is defined to be dR Ch(E) = tr exp(iF/2π). (17.26)

Note that the Chern character involves symmetric functions that are not polynomials but rather are power series, namely f(x1, . . . , xn) = exp(x1) + exp(x2) + ··· + exp(xn). (17.27) Following the above recipe, we see 1 1 f(x , . . . , x ) = n + π + π + π + ··· (17.28) 1 n 1 2! 2 3! 3 17–8 so that 1 1 1 Ch(E) = n + θ + θ + θ + ··· + θ (17.29) 1 2! 2 3! 3 d! d where d is the integer part of dim M/2.

The relations between the elementary symmetric functions and the power sums give corresponding for- mulae for the Chern classes themselves,

c1 = θ1, 1 c = (θ2 − θ ), 2 2 1 2 1 1 1 c = θ − θ θ + θ3. 3 3 3 2 1 2 6 1 (See the Exercises for an algorithmic way to derive these formulae. Note, however, there is no simple formula for writing ck in terms of the θ’s.)

Theorem 17.3.11 The total Chern class c has the following properties.

(i) Naturality If f : N → M is a smooth map, then c(f −1(E)) = f ∗c(E).

(ii) Product formula c(E1 ⊕ E2) = c(E1)c(E2).

n (iii) Normalization Let L → CP be the tautological line bundle. Then ck(L) = 0 for k > 1 and Z c1(L) = − 1. 1 CP

Proof

(i) This follows because of the naturality of the constructions. We can use f ∗(∇) as our connection on −1 ∗ ∗ ∗ f (E) to compute the Chern class. Then F (f (∇)) = f F (∇) and it follows that θk(f (∇)) = ∗ f θk. Since the Chern classes are given by universal (i.e. independent of the particular bundle) polynomials in the θ’s, the result follows.

(ii) If ∇1 and ∇2 are connections on E1 and E2 respectively, we can use ∇1 ⊕ ∇2 as a connection on E1 ⊕E2. The curvature of this connection is then the direct sum of the separate curvatures, leading to c(E1 ⊕ E2) = det(1 + iF (∇1)/2π) det(1 + iF (∇2)/2π) = c(E1)c(E2).

(iii) We proved this in example 16.2.4 and equation (16.17). You only have to believe that the expression for F given there is indeed i/2 times the standard area form on S2 =∼ CP 1 as claimed, and that the area of the unit sphere S2 ⊂ R3 is 4π.



Remark 17.3.12 The above three properties actually characterize the Chern classes, as was proved long R R ∗ ago by Hirzebruch. If the formula P 1 c1(L) = −1 irritates you, you can restate it as P 1 c1(L ) = 1 ∗ C C where L = homC(L, C) is the dual of the tautological line bundle.

In addition to this Theorem, we have the following important properties:

Theorem 17.3.13 (i) If rank E = n, then ck(E) = 0 for k > n. (ii) If E = E0 ⊕ E00 (direct sum of complex vector bundles) where E00 is trivial of rank k, then

cn−k+1(E) = ··· = cn(E) = 0.

∗ (iii) Let E = homC(E, C) be the dual of E. Then

∗ k ck(E ) = (−1) ck(E) . 17–9

Proof

(i) This is really a definition. The point is that the elementary symmetric function σk(x1, . . . , xn) is zero for k > n. (ii) This follows at once from the product formula and the fact that c(trivial) = 1. Indeed,

0 00 0 0 c(E) = c(E )c(E ) = 1 + c1(E ) + ··· + cn−k(E )

shows that cn−k+1(E) = ··· = cn(E) = 0. (iii) Let Ecg → M be the complex vector bundle conjugate to E. That is, Ecg = E as a real vector bundle, but we define scalar multiplication in Ecg so that zv ∈ Ecg equalszv ¯ ∈ E whenever v ∈ Ecg = E and z ∈ C. A choice of Hermitian structure on E reveals that Ecg is isomorphic to E∗. Therefore all we need to show is that

cg k ck(E ) = (−1) ck(E) . A connection ∇ on E can also be viewed as a connection ∇cg on Ecg, since a section of E is also a section of Ecg. (More precisely, the Leibniz condition

∇(fs) = df · s + f · ∇s

for sections s of E and smooth C-valued functions f translates into ∇(fs¯ ) = df¯· s + f¯· ∇s

for sections s of Ecg.) As the connections are “the same”, parallel transport is the same in the two bundles and consequently we may be tempted to write

F (∇) = F (∇cg) ∈ Ω2(M, End(Ecg)) = Ω2(M, End(E)) .

cg cg But there is a little problem here: End(E ) = EndC(E ) and End(E) = EndC(E) are “the same” as real vector bundles, but here we need to view both of them as complex vector bundles, and as such they are again in the conjugate relationship. — When we apply an invariant polynomial of degree k, for example “trace of k-th power”, then we need to express endomorphisms by matrices. cg More precisely, for x ∈ M, any ordered basis for Ex can also serve as an ordered basis for Ex , cg and an endomorphism of Ex can also be viewed as an endomorphism of Ex , but the two matrices describing these endomorphisms are not always the same: the second is the entry-wise conjugate of the first. Therefore F (∇) = conjugate of F (∇cg), iF (∇)/2π = −conjugate of iF (∇cg)/2π . Substituting this equation into appropriate invariant polynomials gives the required relation between cg ∗ ck(E ) = ck(E ) and ck(E).

 Finally for the Chern character, we have

Theorem 17.3.14 The Chern character has the properties

Ch(E1 ⊕ E2) = Ch(E1) + Ch(E2), Ch(E1 ⊗ E2) = Ch(E1) ∧ Ch(E2). (17.30)

Proof The first part of this follows the proof of the corresponding fact about c(E): just use the direct sum connection on E1 ⊕ E2 to compute the LHS.

To deal with the second part, we choose connections ∇1 on E1 and ∇2 on E2. There is an induced connection on the tensor product, given by

∇(e1 ⊗ e2) = (∇1e1) ⊗ e2 + e1 ⊗ (∇2e2) for sections e1 and e2 of E1 and E2 , respectively. Correspondingly, using the Leibniz rule, we have

2 2 2 (d∇) (e1 ⊗ e2) = (d∇) e1 ⊗ e2 + e1 ⊗ (d∇) e2 (17.31) 17–10

2 For F (∇) ∈ Ω (M, End(E1 ⊗ E2)) this means that

F (∇)(x)(v, w) = F (∇1)(x)(v, w) ⊗ id2 + id1 ⊗ F (∇2)(x)(v, w) (17.32) where x ∈ M and v, w ∈ TxM and id1, id2 are the identity endomorphisms of E1 and E2 at x. We need to work out tr(exp(iF (∇)/2π)). This is now a purely algebraic exercise, and we recall the recipe described in the previous section. (See also remark 17.3.15 below.) According to that recipe, we may treat iF (∇1)/2π and iF (∇2)/2π as diagonal matrices with entries x1, . . . , xm and y1, . . . , yn respectively, where m = rank E1 and n = rank E2. Then the formula (17.31) shows that the corresponding matrix in the tensor product is an mn × mn diagonal matrix, with entries xi + yj. Thus the corresponding symmetric function is X X  X  exi+yj = exi eyj i,j

On the RHS we have the product of the symmetric functions corresponding to Ch(E1) and Ch(E2), which proves the result. 

Remark 17.3.15 Added by MW June 2018: After a great deal of thinking I am happy with the above proof, but I feel that many little details (about invariant polynomials) have been left out. Therefore I cooked up the following alternative argument. I am under the impression that equation (17.32) and the properties of the exponential function imply directly

exp(iF (∇)/2π) = exp(iF (∇1)/2π) ∧ exp(iF (∇2)/2π) so that


tr exp(iF (∇)/2π) = tr exp(iF (∇1)/2π)⊗exp(iF (∇2)/2π) = tr exp(iF (∇1)/2π) ∧tr exp(iF (∇2)/2π) .

p q (I have used the following notation and observations: if α ∈ Ω (M, End(E1)) and β ∈ Ω (M, End(E2)), p+q then α ∧ β can be made sense of as an element of Ω (M, End(E1 ⊗ E2)) and we have tr(α ∧ β) = p+q tr(α) ∧ tr(β) in Ω (M). And this is based on the following much more elementary remark: if V1 and V2 are finite dimensional complex vector spaces, A ∈ End(V1) and B ∈ End(V2), then A⊗B ∈ End(V1 ⊗C V2) and we have tr(A ⊗ B) = tr(A)tr(B).)

Example 17.3.16 (The complex tangent bundle of CP n) Let us try to get a grip of the tangent n n+1 bundle of CP . Suppose Lz is a 1-dimensional complex linear subspace in C , representing a point z of CP n. We have seen that any nearby point of CP n can be represented as the graph of a linear map n+1 from Lz to the unitary complement of Lz in C . More intrinsically, we can think of such linear maps n+1 n as elements of Hom(Lz, C /Lz). These linear maps thus represent tangent vectors to CP at z. Now over CP n we have two natural complex vector bundles, the tautological line bundle L and the bundle n+1 n Q whose fibre at z is C /Lz. The above argument identifies T CP with Hom(L, Q). Since the latter is a complex vector bundle, we shall refer to this as the complex tangent bundle of CP n and denote it by T 0CP n. (This is not so ad hoc as it might seem. CP n is an example of a complex manifold, and the tangent bundle of a complex manifold is, in a natural way, a complex vector bundle.) r Write C for the trivial complex vector bundle CP n+1 × Cr → CP n+1. From the exact sequence n+1 0 → L → C → Q → 0 (17.33) of vector bundles on CP n, we may apply hom(L, —) to get n+1 1 M ∗ 0 n 0 → C → L → T CP → 0 . (17.34) j=1 By the product formula n+1 0 n M ∗ n+1 c(T CP ) = c( L ) = (1 + x) (17.35) j=1 ∗ 2 n R where x = c1(L ) ∈ H ( P ) is characterised by 1 x = 1. This calculates the total Chern class of dR C CP the complex tangent bundle T 0CP n).

R 0 1 Remark 17.3.17 If n = 1, we see that 1 c1(T P ) = 2. This is closely related to the fact that the CP C Euler number of CP 1 =∼ S2 is 2. 17–11

17.4 Pontryagin Classes

The total Pontryagin class of a real vector bundle E → M can be defined directly by a formula very similar to that for the total Chern class:

even p(E) = det(1 + F/2π) ∈ HdR (M) , (17.36) where F is the curvature of a connection ∇ on E. This is independent of the connection chosen. Note the absence of a factor i in front of the F . The formal properties of the total Pontryagin class are very similar to those of the total Chern class. In particular there is a product formula p(E1 ⊕ E2) = p(E1)p(E2), and for a trivial vector bundle E we have p(e) = 1. A real vector bundle E → M of rank n determines a complex vector bundle EC → M of the same rank C C C n, where Ex = Ex ⊗R C for every x ∈ M. A connection ∇ on E determines a connection ∇ on E by the same procedure. The curvatures are the same:

2 2 F (∇) = F (∇C) ∈ Ω (M, End(EC)) = Ω (M, End(E)) ⊗ C . (17.37)

C C 2k It follows, by comparing the definition of c(E ) with the definition of p(E), that ck(E ) ∈ HdR(M) ⊗ C k 2k is equal to i times the component of p(E) in HdR(M). But we had already convinced ourselves that C 2k ck(E ) is always in the real part of HdR(M) ⊗ C. This has two consequences which we spell out:

2k Proposition 17.4.1 If k is odd, then the component of p(E) in HdR(M) is zero. If k = 2j is even, then 2k 4j j C the component of p(E) in HdR(M) = HdR(M) is equal to (−1) c2j(E ).

2k Therefore it is customary to suppress the components of p(E) in HdR(M) for odd k, and to write

4j p(E) = 1 + p1(E) + p2(E) + p3(E) + ··· , pj(E) ∈ HdR(M) .

Here is an application taken from Bott and Tu.

Example 17.4.2 Decide whether CP 4 can be embedded smoothly in Rk. If so, denote by N → CP 4 the normal bundle to the embedded CP 4. Then T CP 4 ⊕ N is trivial (the restriction of T Rk to CP 4). Hence 4 p(T CP )p(N) = 1. Now we computed the total Chern class of the complex tangent bundle T 0CP 4. It was c = (1+x)5, where 2 4 4 x is the standard generator of HdR(CP ). The Pontryagin classes of T CP are, up to signs and some re-indexing, the Chern classes of the complexification T CP 4 ⊗C of T CP 4. It is not hard to see that when you take a complex vector bundle E → M, forget the complex structure and complexify the underlying real vector bundle, you get something isomorphic to E ⊕ Ecg as a complex vector bundle. Therefore and by the product formula for Chern classes, the total Chern class of T CP 4 ⊗ C is (1 + x)5(1 − x)5 = (1 − x2)5 = 1 − 5x2 + 10x4

4 2 4 2 4 2 and so p(T CP ) = 1 + 5x + 10x . Hence p(N) = 1 − 5x + 15x , in other words p1(N) = −5x 4 ∗ 4 4 and p2(N) = 15x . Now if you know a few things about HdR(CP ), you will know that x 6= 0, and consequently p2(N) 6= 0. This implies that N must have rank ≥ 4. It follows that k must be at least 12. So CP 4 cannot be smoothly embedded in Rk for k < 12. Indeed we have proved more. Namely, CP 4 cannot be smoothly immersed in Rk for k < 12. (A smooth immersion f : M1 → M2 is a smooth map such that all linear maps Txf : TxM1 → TxM2 are injective. By the implicit function theorem, such a map is locally a smooth embedding, that is, every point x ∈ M1 has an open neighborhood U in M1 such that f|U is a smooth embedding from U to M2. A smooth immersion does not have to be injective, but it has a normal vector bundle just like a smooth embedding.)

Example 17.4.3 Let M be a smooth oriented compact manifold of dimension 4s (with empty boundary). Let J = J(p1, p2,... ) be a polynomial (with rational coefficients) in the Pontryagin classes which is homogeneous of geometric degree 4s. (To determine the geometric degree, a slightly improvised concept, 3 2 2 we assign degree 4k to pk. For example, if s = 7 we could take J(p1, p2,... ) = p1p2 + p1p3 + p3p4.) Evaluating each pk on the tangent bundle TM we obtain

4s J(p1(TM), p2(TM),... ) ∈ HdR(M) . 17–12

This class can be represented by some ψ ∈ Ω4s(M), automatically a closed differential form, and the integral Z ψ ∈ R M is independent of the choice of representative. (We have a preferred choice for ψ if we choose a connection ∇ on TM first.) We call it the characteristic (Pontryagin-)number J[M] associated with M and the polynomial J. If the coefficients of J are integers, then J[M] is an integer, but we are not in a good position to prove this since it would require a comparison between ordinary integral cohomology and deRham cohomology, and another description of the Pontryagin classes as integral cohomology classes. But we are well qualified to make the following observation. If M is the boundary of a compact smooth manifold W (with boundary), then Z J[M] = ψ = 0 M by Stokes’ theorem. This is so because the cohomology class [ψ] = J(p1(TM), p2(TM),... ) is in the image 4s 4s of the restriction map HdR(W ) → HdR(M). More precisely, it is the image of J(p1(TW ), p2(TW ),... ) under that restriction map.

17.5 A serious calculation

The tangent bundle TSn of the sphere Sn ⊂ Rn+1 has a standard Riemannian metric and, following Ricci’s recipe, a standard connection ∇. Here we want to describe this is in local coordinates. Following Ricci’s recipe means: we exploit the fact that TSn is a vector subbundle of a trivial vector bundle Sn × Rn+1 → Sn to make a connection on TSn. We do not rely explicitly on the celebrated Levi- Civita theorem which says that any Riemannian metric on the tangent bundle TM of a smooth manifold determines a preferred connection on TM, characterized by such and such properties. (But I believe there is a theorem saying that the older and easier Ricci recipe for making connections on tangent bundles agrees with the more general Levi-Civita recipe in the situation of a smooth submanifold M of Rk.) It may not be absolutely necessary, but I prefer to work with a local trivialization of TSn which respects the Riemannian metric(s). This requires more thought and preparation. There is the reward that we get an illustration of the theory concerning connections on vector bundles with Riemannian metric.

n n+1 n+1 n+1 For x = (x1, . . . , xn, 0) ∈ R ⊂ R let Kx be the linear map from R to R given by

v 7→ hv, xien+1 − hv, en+1ix .

n+1 (I am writing ej for the standard basis vectors of R .) For example, if x = ej where j ∈ {1, 2, . . . , n}, then Kx is described by an (n + 1) × (n + 1)-matrix which has an entry 1 in column j, row n + 1, and an entry −1 in column n + 1, row j; all other entries are zero. In general, Kx is skew-symmetric:

hKx(v), vi = 0

n+1 for all v ∈ R . This is easy to verify from the definition. Note also that Kx depends linearly on x. Let

ρx = exp(Kx)

n+1 n+1 (matrix exponential). For x 6= 0, the linear map ρx : R → R is a rotation in the plane spanned by x and en+1, by an angle equal to ±kxk. For x = 0, it is the identity. The map x 7→ ρx is smooth (as a map from Rn to the vector space of real (n + 1) × (n + 1)-matrices). Let U = {x ∈ Rn | kxk < ε} (for some fixed small ε) and let E → U be the trivial vector bundle, n+1 0 0 n E = U × R . Let E ⊂ E be the vector subbundle such that Ex = ρx(R ) for x ∈ U. This inherits a Riemannian metric from E. The vector bundle E0 → U is our local model for TSn. More precisely, we have a chart ϕ : U → Sn given 0 n 0 by ϕ(x) := ρx(en+1) and a map E → TS which takes v ∈ Ex to the same v, viewed as an element of n 0 ∼ ∗ n the tangent vector space Tϕ(x)S . This amounts to an isomorphism of vector bundles, E = ϕ (TS ). It respects the Riemannian metrics. 0 The smooth sections sj of E → U defined by

sj(x) = ρx(ej) 17–13 for j = 1, 2, . . . , n form an everywhere orthonormal system. By the Ricci method, the standard connection on E determines a connection on E0, and in fact a connection compatible with the Riemanian metric. We want to write this in the form d + A 1 n 0 (where A ∈ Ω (U; End(R )) using the trivialization of E given by the sections sj together. Let px be 0 the orthogonal projection from Ex to the summand Ex viewed as an endomorphism of Ex. Following the recipe of Ricci the j-th column of A at the point x ∈ U is

−1 p0ρx px · dsj

n+1 −1 ... that is to say, we apply px to the values dsj(x)(v) ∈ R = Ex, and then we apply p0ρ(x) in order to express the result in the basis s1(x), . . . , sn(x). This can also be written in the form

−1 p0 · ρx dρ(x) · ej

−1 −1 because p0ρx px = p0ρx and because sj(x) = ρx(ej). Note that applying p0 on the left means deleting the last row of an (n + 1) × (n + 1)-matrix and applying ·ej on the right means taking the j-th column. This is easy to understand, so our task boils down mainly to calculating

−1 ρx dρ(x) where x ∈ U. Now it is time to make some sacrifices. Because of symmetry considerations we should be happy if we can compute the curvature matrix at the origin 0 ∈ U. For that we need first derivatives of the connection form A at 0. To calculate the connection form itself we may require the first derivatives of x 7→ ρx and −1 −1 also of x 7→ ρx . Therefore it looks as if we need the first and second derivatives of x 7→ ρx and x 7→ ρx at 0 ∈ U. Consequently we write

2 −1 2 ρx = exp(Kx) ≈ I + Kx + (Kx) /2 , ρx = exp(−Kx) ≈ I − Kx + (Kx) /2 .

Since Kx depends linearly on x, these approximations are second order polynomial mappings. We get

1 n+1 n+1 dρ(x) ≈ v 7→ Kv + KxKv/2 + KvKx/2 ∈ alt (R , End(R )) where x ∈ U and v ∈ Rn+1, and  K K − K K  ρ−1 dρ(x) ≈ v 7→ K + v x x v ∈ alt1( n+1, End( n+1)) x v 2 R R where terms involving two or more factors Kx have been dropped. This can also be written in the form  [K ,K ] v 7→ K + v x ∈ alt1( n+1, End( n+1)). v 2 R R Now we must delete the last row and column of all (n + 1) × (n + 1)-matrices in sight to obtain A(x). Then the Kv summand disappears and we can write [K ,K ]  A(x)(v) ≈ v x jk . 2 j,k∈{1,2,...,n} We are pleased to note that this is a skew-symmetric matrix for any fixed x ∈ U. But we also note that A(0) = 0, so that the formula F = dA + A ∧ A simplifies (at 0 ∈ U) to F = dA. Moreover, our expression for A(x)(v) (which is an approximation) is bilinear alternating in the variables x and v; this simplifies the calculation of dA at 0. See remark 17.5.1 just below. Therefore the curvature matrix at 0 ∈ U can be described as the (n × n)-matrix with entries

(v, w) 7→ − [Kv,Kw]jk (17.38) where v, w ∈ Rn (and j, k ∈ {1, 2, . . . , n}).

Remark 17.5.1 At the very end of the above calculation, the following was used. If B : Rn × Rn → R is a skew-symmetric bilinear map, and β ∈ Ω1(Rn) is defined by β(x)(v) = B(v, x), then dβ(0) = −2B. Here d denotes the exterior derivative. (Exercise.) For example, B(v, x) could mean: the (j, k)-entry of the matrix [Kv,Kx]/2. 17–14

17.6 The Euler class and the Pfaffian

The Euler class is a characteristic class for vector bundles which admits a definition in terms of curvature, much like the Pontryagin classes. But there are special requirements. - We need a (smooth) vector bundle E → M of even dimension 2k and we get an Euler class in 2k HdR(M). - The vector bundle has to be oriented. (This condition can be dropped, at the price that the Euler 2k class turns out to be an element of a variant of HdR(M).) - To get a specific differential 2k-form representing the Euler class, we need to equip the vector bundle with a Riemannian metric, and then a connection ∇ which respects the metric. Since we assume a Riemannian metric on E, we can choose a local trivialization (over some open U in M) which respects the Riemannian metric, and we get as usual a description of ∇ in the form d + A where A ∈ Ω1(U; End(R2k). The matrix A (of differential 1-forms) is now skew-symmetric. The Pfaffian Pf is something like a polynomial invariant function, homogeneous of degree k, for (some) matrices X of size 2k × 2k. It satisfies (Pf (X))2 = det(X). But (i) it is only defined for skew-symmetric matrices X; (ii) the invariance property is Pf (BXBT ) = det(B)Pf (X), where B can be any 2k × 2k-matrix. Note that BXBT in (ii) is again a skew-symmetric matrix, so that this property makes some sense. Unfortunately, it does not seem to be what we need; we need BXB−1. But fortunately we use property (ii) mostly in the case where B is an orientation-preserving orthogonal matrix, in which case BT = B−1. In that situation (ii) tells us Pf (BXB−1) = Pf (X). In view of these good properties, we can make the following definitions in a coordinate free spirit. Let V be a real vector space of finite dimension 2k (where k > 0) with an orientation and an inner product (positive definite). Let g : V → V be a linear endomorphism which is skew-symmetric, i.e., we have hg(v), wi = −hv, g(w)i for all v, w ∈ V . Then we can speak of Pf (g) ∈ R, the Pfaffian. (Recipe for this: choose a linear isometry f : V → R2k which preserves orientation. Then take Pf (fgf −1).) If we change the orientation of V , then g is still a perfectly good endomorphism of V , but Pf (g) with the new orientation is −Pf (g) with the old orientation. The name Pfaffian was chosen not by Pfaff, of course, but by Cayley, who also established the fundamental properties. A formula for the Pfaffian as a matrix polynomial is given in Morita’s book, The geometry of characteristic classes. He does not explain where it comes from. Lang, in his old algebra book, explains where the polynomial formula comes from, but does not reveal it! This is very amusing. The formula is (according to Morita) 1 X Pf (X) := sign(σ) x x ··· x 2kk! σ1,σ2 σ3,σ4 σ2k−1,σ2k−2 σ where the sum is taken over all permutations on 2k letters. (I have written σj instead of σ(j) to make it less legible. I bow my head in shame for omitting the factor sign(σ) in an earlier edition.) The equation (Pf (X))2 = det(X) should be easy to deduce from that. The equation Pf (BXBT ) = det(B)Pf (X) follows from (Pf (X))2 = det(X) and some continuity considerations, I hope. (But this works only over the field R and it may not be the best way to do it.) Another way to write the formula for Pf (X): leave out the factor (2kk!)−1 in front of the sum sign and make up for that by summing only over the permutations σ which satisfy

σ1 < σ3 < ··· < σ2k−1 and also σ1 < σ2, σ3 < σ4,..., σ2k−1 < σ2k. In this description, there is one summand for every (unordered) partition of {1, 2,..., 2k − 1, 2k} into k subsets of cardinality 2. 17–15

Definition 17.6.1 The Euler form of E and ∇, with curvature F = F (∇) ∈ Ω2(M; End(E)), is the element Pf (F/2π) ∈ Ω2k(M).

Regarding this definition, remember that we assumed a Riemannian metric and an orientation on E, and ∇ was supposed to respect the metric. This has the consequence that F (∇) is a differential 2-form taking values in the skew-symmetric endomorphisms of E. In terms of a vector bundle trivialization (respecting the orientation and the metric), we may write ∇ = d + A where A ∈ Ω1(M; End(R2k) is skew-symmetric. Then dA + A ∧ A ∈ Ω1(M; End(R2k) is again skew-symmetric, and this is easy to verify.

Theorem 17.6.2 The Euler form of E and ∇ is a closed form.

Proof Choose some x ∈ M. The plan is to look at d(Pf (F )) at the point x. We can do the calculation using a local trivialization (respecting the Riemannian metric and the orientation), so we replace M by an open neighborhood U of M and we may pretend that E is U × R2k. As usual, ∇ can now be written in the form d + A where A ∈ Ω1(U; End(R2k). Then F (∇) = dA + A ∧ A. It is in general unpleasant to have to work with Pf (dA + A ∧ A) or with dPf (dA + A ∧ A). But let us now reconsider our choice of trivialization of E around x. With a little care, we can choose the trivialization (still respecting the metric) so that the matrix A is zero at the distinguished point x ∈ U. Then the calculation is much easier (at the point x) and it is fairly clear that we get zero for dPf (dA + A ∧ A) at the point x. 

2k The cohomology class determined by the Euler form of E and ∇ is the Euler class, eu(E) ∈ HdR(M).

Lemma 17.6.3 The cohomology class eu(E) does not depend on the Riemannian metric selected for E. It changes sign if we reverse the orientation of E.

Proof For the first part: if we have two choices of Riemannian metric on E, we may construct a single Riemannian metric on E × R → M × R which agrees with one of them over M × {0} and with the other 2k over M × {1}. We get eu(E × R) ∈ HdR(M × R). Using the two maps y 7→ (y, 0) and y 7→ (y, 1) from 2k M to M × R, we can pull this back to HdR(M) and we get the two elements competing for the title eu(E). For the second part: this follows from our coordinate free definition of Pf (g) for endomorphisms g : V → V of vector spaces with inner product and orientation. 

Proposition 17.6.4 If E admits an everywhere nonzero section s : M → E, then

2k eu(E) = 0 ∈ HdR(M) .

Proof If there is such a section, then we can write E as a Whitney sum E] ⊕ L, where L is a trivial real line bundle, L = M × R. Then we can choose a Riemannian metric on E]. On L, we have a preferred Riemannian metric (so we don’t choose anything). We also choose a connection ∇] on E] compatible with the Riemannian metric, and the obvious (flat) connection on L. Putting the two together, we get a connection ∇ on E. With this particular connection, it is easy to see that the Euler form of E and ∇ is strictly zero. [ From a local trivialization of E], respecting the metric, we get a local trivialization of E and the connection becomes d + A with A ∈ Ω1(M; End(R2k), where the last row and last column of A are filled with zeros. It follows that last row and last column of F = dA + A ∧ A are also filled with 2k zeros. Then it follows that Pf (F ) is zero, as an element of Ω (M). ] 

Remark 17.6.5 The square of the Euler form of E → M and ∇ is the Pontryagin form of E and ∇, in 4k 2 4k Ω (M). It follows that (eu(E) = pk(E) ∈ H (M). (We are still assuming that E has fiber dimension 2k, that E and M are oriented, and more: E equipped with Riemannian metric, ∇ an orthogonal connection.) Proof : (Pf (F/2π))2 = det(F/2π). The right-hand side of this equation is precisely the contribution to det(1 + F/2π) coming from Ω4k(M).

17.7 Gauss-Bonnet and Poincar´e-Hopf

(Under construction.) 17–16

Let E → M be a smooth vector bundle, where M is oriented, closed (i.e., compact and with empty boundary) and E is also oriented as a vector bundle. If the fiber dimension of E agrees with the dimension n of M, then the Euler number of E can be defined. It is an integer. To determine the Euler number, choose a smooth section s : M → E which is transverse to the zero section M,→ E. (Such a thing exists by transversality theory.) Then there will only be finitely many solutions to s(x) = 0, and each has a sign ±1 attached to it. Counting these solutions with their signs gives the Euler number of E. (See also section 8.5.) [The sign of a solution x is determined as follows. Let ζ : M → E be the zero section. The differential of s at x is an injective linear map a : TxM → T(x,0)E and the differential of ζ at x is an injective linear map

b : TxM → T(x,0)E.

Therefore im(a) and im(b) are n-dimensional linear subspaces of the 2n-dimensional vector space T(x,0)E. By assumption (transversality), we have im(a) + im(b) = T(x,0)E. This leads to a linear isomorphism im(a) → T(x,0)/im(b) of n-dimensional vector spaces. These are both oriented (the target T(x,0)/im(b) can be identified with the vertical subspace Ex = T0Ex of T(x,0)). If this linear isomorphism from im(a) to T(x,0)/im(b) is orientation preserving, the multiplicity is +1; if not, then −1. ]

Lemma 17.7.1 The Euler number of E is well defined, i.e., does not depend on the choice of a section s : M → E transverse to the zero section.

Proof It is convenient to read the provisional definition of the Euler number just given as follows. We choose a section s : M → E of E → M transverse to the zero section. We obtain

s−1(zero section of E) = {x ∈ M | s(x) = 0} which is a 0-dimensional compact smooth oriented manifold (here a submanifold of M). Then we pass to the oriented bordism class of that. (The bordism group of 0-dimensional compact smooth oriented manifolds is isomorphic to Z. See section 8.3.) Suppose that s0 and s1 are two smooth sections of E → M which are both transverse to the zero section. For the vector bundle E × [0, 1] −→ M × [0, 1] we can find a section

s[0,1] : M × [0, 1] −→ E × [0, 1] which is transverse to the zero section, agrees with s0 on M × {0} and agrees with s1 on M × {1}. Then the preimage of the zero section under s[0,1] is a compact smooth oriented manifold L of dimension 1 whose oriented boundary is the disjoint union

−1 −1 s1 (zero section of E) t − s0 (zero section of E). (The minus sign means: reverse orientation.) This means that L is an oriented bordism from the oriented −1 −1 0-dimensional manifold s0 (zero section of E) to s1 (zero section of E). 

Proposition 17.7.2 If the dimension of M (which is also the fiber dimension of E → M) is odd, then the Euler number of E is zero.

Proof Let s : M → E be a smooth section which is transverse to the zero section. Then −s : M → E is also a smooth section which is transverse to the zero section. Each solution of s(x) = 0 with multiplicity +1 can be viewed as a solution of −s(x) = 0 with multiplicity−1, and vice versa. (We are using the fact that −id : V → V is orientation reversing if V is an odd dimensional real oriented vector space.) Since we may use s or −s to calculate the Euler number, it follows that the Euler number is zero. 

Theorem 17.7.3 Let E → M be an oriented vector bundle of fiber dimension n on an oriented closed smooth manifold of dimension n. If n is even, we have Z Euler number of E = eu(E) . M 17–17

This can be said to be the Gauss-Bonnet theorem in all even dimensions, especially if we use a definition of eu(E) relying on connections and curvature. Proof It turns out that the theorem is easier to prove in a more general formulation which allows non- compact manifolds M. To this end, suppose that E → M is an oriented vector bundle of fiber dimension n on an oriented smooth manifold M of dimension n. We do not want to assume that M is compact, but we need some additional information to make up for possible noncompactness. The additional information is a smooth section s : M → E of E such that the set {x ∈ M | s(x) = 0} is compact in M. (Much of this additional information will turn out to be unnecessary information, but it is convenient to carry it around for now.)

In this situation we define the Euler number of (E, s) as follows. Choose a smooth section s1 : M → E which is transverse to the zero section and which agrees with s outside a compact subset of M. Count the solutions of s1(x) = 0 with their multiplicities ±1, as usual. This gives the Euler number. It is well defined (independent of the choice of s1) by the same arguments that we employed in the compact case. We can also make sense of the integral Z eu(E, s). M The problem here, at first sight, is that M can be noncompact. Therefore we should define the Euler form of E as an element of Ω2k(M) with compact support. This can be done as follows. We choose a Riemannian metric on E such that, for all y ∈ M outside a compact set C ⊂ M, s(y) ∈ Ey has length 1. ⊥ For such y, the vector space Ey is then an orthogonal sum Ly ⊕Ly where Ly = R·s(y) is the 1-dimensional linear subspace spanned by s(y) ∈ Ey . We choose a neighborhood of C with compact closure C1 in M and a connection ∇ on E which is compatible with the Riemannian metric already chosen and which, on ⊥ the restriction of E to M r C1 , respects the splitting into line bundle L and complement L . Now the Euler form of E and ∇ has support contained in C1 . Therefore Z Euler form of E and ∇ M R is a real number. We take that as the meaning of M eu(E, s). By a straightforward application of Stokes’ theorem to the manifold M × [0, 1], that real number is well defined, i.e., does not depend on the choice of Riemannian metric on E and connection ∇ (subject to various conditions) that we had to make. Note that the new definitions agree (for rather trivial reasons) with the old definitions in the situation where M is compact; in that case that additional datum s is entirely superfluous. Now the plan is to show Z Euler number of (E, s) = eu(E, s) (17.39) M using the new definitions (in which M is allowed to be noncompact, but the zero set of s is required to be compact). We will do so mainly by observing that some good properties hold. R (i) Both the Euler number of (E, s) and the integral M eu(E, s) are invariant under homotopies of s, say (st)t∈[0,1] where each st is a smooth section of E → M depending smoothly on t; but here we require that the set {(x, t) ∈ M × [0, 1] | st(x) = 0} be compact. R (ii) Both the Euler number of (E, s) and the integral M eu(E, s) are invariant under certain restrictions. Namely, if U is an open subset of M which contains the compact set {x ∈ M | s(x) = 0}, then the Euler number of (E|U , s|U ) is equal to the Euler number of (E, s). Similarly, the numbers R R M eu(E, s) and U eu(E|U , s|U ) are the same.

(iii) If M is a disjoint union M1 ∪ M2 (both M1 and M2 open in M), then X Euler number of (E, s) = Euler number of (E|Mj , s|Mj ) j=1,2

and similarly Z X Z eu(E, s) = eu(E|Mj , s|Mj ) . M j=1,2 Mj

These properties are easy to establish. (This is left to the reader.) We can immediately draw the following conclusion: 17–18

(iv) it is enough to establish (17.39) in the case where M = Rn (even n) with the standard orientation, E is the trivial (oriented) vector bundle with fiber Rn and s is tranverse to the zero section, and the only solution to s(x) = 0 is x = (0, 0,..., 0) ∈ Rn. Reasons for the reduction (iv): by (i), we may assume that s is transverse to the zero section; then the set T := {x ∈ M | s(x) = 0} is finite. For each x ∈ T we may choose an open neighborhood Ux in M n such that Ux is (oriented) diffeomorphic to R , and in such a way that the Ux for x ∈ T are pairwise disjoint. Then, using (ii), we may altogether replace M by the disjoint union of the Ux for x ∈ T . By (iii), it is enough to investigate each Ux (and the restriction of E to it) separately. — In the situation described in (iv), where M = Rn etc., the Euler number is clearly ±1. The integral R M eu(E, s) is not easy to calculate, but we can say that up to homotopy, as in (ii), there are only two cases to be investigated. These can be described as follows: E = Rn × Rn and M = Rn and E → M is the first projection, and s : M → E is a map of the form

n n n R 3 x 7→ (x, A(x)) ∈ R × R where A is an invertible linear map from Rn to Rn. We can treat this multitude of cases as just “two cases” because, in view of (ii), we care only whether det(A) > 0 or det(A) < 0. (The general linear group GL(n, R) has exactly two connected components.) Suppose that in the case where det(A) > 0 we get a real number c for the integral Z eu(E, s). M Then, in the case where det(A) < 0, we will get −c for that integral. This is easy to show by a symmetry consideration. Therefore, using (i),(ii),(iii) and (iv) we obtain Z eu(E, s) = c · (Euler number of (E, s)). M In words, the integral of the Euler class is proportional to the Euler number, with proportionality constant c. And by the above reasoning, this is truly valid for all M and E and s, subject only to conditions as in (17.39). It remains to show that c = 1. To establish this we can look at one particular case where we know that the Euler number of (E, s) is nonzero. We choose the case where M = Sn and E → M is the tangent bundle of Sn. In this case the Euler number is 2 (since n is even). Therefore we want to show that Z eu(TSn) = 2 . Sn Here we can use section 17.5 and specifically formula (17.38). In the notation there, the tangent space at 0 ∈ U is identified with the tangent space of Sn at (0, 0,..., 0, 1) by means of the chart ϕ, where ϕ(x) = ρx(en+1), and more precisely, by the differential of ϕ at 0. That differential is a linear isomorphism which respects the standard inner products and the standard orientations. We described the entry in row j and column k of the curvature matrix as

(v, w) 7→ − [Kv,Kw]jk

n 2 n where v, w ∈ R . This simplifies to dxj ∧ dxk ∈ alt (R ). Substituting this for Fjk in the expression Pf (F/2π) for the Euler form we get therefore 1 n! ω 2n/2(n/2)!(2π)n/2

n where ω ∈ alt (Rn) is the standard volume form. Therefore we have Z n! 1 · 3 ··· (n − 3) · (n − 1) eu(TSn) = (volume of Sn) = (volume of Sn) = 2 . n/2 n/2 n/2 Sn 2 (n/2)!(2π) (2π) n Here I have used a well known formula for the volume of S in the case of even n. 

Theorem 17.7.4 Let M be an oriented smooth closed n-dimensional manifold. Let E → M be the cotan- gent bundle, dual of the tangent bundle. Then the Euler number of E agrees with the Euler characteristic of M. 17–19

(Here it is assumed that you know a definition of Euler characteristic in terms of homology or trian- gulations; not the definition in section 8.5., which would make the claim a tautology.) That statement probably also has a name, which could be: Poincar´e-Hopf index theorem. By combining the Gauss- Bonnet theorem as stated above with the Poincar´e-Hopfindex theorem, we learn that the Euler number of T ∗M alias TM agrees with the Euler characteristic of M in the even-dimensional oriented case. (In the odd-dimensional oriented case, both are zero.) The separation which I am making between Gauss-Bonnet and Poincar´e-Hopfmay not be 100% histori- cally correct. I am making it in order to emphasize that one of these results is about curvature of vector bundles (and can therefore be stated in some generality) whereas the other is not about curvature, and not about general vector bundles, but only about cotangent bundles. Proof (of theorem 17.7.4, using Morse theory.) By an advanced transversality theorem, we can find a smooth function f : M → R such that df, viewed as a section of the cotangent bundle T ∗M → M, is transverse to the zero section. (Such as function is called a Morse function, after Marston Morse.) The points x ∈ M where df is zero are the critical points of f. There are finitely many of them, say z1, . . . , zr ∈ M , and (by perturbing f slightly and renumbering the critical points if necessary) we can assume that f(z1) < f(z2) < ··· < f(zr) .

The transversality assumption means that, in local coordinates for M about a critical point zj , the second derivative matrix for f at zj is nondegenerate. At the same time it is a symmetric matrix, and consequently it will have only real nonzero eigenvalues, of which p(j) positive and q(j) negative, so that p(j) + q(j) = n. The number q(j) is called the index of the critical point zj of the Morse function f. For example, since we have an absolute minimum of f at z1 , the index q(1) is 0, and since we have an absolute maximum at zr , the index q(r) is n.

The celebrated Morse lemma states that we can choose local coordinates about each critical point zj so that in these coordinates we have

p(j) n  1 X X f(x , . . . , x ) = x2 − x2 . 1 n 2  k k k=1 k=p(j+1)

n (More correctly stated, there exist an open subset Uj of R containing the origin, and a chart ϕj : Uj → M taking 0 to zj , such that fϕ can be described by such a pretty formula.) The original proof by Morse was not easy, but later Palais found a very elegant proof (of a more general statement), so that this is now called the Morse-Palais lemma. (See for example Lang’s book Differential manifolds.) Using a partition of unity we can choose a Riemannian metric on TM (also known as Riemannian metric on M) such that ϕj : Uj → M respects the Riemannian metrics (where Uj has the standard Riemannian n metric as an open subset of R ). Of course it may be necessary to replace Uj by a smaller open subset, still containing the origin. Now that we have a Riemannian metric on M, it makes sense to speak of grad(f), the gradient vector field of f. It is a smooth vector field on M. It is also bounded, so that its integral curves have the form γ : R → M (no explosions). Note in passing that for each critical point zj there is a constant integral curve of grad(f), with constant value zj . By construction, the vector field grad(f) has a pretty description in our preferred local coordinates about each critical point zj , which is

grad(f)(x1, . . . , xn) = (x1, x2, . . . , xp(j), −xp(j)+1,..., −xn) . (17.40)

For c ∈ R let Mc ⊂ M be the preimage under f of the interval (−∞, c]. For example Mc = ∅ if c < f(z1) and Mc = M if c ≥ f(zr). The idea of Morse theory is to ask: how does the homotopy type of Mc change as c runs from −∞ to +∞. Along these lines, the most important statements for us are as follows.

(i) If c, d ∈ R and c < d, and if the half-open interval (c, d ] contains no critical values of f, then the inclusion Mc → Md is a homotopy equivalence. Indeed, Mc is a deformation retract of Md .

(ii) If c, d ∈ R and c < d, where d = f(zj) is a critical value, and the interval [ c, d ) contains no (other) critical values of f, then the inclusion Mc → Md is not a homotopy equivalence but it extends to a map k Mc ∪g D −→ Md k−1 k which is a homotopy equivalence. Here g : S = ∂D −→ ∂Mc ⊂ Mc serves as attaching map for k a k-cell. (The space Mc ∪g D can also be described as the mapping cone of g.) The number k is the index of the critical point zj , also known as q(j). 17–20

Before getting into the details of this, let us note that (i) and (ii) imply the statement that we want, equality between the Euler number of TM and the Euler characteristic of M. Namely, we can choose real numbers c0, c1, . . . , cr in such a way that

c0 < f(z1) < c1 < f(z2) < ··· < cr−1 < f(zr) < cr .

Then, from the long exact sequence in homology of the pair

(Mcj+1 ,Mcj )

q(j) we deduce that Euler characteristic of Mcj+1 minus Euler characteristic of Mcj is equal to (−1) , so that the Euler characteristic of M is X (−1)q(j). j

∗ q(j) But this is exactly the Euler number of T M, since (−1) happens to be the multiplicity of zj as a zero of the section df : M → T ∗M.—

Sketch proof of (i). A homotopy (ht : Md → Md)t∈[0,1] can be defined as follows. For x ∈ Mc and all t ∈ [0, 1] let ht(x) = x. For x ∈ Md r Mc and t < 1 let ht(x) be the unique point y which is on the same trajectory of grad(f) as x and which has f(y) = (1 − t)f(x) + tc. That definition of ht(x) works also in most cases where t = 1 and x ∈ Md r Mc. If there are cases where it does not work, then c = f(zj) for some j and we must define h1(x) = zj, which is also the limit of the hs(x) for s → 1. The homotopy (ht) is a strong deformation retraction, so that Mc turns out to be a strong deformation retract of Md .

Sketch proof of (ii). We are assuming that d = f(zj). If j = 1, then Md is a single point and Mc = ∅ and k = 0, so the statement is true. From now on we assume j > 1. We are interested in the integral curves of grad(f) which approach the critical point zj from below, i.e., integral curves γ : R → M of grad(f) such that lim γ(t) = zj t→+∞ while f(γ(t)) < f(zj) = d for all t. By the description (17.40) of grad(f), each of these integral curves must approach zj along a straight line segment, in the coordinates of (17.40). Therefore these integral curves are in canonical bijection with elements of Sq(j)−1, the unit sphere in

q(j) p(j) q(j) n 0 × R ⊂ R × R = R ,

q(j)−1 and we write v ↔ γv to indicate that relationship, where v ∈ S . In this way we obtain a smooth embedding q(j)−1 −1 g : S −→ f (c) = ∂Mc ⊂ M q(j)−1 by taking v ∈ S to the unique point in ∂Mc which intersects the curve γv . Moreover, we obtain an inclusion q(j) Mc ∪g D −→ Md (17.41) q(j) (which extends the inclusion Mc → Md) by mapping points of the form tv ∈ D , where kvk = 1 and t ∈ (0, 1], to γv(s), where s satisfies f(γv(s)) = (1 − t)d + tc; in the case t = 0 we map tv to zj ∈ Md. The image of this inclusion is the union of Mc , the (images of the) curves γv and the single point zj . It remains to show that (17.41) is a homotopy equivalence. I believe an honest (therefore probably tedious) way to organize it is as follows. We write q(j) = k and we think of that inclusion map as a map of the form A0/B0 → A1/B1 induced by an injective (or inclusion) map of pairs (A0,B0) → (A1,B1), where

k−1 k−1 k−1 A0 := Mc ∪g S × [c, d],B0 := S × {d},A1 := Mc ∪ ∂Mc × [c, d],B1 := g(S ) × {d}.

k−1 k−1 (The notation is informal; in the description of A0, the subspace S × {c} of S × [c, d] is to be k−1 identified with g(S ) ⊂ ∂Mc ⊂ Mc , and similarly in the description of A1 , the subspace ∂Mc × {c} of ∂Mc × [c, d] is to be identified with ∂Mc ⊂ Mc .) It is clear that A0 → A1 and B0 → B1 are homotopy equivalences. Because the inclusions B0 → A0 and B1 → A1 have the homotopy extension property, we can conclude that the map (A0,B0) → (A1,B1) is a homotopy equivalence of pairs. It follows that the induced map of quotients A0/B0 → A1/B1 is a homotopy equivalence.  17–21

17.8 A theorem of Bott on foliations and characteristic classes

Definition 17.8.1 Let M be a smooth manifold of dimension n.A smooth foliation of codimension q on M is (represented by) a smooth atlas for M (contained in the maximal atlas which defines the smooth structure of M) with charts ϕα : Uα → M subject to the following conditions.

q n−q - Each Uα has the form Uα,1 × Uα,2 where Uα,1 is open in R and Uα,2 is open in R .

- For all α, β and every x ∈ im(ϕα)∩im(ϕβ) there exists an open neighborhood Wx ⊂ im(ϕα)∩im(ϕβ) of x such that the partition of Wx into subsets

Wx ∩ ϕα(y × Uα,2)

(where y ∈ Uα,1) agrees with the partition of Wx into subsets

Wx ∩ ϕβ(z × Uβ,2)

(where z ∈ Uβ,1).

Two such atlases, say F1 and F2, are said to be equivalent (define the same foliation) if F1 ∪ F2 still satisfies the conditions.

Definition 17.8.2 In the notation of the previous definition, a continuous path f : [0, 1] → M is vertical (with respect to the foliation F) if for every t ∈ [0, 1] there exists a neighborhood [a, b] of t in [0, 1] such that f|[a,b] can be written as ϕα ◦ g where g :[a, b] → Vα × Wα is constant in the Vα -direction. Elements x, y of M are said to be in the same leaf of the foliation if there is a vertical path f : [0, 1] → M such that f(0) = x and f(1) = y. This relation, in the same leaf, is an equivalence relation; the equivalence classes are the leaves of the foliation.

Here is some vocabulary which we will need in a moment. A smooth submersion f : X → Y is a smooth map such that, for every z ∈ X, the differential TzX → Tf(z)Y is a surjective linear map. A smooth immersion f : X → Y is a smooth map such that, for every z ∈ X, the differential TzX → Tf(z)Y is an injective linear map. The implicit function gives important information about the local behavior of such maps. For example, a smooth immersion f : X → Y need not be injective, but every z ∈ X has an open neighborhood Vx such that f restricted to Vx is a smooth embedding (in particular, injective). We make a careful distinction between injective smooth immersions and smooth embeddings. By defini- tion (of embedding), the image of a smooth embedding f : X → Y is a smooth submanifold of Y . The image of an injective immersion does not have to be a submanifold.

Example 17.8.3 Smooth submersion.

Example 17.8.4 Foliated torus.

Example 17.8.5 Everywhere nonzero smooth vector field.

Proposition 17.8.6 For each leaf L of the foliation, there exist a smooth connected manifold L\ of dimension n − q and an injective smooth immersion L\ → M whose image is exactly L. (There should also be a uniqueness statement here, but we do not have the time for it.)

Proof Without loss of generality, each Uα (from the foliation atlas) is connected. For fixed x ∈ L we can choose a chart ϕα : Uα → M from the foliation atlas such that x = ϕα(a, b) where a ∈ Uα,1 and b ∈ Uα,2 . Then the map Uα,2 → L given by y 7→ ϕα(a, y) can be used as a chart for the set L. These charts make up a smooth atlas for L. But: the topology on L determined by this atlas need not be in agreement with the topology of L as a subspace of M. Therefore we write L\ for L with the topology determined by the atlas. We want to know, at least, that L\ is Hausdorff. Indeed it is metrizable. Why? Without loss of generality, M is connected. We can equip TM with a Riemannian metric. Then M has an (honest) metric in which the distance between two points p, q is the infimum of the lengths of smooth paths from p to q in M. If we restrict the metric on M to obtain a metric on L, then this may not agree with the topology of L\ 17–22 determined by the atlas which we specified. But there is another metric on L as follows: the distance between two points p, q in L is the infimum of the lengths of smooth vertical paths from p to q in M. This metric on L is compatible with the topology of L\ determined by the atlas. For the injective immersion L\ → M we can now simply take the composition L\ → L,→ M, where the first arrow is the identity map (as a map of sets). This is also continuous, but as we have noted it need not be a homeomorphism (if L is viewed as a subspace of M).  Observations regarding space of leaves.

Remark 17.8.7 Foliation as a local equivalence relation.

Definition 17.8.8 The vertical tangent bundle T F of the foliation F on M is a vector subbundle of the tangent bundle TM. Briefly, the fiber of T F at x ∈ M is the linear subspace of TxM consisting of the tangent vectors v which are tangent to the unique leaf L through x. The normal bundle NF of the foliation F on M is the quotient vector bundle T M/T F on M.

This definition of the vertical tangent bundle T F can be made more precise. Let ϕα : Uα → M be a chart from the foliation atlas. Since Uα is a product Uα,1 × Uα,2 , the tangent bundle TUα → Uα has a ∗ distinguished vector subbundle of fiber dimension n − q which we can describe as pr2(TUα,2), where pr2 is the projection from Uα,1 × Uα,2 to Uα,2. We may call it the vertical subbundle of TUα. The image of this under T (ϕα): TUα → TM is exactly T F restricted to im(ϕα). In other words, we have a convincing chartwise description of T F as a vector subbundle of TM.

Theorem 17.8.9 The normal bundle NF → M of the foliated manifold (M, F) admits a connection ∇ which is vertically flat.

This means the following. Let a, b be two points in the same leaf of F and let u, v : [0, 1] → M be two vertical paths from a to b. If u is vertically smoothly homotopic to v by a homotopy which is stationary on 0 and 1, then the parallel transport isomorphisms NFa → NFb determined by ∇ and the paths u and v, respectively, are the same. Proof (Sketch.) We will construct chartwise solutions to this, and then piece the solutions together using a partition of unity. So let ϕα : Uα → M be a chart from the foliation atlas. Let us write gα for the composition −1 ϕα pr1 M ⊃ im(ϕα) / Uα / Uα,1 This is important to us because the vector bundle

∗ gα(TUα,1) is canonically identified with the restriction of NF to im(ϕα). We now choose a connection ∇α on TUα,1 ∗ (just any) and pull it back under gα to obtain a connection gα(∇α) on

NF|im(ϕα) . By construction, this has the property of being vertically flat. But the following observation is more remarkable. Let w : [0, 1] → im(ϕα) ∩ im(ϕβ) be a (smooth) vertical path. Then the parallel transport ∗ ∗ in NF along this path for the connections gα(∇α) and g (∇β) is the same. A better formulation:

∗ ∗ ∗ ∗ w (gα(∇α)) = w (gβ(∇β)) ; these are connections on a vector bundle of fiber dimension q on the interval [0, 1]. ∗ [ An attempt to explain this better. The identification of gα(TUα,1) with the restriction of NF to im(ϕα) means that the restriction of NF to any fiber of gα is trivialized, that is to say, equipped with an isomorphism to a vector bundle of the form “base space times vector space”. This trivialization over each ∗ α fiber of gα already determines the parallel transport along vertical paths using the connection gα(∇ ). Therefore the parallel transport along vertical paths does not depend on our choice of ∇α; it depends only ∗ on the identification of gα(TUα,1) with the appropriate restriction of NF, as stated. Therefore it suffices ∗ to establish some compatibility between the identification of gα(TUα,1) with the appropriate restriction ∗ of NF and the identification of gβ(TUβ,1) with the appropriate restriction of NF.] 17–23

Now choose a partition of unity on M subordinate to the open cover of M by subsets im(ϕα). Use this to piece the connections ∇α together; the result is a connection ∇ on all of NF. By the remarkable observation, for a vertical smooth path w : [0, 1] → M, the parallel transport in NF along w using ∗ connection ∇ agrees with the parallel transport in NF along w using any of the connections gα(∇α), where applicable. (This is applicable when the restriction of w to some subinterval of [0, 1] lands in ∗ im(ϕα).) This has the consequence that ∇ is still vertically flat, since each gα(∇α) was vertically flat. 

Corollary 17.8.10 Suppose that M is equipped with a smooth codimension q foliation F. Let NF → M be the normal bundle of the foliation and let Q(p1, p2, p3,... ) be a polynomial in the Pontryagin classes, homogeneous of geometric degree 4k. If 4k > 2q, then

4k Q(p1(NF), p2(NF),... ) = 0 ∈ HdR(M).

(For the definition of the geometric degree, see example 17.4.3.) Proof Choose a connection ∇ for the vector bundle NF → M which is vertically flat, as in the theorem. Fix some x ∈ M and choose an ordered basis for the q-dimensional vector space NFx. The curvature of 2 ∇ at x is then given by a (q × q)-matrix with entries λjk ∈ alt (TxM). Due to the vertical flatness, each entry λjk satisfies the following:

λjk(v, w) = 0 if v, w ∈ T Fx ⊂ TxM.

Therefore, if we apply invariant polynomials to these entries, take products etc., we will always get ∗ elements in the graded subalgebra Kx of alt (TxM) generated by alternating forms τ in degree 2 which satisfy the above condition, τ(v, w) = 0 whenever v, w ∈ T Fx ⊂ TxM. It is an exercise (homework) to 2s show that this subalgebra has the property Kx ∩ alt (TxM) = 0 whenever 2s > 2q. 

In other accounts, it is often corollary 17.8.10 which is called Bott’s theorem on foliations etc. . In such a case the statement which is called theorem 17.8.9 here can be expected to occupy a position of obscurity somewhere in the proof of the other statement (which we have called a corollary).