Fibre Bundles

Andrew Kobin

Spring 2017 / Spring 2018 Contents Contents

Contents

0 Introduction 1

1 Fibre Bundles 2 1.1 Locally Trivial Fibre Bundles ...... 2 1.2 Principal Bundles ...... 5 1.3 Classifying Spaces and Universal Bundles ...... 10 1.4 Reduction of the Structure Group ...... 16 1.5 Gauge Groups ...... 19

2 Vector Bundles 22 2.1 Vector and Tangent Bundles ...... 22 2.2 Sections ...... 25 2.3 The Induced Bundle ...... 28 2.4 Bundle Operations ...... 30

3 Characteristic Classes 34 3.1 Metrics on Vector Bundles ...... 34 3.2 Stiefel-Whitney Classes ...... 35 3.3 The Universal Bundle ...... 44 3.4 Construction of Stiefel-Whitney Classes ...... 51 3.5 Oriented Bundles ...... 53 3.6 Chern Classes ...... 55 3.7 Pontrjagin Classes ...... 57 3.8 General Characteristic Classes ...... 58

4 K-Theory 70 4.1 K-Theory and Reduced K-Theory ...... 71 4.2 External Products in K-theory ...... 73 4.3 Bott Periodicity ...... 75 4.4 Characteristic Classes in K-Theory ...... 82

5 Chern-Weil Theory 85 5.1 Connections on a Vector Bundle ...... 86 5.2 Characteristic Classes from Curvature ...... 92 5.3 The Levi-Civita Connection ...... 100

6 Index Theory 103 6.1 Differential Operators ...... 103 6.2 Hodge Decomposition ...... 108 6.3 The Index Theorem ...... 110 6.4 Dolbeault Cohomology ...... 113

i 0 Introduction

0 Introduction

This document follows two courses in fibre bundles taught respectively by Dr. Slava Krushkal in Spring 2017 and Dr. Thomas Mark in Spring 2018 at the University of Virginia. The main concepts covered are:

ˆ Locally trivial fibre bundles

ˆ Principal bundles and classifying spaces

ˆ Vector bundles and their important properties

ˆ Characteristic classes, including Stiefel-Whitney, Chern and Pontryagin classes

ˆ Orientability

ˆ Euler class

ˆ Elements of K-theory

ˆ Connections and Chern-Weil theory

ˆ The Atiyah-Singer index theorem.

Standard references include: Milnor-Stasheff’s Characteristic Classes, Hatcher’s Vector Bun- dles and K-Theory and Husem¨oller’s Fibre Bundles.

1 1 Fibre Bundles

1 Fibre Bundles

1.1 Locally Trivial Fibre Bundles

Definition. A bundle is a triple (E, B, p) consisting of topological spaces E,B and a con- tinuous map between them, p : E → B. We call E the total space of the bundle and B the base space.

Definition. A morphism of bundles (E1,B1, p1) → (E2,B2, p2) is a pair of maps f : ˜ B1 → B2, f : E1 → E2, making the diagram

f˜ E1 E2

p1 p2 f B1 B2

−1 ˜ −1 commute and preserving fibres, that is, for any x ∈ B1 and y ∈ p1 (x), f(y) ∈ p2 (f(x)). Example 1.1.1. When f is the identity B → B, we say the bundles and morphisms of bundles are over B. p Example 1.1.2. Let E −→ B be a bundle and A ⊆ B be a subspace. Then the restriction −1 p|A : E|A := p (A) → A is a bundle over A and the natural inclusion E|A ,→ E induces a bundle morphism

E|A E

p|A p

A B

Example 1.1.3. Let F be any space. The trivial bundle over B with fibre F is the canonical projection p : B × F → B. Definition. A locally trivial fibre bundle over B with fibre F , hereafter shortened to a p fibre bundle, is a bundle E −→ B that is locally isomorphic to a trivial bundle with fibre F . That is, there exists a collection of open sets {Ui} together with bundle isomorphisms ∼ ϕi : E|Ui = Ui × F , called local trivializations. p Given a fibre bundle E −→ B with locally trivial open sets {Ui}, there is a choice of −1 isomorphism ϕi : E|Ui → Ui for each Ui. For each x ∈ Ui, p (x) is homeomorphic to F via −1 ϕi, so when x ∈ Ui ∩Uj, there are two different ways to identify p (x) with F . In particular, −1 ϕj ◦ ϕi :(Ui ∩ Uj) × F → (Ui ∩ Uj) × F is a homeomorphism and induces a homeomorphism F → F . In other words, for each pair of open sets Ui,Uj, there is a function

gij : Ui ∩ Uj −→ Homeo(F )

2 1.1 Locally Trivial Fibre Bundles 1 Fibre Bundles

called a transition map. One can show that the gij are continuous. Note that the gij satisfy a few nice conditions, namely that gii = 1, gijgji = 1 and gijgjkgki = 1 pointwise on F . In fact, the data of a covering {Ui} and continuous transition maps {gij} satisfying these conditions are enough to determine a fibre bundle.

Proposition 1.1.4. Let B and F be spaces and suppose there is an open cover {Ui} of B and continuous maps gij : Ui ∩ Uj → Homeo(F ) satisfying:

(a) For all x ∈ Ui, gii(x) = 1, where 1 is the identity map F → F .

(b) For all x ∈ Ui ∩ Uj, gij(x)gji(x) = 1.

(c) For all x ∈ Ui ∩ Uj ∩ Uk, gij(x)gjk(x)gki(x) = 1.

Then there exists a fibre bundle over B with fibre F which is trivial over the Ui and has the gij as its transition maps. Moreover, such a bundle is unique up to isomorphism of bundles over B.

Proof. (Sketch) Define the total space of the bundle by Y E := (Ui × F )/ ∼ i

where ∼ is the equivalence relation identifying (x, y) ∈ Ui ×F and (x, gij(x)(y)) ∈ Uj ×F for any x ∈ Ui ∩ Uj and y ∈ F . The bundle map p : E → B is given by the natural projection on each disjoint piece, which is well-defined on all of E by (a) – (c) and is continuous since p the gij are continuous. It is straightforward to check that E −→ B is a locally trivial bundle with fibre F having transition maps gij. p˜ For uniqueness, suppose Ee −→ B is another bundle with gij as its transition maps. For −1 each Ui, there is a fibre-preserving homeomorphismp ˜ (Ui) → Ui × F . Define a bundle map −1 Ee → E by sending a pointx ˜ ∈ p (Ui) to the corresponding point in Ui × F ⊆ E. One now checks that this is a bundle isomorphism.

p1 p2 Given two bundles E1 −→ B and E2 −→ B with fibre F and local trivializations {(Ui, ϕi)} and {(Vj, ψj)}, respectively, when can we tell if they are isomorphic? First, we may refine the covers {Ui} and {Vj} by taking intersections until each bundle is trivial over both collections. Thus we may assume both bundles are trivial over {Ui}. Suppose f : E1 → E2 is an isomorphism of bundles over B. Then each Ui corresponds to a local isomorphism of bundles

−1 fi := ψi ◦ f ◦ ϕi : Ui × F → E1|Ui → E2|Ui → Ui × F.

In fact, this gives us a necessary and sufficient condition for f to induce a bundle isomorphism.

Lemma 1.1.5. The map f : E1 → E2 is an isomorphism of bundles over B if and only if there exist local bundle isomorphisms fi : Ui × F → Ui × F that are compatible with the −1 −1 transition maps ϕj ◦ ϕi and ψj ◦ ψi , i.e. the following diagram commutes:

3 1.1 Locally Trivial Fibre Bundles 1 Fibre Bundles

fi (Ui ∩ Uj) × F (Ui ∩ Uj) × F

−1 −1 ϕj ◦ ϕi ψj ◦ ψi

fj (Ui ∩ Uj) × F (Ui ∩ Uj) × F

Example 1.1.6. Let M ⊆ RN be a smooth manifold and define the tangent bundle to M by N N TM = {(x, v) ∈ R × R | x ∈ M, v ∈ TxM}. The projection p : TM → M, (x, v) 7→ x gives a fibre bundle structure on TM with fibre −1 ∼ n p (x) = TxM = R if M is an n-manifold. If {(Ui, ϕi)} is an atlas of coordinate charts for the manifold structure on M, then the derivatives of their transition maps,

−1 gij : x 7−→ dϕi(x)(ϕj ◦ ϕi ),

n define the transition maps gij : Ui ∩ Uj → Homeo(R ) of the tangent bundle. Since these n derivatives are linear, the gij in fact have target GLn(R) ⊂ Homeo(R ). This reflects the fact that TM is a vector bundle (see Chapter 2). For example, let M = S2 be the 2-sphere. We may view S2 = CP 1, the complex projective line, which has a covering by two coordinate charts

U0 = {[z0, z0] | z0 6= 0} and U1 = {[z0, z1] | z1 6= 0}

z1 z0 with homeomorphisms ϕ0 : U0 → , [z0, z1] 7→ and ϕ1 : U1 → , [z0, z1] 7→ . Then C z0 C z1 −1 1 the transition maps ϕj ◦ ϕi : C → C are both given by λ 7→ λ . Thus the transition maps 2 1 g01, g10 for the tangent bundle TS are each given by λ 7→ − λ2 , viewed as a 2×2 matrix with × × × real entries in GL2(R), or a scalar element of GL1(C) = C . Notice that g01 : C → C is homotopic to a map S1 → S1 of degree −2. It is no accident that S2 has Euler characteristic 2 – we will see this connection explicitly borne out by the Euler class (see Section 3.5). Let G be a topological group and F a space on which G acts by homeomorphisms, i.e. such that there is a continuous homomorphism ρ : G → Homeo(F ). Definition. A fibre bundle with fibre F has structure group G if it has transition functions gij : Ui ∩ Uj → G. Example 1.1.7. Fibre bundles with fibre F = Rn (resp. F = Cn) and structure group G = GLn(R) (resp. GLn(C)) are called real (resp. complex) vector bundles (see Chapter 2). Example 1.1.8. A bundle with fibre F = Sn and structure group G = O(n + 1) is called an (orthogonal) sphere bundle. Example 1.1.9. The trivial structure group G = {1} corresponds to the trivial bundle over any base (with any fibre). p Given a bundle E −→ B with fibre F and structure group G, there may be other spaces on which G acts, so that the same transition functions gij : Ui ∩ Uj → G produce bundles with those other spaces as fibres.

4 1.2 Principal Bundles 1 Fibre Bundles

p Example 1.1.10. Let E −→ B be a sphere bundle with fibre Sn and structure group O(n+1). 0 n+1 0 p Then O(n + 1) ⊆ GLn+1(R) naturally acts on all of R , so it determines a bundle E −→ B with fibre Rn+1 and the same transition maps (and E0 is in fact a vector bundle over B). Conversely, a bundle E0 with fibres Rn+1 and structure group O(n + 1) determines a “unit sphere bundle” E inside E0, with fibres Sn.

n Example 1.1.11. For a bundle with fibre R and structure group GLn(R), the natural n action of GLn(R) on R preserves a vector space structure on each fibre. In this way, any bundle with this data becomes a vector bundle (see Chapter 2).

The takeaway from these examples is that if G acts on a space F and preserves a certain structure on F , then bundles with structure group G inherit this structure.

1.2 Principal Bundles

Continuing from the discussion in the previous section, any group G has a natural action on itself by left multiplication. Therefore, for any base B with covering {Ui} and transition maps gij : Ui ∩ Uj → G, there is a canonically determined bundle P → B with fibre G, structure group G and transition maps gij. The bundle P → B is called a G-principal bundle over B.

Definition. A principal G-bundle is a space P with a free right action of G such that the quotient map p : P → P/G defines a fibre bundle over the base B = P/G with G-equivariant −1 local trivializations ϕi : p (Ui) → Ui × G. Lemma 1.2.1. A principal G-bundle P → B = P/G is equivalent to the data of a base B, a covering {Ui} and transition maps gij : Ui ∩ Uj → G. Remark. Note that the fibres G in a G-principal bundle P → B need not have a group structure (compatible with the bundle structure); they are in general only G-sets. A principal bundle whose fibres have a compatible group structure is sometimes called a group bundle.

Let p : P → B be a principal G-bundle and F a left G-space, via a continuous homomor- phism ρ : G → Homeo(F ). Then p defines a fibre bundle E(P,ρ) → B with fibre F , called an associated bundle.

Definition. Let P → B be a principal G-bundle and ρ : G → Homeo(F ) a continuous homomorphism. The associated bundle for this data is the fibre bundle E(P,ρ) → B with total space E(P,ρ) := P ×G F = {(y, t) ∈ P × F | (y · g, t) ∼ (y, ρ(g) · t)} and local trivializations

E(P,ρ)|Ui = (Ui × G) ×G F −→ Ui × F [(x, h), t] 7−→ (x, ρ(h−1) · t).

5 1.2 Principal Bundles 1 Fibre Bundles

Example 1.2.2. For G = GLn(R), a principal G-bundle is the same thing as a vector bundle with fibre Rn via the associated bundle operation

{principal GLn(R)-bundles} −→ {n-dimensional vector bundles} n (P → B) 7−→ (P ×GLn(R) R → B). Conversely, let V be an n-dimensional real vector space and let F (V ) be the set of ordered bases, or n-frames, of V . Then GLn(R) acts freely and transitively on F (V ). Given a vector ∼ n bundle E → B with fibre V = R , let F (E) denote the frame bundle associated to E:

F (E) := {(x, e1, . . . , en) | x ∈ B, (e1, . . . , en) ∈ F (V )}.

Then by construction there is a free right action of GLn(R) on F (E), so F (E) is a principal n GLn(R)-bundle. Moreover, one can check that the assignments P 7→ P ×GLn(R) R and E 7→ F (E) are inverses. Example 1.2.3. A similar proof to that in Example 1.2.2 using orthogonal frames shows that there is a bijective correspondence between principal O(n)-bundles and n-dimensional vector bundles with inner product. p Definition. A section of a fibre bundle E −→ B is a map s : B → E such that p ◦ s = idB. p Lemma 1.2.4. A principal G-bundle P −→ B admits a section s : B → P if and only if P is a trivial G-bundle. Proof. ( ⇒ = ) If P ∼= B ×G then any function f : B → G, for example the constant function b 7→ e ∈ G, defines a section x 7→ (x, f(x)). ( =⇒ ) If s : B → P is a section, define

ϕ : P −→ B × G y 7−→ (p(y), g(y))

where g(y) ∈ G is the unique element such that y = s(p(y)) · g(y). Since G acts freely on P , this map is well-defined. Moreover, it is easy to check ϕ is G-equivariant. Finally, the continuity of ϕ follows from the lemma below. Lemma 1.2.5. For a principal G-bundle P → B, define

∗ P = {(y1, y2) ∈ P × P | y2 = y1g for some g ∈ G}.

∗ Then the map τP : P → G, (y1, y1g) 7→ g is continuous. Proof. Locally, P ∼= U × G so because continuity is a local question, we may assume P = −1 U × G. If y1 = (x, g1) and y2 = (x, g2), then τP is given by τP (y1, y2) = g1 g2. This is evidently the composition

q (U × G) × (U × G) → G × G −→ G

−1 where the first map is projection in each coordinate and q :(g1, g2) 7→ g1 g2. Since G is a topological group, each of these is continuous so τP is continuous.

6 1.2 Principal Bundles 1 Fibre Bundles

Definition. Let P be any free right G-space. We say P admits local sections if there is an open cover {Ui} of B = P/G such that the quotient map p : P → P/G has the property

−1 that each p|p (Ui) : P |Ui → Ui has a section, called a slice. The set of G-orbits of a slice is called a tube. Corollary 1.2.6. A free right G-space P is a principal G-bundle over B = P/G if and only ∗ if P admits local sections and τP : P → G is continuous. Example 1.2.7. Let G be a topological group and H a subgroup (which is a topological group with the subspace topology). Then H acts on G by right multiplication, giving a fibre bundle G → G/H with fibres H. By Corollary 1.2.6, this is a principal H-bundle if and only if there exists a slice for the H-action through the identity e ∈ G. (The map τP is guaranteed to be continuous in this case. Moreover, by homogeneity, it is enough to have a slice through the identity.) Proposition 1.2.8. Let G be a topological group, H ⊆ G a subgroup and suppose there exist subsets T ⊆ G and W ⊆ H, each containing e and with W open, such that multiplication m : T × W → TW ⊆ G is a homeomorphism onto its image. Then G → G/H is a principal H-bundle. Proof. By Example 1.2.7, it’s enough to find a slice through e ∈ G. Write W = H ∩ V for some open set V ⊆ G. Since e ∈ V and G is a topological group, there is an open set U ⊆ V containing e and satisfying U −1U ⊆ V . Set S = T ∩ U and consider the multiplication map m : S × H → SH ⊆ G. By hypothesis, restriction of m to S × W is a homeomorphism, but since H is a topological group, this means m is a local homeomorphism on all of S × H. −1 Moreover, if (s1, h1) and (s2, h2) have the same image under m, then s1h1 = s2h2, so s2 s1 = −1 −1 −1 −1 −1 −1 h2h1 . But s2 s1 ∈ U U ⊆ V and h2h1 ∈ H, so s2 s1 = h2h1 ∈ V ∩ H = W . Since m is a homeomorphism on S × W , it follows that (s1, h1) = (s2, h2). Thus m is a one-to-one local homeomorphism, hence a global homeomorphism. This gives us a slice SH → S × H = p−1(SH) through e as required. Corollary 1.2.9. Let G be a Lie group and H ⊆ G a closed subgroup. Then G → G/H is a principal H-bundle. Proof. A closed subgroup of a Lie group is itself a Lie group, so the tangent space at the identity TeH is a vector subspace of TeG. Let V be the orthogonal complement of TeH ⊆ TeG and define a map f : V × H −→ G (v, h) 7−→ exp(v)h, where exp : V → G is the restriction of the exponential map to V . Then the derivative of ∼ f is equal to the identity V × TeH → V × TeH = G, so f is a local diffeomorphism. Thus there exist open sets T ⊆ V and W ⊆ H, with e ∈ W , such that f| is a diffeomorphism. e Te×W Taking T = exp(Te) ⊆ G, Proposition 1.2.8 implies G → G/H is a principal H-bundle. Proposition 1.2.10. Let H ⊆ G be any subgroup of a topological group for which G → G/H is a principal H-bundle. Then for every subgroup K ⊆ H, the natural map G/K → G/H is a fibre bundle with fibre H/K.

7 1.2 Principal Bundles 1 Fibre Bundles

Example 1.2.11. Let k < n be integers and consider the inclusion of orthogonal groups O(n − k) ,→ O(n) given by the “lower diagonal inclusion” ! Ik 0 A 7→ 0 A .

The Lie group O(n) acts (on the right) on the set of orthonormal k-tuples of vectors in any n-dimensional inner product space with stabilizers O(n − k). Consider the Stiefel manifold

n Vn,k := {orthonormal k-tuples in R }. ∼ Then Vn,k = O(n)/O(n − k) and by Corollary 1.2.9, the quotient map O(n) → Vn,k is a n principal bundle with fibre O(n − k). Similarly, the Grassmannian manifold Grk(R ) = Vk,n/O(k) has a canonical principal given as follows. Take G = O(n), H = O(n − k) × O(k) and K = O(n − k). Then H/K = O(k) so by Proposition 1.2.10 and Corollary 1.2.9, the n quotient map Vn,k → Grk(R ) is a principal bundle with fibre O(k). (See Section 3.3 for further details.)

Lemma 1.2.12. Suppose P1 → B and P2 → B are principal G-bundles and f : P1 → P2 is a morphism of G-bundles over B. Then f is an isomorphism.

Proof. Locally, f looks like U × G → U × G, (x, g) 7→ (x, f0(x)g) where f0 is the function U → G defined uniquely by f(x, e) = (x, f0(x)) (and extended by G-equivariance). The −1 −1 inverse on U × G is then f :(x, h) 7→ (x, f0(x) h), so f is a local homeomorphism. Since a bundle morphism preserves fibres, f is also a global homeomorphism, hence a bundle isomorphism.

p Definition. Let P −→ B be a principal G-bundle and f : B0 → B be a continuous map. The pullback bundle of p along f (or the induced bundle) is the bundle f ∗p : f ∗P → B0 with total space f ∗P := {(b0, y) ∈ B0 × P | f(b0) = p(y)}. In other words, f ∗P is the pullback of f and p in the category of principle G-bundles over B:

f ∗P P

f ∗p p f B0 B

Example 1.2.13. If i : A,→ B is a subspace, the pullback i∗P of any bundle P → B coincides with the restriction P |A → A of Example 1.1.2. Example 1.2.14. If f : B0 → B is a constant map, then every pullback bundle along f is trivial.

8 1.2 Principal Bundles 1 Fibre Bundles

q p Example 1.2.15. Suppose Q −→ B0 and P −→ B are principal G-bundles and F : Q → P is a morphism of principal G-bundles, lying over f : B0 → B. Then it follows from Lemma 1.2.12 that Q ∼= f ∗P . From now on, we will assume B is paracompact, so that it admits partitions of unity. Let I = [0, 1] and r : B × I → B × {1} the natural projection.

Lemma 1.2.16. Let a, b ∈ R, A be a space and P → A × [a, b] be a principal G-bundle. If for a ≤ c ≤ b, P |A×[a,c] and P |A×[c,b] are both trivial, then so is P .

Proof. Write B1 = A×[a, c],B2 = A×[c, b] and let ϕi : P |Bi → Bi ×G be local trivializations −1 for i = 1, 2. Over A × {c}, the composition H := ϕ2 ◦ ϕ1 : A × {c} × G → A × {c} × G is of the form H(a, c, g) = (a, c, h(a)g) for some h : A → G. Extend H across all of B1 by

H(a, t, g) = (a, t, h(a)g) for a ≤ t ≤ c. Then H ◦ ϕ1 and ϕ2 agree on P |B1 ∩ P |B2 = P |A×{c}. Hence these give a well-defined isomorphism of bundles P → (B1∪B2)×G = A×[a, b]×G. Lemma 1.2.17. Suppose P → B × I is a principal G-bundle. Then there exists a cover

{Ui} of B such that each P |Ui×I is trivial. Proof. Since P is locally trivial and I is compact, for each b ∈ B there is a sequence 0 = t0 < t1 < ··· < tn = 1 and neighborhoods Ui of b such that each P |U ×[t ,t ] is trivial. T i j j+1 Set U(b) = Ui. Then by Lemma 1.2.16, P |U(b)×I is trivial.

∼ ∗ Theorem 1.2.18. If P → B × I is a principal G-bundle, then P = r (P |B×{1}).

Proof. Set Pt = P |B×{t} for each 0 ≤ t ≤ 1. By Lemma 1.2.17, there exists a cover {Ui} of

B with each P |Ui×I → Ui × I all trivial. Take a partition of unity subordinate to the Ui, i.e. a collection of functions ηi : B → I with ηi supported on Ui and max ηi ≡ 1 on B. Let hi :(Ui × I) × G → P |Ui×I be local trivializations. By paracompactness, for each b ∈ B, there is a neighborhood U(b) meeting finitely many of the Ui. Let S be the index set of the Ui equipped with some ordering and for each i ∈ S, define a function

ri : B × I −→ B × I

(b, t) 7−→ (b, max{ηi(b), t}).

Let fi : P → P be the function that is the identity outside Ui × I and is given by

fi(hi(x, t, g)) = hi(ri(x, t), g) on Ui × I (which makes sense because every element of Ui is of the form hi(x, t, g) for some x ∈ Ui, t ∈ I, g ∈ G). These define a map f : P → P which is the “composition over S” of all the fi in the order given by the ordering on S, which is well-defined since for each b ∈ B, U(b) meets only finitely many of the Ui and therefore only finitely many of the fi are not the identity on U(b). By the same token, r : B × I → B × {1} is the composition over S of the ri. Since max ηi(b) = 1 for all b ∈ B, the map f determines a bundle morphism P → P1. ∼ ∗ Finally, Lemma 1.2.12 implies that P = r P1 as desired.

9 1.3 Classifying Spaces and Universal Bundles 1 Fibre Bundles

Corollary 1.2.19. If i0, i1 : B → B × I are the inclusions of B × {0} and B × {1}, ∗ ∼ ∗ respectively, and P → B × I is a principal G-bundle, then i0P = i1P as bundles over B. 0 Corollary 1.2.20. If f0, f1 : B → B are homotopic and P → B is a principal G-bundle, ∗ ∼ ∗ 0 then f0 P = f1 P as bundles over B . 0 ∗ Proof. Suppose F : B × I → B is a homotopy between f0 and f1. Then P0 × I = F P = P1 × I, f0 = F ◦ i0 and f1 = F ◦ i1, so by Corollary 1.2.19,

∗ ∗ ∗ ∗ ∗ ∗ f0 P = i0F P = i1F P = f1 P.

Corollary 1.2.21. If B is a contractible space, then any principal G-bundle over B is trivial. Corollary 1.2.22. Suppose E → B is an arbitrary fibre bundle with fibre F and structure group G. Then the previous results hold for E, namely:

∗ ∼ ∗ (1) If Ee → B × I is a fibre bundle with structure group G, then i0Ee = i1Ee as bundles over B.

0 ∗ ∼ ∗ 0 (2) If f0, f1 : B → B are homotopic, then f0 E = f1 E as bundles over B . (3) If B is contractible, then E ∼= B × F . Proof. Apply the previous results to the associated bundle construction on E.

For a space X and a group G, let BunG(X) denote the set of isomorphism classes of principal G-bundles over X.

Corollary 1.2.23. The assignment X 7→ BunG(X) is a contravariant homotopy functor Top → Set.

We may ask if the functor BunG(−) is representable. This leads to the notion of a classifying space for principal G-bundles.

1.3 Classifying Spaces and Universal Bundles

p Definition. A principal G-bundle P −→ B is a universal bundle for G-bundles if for each space X, the map

[X,B] 7−→ BunG(X) (f : X → B) 7−→ (f ∗P → X) is a bijection. For a bundle Q → X, a map f : X → B such that f ∗P ∼= Q is called a classifying map for Q.

Remark. Any arbitrary G-bundle P → B induces a natural transformation αP :[−,B] → BunG(−) since pullback is natural, so saying that P is a universal G-bundle is equivalent to saying that αP is a natural isomorphism.

10 1.3 Classifying Spaces and Universal Bundles 1 Fibre Bundles

Lemma 1.3.1. If a universal bundle exists for a group G, then it is unique up to homotopy equivalence. Proposition 1.3.2. Suppose p : E → B is a fibre bundle with contractible fibre F . Then p is a homotopy equivalence. Moreover, p admits a section and any two sections of p are homotopic through sections. Theorem 1.3.3. A principal G-bundle P → B is a universal bundle for G-bundles if and only if P is contractible. p q Proof. ( ⇒ = ) Suppose P −→ B is a G-bundle with P contractible. Let Q −→ X be any principal G-bundle. By definition, P is a right G-space, but we may view it as a left G-space via ρ(g)p := pg−1 for any p ∈ P, g ∈ G. This data determines an associated bundle E = E(P,ρ) = Q×G P , so that E → X is a fibre bundle with fibre P . In particular, since the fibre is contractible, Corollary 1.2.22 and Lemma 1.2.4 say that E admits a section, but a section of an associated bundle corresponds to a maps ˜ : Q → P satisfyings ˜(yg) = ρ(g)−1s˜(y) =s ˜(y)g for all y ∈ Q, g ∈ G. This shows thats ˜ is a G-equivariant map Q → P taking fibres to fibres, so it also induces a map f : X → B. By Lemma 1.2.12, Q ∼= f ∗P , which shows that [X,B] → BunG(X) is surjective. On the other hand, any map f : X → B such that f ∗P ∼= Q comes from a section X → E and any two such sections are homotopic through sections by Proposition 1.3.2. Thus any two maps f, f 0 : X → B inducing isomorphisms ∗ ∼ 0∗ ∼ f P = Q, f P = Q are homotopic, so [X,B] → BunG(X) is injective. Hence P → B is a universal bundle. ( =⇒ ) We will construct universal bundles EG → BG for all G and show that their total spaces are contractible. Once that is done, the proof will be complete. Definition. Let EG → BG be a universal bundle for a group G. The space BG is called a classifying space for G.

n n For G = O(k), the kth orthogonal group, recall the principal bundle Vk(R ) → Grk(R ) with fibre O(k). Proposition 1.3.4. For any n ≥ 1, the inclusion O(n) ,→ O(n + 1) (resp. SO(n) ,→ SO(n + 1)) given by ! In 0 A 7→ 0 A induces maps πi(O(n)) → πi(O(n + 1)) (resp. πi(SO(n)) → πi(SO(n + 1))) which are isomorphisms for i ≤ n − 2 and surjective for i = n − 1. Similarly, the inclusion U(n) ,→ U(n + 1) (resp. SU(n) ,→ SU(n + 1)) induce πi(U(n)) → πi(U(n + 1)) (resp. πi(SU(n)) → πi(SU(n + 1))) which are isomorphisms for i ≤ 2n − 1 and surjective for i = 2n.

n n+1 n Proof. The map O(n + 1) → S ⊆ R is a fibre bundle with fibre O(n) and πi(S ) = 0 for all i < n, so the result follows from the long exact sequence in homotopy groups

n n · · · → πi+1(S ) → πi(O(n)) → πi(O(n + 1)) → πi(S ) → · · ·

11 1.3 Classifying Spaces and Universal Bundles 1 Fibre Bundles

Corollary 1.3.5. For all k ≤ n, O(n − k) ,→ O(n) (resp. SO(n − k) ,→ SO(n)) induces maps πi(O(n − k)) → πi(O(n)) (resp. πi(SO(n − k)) → πi(SO(n))) which are isomorphisms for all i ≤ n − k − 2 and surjective for i = n − k − 1. A similar statement holds for U(n) and SU(n).

n n Corollary 1.3.6. The Stiefel manifold Vk(R ) has πi(Vk(R )) = 0 for i ≤ n − k − 1. Simi- n n larly, the complex Stiefel manifold Vk(C ) (defined using unitary k-frames) has πi(Vk(C )) = 0 for i ≤ 2(n − k).

n n Corollary 1.3.7. Vk(R ) → Grk(R ) is universal for O(k)-bundles over CW-complexes of dimension at most n − k − 1.

∞ S∞ n Let Vk = Vk(R ) := n=1 Vk(R ) be the infinite Stiefel manifold with the direct limit n n+1 ∞ topology, using the standard inclusions Vk(R ) ,→ Vk(R ). Similarly, let Grk = Grk(R ) := S∞ n n=1 Grk(R ) be the infinite Grassmannian manifold.

Theorem 1.3.8. Vk → Grk is a universal bundle for principal O(k)-bundles.

Proof. Note that each Vk admits a CW-structure that is compatible with the inclusions n n+1 Vk(R ) ,→ Vk(R ) (see the Schubert cell description in the proof of Theorem 3.3.9), so this induces an infinite CW-structure on Vk. Further, by Corollary 1.3.6, πi(Vk) = 0 for all i so by Whitehead’s theorem in homotopy theory, Vk is contractible. Hence by Theorem 1.3.3, Vk → Grk is universal for O(k)-bundles. Here’s an alternate proof that Vk is contractible. Define the ‘even’ and ‘odd’ embeddings of a Euclidean space into the space with twice its dimension:

n 2n i1 : R ,→ R , (x1, . . . , xn) 7→ (x1, 0, x2, 0, . . . , xn, 0) n 2n i2 : R ,→ R , (x1, . . . , xn) 7→ (0, x1, 0, x2,..., 0, xn).

n 2n n These induce inclusions Vk(R ) ,→ Vk(R ). Let {e1, . . . , en} be the standard basis for R and define a homotopy

n 2n F : Vk(R ) × I −→ Vk(R ) ((v1, . . . , vn), t) 7−→ (1 − t)i1(v1, . . . , vn) + ti2(v1, . . . , vn).

n n+1 One can check that F commutes with the inclusions Vk(R ) ,→ Vk(R ), and so induces Fe : Vk × I → Vk, a homotopy from the constant map induced by i2 to the identity map induced by i1. Corollary 1.3.9. For every compact Lie group G, there exists a universal bundle EG → BG with EG contractible.

Proof. Any compact Lie group G embeds as a closed subgroup of an orthogonal group O(k) (this is equivalent to saying every such G has a finite dimensional faithful representation with ∞ an inner product). In particular, there is a right G-action on Vk = Vk(R ), so Vk → Vk/G is a principal bundle with fibre B. By Theorem 1.3.8, Vk is contractible so Vk → Vk/G is universal.

12 1.3 Classifying Spaces and Universal Bundles 1 Fibre Bundles

Notice that for any compact Lie group G, an embedding G,→ O(k) induces a map BG = Vk/G → Vk/O(k) = Grk. This suggests the following construction of classifying spaces due to Milnor. For arbitrary spaces X and Y , define their join X ∗ Y by

X ∗ Y := {t0x + t1y | ti ∈ [0, 1], x ∈ X, y ∈ Y } = X × Y/ ∼ where (x, 0, y) ∼ (x, 0, y0) and (x, 1, y) ∼ (x0, 1, y) for all x, x0 ∈ X, y, y0 ∈ Y . More generally, the join of a collection of spaces X0,...,Xn is given by

( n n ) X X X0 ∗ · · · ∗ Xn := tixi : xi ∈ Xi, ti ∈ [0, 1], ti = 1 = X0 × · · · × Xn × ∆n/ ∼ i=0 i=0 where ∆n is the standard n-simplex and ∼ is analogous to the above. Then the inclusions Xi ,→ X0 ∗ · · · ∗ Xn as the ‘faces’ of the simplex are nullhomotopic. Let X1 = X and inductively define Xn = Xn−1 ∗ X. Then the sequence X1,X2,X3,... becomes increasingly ‘connected’ in the sense that their homotopy groups eventually vanish.

Proposition 1.3.10. For any space X, let Xn = X ∗ · · · ∗ X be the n-fold join of X. Then S∞ the infinite join X∞ = n=1 Xn is contractible.

Given a topological group G, let G∞ be the infinite join of G as above. By Proposi- tion 1.3.10, G∞. Moreover, we can define a G-action on G∞ by:

∞ ! ∞ X X tigi · g = tigig. i=0 i=0

(Note that the formal sums have finitely many ti nonzero, so they are in reality honest sums.) It’s easy to see that this action is free.

Lemma 1.3.11. G∞ → G∞/G is a principal G-bundle. Proof. Consider the ‘coordinate functions’

Ti : G∞ −→ [0, 1] ∞ X tjgj 7−→ ti. j=0

Then by construction, the Ti descend to G∞/G. We claim that G∞ → G∞/G is trivial over 1 each Ui = Ti ((0, 1]). Define

ϕi : Ui × G −→ G∞ " ∞ # ! ∞ X X −1 tjgj , g 7−→ tjgjgi g. j=0 j=0

hP∞ i Then ϕi is independent of the choice of representative of j=0 tjgj and it’s easy to check that ϕi is a bundle isomorphism.

13 1.3 Classifying Spaces and Universal Bundles 1 Fibre Bundles

There are some technicalities present with our construction:

(1) Is the G-action on G∞ continuous? Note that we have projections −1 πi : p (Ui) −→ G ∞ X tjgj 7−→ gi j=0

and the map m : G∞ × G → G∞ representing the G-action composed with any of the Ti or πi is continuous, but the topology on G∞ is different from the one induced by the Ti, πi. So instead, we may take G∞ to have this topology, i.e. the coarsest so that all Ti and πi are continuous. But then one still has to check that G∞ is contractible with respect to this topology – see Dold for the proof of this.

(2) Is G∞/G paracompact? Milnor showed that the answer is no in general, but the cover {Ui} of G∞/G constructed above is numerable so the rest of the results in Section 1.2 still apply. Setting these issues aside, we have proven:

Corollary 1.3.12. For all groups G, the bundle G∞ → G∞/G is a universal G-bundle. Corollary 1.3.13. Let G → H be a continuous homomorphism of topological groups. Then there is a morphism of principal bundles EG EH

BG BH

That is, G 7→ BG is a functor Top → Group. Theorem 1.3.14. A continuous homomorphism α : G → H induces a natural transforma- tion BunG(−) → BunH (−). Moreover, this is a natural isomorphism if and only if α is a homotopy equivalence.

Proof. By definition, BunG(−) is represented by BG so Corollary 1.3.13 implies that α induces a map Bα : BG → BH, and hence a function

(Bα)∗ BunG(−) = [−,BG] −−−→ [−,BH] = BunH (−). For any map f : X → Y , the diagram

(Bα)∗ [X,BG] [Y,BG]

f ∗ f ∗

[Y,BG] [Y,BH] (Bα)∗

14 1.3 Classifying Spaces and Universal Bundles 1 Fibre Bundles

commutes by associativity of composition. Thus (Bα)∗ is a natural transformation. Now (Bα)∗ is a bijection [X,BG] → [X,BH] for all spaces X if and only if Bα is a homotopy equivalence, so it suffices to show Bα is a homotopy equivalence precisely when α is one. Consider the pullback EHg := (Bα)∗EH:

EHg EH

BG BH Bα

Then Bα is a homotopy equivalence if and only if EHg is universal for H-bundles (or equiv- alently by Theorem 1.3.3, EHg is contractible). On the other hand, let Eα : EG → EH be the map induced on total spaces by α : G → H. Then EHg is a universal H-bundle if and only if there is a G-bundle morphism

EG EHg

BG

If Eα is a homotopy equivalence EG → EHg, it induces a homotopy equivalence on each fibre, but this is just α on each fibre so α is itself a homotopy equivalence. Conversely, if α is a homotopy equivalence, then over each locally trivial open set U ⊆ BG, Eα has the form id × α : U × G → U × H. So Eα is locally a fibrewise homotopy equivalence and one can prove that this implies Eα is globally a homotopy equivalence. Hence Eα is a homotopy equivalence if and only if α is one, so we are done.

Remark. For any group G, there is a fibration EG → BG with fibre G, giving rise (by homotopy theory) to a fibration sequence

· · · → ΩG → ΩEG → ΩBG → G → EG → BG.

Then for any topological space X, there is an exact sequence of sets

··· [X, ΩG] → [X, ΩEG] → [X, ΩBG] → [X,G] → [X,EG] → [X,BG] = BunG(X).

Since EG is contractible, [X, ΩkEG] = 0 for all k ≥ 0 so there is a bijection [X, ΩkBG] ↔ k−1 n ∼ [X, Ω G]. In particular, taking X = S gives isomorphisms of groups πn(ΩBG) = πn(G) for all n, so by Whitehead’s theorem, ΩBG → G is a homotopy equivalence. Here is another perspective.

Lemma 1.3.15. For a topological group G, let PBG be the path space of BG. Then there exists a bundle morphism

15 1.4 Reduction of the Structure Group 1 Fibre Bundles

EG PBG

BG

where PBG → BG, γ 7→ γ(1) is the endpoint fibration.

Proof. Since EG is contractible (Theorem 1.3.3), p : EG → BG extends to a mapp ˜ : CEG → BG where CEG = EG ∧ I is the cone on EG. By adjointness,p ˜ determines a mapp ¯ : EG → PBG given explicitly byp ¯(y)(t) =p ˜(t, y). Sincep ˜ extends p, for t = 1 we havep ¯(y)(1) =p ˜(1, y) = p(y) so indeed the diagram above commutes. Moreover, since EG is contractible,p ¯ is a homotopy equivalence on fibres and thus PBG → BG is, too.

Corollary 1.3.16. There is a homotopy equivalence G → ΩBG.

Proof. Suppose P → X is a principal G-bundle classified by a map f : X → BG. Then P is the homotopy fibre of f, and f is the pullback of PBG → BG. Thus we have two bundle morphisms f ∗PBG PBG P EG

and p

X BG X BG f f

Since f and p are both homotopy equivalences, we get f ∗PBG ' P = f ∗EG.

1.4 Reduction of the Structure Group

Let α : G → H be a continuous homomorphism of topological groups and suppose PG → X is a principal G-bundle. Then by the results in Section 1.3, α induces a principal H-bundle over X in two ways:

(1) Composition of a classifying map fG : X → BG for PG with the induced map Bα : ∗ BG → BH gives an H-bundle PH = (Bα ◦ fG) EH. (2) The map α determines a G-space structure on H via ρ(g)h = α(g)h, so there is an associated bundle PG ×G H → X with fibre H.

We may ask the converse of this question: when is an H-bundle PH → X induced by a G-bundle over X? There are two perspectives on this question, mirroring the perspectives above:

(1) A classifying map fH : X → BH for PH may by induced by composition of some fG with Bα:

16 1.4 Reduction of the Structure Group 1 Fibre Bundles

BG

fG Bα

X BH fH

(2) PH may be an associated bundle for some PG → X. Example 1.4.1. If α : G → H is the inclusion of a closed subgroup, the universal G- bundle is given by EH → EH/G (EH is contractible and EH → EH/G has fibre G by Proposition 1.2.8). Thus composition with Bα gives a fibre bundle EH/G → BH with fibre H/G, but not necessarily a principal bundle.

Example 1.4.2. If α : G → H is a quotient map, then H ∼= G/K where K = ker α, and EG → EG/K = EK is a principal K-bundle, thus universal since EG is contractible. Also, BK → BG is a principal bundle with fibre G/K = H classified by the map Bα : BG → BH. This is because Eα : EG → EH is given by G → G/K on fibres, so it factors through EG/K and thus EG/K = (Bα)∗EH.

Lemma 1.4.3. For a quotient map α : G → H = G/K, BK is the homotopy fibre of the associated map Bα : BG → BH.

These examples suggest answers to the reduction question in the form posed in (1) above: if α : G → H is a closed inclusion, we have a fibration sequence B(H/G) → BG → BH, whereas if α is a quotient map, BK → BG → BH is the corresponding fibration sequence. In the former case, for every X there is an exact sequence

[X, H/G] → [X,G] → [X,H] → [X,B(H/G)] → [X,BG] → [X,BH].

This proves:

Proposition 1.4.4. Let α : G → H be the inclusion of a closed subgroup. Then a principal H-bundle PH → X classified by f : X → BH is induced by a principal G-bundle over X if and only if the map Bα ◦ f : X → B(H/G) is nullhomotopic.

Example 1.4.5. Let G = SO(n),H = O(n) and α : SO(n) ,→ O(n) the standard inclusion. Here, principal O(n)-bundles identify canonically with real vector bundles of dimension n (Example 1.1.10) and SO(n)-bundles identify with real oriented vector bundles of dimension n. Reduction of the structure group in this case just amounts to determining if a bundle is orientable or not. By Proposition 1.4.4, an O(n)-bundle E → X is orientable (that is, reduces to an SO(n)-bundle) if and only if

X → BO(n) → B(O(n)/SO(n)) = B(Z/2Z) is nullhomotopic. Notice that E(Z/2Z) may be taken to be any contractible space with a free Z/2Z-action, so in particular we may take E(Z/2Z) = S∞, so that B(Z/2Z) = S∞/Z/2Z =

17 1.4 Reduction of the Structure Group 1 Fibre Bundles

RP ∞. Usefully, RP ∞ = K(Z/2Z, 1), an Eilenberg-Maclane space, which by definition means for any space X,

1 [X,B(Z/2Z)] = [X,K(Z/2Z, 1)] = H (X; Z/2Z).

Consequently, X → B(Z/2Z) is nullhomotopic if and only if (Bα◦f)∗ω = 0 in H1(X; Z/2Z), • ∞ ∼ where ω is a generator of the cohomology ring H (RP ; Z/2Z) = Z/2Z[ω]. This class ∗ 1 ω1(E) := (Bα ◦ f) ω ∈ H (X; Z/2Z) is our first example of a characteristic class, called the first Stiefel-Whitney class, which we develop in Section 3.2.

Let’s return to perspective (2) on the structure group reduction question, which asks if PH → X is the associated bundle for some PG → X. If α : G → H is the inclusion of a closed subgroup, then PH reduces to a G-bundle PG → X if and only if PH = PG ×G H for some space PG with a free left G-action. Proposition 1.4.6. Let α : G → H be the inclusion of a closed subgroup. Then a principal H-bundle PH → X reduces to a G-bundle if and only if PH /G → X has a section.

Proof. ( =⇒ ) If PH = PG ×G H for some free left G-space PG, then PH /G = (PG ×G H)/G = PG ×G (H/G), so

X := PH /H = (PG ×G H)/H

= PG ×G (H/H)

= PG ×G (G/G) = PG,

so let s : X → PH /G be the map corresponding to PG → PG ×G (H/G). This defines a section by construction. ∗ ( ⇒ = ) Suppose s is a section of PH /G → X = PH /H. Let Q = s (PH → PH /G) be the induced bundle:

Q PH

X PH /G

This determines a map γ : Q ×G H → PH , γ([q, h]) =s ˜(q)(h), wheres ˜ is the lift of s as in the proof of Theorem 1.3.3. It’s easy to check that γ is G-equivariant, so by Lemma 1.2.12, ∼ Q ×G H = PH . This gives another proof of Lemma 1.2.4: a principal G-bundle P → B is trivial if and only if P = P/{1} admits a section.

Corollary 1.4.7. For any closed subgroup G ≤ H, the quotient H/G is contractible if and only if every H-bundle reduces to a G-bundle.

18 1.5 Gauge Groups 1 Fibre Bundles

1.5 Gauge Groups

This section follows a talk given by Peter Johnson for Dr. Mark’s class.

Definition. The gauge group of a principal G-bundle p : P → B, denoted AutB(P ), is the group of principal G-bundle isomorphisms P P

B

We endow AutB(P ) with the subspace topology inherited from the compact-open topol- ogy on MapG(P,P ). One can check that this makes AutB(P ) into a topological group. This means MapG(P,EG) is nonempty. Moreover, AutB(P ) acts freely on the right on MapG(P,EG) via σf = σ ◦ f.

Question. What is the classifying space for AutB(P )-bundles?

Let Clp(B,BG) be the set of classifying maps B → BG for p : P → B. By Corol- lary 1.2.20, all elements of Clp(B,BG) are homotopic. It follows that MapG(P,EG) is path-connected (it is also nonempty by the existence of classifying maps). ∼ Lemma 1.5.1. For any principal G-bundle p : P → B, MapG(P,EG)/ AutB(P ) = Clp(B,BP ) as topological spaces. Proof. Define a map

S : MapG(P,EG) −→ Clp(B,BG) σ 7−→ σ¯ whereσ ¯ is the map induced by the G-equivariant σ under the identification B = P/G and

BG = EG/G. Suppose f ∈ AutB(P ) and σ ∈ MapG(P,EG). Then there is a G-equivariant commutative diagram f σ P P EG

p p id σ¯ B B BG

It follows that σf =σ ¯, i.e. the map S descends to the quotient MapG(P,EG)/ AutB(P ). Now suppose σ1, σ2 ∈ MapG(P,EG) such thatσ ¯1 =σ ¯2. Then there exists some map α : P → G such that for all x ∈ P ,

σ1(x) = σ2(x)α(x) = σ2(xα(x)).

Then the map f : P → P, x 7→ xα(x) lies in AutB(P ). One can also prove that S is a ∼ homeomorphism, so MapG(P,EG)/ AutB(P ) = Clp(B,BP ) as spaces.

19 1.5 Gauge Groups 1 Fibre Bundles

Theorem 1.5.2. For all principal G-bundles p : P → B, MapG(P,EG) → Clp(B,BG) is universal for AutB(P )-bundles.

Proof. It is straightforward to prove that MapG(P,EG) → Clp(B,BG) is a locally trivial fibre bundle with fibre AutB(P ). Moreover, the gauge group acts freely on MapG(P,EG), so we indeed have a principal AutB(P )-bundle and we need only prove MapG(P,EG) is contractible (by Theorem 1.3.3). Consider the product bundle

p × id : P × MapG(P,EG) −→ B × MapG(P,EG).

By our observations above, MapG(Q, EG) is nonempty and path-connected for any bundle Q, so in particular this is true for ∼ MapG(P × MapG(P,EG),EG) = Map(MapG(P,EG), MapG(P,EG))

(this identification is by the adjoint property of × and Map). Therefore there is a path in

Map(MapG(P,EG), MapG(P,EG)) from the identity map to a constant map, which proves MapG(P,EG) is contractible. Now assume the topological group G is abelian, so that the multiplication map m : G × G → G is a continuous homomorphism. This determines a map Bm : BG × BG → BG and similarly, inversion i : G → G and the identity e : ∗ → G determine maps Bi : BG → BG and Be : ∗ → BG. Together, these give BG the structure of an abelian topological group. Repeating this procedure gives a sequence of abelian topological groups

B, BG, B2G, B3G, . . . where BnG = B(Bn−1G). By Corollary 1.3.16, G ∼= ΩBG so for G abelian, we get G ∼= ΩnBnG for all n ≥ 0.

Lemma 1.5.3. If G is any abelian group with the discrete topology, then for all n ≥ 0, BnG = K(G, n).

Proof. Let m ≥ 0. Then

n m n 0 m n πm(B G) = [S ,B G] = [S , Ω B G] by adjointness = [S0, Ωm−nΩnBnG] = [S0, Ωm−nG] ( m−n G, m = n = [S ,G] = πm−n(G) = 0, m 6= n.

This proves:

Theorem 1.5.4. For all abelian G and n ≥ 0, K(G, n) is an abelian topological group and BK(G, n − 1) = K(G, n).

20 1.5 Gauge Groups 1 Fibre Bundles

∼ Lemma 1.5.5. If G is abelian, then for any principal G-bundle p : P → B, AutB(P ) = Map(B,G). In particular, the gauge group does not depend on the total space B.

Thus the gauge group AutB(K(G, n)) is well-defined for all abelian G and n ≥ 0. This group can be viewed as: ∼ ∼ n AutB(K(G, n)) = Map(B,K(G, n)) = [B,K(G, n)] = H (B; G). Theorem 1.5.6 (Thom). For all spaces B and K(G, n), Map(B,K(G, n)) is homotopy equivalent to n Y K(Hn−q(B; G), q). q=0 Proof. We induct on n. When n = 0, K(G, 0) = G and K(H0(B; G), 0) =∼= G so they are in fact homeomorphic. For n > 0, Y Map(B,K(G, n)) = B(Map(B,K(G, n − 1))) [B,K(G,n)] where the product is indexed over the connected components of Map(B,K(G, n)), repre- sented by elements of [B,K(G, n)]. By induction,

n−1 ! Y Map(B,K(G, n)) ' K(Hn(B; G), 0) × B K(Hn−q−1(B; G), q) q=0 n Y = K(Hn−q(B; G), q). q=0 Hence the statement holds by induction. As a beautiful consequence, we derive a proof of the K¨unnethformula from algebraic topology. Corollary 1.5.7 (K¨unnethFormula). For any CW-complexes X and Y and any abelian group G, there isomorphisms n M Hn(X × Y ; G) ∼= Hq(X; Hn−q(Y ; G)). q=0 Proof. By definition of K(G, n), Hn(X × Y ; G) = [X × Y,K(G, n)] = [X, Map(Y,K(G, n))] by adjointness n Y = [X,K(Hn−q(Y ; G), q)] by Theorem 1.5.6 q=0 n M = Hq(X; Hn−q(Y ; G)). q=0

21 2 Vector Bundles

2 Vector Bundles

2.1 Vector and Tangent Bundles

A vector bundle is a special type of fibre bundle in which the fibre has the structure of a (Euclidean) vector space. The definition of a vector bundle is also quite similar to that of a manifold, in that there are certain “local triviality” and compatibility conditions placed on the space. Recall: Definition. A topological (resp. smooth) n-dimensional manifold is a space M such that for any point p ∈ M, there is a neighborhood U ⊂ M containing p which is homeomor- phic (resp. diffeomorphic) to an open subset of Rn. We introduce the definition of a vector bundle in a parallel fashion. Definition. A topological (resp. smooth) n-dimensional vector bundle is an object ξ con- sisting of the following data: a total space E, a base B and a continuous (resp. smooth) map π : E → B, where E and B are topological (resp. smooth) manifolds, such that for −1 each p ∈ B, the fibre Fp := π (p) is a vector space of dimension n. Moreover, for each p ∈ B, there must exist a neighborhood U ⊆ B containing p and a homeomorphism (resp. diffeomorphism) h : U × Rn → π−1(U) making the following diagram commute:

h −1 U × Rn π (U)

proj π

U

n −1 and for all x ∈ U, h|{x}×Rn : {x} × R → π (x) is a vector space isomorphism. Alternatively, one can define a manifold from a topological space M using transition maps (which are either continuous or smooth, depending on if the manifold is topological or smooth). This has an analagous interpretation for vector bundles. Given two overlapping neighborhoods U, V ⊆ B, consider the local trivializations

hU hV (U ∩ V ) × Rn π−1(U ∩ V ) (U ∩ V ) × Rn

proj π proj U ∩ V

The vector space isomorphism condition then guarantees that for any x ∈ U ∩ V , the −1 hU hV composition {x} × Rn −→ π−1(x) −−→ {x} × Rn is a vector space isomorphism, i.e. an element of GLn(R). Such a map is called a clutching map, or a transition map. What’s more, this defines a continous (resp. smooth) map U ∩ V → GLn(R). We will see in Section 2.4 that this is enough data to reconstruct the original vector bundle.

22 2.1 Vector and Tangent Bundles 2 Vector Bundles

Examples.

1 The trivial bundle of dimension n over any manifold B consists of E = B × Rn and the n −1 n ∼ n projection map π : B × R → B, (p, v) 7→ p. For each b ∈ B, π (p) = {b} × R = R . The local triviality condition in the definition of a vector bundle intuitively says that every vector bundle locally looks like the trivial bundle. Notice that there is a natural projection onto the fibre: B × Rn → Rn, (p, v) 7→ v. In fact, the existence of such a projection characterizes the trivial bundle; that is, any nontrivial bundle (in the sense of bundle equivalence to be defined in a moment) does not have a projection onto the fibre.

2 Perhaps the most important vector bundle is the tangent bundle p : TM → M of a smooth manifold M (see Example 1.1.6).

3 The canonical or tautological bundle is an important example of a vector bundle over RP n, defined as follows. The total space is given by

n n+1 E = {(x, v) ∈ RP × R : v = λx for some scalar λ ∈ R}.

In other words, viewing RP n as the quotient of the unit sphere Sn ⊂ Rn+1 by the antipodal action, a point (x, v) ∈ E is determined by a pair of points x, −x on the n-sphere and a vector v lying on the line between them. Here, the projection map π : E → RP n is given by π(x, v) = x. Let us show that this information determines the structure of a vector bundle over n n n RP , denoted by γ1 . Take a chart U ⊂ S that is small enough so that U ∩ −U = ∅. n n 0 0 If p : S → RP is the double cover, then p|U : U → U = p(U) and p|−U : −U → U are diffeomorphisms. We construct a trivialization of U 0 by defining

0 −1 0 h : U × R −→ π (U ) (¯x, t) 7−→ (¯x, tx)

where x is the point in p−1(¯x) = {x, −x} lying in U. (The subtlety with this definition is that one may choose −x ∈ U instead, still have p(−x) =x ¯, but obtain a different trivialization.) Since U and −U don’t overlap, h is a diffeomorphism. Further, h clearly commutes with π and is an isomorphism on fibres, so h is a local trivialization.

Proposition 2.1.1. For any smooth manifold M of dimension n, the tangent bundle TM is a smooth vector bundle of dimension n.

Proof. We need only check the local triviality condition, since the projection π : TM → M is already defined. Fix p ∈ M and choose a coordinate chart U ⊆ M of p; this comes equipped with a diffeomorphism ϕ : U → ϕ(U) ⊆ Rn. Define h : U × Rn → π−1(U) by −1 −1 h(x, v) = (x, Dϕ(x)ϕ (v)), where Dyϕ is the differential of the inverse diffeomorphism −1 −1 ϕ : ϕ(U) → U at a point u ∈ ϕ(U). By definition, TxM = im(Dϕ(x)ϕ ) for any x ∈ U, n −1 so h is well-defined. Moreover, for each x, h|{x}×Rn : {x} × R → π (x) is an isomorphism since ϕ and ϕ−1 are diffeomorphisms. This completes the proof.

23 2.1 Vector and Tangent Bundles 2 Vector Bundles

Vector bundles are used in modern differential topology to distinguish between smooth manifolds. We next define the notion of equivalence of vector bundles.

Definition. Given two vector bundles π1 : E1 → B and π2 : E2 → B over the same base B, we say (E1, B, π1) and (E2, B, π2) are isomorphic vector bundles if there exists a −1 −1 diffeomorphism ϕ : E1 → E2 such that for all b ∈ B, ϕ|π−1 (b): π1 (b) → π2 (b) is a vector ∼ 1 space isomorphism. More succinctly, (E1, B, π1) = (E2, B, π2) if and only if the following diagram commutes: ϕ E1 E2

π1 π2

B

Example 2.1.2. We claim that the tangent bundle TS1 to the unit circle is isomorphic to the trivial bundle over S1. To show this, we must construct a diffeomorphism ϕ : S1 × R → TS1 making the diagram commute: ϕ S1 × R TS1

proj π

S1

1 1 1 Given any point p = (x, y) ∈ S , TpS is spanned by (−y, x) ∈ TpS . Define the map

1 1 2 2 ϕ : S × R −→ TS ⊂ R × R ((x, y), t) 7−→ ((x, y), t(−y, x)).

This is clearly smooth and one-to-one, and restricts on each {p} × R to a diffeomorphism 1 1 with TpS . Therefore TS is the trivial bundle. Example 2.1.3. One may be tempted to think that all vector bundles are isomorphic to the trivial bundle. Consider the tautological bundle over RP n:

n n+1 n E = {(x, tx) ∈ RP × R | x ∈ S , t ∈ R}.

Then each point x ∈ RP n may be viewed as a line in Rn+1 through the origin, so π−1(x) 1 ∼ 1 consists of all vectors in that line defined by x. In the case n = 1, RP = S and E looks like a quotient of an infinite rectangle:

E

R

24 2.2 Sections 2 Vector Bundles

Then E is homeomorphic to the M¨obiusband. The line bundle E → S1 cannot be the trivial bundle over S1 because E is not diffeomorphic to S1 × R (e.g. S1 × R is orientable and the 1 ∼ 1 M¨obiusband is not). Hence we have exhibited a nontrivial vector bundle over RP = S . In general, the tautological bundle π : E → RP n is nontrivial, as we show in Section 2.2. Definition. A manifold M whose tangent bundle TM is isomorphic to the trivial bundle is called parallelizable.

In Example 2.1.2 we showed that S1 is parallelizable. It turns out that for n ≥ 2 except n = 3, 7, Sn is not parallelizable. We will prove this in the next section.

2.2 Sections

Definition. Given a vector bundle ξ : E −→π B, a section of the bundle is a smooth map s : B → E such that π ◦ s = idB. The set of all sections of ξ is denoted Γ(ξ), or Γ(B,E) when the map π is understood.

Definition. When E = TM is the tangent bundle over a manifold M, a section s : M → TM is instead called a smooth tangent vector field on M.

−1 Example 2.2.1. The trivial section s0 : B → E, p 7→ 0 ∈ π (p) is smooth for any vector bundle π : E → B. In particular, sections exist for every vector bundle over B. One can show that the space of sections for any bundle of nonzero dimension is uncountable. The existence of a zero section is an important characteristic of vector bundles which does not generalize to fibre bundles. What’s more, any smooth vector bundle E −→π B is a homotopy equivalence between E and B – the image of B under the 0 section is a strong deformation retract of E.

Example 2.2.2. The sections of the trivial bundle B × Rn → B are precisely all of the smooth functions B → Rn, that is, Γ(B,B × Rn) = C∞(B, Rn). An interesting question is: given a bundle ξ, does there exist a nonvanishing section, i.e. a section s : B → E for which s(x) 6= 0 for all x ∈ B? More generally:

Definition. Given an n-bundle ξ and a number 1 ≤ k ≤ n, we say sections s1, . . . , sk ∈ Γ(ξ) are nowhere linearly dependent, or simply linearly independent sections, if for every −1 x ∈ B, s1(x), . . . , sk(x) are linearly independent in π (x). Then nowhere vanishing is the same as nowhere linearly dependent for k = 1. The existence of k linearly independent sections of a vector bundle is an important one, and one that can be answered with the information encoded in Stiefel-Whitney classes, to be introduced later.

Example 2.2.3. There is always a nonvanishing section on any trivial bundle B × Rn → B, for n ≥ 1, since for a fixed nonzero vector v ∈ Rn, s(b) = (b, v) is smooth. We have the following analogue of Lemma 1.2.12.

25 2.2 Sections 2 Vector Bundles

Lemma 2.2.4. Suppose f : E1 → E2 is a smooth map of vector bundles over B that makes the following diagram commute:

f E1 E2

π1 π2

B

Then if f is an isomorphism on each fibre, it is a bundle isomorphism.

Proof. Fix p ∈ B and take local trivializations U1,U2 ⊆ B of E1,E2 about p. Then both n bundles are trivial over U1 ∩ U2, i.e. there exist diffeomorphisms h1 :(U1 ∩ U2) × R → −1 n −1 π1 (U1 ∩ U2) and h2 :(U1 ∩ U2) × R → π2 (U1 ∩ U2) commuting with the projections. Consider the diagram f n h1 −1 −1 h2 n (U1 ∩ U2) × R π1 (U1 ∩ U2) π2 (U1 ∩ U2) (U1 ∩ U2) × R

π1 π2

U1 ∩ U2 ⊆ B

−1 n n Define the composition h = h2 ◦f ◦h1 :(U1 ∩U2)×R → (U1 ∩U2)×R . Then by hypothesis, h is smooth, commutes with the projection map and is a vector space isomorphism on each n fibre {x} × R for x ∈ U1 ∩ U2. Fixing such an x, this map is given by h(x, v) = (x, M(x) · v) for some matrix M(x) ∈ Mn(R), whose entries vary smoothly in terms of x. Moreover, since h is an isomorphism on fibres, det M(x) 6= 0 and there is even a formula for M(x)−1 which is also smooth in each entry in terms of x. Hence h is a diffeomorphism, so f is as well. We have the following analogue of Lemma 1.2.4.

Theorem 2.2.5. An n-dimensional vector bundle is trivial if and only if it admits n linearly independent sections.

n n Proof. ( =⇒ ) The trivial bundle B × R → B admits sections si : B → B × R , x 7→ (x, ei), n where ei is the ith standard basis vector in R . These are clearly linearly independent at every π point in the base. For any trivial bundle E −→ B with bundle isomorphism ϕ : B × Rn → E, set ti = si ◦ ϕ : B → E. These are n linearly independent sections, since ϕ B × Rn E

π

B

26 2.2 Sections 2 Vector Bundles

commutes and ϕ is an isomorphism on each fibre. ( ⇒ = ) Conversely, suppose E −→π B is an n-dimensional bundle with linearly independent sections s1, . . . , sn. Define a map

n ψ : B × R −→ E n X (x, v1, . . . , vn) 7−→ visi(x). i=1

Clearly ψ is a vector space isomorphism on each fibre of the bundle, since the si(x) are linearly independent, and the following diagram commutes: ψ B × Rn E

π

B

Thus by Lemma 2.2.4, ψ is a bundle isomorphism, so E is trivial.

Example 2.2.6. (Parallelizable spheres) Consider Sm for m ≥ 1. It turns out that only a select few spheres are parallelizable. For instance, S1 has a nonvanishing vector field:

F (x, y) = (−y, x)

Thus by Theorem 2.2.5, S1 is parallelizable (see to Example 2.1.2). However, the hairy ball theorem states that S2 has no nonvanishing tangent vector fields, so S2 is not parallelizable. This is related to the fact that χ(S2) = 2 and the Euler class of the tangent bundle TS2 → S2 is nontrivial. A similar proof shows that all even-dimensional spheres are not parallelizable. This suggests only look at odd-dimensional spheres to find nonvanishing sections, since any such sphere has Euler characteristic 0. For example, consider S3. As a subset of R4, this 3 4 2 2 2 2 3 is given by: S = {(x1, x2, x3, x4) ∈ R | x1 + x2 + x3 + x4 = 1}. To show S is parallelizable, we exhibit three linearly independent sections:

s1(x1, x2, x3, x4) = (−x2, x1, −x4, x3)

s2(x1, x2, x3, x4) = (−x3, −x4, x1, x2)

s3(x1, x2, x3, x4) = (−x4, −x3, x2, x1).

27 2.3 The Induced Bundle 2 Vector Bundles

One easily checks that these are smooth and linearly independent. Therefore by Theo- rem 2.2.5, S3 is parallelizable. Unfortunately, this technique does not work for odd higher dimensional spheres – in fact, S7 is the only other parallelizable sphere!

n n Example 2.2.7. Here we prove that the tautological line bundle γ1 over RP is nontrivial for all n. We have shown this already for RP 1 (see Example 2.1.3), but the general case is n straightforward. By Theorem 2.2.5, it suffices to show there is no nonvanishing section of γ1 . Suppose s : RP n → E is such a section. Composing with the double cover p : Sn → RP n gives a smooth map s ◦ p : Sn → RP n × Rn+1. Now a map is smooth if and only if each component is smooth, so we have

s ◦ p(x) = (¯x, t(x)x) for some smooth function t(x).

If s is nonvanishing, t(x) 6= 0 for all x ∈ Sn. However, since p(−x) = p(x) =x ¯, we have (¯x, −tx) = s ◦ p(−x) = s ◦ p(x) = (¯x, tx), and so t(x) = −t(x) which is only possible if n t(x) = 0. Therefore no nonvanishing section exists, so γ1 is nontrivial.

2.3 The Induced Bundle

Definition. A bundle map (or a morphism of (vector) bundles) is a pair of smooth maps between vector bundles of the same dimension

f E1 E2

π1 π2 f¯ B1 B2

making the diagram commute, and such that f maps the fibres of π1 isomorphically to the fibres of π2. Example 2.3.1. If f¯ : B → B is the identity, then by Lemma 2.2.4, any bundle map f : E1 → E2 over B must be a bundle isomorphism. Definition. Let ξ : E −→π B be a vector bundle and suppose f : B0 → B is a smooth map. 0 Then the induced bundle of ξ by f is a bundle f ∗ξ : E0 −→π B0 consisting of total space

E0 = {(x, y) ∈ B0 × E | f(x) = π(y)} and projection π0(x, y) = x. Proposition 2.3.2. For any ξ : E −→π B and f : B0 → B, the induced bundle f ∗ξ is a vector bundle over B0. Proof. Fix x ∈ B0. Then

(π0)−1(x) = {(x, y) ∈ E0 | π(y) = f(x)} ∼= {y ∈ E | π(y) = f(x)} = π−1(f(x))

28 2.3 The Induced Bundle 2 Vector Bundles

which is a vector space by hypothesis. Hence each fibre of f ∗ξ is a vector space. To show f ∗ξ is a bundle, we must find a neighborhood U 0 ⊆ B0 over which f ∗ξ is trivial. For f(x) ∈ B, we know there is a neighborhood U ⊆ B and a diffeomorphism h : π−1(U) → U × Rn which is a bundle isomorphism over U. Set U 0 = f −1(U), which is an open neighborhood of x ∈ B0. Then (π0)−1(U 0) = {(x, y) ∈ E0 | x ∈ U 0, y ∈ π−1(U)} so the map 0 0 −1 0 0 n h :(π ) (U ) −→ U × R (x, y) 7−→ (x, p(h−1(y))) is well-defined, where p : U × Rn → Rn is projection. By construction, h0 is a bundle isomorphism over U 0 (there is even an easy formula for the inverse), so U 0 is a trivialization of f ∗ξ over x ∈ B0. Example 2.3.3. If ξ : E → B is a trivial bundle, then any pullback f ∗ξ along f : B0 → B is a trivial bundle over B0. Indeed, the triviality of ξ means we can pick U = B in the proof of Proposition 2.3.2, and then U 0 = B0, so the entire bundle is trivial. Example 2.3.4. Every trivial bundle is a pullback:

n B × Rn {x} × R

ξ ξ0 f B {x}

Let f : B → {x} be the unique map from B to a point space and ξ0 the n-bundle over a ∗ point. Then f ξ0 = ξ. 0 Lemma 2.3.5. Suppose ξ : E −→π B and η : E0 −→π B0 are vector bundles and f E E0

π1 π2 g B B0 is a bundle map. Then η is isomorphic to the induced bundle g∗ξ. 00 Proof. By Lemma 2.2.4, it suffices to construct a bundle map η → g∗ξ. Let g∗ξ : E00 −→π B0 be the induced bundle and define a map ϕ : E0 → E00 by ϕ(y) = (π0(y), f(y)). By definition of the induced total space, ϕ(y) ∈ E00. It is immediate that ϕ makes the diagram ϕ E0 E00

π0 π00

B0

commute and is an isomorphism on fibres. Therefore ϕ is a bundle map.

29 2.4 Bundle Operations 2 Vector Bundles

2.4 Bundle Operations

π1 π2 Definition. Given two bundles ξ1 : E1 −→ B1 and ξ2 : E2 −→ B2, one forms the (Cartesian) π1×π2 product bundle ξ1 × ξ2 : E1 × E2 −−−→ B1 × B2. Here, for a point p = (p1, p2), the fibre −1 −1 −1 ∼ −1 −1 is (π1 × π2) (p) = π1 (p1) × π2 (p2) = π1 (p1) ⊕ π2 (p2). Lemma 2.4.1. Given two manifolds M and N, the tangent bundle T (M ×N) is isomorphic to TM × TN as vector bundles.

Example 2.4.2. Consider the torus X ∼= S1 × S1. The tangent bundle to the torus is a product bundle TX = TS1 × TS1. Since S1 is parallelizable, TS1 is trivial and therefore so is TS1 × TS1, so the torus is also parallelizable.

We would like to define a product of bundles over a common base B that yields a bundle over B. For this, we utilize the induced bundle construction from Section 2.3.

π1 π2 Definition. Given two bundles ξ1 : E1 −→ B and ξ2 : E2 −→ B over the same base B, the direct sum, or Whitney sum of ξ1 and ξ2, denoted ξ1 ⊕ ξ2, is the induced bundle of the product bundle over the diagonal map d : B → B × B, that is,

∗ E1 ⊕ E2 = d (E1 × E2) E1 × E2

∗ π1 ⊕ π2 = d (π1 × π2) π1 × π2 d B B × B

Categorically, there are many ways of obtaining new vector spaces from old ones using functors. Examples of functors on vector spaces include: direct sum (⊕), tensor product (⊗), Vk ∗ ∗ exterior power ( ) and the vector space dual (V ), itself a special case (V = Hom(V, R)) of the Hom functor (Hom(V,W )). It turns out that these examples are smooth functors. It is a general principle that any smooth functor of vector spaces determines an analagous construction of vector bundles. It is useful to expand upon our definition of vector bundles in terms of transition maps in order to construct further bundle operations. The basic idea is that transition maps arise from overlapping charts. Specifically, let {Ui} be a locally trivial covering of B with respect π n −1 to some bundle ξ : E −→ B and let hi :(Ui × R → π (Ui) be the local trivializations. Then n for each pair of overlapping charts Ui ∩ Uj 6= ∅, there is a homeomorphism (U ∩ V ) × R → (U ∩ V ) × Rn, which reduces to a vector space isomorphism on each fibre {x} × Rn, induced by the diagram

hj n hi −1 n (Ui ∩ Uj) × R π (Ui ∩ Uj) (Ui ∩ Uj) × R

π

Ui ∩ Uj

30 2.4 Bundle Operations 2 Vector Bundles

This defines a smooth map hij : Ui ∩ Uj → GLn(R) with inverse hji, by which we mean for −1 each x ∈ Ui ∩ Uj, hji(x) = hij(x) . Conversely, given a collection of charts {Ui} and smooth maps on the intersections hij : π Ui ∩ Uj → GLn(R), we can reconstruct the original bundle E −→ B. Explicitly, set

a n E = (Ui × R )/ ∼ i

n n where ∼ is the equivalence relation (p, ~x) ∼ (p, hij(~x)) for (p, ~x) ∈ (Ui ∩ Uj) × R ⊆ Ui × R . n The projection π is induced by Ui ×R → Ui for each chart Ui. Local triviality is guaranteed by construction, so E −→π B is the desired bundle. There is a subtlety in this construction. For three locally trivial charts U1,U2 and U3 of a vector bundle with U1 ∩ U2 ∩ U3 6= ∅, we must have h31h23h12 = 1 in GLn(R). This comes −1 −1 −1 from the fact that on U1 ∩ U2 ∩ U3,(h1 ◦ h3) ◦ (h3 ◦ h2) ◦ (h2 ◦ h1) = id. However, such a condition is not necessarily guaranteed when constructing a bundle from abstract transition maps. We will implicitly assume the condition is always met.

1 1 1 Example 2.4.3. Consider the tautological line bundle γ1 : E → RP = S . Let’s instead ∗ view this bundle as a family of maps into GL1(R) = R r {0} = R . Take two overlapping charts U and V :

U ) ) 1

) S )

V

−1 ∼ −1 ∼ Then π (U) = U × R and π (V ) = V × R, and U ∩ V = A ∪ B, the union of two disjoint ∗ open intervals on the circle. We can define the map hUV : U ∩ V → R by our choices of sending A and B each to either R>0 or R<0. Thus the equivalence classes of line bundles 1 over S are determined by whether the maps hUV send the disjoint pieces of U ∩ V into the same connected component of R∗ or the opposite components. Hence the only line bundles over the circle are, up to isomorphism, the tautological and trivial bundles.

Definition. A structure group for a vector bundle ξ with fibres V is a subgroup G of GL(V ) such that all transition maps of ξ take values in G.

Example 2.4.4. Let E −→π S2 be a two-dimensional bundle over the 2-sphere. Consider an open covering U ∪ V = S2 with U as the northern hemisphere and V as the southern hemisphere.

31 2.4 Bundle Operations 2 Vector Bundles

V

U

Then U ∼= D2,V ∼= D2 and U ∩ V is homotopy equivalent to the circle S1. The total space 2 ` 2 of any bundle may be constructed by E = (U × R ) (V × R )/ ∼ where the gluing is 1 induced by the single transition map hUV : S → GL2(R). In fact, the special orthogonal 2 group SO(2) ⊂ GL2(R) is a structure group for any 2-bundle over S . Identifying SO(2) with S1 ⊂ R2, any transition map may be viewed as a map h : S1 → S1. Of course, up to homotopy, any such map can be identified with an integer n, its winding number. For each n ∈ Z, let  cos x sin x h (x) = n − sin x cos x 1 1 be the corresponding transition map hn : S → S = SO(2). We will show that these hn give non-isomorphic 2-bundles over S2. To construct new bundles from old ones, suppose ξ and η are vector bundles over B, of respective dimensions m and n, with some specified transition maps. By refining the charts of each bundle if necessary, we may assume there is a collection of charts {U} over which both bundles are trivial. Denote the transition maps for ξ by gUV : U ∩ V → GLm(R) and for η by hUV : U ∩ V → GLn(R). Now any functorial operation on vector spaces defines an operation on vector bundles. Explicitly, if F is a functor on the category of pairs of vector spaces (W1,W2), then for any bundles ξ with fibres W1 and η with fibres W2, there is a vector bundle F(ξ, η) induced by the transition maps F(gUV , hUV ): U ∩ V −→ GL(F(W1,W2)). Example 2.4.5. The Whitney sum ξ ⊕ η over a base B may be constructed from the direct sum functor, with transition maps

(g ⊕ h)UV := gUV ⊕ hUV : U ∩ V −→ GLm+n(R). It is immediate that these maps are smooth, so they indeed define a bundle. One can show that this corresponds with the Whitney sum construction defined previously. Definition. The tensor bundle of ξ and η over B is the bundle ξ ⊗η induced by the tensor product functor, given by transition maps

(g ⊗ h)UV := gUV ⊗ hUV : U ∩ V −→ GLmn(R). Definition. The dual bundle of a vector bundle ξ is the bundle ξ∗ induced by the dual functor (−)∗ = Hom(−, R), using the matrix transpose as transition maps:

∗ t gUV := gUV : U ∩ V −→ GLn(R).

32 2.4 Bundle Operations 2 Vector Bundles

More generally, we define:

Definition. The Hom bundle HomB(ξ, η) between two vector bundles ξ and η over B is ∼ ∗ the bundle induced by the Hom functor Hom(W1,W2) = W1 ⊗ W2, defined explicitly by the transition maps t Hom(g, h)UV := gUV ⊗ hUV : U ∩ V −→ GLmn(R). ∗ ∗ In other words, HomB(ξ, η) = ξ ⊗ η, where ξ is the dual bundle. Definition. Let ξ be an n-dimensional vector bundle over B and k any nonnegative integer. Then the exterior power functor Vk defines the kth exterior power bundle Vkξ which is given by transition maps ^k gUV : U ∩ V −→ GL n (R). (k) Example 2.4.6. If ξ and η are line bundles over B, then ξ ⊗ η is again a line bundle over B. This in fact defines a group structure on the set of equivalence classes of line bundles over B. 1 Take for example the two line bundles over S : the trivial bundle ξ0 and the tautological 1 1 bundle γ1 . Let S be covered by {U, V } as in Example 2.4.3. Then the tensor product on ∗ transition maps gUV ⊗ hUV : U ∩ V → GL1(R) = R is just given by scalar multiplication:

(gUV ⊗ hUV )(x) = gUV (x)hUV (x).

It follows that

∼ 1 ∼ 1 1 ∼ 1 1 1 ∼ ξ0 ⊗ ξ0 = ξ0, ξ0 ⊗ γ1 = γ1 , γ1 ⊗ ξ0 = γ1 and γ1 ⊗ γ1 = ξ0.

Hence the group of line bundles over S1 is isomorphic to Z/2Z.

33 3 Characteristic Classes

3 Characteristic Classes

3.1 Metrics on Vector Bundles

There is a natural Euclidean metric on any finite dimensional vector space V given by fixing n Pn an isomorphism ϕ : V → R and setting hv, wi = i=1 ϕ(v)iϕ(w)i, where xi denotes the ith component of a vector x ∈ Rn. To extend the Euclidean metric over the fibres of a vector bundle, we need a slightly different perspective.

Proposition 3.1.1. For a finite dimensional vector space V , there is a bijective correspon- dence between inner products h·, ·i and (positive definite) quadratic forms µ : V → R. Proof. For an inner product h·, ·i on V , a quadratic form is given by µ(v) := hv, vi. Con- 1 versely, a quadratic form µ determines an inner product hv, wi := 2 (µ(v +w)−µ(v)−µ(w)). One now checks that these assignments are inverses.

Definition. A Euclidean metric on a vector bundle ξ : E −→π B is a smooth map µ : E → R such that µ gives a positive definite quadratic form on each fibre π−1(x). Corollary 3.1.2. A Euclidean metric on a vector bundle determines an inner product on every fibre.

Example 3.1.3. A metric on the tangent bundle TM to a manifold M is equivalent to a choice of a Riemannian metric on M.

Proposition 3.1.4. Every vector bundle has a Euclidean metric.

π Proof. Let ξ : E −→ B be a vector bundle bundle. Fix a locally trivial covering {Ui} of B such that there is an isomorphism hi making h n i −1 Ui × R π (Ui)

π

Ui

commute for each Ui. This determines a partition of unity {ρi} of B, i.e. a collection of P maps ρi : B → R>0 such that ρi is supported on Ui and i ρi ≡ 1 on B. For each Ui, put n the product quadratic form on Ui × R ; that is, set

n 0 X 2 µi(x,~v) = vj j=1

n for any (x,~v) ∈ Ui × R . This determines a quadratic form on each fibre given by

0 −1 µi = hi ◦ µi : π (x) −→ R>0

34 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

when x ∈ Ui. If x 6∈ Ui, set µi(x) = 0. We extend this to a map on the entire total space by

µ : E −→ R>0 X x 7−→ ρi(π(x))µi(x) i

It is immediate that µ is a positive definite quadratic form on each fibre π−1(x), and so defines a Euclidean metric on ξ.

π1 π2 Definition. A subbundle of a vector bundle η : E1 −→ B is a bundle ξ : E2 −→ B such −1 −1 that each fibre π2 (x) ⊆ π1 (x) is a vector subspace, and the induced diagram commutes:

E1 E2

π1 π2 B

This is denoted ξ ⊆ η.

π2 π1 Definition. Let ξ : E2 −→ B be a subbundle of η : E1 −→ B. Then the orthogonal complement of ξ in η with respect to a Euclidean metric µ on η is the bundle ξ⊥ whose −1 ⊥ −1 −1 fibres are π2 (x) , the orthogonal complement of π1 (x) in π2 (x), for each x ∈ B. Theorem 3.1.5. Given a subbundle ξ ⊆ η, the orthogonal complement ξ⊥ is a subbundle of η. Further, ξ ⊕ ξ⊥ ∼= η as bundles. Definition. Let M m be a submanifold of N n. Then the normal bundle of M in N is the subbundle orthogonal to TM in TN|M :

n ⊥ VM := (TM) ⊆ TN|M .

n n R ∼ n n Corollary 3.1.6. Let M be a submanifold of R . Then TM ⊕ VM = (R × R )|M , the trivial n-bundle over Rn restricted to M.

3.2 Stiefel-Whitney Classes

Given a vector bundle ξ over B, there is an assignment of cohomology classes in each Hi(B; Z/2Z) satisfying a set of axioms which uniquely characterize these so-called Stiefel- Whitney classes.

Definition. A set of Stiefel-Whitney classes for a vector bundle ξ over B is a choice i of ωi = ωi(ξ) ∈ H (B; Z/2Z) for each i ≥ 0 and a total Stiefel-Whitney class ω(ξ) = ω0 + ω1 + ω2 + ... which satisfy:

(0) If the dimension of the fibres of ξ is n, then ωi(ξ) = 0 for all i > n.

0 (1) ω0(ξ) = 1 in H (B; Z/2Z).

35 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

0 ∗ (2) The ωi are natural with respect to pullbacks; that is, for any map f : B → B, ωi(f ξ) = ∗ f ωi(ξ). (3) For any other bundle η over B,

∞ X Y ω(ξ ⊕ η) = ω(ξ) · ω(η) = ωj(ξ)ωk(η). i=0 j+k=i

1 1 1 1 1 (4) For the tautological bundle γ1 over RP , ω1(γ1 ) 6= 0 in H (RP ; Z/2Z).

Axiom 4 is necessary to rule out the assignment ωi = 0 for each i, as we want to study a nontrivial set of cohomology classes. In Sections 3.3 and 3.4, we will construct the Stiefel-Whitney classes for any vector bundle ξ explicitly and prove that the axioms uniquely determine these.

Remark. By Axiom 3, ω(ξ1 ⊕ ξ2) = ω(ξ1)ω(ξ2) for any bundles ξ1, ξ2 but by definition ∗ of the Whitney sum, ξ1 ⊕ ξ2 = d (ξ1 × ξ2) where ξ1 × ξ2 is the product bundle. Thus ∗ ω(ξ1)ω(ξ2) = d ω(ξ1 × ξ2). Consider the canonical projections

p1 B1

B1 × B2

p2 B2

∗ ∗ ∗ ∗ ∗ ∼ Then p1ξ1 ⊕ p2ξ2 = d (p1ξ1 × p2ξ2) = ξ1 × ξ2 and thus

∗ ∗ ω(ξ1 × ξ2) = p1ω(ξ1)p2ω(ξ2) = ω(ξ1) × ω(ξ2), where × is the algebraic cross product.

n Example 3.2.1. Let ξ0 be the trivial bundle M × R → M over any manifold M. By Example 2.3.4, ξ0 is a pullback of a bundle over a point space, so it follows from Axiom 2 that ω(ξ0) = 1. For any manifold M, we will write ω(M) = ω(TM).

n n R Proposition 3.2.2. For any submanifold M ⊆ R , ω(M)ω(VM ) = 1.

n R n Proof. By Corollary 3.1.6 and Axiom 3, ω(M)ω(VM ) = ω(M × R ) = 1.

n+1 n n+1 R Example 3.2.3. View S ⊂ R and consider the normal bundle VSn . Then the function n+1 R n x 7→ ~x is a nonvanishing section of VSn , so by Theorem 2.2.5, the normal bundle over S is trivial. Thus by Axiom 3 and Proposition 3.2.2,

n+1 n R n n 1 = ω(TS )ω(VSn ) = ω(TS ) · 1 = ω(TS ).

Therefore we have proven that the total Stiefel-Whitney class of any sphere is trivial.

36 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

Let ω = ω(ξ) be the total Stiefel-Whitney class of a bundle ξ over B. By Axiom 1, ω0 = 1 so we can write ω = 1 + ω1 + ω2 + ...

We can view this sum as a formal power series ω = 1 + x, where x = ω1 + ω2 + ..., so that there exists a formal inverse

−1 2 2 ω¯ := ω = 1 − x + x − ... = 1 − (ω1 + ω2 + ...) + (ω1 + ω2 + ...) + ...

Alternatively, there is a recursive formula for theω ¯i inω ¯ =ω ¯0 +ω ¯1 +ω ¯2 + ... coming from the relation ω · ω¯ = 1:  ω¯0 = 1  X ω¯i = ωjω¯k for i ≥ 1.  j,k>0 j+k=i

(Here, we are using the fact that the ωi are elements of Z/2Z to eliminate signs everywhere.)

n Rn Example 3.2.4. For any manifold M ⊆ R , ω(M) = ω(VM ).

Example 3.2.5. We will compute the Stiefel-Whitney classes for RP n using our knowledge of ω(TSn). Recall that the cohomology ring of RP n with Z/2Z coefficients is a truncated polynomial ring • n ∼ n+1 H (RP ; Z/2Z) = F2[a]/(a ) 1 where a is an indeterminate. By definition, the tautological bundle γn is a subbundle of the n+1 n ⊥ 1 trivial bundle ξ0 on RP . Let γ denote the orthogonal complement of γn in the trivial 1 bundle. We are interested in computing the Stiefel-Whitney class ω = ω(γn). By Axiom 0, 1 1 n ωi = 0 for all i > dim γn = 1, so we have ω = 1 + ω1. Viewing RP ⊂ RP , we actually have a bundle map

1 1 E(γ1 ) E(γn)

i n RP 1 RP

1 ∼ which is an isomorphism on each fibre (i.e. a bundle morphism). By Lemma 2.3.5, γ1 = ∗ 1 ∗ 1 1 1 i (γn), the induced bundle. Then by Axiom 2, i ω1(γn) = ω1(γ1 ) but by Axiom 4, ω1(γ1 ) 6= 0. ∗ 1 1 1 Thus i ω1(γn) 6= 0, so ω1(γn) 6= 0. Hence we deduce that ω(γn) = 1+a where a is a generator of H1(RP n; Z/2Z). Next, by the formulas for inverses of Stiefel-Whitney classes, we have

⊥ 1 −1 ω(γ ) = ω(γn) = (1 + a) = 1 − a + a2 − ... ± an 2 n = 1 + a + a + ... + a using Z/2Z-coefficients.

n ∼ 1 ⊥ Lemma 3.2.6. There is a bundle isomorphism T RP = Hom(γn, γ ).

37 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

Proof. View RP n as the quotient of Sn by the antipodal action ∼. For any x ∈ Sn, a tangent n 0 n vector ~v ∈ TxS is of the form ~v = α (0) for some curve α(t) in S with α(0) = x. Passing n n n to the quotient S / ∼, ~v ∈ TxS is identified with −~v ∈ T−xS . So we see that there is n ∼ n n a correspondence T RP = {((x, −x), (~v, −~v)) | x ∈ S ,~v ∈ TxS }. On the other hand, n ⊥ n+1 n ∼ ⊥ TxS = Lx , where Lx is the line in R through x and −x. Thus Tx¯RP = Hom(Lx,Lx ). 1 ⊥ This defines the isomorphism on fibres, since Lx is the fibre of x in γn and by definition Lx is the fibre in γ⊥. This defines a map

n 1 ⊥ ϕ : T RP −→ Hom(γn, γ ) (¯x,~v) 7−→ t~v

⊥ where t~v is the transformation Lx → Lx , ~x 7→ ~v. It is routine to verify that ϕ is a smooth map, so by Lemma 2.2.4, ϕ is a bundle isomorphism.

n ∼ 1 ⊥ Now that we have T RP = Hom(γn, γ ), take the Whitney sum with the trivial line 1 1 ∼ 1 bundle Hom(γn, γn) = ξ0 – here, triviality follows from the existence of the nonzero section 1 1 id : γn → γn – on both sides: n 1 ∼ 1 ⊥ 1 1 T RP ⊕ ξ0 = Hom(γn, γ ) ⊕ Hom(γn, γn) n ∼ 1 ⊥ 1 =⇒ T RP = Hom(γn, γ ⊕ γn) since Hom commutes with direct sums n ∼ 1 n+1 =⇒ T RP = Hom(γn, ξ0 ) by definition of the orthogonal complement n+1 ∼ M 1 1 = Hom(γn, ξ0 ) i=1 n+1 ∼ M 1 ∗ = (γn) by definition of the dual bundle i=1 n+1 ∼ M 1 = γn, i=1 where in the last step, we use the fact that each fibre is an inner product space and thus n ∼ Ln 1 isomorphic to its dual. Applying Axiom 3 to the result T RP = i=1 γn gives us

n 1 n+1 n+1 n ω(RP ) = ω(γn) = (1 + a) = 1 + (n + 1)a + ... + (n + 1)a . (At the end, we use Axiom 0: an+1 = 0.) Example 3.2.7. For the projective plane, the above calculations show that

2 3 2 2 ω(RP ) = (1 + a) = 1 + 3a + 3a ≡ 1 + a + a (mod 2).

In particular, RP 2 has nontrivial Stiefel-Whitney classes in every dimension. There is a generalization of Proposition 3.2.2 to immersed manifolds.

Theorem 3.2.8. Let M be an m-manifold and f : M → Rn an immersion. Then TM is ∗ n n n a subbundle of the pullback f ξ0 of the trivial n-bundle ξ0 over R , and ω(M)ω(Vf ) = 1, ⊥ ∗ n where Vf = (TM) as a subbundle of f ξ0 .

38 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

Proof. For any map f : M → Rn, we have a commutative diagram df TM Rn × Rn

f M Rn

n which is dxf : TxM → {f(x)} × R on each tangent space. So if f is an immersion, TM n naturally embeds as a subbundle of ξ0 |f(M), the trivial n-bundle restricted to f(M). Under the pullback

∗ n n f E(ξ0 ) E(ξ0 )

f M Rn

∗ n we also have TM ⊆ f ξ0 as a subbundle. Hence by Theorem 3.1.5 and the Stiefel-Whitney axioms, ∗ n ∗ n ∗ ω(M)ω(Vf ) = ω(f ξ0 ) = f ω(ξ0 ) = f (1) = 1.

Corollary 3.2.9. If an n-dimensional manifold M can be immersed into Rn+1, then for i each i, ωi(M) is the i-fold cup product (ω1(M)) .

n+1 n+1 ∼ Proof. Suppose i : M → R is an immersion. Then by Theorem 3.2.8, R |M = TM ⊕Vf ⊥ as bundles and consequently ω(M) · ω(Vf ) = 1. Here, Vf = (TM) is a line bundle, so ω(Vf ) = 1 + ω1(Vf ). Hence we have

1 = ω(M) · ω(Vf ) = ω(M)(1 + ω1(Vf )) 2 n n =⇒ ω(M) = 1 − ω1(Vf ) + ω1(Vf ) − ... + (−1) ω1(Vf ) by “geometric series” 2 n = 1 + ω1(Vf ) + ... + ωn(Vf ) (mod 2).

i i Therefore ω1(M) = ω1(Vf ) and for each i > 1, ωi(M) = ω1(Vf ) = ω1(M) .

Example 3.2.10. We will show that there does not exist an immersion RP 4 → R6. By Example 3.2.5, the total Stiefel-Whitney class of RP 4 is

4 5 2 3 4 5 ω(RP ) = (1 + a) = 1 + 5a + 10a + 10a + 5a + a ≡ 1 + a + a4 (mod 2),

where a is a generator of H1(RP 4; Z/2Z). Suppose f : RP 4 → R6 is an immersion. By 6 ∼ 4 Theorem 3.2.8, R |RP 4 = T RP ⊕ Vf but Vf is a 2-bundle so ω(Vf ) = 1 + ω1 + ω2 for Stiefel- • 4 4 Whitney classes ω1, ω2 ∈ H (RP ; Z/2Z) for the bundle Vf . Then ω(RP ) · ω(Vf ) = 1 which

39 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

can be written

4 1 = ω(RP ) · ω(Vf ) 4 = (1 + a + a )(1 + ω1 + ω2) 4 4 4 = 1 + (a + ω1) + (aω1 + ω2) + aω2 + a + a ω1 + a ω2.

This implies a4 = 0 in H4(RP 4; Z/2Z) which is false. Therefore no such immersion exists.

Example 3.2.11. By a similar argument, we can show that RP 9 does not admit an immer- sion into R15. Using Example 3.2.5, we have

9 2 8 1 = ω(RP )ω(Vf ) = (1 + a + a )(1 + ω1 + ω2 + ... + ω6), so it follows from an elementary calculation that

2 4 6 ω(Vf ) = 1 + a + a + a .

As before, this implies RP 9 does not immerse into RN for N < 16. By Whitney’s embedding theorem, RP 9 does immerse into R17, so it is an interesting question whether RP 9 immerses into R16. Theorem 3.2.12. Let ξ be an n-dimensional bundle over B. If ξ has a nonvanishing section, n then ωn(ξ) = 0 in H (B; Z/2Z). Proof. Let s : B → E be a nonvanishing section. Then for each x ∈ B, s(x) 6= 0 in π−1(x) so −1 −1 this spans a linear subspace Lx ⊆ π (x). Since s is smooth, it follows that the Lx ⊆ π (x) define a 1-dimensional subbundle η: E(η) E

π

B

Further, s also defines a nonvanishing section of η, so η is a trivial line bundle and thus ω(η) = 1. Fix a Euclidean metric on ξ, so that η⊥ is defined as a subbundle of ξ. Then by Theorem 3.1.5, η ⊕ η⊥ ∼= ξ. By the Stiefel-Whitney axioms,

ω(ξ) = ω(η)ω(η⊥) = 1 · ω(η⊥) = ω(η⊥),

⊥ ⊥ but since the dimension of η is n − 1, ωn(η ) = 0. Hence ωn(ξ) = 0.

Corollary 3.2.13. If n is even, then RP n is not parallelizable.

n Proof. By our calculations in Example 3.2.5, ωn(RP ) = n + 1 6= 0 (mod 2), so Theo- rem 3.2.12 implies T RP n has no nonvanishing section. Hence by Theorem 2.2.5, T RP n is nontrivial.

40 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

n n Example 3.2.14. Notice that for even n, ω(S ) = 1 and therefore ωn(S ) = 0. Thus we cannot use the same argument to show that spheres are not parallelizable. We will introduce the Euler class to study TSn more effectively.

Example 3.2.15. Notice that for RP 2, the fact that ω(RP 2) 6= 1 does not preclude the existence of an immersion into R3. In fact, there does exist an immersion RP 2 → R3 called Boy’s surface (discovered in 1901). This is a special case of a more general result:

Theorem 3.2.16 (Immersion Conjecture, proved by R. Cohen). Let M be a smooth, compact n-manifold. Consider the dyadic expansion of n,

i1 i2 ik n = 2 + 2 + ... + 2 for i1 < i2 < . . . < ik.

Let α(n) = k, the number of 1’s in this expansion of n. Then there exists an immersion M → R2n−α(n). Massey also showed that this 2n − α(n) bound is sharp. Assume M is a smooth, closed, compact n-manifold. Then from algebraic topology we n ∼ know that H (M; Z/2Z) = Z/2Z. The cup product gives a pairing

n n H (M; Z/2Z) × H (M; Z/2Z) −→ Z/2Z (α, c) 7−→ α ∩ c

and in particular, Poincar´eduality can be realized as a map

n H (M; Z/2Z) −→ Z/2Z α 7−→ α ∩ [M],

where [M] ∈ Hn(M; Z/2Z) is the fundamental class of M.

Definition. The ith Stiefel-Whitney number of M is the element ωi(M) ∩ [M] in Z/2Z.

ε1 εn More generally, let ε1, . . . , εn ≥ 0 be integers and consider the cohomology class ω1 ··· ωn . n When ε1 + 2ε2 + ... + nεn = n, this belongs to the top cohomology H (M; Z/2Z) and so the integer ε1 εn ω1 ··· ωn ∩ [M] ∈ Z/2Z

is defined. We call this integer the Stiefel-Whitney number for the sequence ε1, . . . , εn. Theorem 3.2.17. For any smooth, closed, compact n-manifold M, the Stiefel-Whitney num- ε1 εn bers ω1 ··· ωn ∩ [M] are 0 for all sequences ε1, . . . , εn satisfying ε1 + ... + nεn if and only if M = ∂W for some (n + 1)-manifold W .

Proof. Assume M is the boundary of a manifold W . Then TM is a subbundle of the W ⊥ restriction TW |M and VM = (TM) is a line bundle over M. In fact, since M = ∂W , there is a consistent choice of an outward unit normal vector at every point of M ⊂ W , and thus W W a nonvanishing section of VM . So VM is a trivial line bundle by Theorem 2.2.5. Therefore Theorem 3.1.5 implies that ω(M) = ω(TW |M ).

41 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

Now take the fundamental class [W ] ∈ Hn+1(W, M; Z/2Z). Recall that ∂∗[W ] = [∂W ] = [M] in Hn(M; Z/2Z). Then by duality for manifolds with boundary, we have

ε1 εn ε1 εn ω1 ··· ωn ∩ [M] = (ω1 ··· ωn ) ∩ ∂∗[W ] ε1 εn = δ(ω1 ··· ωn ) ∩ [W ].

ε1 εn ∗ To prove the Stiefel-Whitney number is 0, it suffices to show ω1 ··· ωn ∈ im i , since we have an exact sequence

∗ n i n δ n+1 H (W ; Z/2Z) −→ H (M; Z/2Z) −→ H (W, M; Z/2Z).

∗ However, notice that if i : M,→ W is the natural embedding, then we have TW |M = i (TW ), the pullback of the tangent bundle of W . Thus by naturality of Stiefel-Whitney classes, for each j ≥ 0, ∗ ∗ ωj(TM) = ωj(TW |M ) = ωj(i TW ) = i ωj(TW ). ∗ Hence each ωj ∈ im i so the proof is complete. (The converse follows from Thom’s theorem.)

Corollary 3.2.18. For n even, RP n is not the boundary of any (n + 1)-manifold.

Proof. If n is even, ω(RP n) = 1+(n+1)a+...+(n+1)an by Example 3.2.5, so in particular n n n n ωn = (n + 1)a = a (mod 2). Thus ωn is a generator of H (RP ; Z/2Z). It follows that n n ωn ∩ [RP ] 6= 0 so by Theorem 3.2.17, RP does not bound a manifold.

Definition. Two n-manifolds M1 and M2 are called cobordant if there exists a compact ` (n + 1)-manifold W such that ∂W = M1 M2.

Corollary 3.2.19. Two compact, closed n-manifolds M1 and M2 have the same Stiefel- Whitney numbers for all sequences ε1, . . . , εn such that ε1 + ... + nεn = n if and only if they are cobordant. ` Proof. This follows directly from Theorem 3.2.17 with M = M1 M2, using the fact that ωi(M) ∩ [M] = ωi(M1) ∩ [M1] − ωi(M2) ∩ [M2] for all i.

Let Mn be the set of cobordism classes of smooth closed n-dimensional manifolds.

Lemma 3.2.20. For any n ≥ 1, Mn is a group under disjoint union.

Proof. Let [M] denote the cobordism class of a smooth closed manifold M and αM ∈ n Hn(M; Z/2Z) the (unoriented) fundamental class of M. Also ε = (ε1, . . . , εn) ∈ N0 will denote a sequence which is assumed to satisfy ε1 + 2ε2 + ... + nεn = n. For such a sequence, ε1 εn we let sM (ε) = (ω1(M) ∪ · · · ∪ ωn(M) ) ∩ αM ∈ Z/2Z denote the Stiefel-Whitney number of the sequence ε. First, for [M], [N], [P ] ∈ Mn, we have a cobordism

42 3.2 Stiefel-Whitney Classes 3 Characteristic Classes

M N P

M N P

which shows associativity. The fact that [∅] is the zero class is obvious from the definition of addition using disjoint union. Finally, every class [M] is its own inverse since W = M ×[0, 1] gives a cobordism between M × {0} ` M × {1} and the empty set: M M

∅

Therefore Mn is a group, and even a 2-group since [M] + [M] = 0 for all M.

Theorem 3.2.21. Mn is a finite group.

Proof. Corollary 3.2.19 says that M and N are cobordant if and only if sM (ε) = sN (ε) for n all sequences ε = (ε1, . . . , εn) ∈ N0 satisfying ε1 + 2ε2 + ... + nεn = n. Of course, there are only a finite number of these sequences satisfying the given identity to begin with, and for each sequence ε we may specify a 0 or a 1 for its Stiefel-Whitney number. Therefore there are only a finite number of choices s(ε) ∈ Z/2Z for all sequences ε, so by Corollary 3.2.19 there are a finite number of distinct cobordism classes of n-manifolds.

Example 3.2.22. For n = 4, we exhibit 4 distinct cobordism classes of 4-manifolds. The trivial class corresponds to Stiefel-Whitney numbers s0(ε) = 0 for all admissible ε – this is Theorem 3.2.17 – so to show any other cobordism classes are nontrivial, it will suffice to compute nonzero Stiefel-Whitney numbers on any of the following sequences:

(4, 0, 0, 0), (2, 1, 0, 0), (1, 0, 1, 0), (0, 2, 0, 0) and (0, 0, 0, 1).

4 4 4 Let’s begin with the class [RP ] ∈ M4. By Example 3.2.5, ω(RP ) = 1 + a + a , where a is a generator of H1(RP 4; Z/2Z). In particular, we have Stiefel-Whitney numbers:

4 4 4 4 sRP 4 (0, 0, 0, 1) = ω4(RP ) ∩ αRP 4 = 1 since ω4 = a generates H (RP ; Z/2Z) 4 2 sRP 4 (0, 2, 0, 0) = ω2(RP ) ∩ αRP 4 = 0 since ω2 = 0.

43 3.3 The Universal Bundle 3 Characteristic Classes

Since s(0, 0, 0, 1) = 1, [RP 4] is a nontrivial cobordism class. Next, consider [RP 2 × RP 2]. By the (external) product formula for Stiefel-Whitney classes,

2 2 2 2 ω(RP × RP ) = ω(RP ) × ω(RP ) = (1 + a) × (1 + b),

∗ ∗ 2 2 2 but by definition, (1+a)×(1+b) = p1(1+a)∪p2(1+b), where p1, p2 : RP ×RP → RP are the canonical projections and a (resp. b) generates H1(RP 2; Z/2Z) for one of the copies of ∗ ∗ 1 2 2 ∗ ∗ Z/2Z. Set c = p1a = p2b ∈ H (RP ×RP ; Z/2Z). Now we really have p1(1+a)∪p2(1+b) = 1 + c + c + c2 ≡ 1 + c2 (mod 2). Thus we can compute the same Stiefel-Whitney numbers again for this cobordism class:

2 2 s(RP 2)2 (0, 0, 0, 1) = ω4((RP ) ) ∩ α(RP 2)2 = 0 ∩ α(RP 2)2 = 0 2 2 2 2 s(RP 2)2 (0, 2, 0, 0) = ω2((RP ) ) ∩ α(RP 2)2 = c ∩ α(RP 2)2 = 1.

The latter tells us that [RP 2 × RP 2] is nontrivial in the cobordism group, while the former 2 2 4 4 says that [RP ×RP ] 6= [RP ]. Finally, since every class in M4 is its own inverse and [RP ] and [RP 2 × RP 2] are nontrivial and distinct, we get a fourth nontrivial class by adding these 4 2 2 4 ` 2 2 together: [RP ] + [RP × RP ] = [RP (RP × RP )].

3.3 The Universal Bundle

In this section we construct a “universal bundle” γn of dimension n such that every n- dimensional vector bundle ξ over B is induced by a map from B into the base of γn, with cer- ∗ n tain loose conditions assumed for B. Then by the Stiefel-Whitney axioms, ωi(ξ) = f ωi(γ ) for any such map f, so to explicitly construct the Stiefel-Whitney classes, we construct a set of generators for the cohomology ring of the base of γn and show that they satisfy the axioms. 1 n k The tautological line bundle γn over RP generalizes to a “canonical bundle” γn of the Grassmannian manifold of the appropriate dimensions. We recall the construction of this manifold here.

Definition. For n ≥ 1 and any vector space V , the Grassmannian manifold Grn(V ) is the set of all n-dimensional vector subspaces of V .

n+k We will particularly be focused on the Grassmannian Grn(R ) for k ≥ 0. n+1 n Example 3.3.1. For any n, Gr1(R ) = RP , projective n-space. n+k Definition. For n ≥ 1, k ≥ 0, the Stiefel manifold Vn(R ) is defined as the set of all n-frames in Rn+k, i.e. the set of all collections of n linearly independent vectors in Rn+k. n+k n+k There is a natural surjection p : Vn(R ) → Grn(R ) sending an n-frame to the subspace of Rn+k it spans. n+k n(n+k) Proposition 3.3.2. Vn(R ) is an open submanifold of R for any n, k. Definition. The determinant function det : Rn(n+k) = Rn × Rn+k → Rn+k is a continuous n+k −1 n+k function, and clearly Vn(R ) = det (R r {0}).

44 3.3 The Universal Bundle 3 Characteristic Classes

n+k n+k n+k Now the map p : Vn(R ) → Grn(R ) defines the quotient topology on Grn(R ): a n+k −1 n+k set U ⊆ Grn(R ) is open if and only if p (U) is open in Vn(R ). n+k Lemma 3.3.3. Grn(R ) is a compact manifold of dimension nk. n+k Proof. Given a point V ∈ Grn(R ), we need to define an open neighborhood U of V and a diffeomorphism ϕ : U → ϕ(U) ⊆ Rnk. By definition, V is an n-dimensional subspace of Rn+k, so it has a k-dimensional orthogonal complement V ⊥ such that Rn+k = V ⊕ V ⊥. Define the open neighborhood

n+k ⊥ U = {X ⊆ R | X is an n-dimensional subspace and X ∩ V = 0}. Note that there is a one-to-one correspondence

⊥ ∼ nk U ←→ Hom(V,V ) = R X 7−→ (L : V → V ⊥)

where L is the linear map between V and its complement such that X is the graph of L. To show that the correspondence is a diffeomorphism, we show that the composition n+k ⊥ Vn(R ) → U → Hom(V,V ) is smooth. For X ∈ U, choose a basis {w1, . . . , wn} of X n+k which is an n-frame in Vn(R ). Then for each 1 ≤ i ≤ n, we may write wi = vi + L(vi) where {v1, . . . , vn} is a basis for V . Notice that L depends smoothly on the entire n-frame {w1, . . . , wn} because the matrix entries of L can be expressed in terms of the two bases {vi} and {wj}. Conversely, the n-frames {w1, . . . , wn} also depend smoothly on such L, so the bijection U ↔ Hom(V,V ⊥) is in fact smooth. One now shows that the transition maps on overlaps between these charts U are smooth, and this completes the proof.

n+k Remark. An alternative proof that Grn(R ) is a smooth manifold comes from Lie theory. n+k n+k Since Grn(R ) is the set of n-planes in R , the Lie group GLn+k(R) acts transitively on n+k the Grassmannian as a set. For an n-plane X ∈ Grn(R ), the isotropy subgroup of this action is: Stab(X) = {M ∈ GLn+k(R) | M · X = X}. With respect to the decomposition Rn+k = X ⊕ X⊥, we can write any M ∈ Stab(X) as a block diagonal matrix  A B  M = . 0 D

n+k In the topology on GLn+k(R), these matrices form a closed subgroup. Therefore Grn(R ) is a homogeneous space for GLn+k(R), so it follows from the theory of Lie groups that there n+k is a unique topology and a unique smooth structure on Grn(R ) with respect to which the action of GLn+k(R) is smooth. n+k n Definition. The canonical n-bundle over Grn(R ) is the bundle γn+k with total space

n n+k n+k n+k E(γn+k) = {(X,~v): X ∈ Grn(R ),~v ∈ X} ⊆ Grn(R ) × R

n and projection π :(X,~v) → X. The topology on E(γn+k) is the subspace topology inherited n+k n+k from Grn(R ) × R .

45 3.3 The Universal Bundle 3 Characteristic Classes

n Lemma 3.3.4. γn+k is a vector bundle. Proof. We construct a local trivialization over each chart U defined in the previous proof. n+k ⊥ ∼ n Fix V ∈ Grn(R ) and U = {X : X ∩ V = 0}. Then V = R , so fixing an isomorphism once and for all, we may define h : U × V −→ π−1(U) (X,~v) 7−→ (X,~v + L(~v)), where L ∈ Hom(V,V ⊥) is the linear map having graph X. On a fibre, {X} × V → π−1(X) is given by ~v 7→ ~v + L(~v) which is linear and whose inverse is given by orthogonal projection onto V . Further, h is defined in terms of L ∈ Hom(V,V ⊥) which was shown to be smooth n in the previous proof. Therefore h is a local trivialization of γn+k. Many of the next results were proven in greater generality in Section 1.2; however we give the proofs here in terms of vector bundles since they are integral to understanding the structure of the universal bundle for vector bundles. Lemma 3.3.5. Let ξ be an n-dimensional bundle over a compact m-manifold B. Then ∼ ∗ n N ξ = f γN for some map f : B → Grn(R ). Proof. We aim to construct a map F : E(ξ) → RN for N large enough so that each restriction N F |π−1(x) is an injective linear map. Given such an F , we then define f : B → Grn(R ) by f(x) = F (π−1(x)). This yields a bundle map

n E(ξ) E(γN )

f N B Grn(R )

∼ ∗ n Then Lemma 2.3.5 implies that ξ = f γN as desired. Since B is compact, there is a finite cover {U1,...,Uk} of B such that ξ is trivial over each n −1 n n Ui. Let hi : Ui ×R → π (Ui) be the local trivialization of ξ over Ui and let p : Ui ×R → R −1 −1 n n be the natural projection map. This determines a map gi = p◦hi : π (Ui) → Ui×R → R . Fix a partition of unity {λi} subordinate to {U1,...,Uk}, with λi ≡ 0 on B r Ui and P i λi ≡ 1. Then we extend the gi maps to the total space by settingg ˜i(x,~v) = λi(x)gi(x,~v) for any (x,~v) ∈ E(ξ). Finally, define n n nk f : E −→ R ⊕ · · · ⊕ R = R (x,~v) 7−→ (˜g1(x,~v),..., g˜k(x,~v)).

For each point (x,~v) ∈ E, x is in the support of some λi, 1 ≤ i ≤ k, so λi(x) 6= 0, and then −1 n nk g˜i maps π (x) ⊂ E(ξ) isomorphically to a copy of R ⊂ R . Hence f is as required. We want to consider infinite Euclidean space ∞ R = {(x1, x2,...) | xi ∈ R, finitely many xi are nonzero} with the direct limit topology given by the direct system R ,→ R2 ,→ R3 ,→ · · · That is, U ⊆ R∞ is open (resp. closed) if and only if U ∩ Rm is open (resp. closed) for all m ≥ 1.

46 3.3 The Universal Bundle 3 Characteristic Classes

n+k Definition. The nth infinite Grassmannian is the direct limit of the Grn(R ) directed by inclusion: ∞ ∞ [ n+k Grn(R ) := Grn(R ). k=0

∞ Definition. The canonical n-bundle over Grn(R ) is called the universal n-bundle, writ- ten γn. The topology is taken to be the direct limit topology induced by the direct system n n+1 Grn(R ) ,→ Grn(R ) ,→ · · ·

n+k ∞ Since each Grn(R ) embeds into Grn(R ), we have: Proposition 3.3.6. Every n-bundle ξ over B is isomorphic to the pullback f ∗γn for some ∞ map f : B → Grn(R ).

Note that there is a map E(γn) → R∞, given by (X,~v) 7→ ~v, which is injective on each fibre.

Definition. Two bundle maps f0, f1 : ξ → η are said to be bundle homotopic if there exists a smooth map ft : E(ξ)×[0, 1] → E(η) extending f0 and f1 such that for each t ∈ [0, 1], ft : E(ξ) × {t} → E(η) is a bundle map. Such a map ft is called a bundle homotopy.

n Lemma 3.3.7. Any two bundle maps f0, f1 : ξ → γ are bundle homotopic.

Proof. Recall in the proof of Lemma 3.3.5 we constructed a map F : E(ξ) → RN ⊆ R∞ N ∞ which is injective on each fibre and, from this, constructed f : B → Grn(R ) ⊆ Grn(R ) by sending x to f(x) = F (π−1(x)). Conversely, given a bundle map E(ξ) E(γn)

∞ B Grn(R ) we get a map E(ξ) → R∞ by composing with the map E(γn) → R∞ discussed above. It is immediate that such a map is injective on fibres. Now given two bundle maps E(ξ) E(γn)

f , f 0 1 ∞ B Grn(R )

∞ let F0,F1 : E(ξ) → R be the corresponding maps from the previous paragraph. By the ∞ ∞ above, it suffices to construct Ft : E(ξ) × [0, 1] → R such that each Ft : E(ξ) × {t} → R is linear and injective on fibres. If we define Ft = (1 − t)F0 + tF1 then it is immediate that this map is smooth and linear. The only way Ft fails to be injective on fibres is if there is some nonzero vector ~v ∈ E(ξ) such that F0(~v) = −F1(~v). However, consider the two

47 3.3 The Universal Bundle 3 Characteristic Classes

∞ ∞ linear maps L0,L1 : R → R given by L0(ei) = e2i and L1(ei) = e2i−1, where ei is the ∞ ith standard basis vector in R . Geometrically, L0 and L1 “separate” the components of ∞ R into subspaces with trivial intersection. It is clear that F0 is homotopic to L0 ◦ F0 and ∞ moreover, F0(~v) 6= −L0(F0(~v)) for any nonzero ~v since there is a largest index of F0(~v) ∈ R with a nonzero component. Hence F0 ' L0 ◦ F0 through a family of maps that are linear and injective on fibres. Similarly, F1 ' L1 ◦ F1 through such a family of maps. It follows that L0 ◦ F0 and L1 ◦ F1 are homotopic through a family of maps that are injective on each fibre, but clearly L0 ◦ F0(~v) 6= −L1 ◦ F1(~v) for any ~v 6= 0, showing we can reduce the general case to this case up to homotopy.

∞ ∞ Corollary 3.3.8. For any n ≥ 1, the bundle Vn(R ) → Grn(R ) is universal for O(n)- bundles.

Thus we have shown that every n-bundle ξ over a paracompact base B determines a ∞ ∼ ∗ n well-defined homotopy class of maps f : B → Grn = Grn(R ) such that ξ = f γ . So the existence of Stiefel-Whitney classes for B rests on their construction in the cohomology ring of Grn.

• ∼ Theorem 3.3.9. For any n ≥ 1, H (Grn; Z/2Z) = Z/2Z[ω1, ω2, . . . , ωn], the polynomial i ring in ωi ∈ H (Grn; Z/2Z). Moreover, assuming the existence of Stiefel-Whitney classes, there is a unique choice of ωi satisfying the Stiefel-Whitney axioms for the universal bundle, n that is, ωi = ωi(γ ).

N Proof. We first describe a CW-structure on Grn(R ) and pass to the direct limit to obtain N a structure on Grn. Group n-planes X ∈ Grn(R ) into cells as follows. For each 1 ≤ i ≤ N, i N consider R ,→ R viewed as (x1, . . . , xi, 0,..., 0) and the sequence of spaces X ∩ R,X ∩ R2,...,X ∩ RN = X. Then

2 N dim(X ∩ R), dim(X ∩ R ),..., dim(X ∩ R ) = n is a nondecreasing sequence of N integers between 0 and n. Since N ≥ n and the dimensions increase by at most one in each step, we must have k places in the sequence where the dimension doesn’t change. On the other hand, given a sequence σ = (σ1, . . . , σn) of n integers 1 ≤ σi ≤ N, we define the Schubert symbol for σ:

N σi σi−1 e(σ) = {X ∈ Grn(R ) : dim(X ∩ R ) = i, dim(X ∩ R ) = i − 1}.

In other words, σ encodes the places in the sequence dim(X ∩R),..., dim(X ∩RN ) where the dimension jumps, for each X ∈ e(σ). Each e(σ) is an open CW-cell. To discuss closed cells, it is convenient to reformulate them in terms of (orthogonal) frames. Given an n-plane X ⊂ N σ1 σ2 σn R , there is a unique orthonormal basis {e1, . . . , en} for X which lies in H ×H ×· · ·×H , j N N where H = {(x1, . . . , xk, 0,..., 0) ∈ R : xk > 0} is the kth open half-plane in R . o N Conversely, any such frame determines X ∈ e(σ). Let Vn (R ) be the set of orthonormal n-frames in RN and define

0 o N σ1 σ2 σn e (σ) = Vn (R ) ∩ (H × H × · · · × H )

48 3.3 The Universal Bundle 3 Characteristic Classes called the Schubert cell for σ. By the above description, e0(σ) ∼= e(σ). One can show that 0 e (σ) is a closed CW-cell of dimension (σ1 − 1) + (σ2 − 2) + ... + (σn − n) and that the set 0 {e (σ) | σ = (σ1, . . . , σn)} forms a CW-structure on Grn. • ∼ • To prove H (Grn; Z/2Z) = Z/2Z[ω1, . . . , ωn], we first prove Z/2Z[ω1, . . . , ωn] ⊆ H (Grn; Z/2Z). Consider the bundle map

1 1 E(γ1 ) E(γ )

i ∞ RP 1 RP

1 ∗ 1 By the naturality axiom, ω(γ1 ) = i ω(γ ) so assuming the Stiefel-Whitney classes exist, we are forced to have ω(γ1) = 1 + a for the generator a ∈ H1(RP ∞; Z/2Z). Define the product 1 1 ∞ ∞ bundle ξ = γ × · · · × γ over RP × · · · RP . Then dim ξ = n so by Proposition 3.3.6, | {z } | {z } n n ∞ ∞ n 1 1 there is a map f : RP × · · · × RP → Grn inducing ξ from γ . Thus ω(γ × · · · × γ ) = f ∗ω(γn). Now by the K¨unnethformula,

• ∞ ∞ ∼ • ∞ • ∞ H (RP × RP ; Z/2Z) = H (RP ; Z/2Z) ⊗ · · · ⊗ H (RP ; Z/2Z) ∼ = Z/2Z[a1] ⊗ · · · ⊗ Z/2Z[an] ∼ = Z/2Z[a1, . . . , an]. By the remark following the definition of the Stiefel-Whitney axioms in Section 3.2, the total Stiefel-Whitney class of γ1 × · · · × γ1 can be computed as an algebraic cross product:

ω(γ1 × · · · × γ1) = ω(γ1) × · · · × ω(γ1) ∗ 1 ∗ 1 ∞ n ∞ = p1ω(γ ) × · · · × pnω(γ ) where pi :(RP ) → RP = (1 + a1) ··· (1 + an)

= 1 + (a1 + ... + an) + (a1a2 + a1a3 + ... + an−1an) + ... + a1 ··· an.

Notice that these are just the sum of the elementary symmetric functions of degree 0 through degree n in the ai. By linear algebra, the elementary symmetric functions are algebraically independent, and thus there are no polynomial relations among the Stiefel-Whitney classes of ξ = γ1 × · · · × γ1. Further, ξ = f ∗γn implies that there are no polynomial relations among n • the ωi(γ ) either. This proves that Z/2Z[ω1, . . . , ωn] ⊆ H (Grn; Z/2Z). Next, observe that any d-cell e(σ) in the CW-structure we have defined on Grn satisfies d = (σ1 − 1) + ... + (σn − n), and thus every d-cell determines a partition of d, but clearly this correspondence is one-to-one. Hence the number of d-cells, and hence the rank of d H (Grn; Z/2Z), is equal to the number of partitions d = k1 + . . . ks. On the other hand, k1 ks the degree d part of Z/2Z[ω1, . . . , ωn] consists of elements of the form ω1 ··· ωs where k1 + 2k2 + ... + sks = d. So every such element determines a partition of d, but since every partition can be written in nondecreasing order d = ks+(ks+ks−1)+...+(ks+...+k1), we see that this correspondence is also one-to-one. Hence the d-part of Z/2Z[ω1, . . . , ωn] has rank • • equal to that of H (Grn; Z/2Z), so we conclude that H (Grn; Z/2Z) = Z/2Z[ω1, . . . , ωn].

49 3.3 The Universal Bundle 3 Characteristic Classes

Using this, we show that there is a unique choice of such generators of the cohomology 0 00 ring of Grn satisfying the Stiefel-Whitney axioms. Suppose ωi and ωi are two choices of Stiefel-Whitney classes generating this cohomology ring. If n = 1, we have a bundle map

1 1 E(γ1 ) E(γ )

i ∞ RP 1 RP 1 ∞ ∼ 1 1 0 1 00 1 Then H (RP ; Z/2Z) = H (RP ; Z/2Z). Moreover, ω1(γ1 ) and ω1 (γ1 ) are both nonzero 1 1 ∼ 0 00 in H (RP ; Z/2Z) = Z/2Z, so they must be equal. Hence by naturality, ω1(γ1) = ω1 (γ1). This proves ω0 and ω00 agree on the universal line bundle γ1. For ξ = γ1 × · · · γ1, we get ω0(ξ) = ω00(ξ) from the description of these using the algebraic cross product. Finally, for the universal n-bundle γn, we have f ∗ω0(γn) = ω0(ξ) and f ∗ω00(γn) = ω00(ξ) as above, but since f ∗ is injective on cohomology, we must have ω0(γn) = ω00(γn).

i Corollary 3.3.10. For any bundle ξ over B, if Stiefel-Whitney classes ωi(ξ) ∈ H (B; Z/2Z) exist, then they are unique.

In light of the theory of principal bundles in Chapter 1, a vector bundle p : E → B may be viewed from one of the following equivalent perspectives:

∼ n ˆ A locally trivial fibre bundle with fibre V = R and structure group GL(V ). (For complex bundles, replace R with C.) ˆ A principal O(n)-bundle, by Example 1.1.10. (For complex bundles, replace O(n) with U(n).)

ˆ A vector bundle with a smooth inner product, via the identification of principal O(n)- bundles with such vector bundles. (For complex bundles, replace smooth with Hermi- tian.)

ˆ Up to isomorphism, real vector bundles over B are classified by homotopy classes of ∞ maps B → Grn(R ). Moreover, this is compatible with the associated bundle con- struction, since if F (E) is the frame bundle associated to E → B (see Example 1.2.2) ∞ and f : B → Grn(R ) is a classifying map for F (E), we have

n E = F (E) ×GLn(R) R ∗ ∞ n = f Vn(R ) ×O(n) R ∗ ∞ n = f (Vn(R ) ×O(n) R ) ∗ = f γn

∞ where γn is the tautological bundle over Grn(R ).

50 3.4 Construction of Stiefel-Whitney Classes 3 Characteristic Classes

3.4 Construction of Stiefel-Whitney Classes

To prove that the classes ωi = ωi(ξ) exist at all, we need Steenrod squares and the Thom isomorphism theorem. Recall the following definition from algebraic topology: i Definition. The Steenrod squares for a pair of spaces (X,Y ) are maps Sq : Hn(X,Y ; Z/2Z) → Hn+i(X,Y ; Z/2Z) satisfying (0) Sqi = 0 for all i > n.

0 (1) Sq : Hn(X,Y ; Z/2Z) → Hn(X,Y ; Z/2Z) is the identity for each n ≥ 0. n n (2) Sq : Hn(X,Y ; Z/2Z) → H2n(X,Y ; Z/2Z) is the cup product square, Sq (α) = α ∪ α, for each n ≥ 0. (3) Each Sqi is an abelian group homomorphism.

i i X j i−j (4) For any α, β ∈ H•(X,Y ; Z/2Z), Sq (α ∪ β) = Sq (α) ∪ Sq (β). j=0 (5) The Sqi are natural, that is, for any map f :(X0,Y 0) → (X,Y ), f ∗ Sqi(α) = Sqi(f ∗α) for all α ∈ H•(X,Y ; Z/2Z). Definition. The total Steenrod square is the map • • Sq : H (X,Y ; Z/2Z) −→ H (X,Y ; Z/2Z) ∞ X α 7−→ Sqi(α). i=0 Remark. By Axiom 4, Sq(α ∪ β) = Sq(α) ∪ Sq(β) for any cohomology classes α, β. π Now let ξ : E −→ B be any vector bundle and let E0 be the subset of E consisting of all nonzero vectors in each fibre. Fixing a metric on ξ, in each fibre π−1(x) we can consider all vectors ~v satisfying ||~v|| ≤ 1. This defines a unit disk bundle as a subbundle of ξ: D E

B

Let S = ∂D be the unit sphere bundle which is a subbundle of D. Then there is a deformation retract (E,E0) → (D,S) inducing an isomorphism i ∼ i H (E,E0; Z/2Z) = H (D,S; Z/2Z) for each i ≥ 0. Further, for each x ∈ B, the fibre F = π−1(x) has a set of nonzero vectors F0 = E0 ∩ F and an isomorphism i ∼ i n n−1 H (F,F0; Z/2Z) = H (D ,S ; Z/2Z) ( Z/2Z, i = n ∼= 0, i 6= n.

51 3.4 Construction of Stiefel-Whitney Classes 3 Characteristic Classes

Theorem 3.4.1 (Thom Isomorphism Theorem). There is a unique fundamental class u ∈ n n H (E,E0; Z/2Z) which restricts to the nontrivial class in each fibre H (F,F0; Z/2Z). More- over, for all i ≥ 0, there is an isomorphism

∗ i π i −∪u n+i ϕ : H (B; Z/2Z) −→ H (E; Z/2Z) −−→ H (E,E0; Z/2Z). The map ϕ : α 7→ π∗(α) ∪ u is called the Thom isomorphism. Notice that we must have u = ϕ(1) where 1 is the generator of H0(B; Z/2Z). Theorem 3.4.2. For any vector bundle ξ : E −→π B, the classes

−1 i i ωi(ξ) := ϕ Sq (u) ∈ H (B; Z/2Z) satisfy the Stiefel-Whitney axioms.

Proof. Most of these follow from the axioms for Steenrod squares. For example, axiom 0 for Steenrod squares implies ωi = 0 for i > n. Likewise, naturality of the ωi follows from the naturality of Steenrod squares and naturality of the Thom isomorphism. Thus we have axioms (1), (3) and (4) to check for the Stiefel-Whitney classes. −1 0 −1 (1) By definition, ω0(ξ) = ϕ Sq (ϕ(1)) = ϕ ϕ(1) = 1 since ϕ is an isomorphism. −1 i (3) Consider two bundles ξ : E1 → B1 and η : E2 → B2. Then ωi(ξ) = ϕ Sq (u1) m −1 i where u1 ∈ H (E1, (E1)0) is the fundamental class for ξ and ωi(η) = ϕ Sq (u2) where n u2 ∈ H (E2, (E2)0) is the fundamental class for η. We have

m+n m+n u1 × u2 ∈ H (E1 × E2, (E1)0 × E2 ∪ E1 × (E2)0) = H (E1 × E2, (E1 × E2)0).

m+n But the fundamental class u ∈ H (E1 × E2, (E1 × E2)0) is the unique class restricting −1 nontrivially to each fibre (F,F0). On F = (π1 × π2) (x), we see that u1 × u2 restricts −1 −1 nontrivially, since F = F1 × F2 = π1 (x) × π2 (x), so uniqueness implies u = u1 × u2. Finally, by properties of the Steenrod squares we get Sq(u1 × u2) = Sq(u1) × Sq(u2), where × is the algebraic cross product. The third Stiefel-Whitney axiom now follows from the first remark in Section 3.2. 1 1 1 1 (4) Consider ω1(γ1 ) ∈ H (RP ; Z/2Z), where γ1 is the tautological line bundle. For the 1 fundamental class u ∈ H (E,E0; Z/2Z), we have

−1 1 −1 ω1 = ϕ Sq (u) = ϕ (u ∪ u).

i ∼ i ∼ 1 By the remarks above, H (E,E0) = H (M, ∂M), where M is the M¨obiusband and ∂M = S is its boundary circle. Under this identification, u corresponds to the core circle S1 ,→ M. Using intersection products, u ∪ u corresponds to the intersection of two circles in general position in M, which must consist of exactly one point. Hence u ∪ u 6= 0, so ωi 6= 0. Another way to see this is through the excision isomorphism:

i ∼ i 2 2 ∼ i 2 H (M, ∂M) = H (RP ,D ) = H (RP )

(since D2 is contractible in the last step). Thus u must correspond to a 6= 0 in H1(RP 2), 2 2 2 2 and since a 6= 0 in H (RP ; Z/2Z), we must have u ∪ u 6= 0 in H (E,E0; Z/2Z).

52 3.5 Oriented Bundles 3 Characteristic Classes

3.5 Oriented Bundles

Recall the definition of an orientation on a manifold M. First, on TpM for p ∈ M, an 0 orientation is an equivalence class of ordered bases for TpM, where two bases B and B are said to be equivalent if the change-of-basis matrix B → B0 has positive determinant. Since det takes values in R r {0}, there are two equivalence classes and thus two orientations S on each TpM. We say M is orientable if there is an atlas M = Ui with diffeomorphisms n −1 ϕi : Ui → ϕi(Ui) ⊆ R such that the transition maps ϕij = ϕj ◦ϕi : ϕi(Ui∩Uj) → ϕj(Ui∩Uj) preserve orientation. That is, for all x ∈ ϕi(Ui ∩ Uj), the differential dxϕij has positive determinant. Equivalently, an orientation on M is a choice of orientation on each tangent space TpM n so that if p ∈ Ui, the differential dpϕi : TpM → R takes the class of bases of TpM to the n standard basis {e1, . . . , en} on R . One perspective is that an orientation on M is really just an orientation on the tangent bundle. This can be generalized to arbitrary vector bundles as follows. Let ξ : E −→π B be a S n −1 vector bundle with local trivializations B = Ui, hi : Ui × R → π (Ui). Definition. An orientation on ξ is a choice of orientation on each vector space π−1(x) such n −1 that if x ∈ Ui, the restriction hi|{x}×Rn : {x} × R → π (x) takes the standard orientation on Rn to the chosen orientation on π1(x). We say ξ is orientable if such an orientation exists. An alternative perspective can be phrased in terms of cohomology. Recall that for F = −1 π (x), with nonzero vectors F0 = F r {0}, we have isomorphisms n ∼ n n n−1 ∼ n−1 n−1 ∼ H (F,F0; Z) = H (D ,S ; Z) = H (S ; Z) = Z. Since (Dn,Sn−1) is homotopy equivalent to (∆n, ∂∆n) for an n-simplex ∆n, we can view n n −−→ −−−−→ ∆ ⊆ R with ordered vertices v0, . . . , vn. This defines an orientation [v0v1,..., vn−1vn] on Rn and also a choice of generator of n ∼ n n ∼ n n n ∼ n n n−1 Hn−1(∂∆ ) = Hn(∆ , ∂∆ ) = H (∆ , ∂∆ ) = H (D ,S ).

n Proposition 3.5.1. An orientation on ξ is a choice of generator for H (F,F0; Z) for each −1 n −1 fibre F = π (x) such that if x ∈ Ui, the local trivialization hi : Ui × R → π (Ui) induces ∗ n n n n hi : H (F,F0; Z) → H ({x}×R , {x}×R0 ; Z) mapping the chosen generator to the standard generator.

There is a version of the Thom isomorphism theorem (3.4.1) for cohomology with Z- coefficients. Theorem 3.5.2 (Thom Isomorphism Theorem for Oriented Bundles). Let ξ be an oriented n n-bundle and specify a generator uF ∈ H (F,F0; Z) for each fibre F . Then there exists a n unique class u ∈ H (E,E0; Z) which restricts to uF on each fibre and such that for all i ≥ 0, the map i −∪u n+i H (E; Z) −−→ H (E,E0; Z) is an isomorphism.

53 3.5 Oriented Bundles 3 Characteristic Classes

Notice that if B is connected, Theorem 3.5.2 says that

n ∼ 0 ∼ 0 ∼ H (E,E0; Z) = H (E; Z) = H (B; Z) = Z

n so the fundamental class u ∈ H (E,E0; Z) is just one of the two generators. Let j :(E, ∅) ,→ (E,E0) be the inclusion and π : E → B be a bundle projection. Then ∗ ∗ n j n π n we have isomorphisms H (E,E0; Z) −→ H (E; Z) ←− H (B; Z). Definition. For an oriented bundle ξ : E −→π B, the Euler class of ξ is e(ξ) := (π∗)−1j∗(u), n where u ∈ H (E,E0; Z) is the fundamental class. Although Steenrod squares are only defined for Z/2Z-coefficients, the top Steenrod square is a cup product, Sqn(α) = α∪α, which is defined for any coefficients. Thus we have a similar definition of the Euler class as for the top Stiefel-Whitney class. Lemma 3.5.3. For a bundle ξ, e(ξ) = ϕ−1(u ∪ u) where u is the fundamental class and n 2n ∗ ϕ : H (B; Z) → H (E,E0; Z) is the map α 7→ π (α) ∪ u. Proof. Applying ϕ to ϕ−1(u∪u) of course gives u∪u, while for (π∗)−1j∗(u) it gives ϕ(π∗)−1j∗(u) = j∗(u) ∪ u by definition of ϕ. Then by properties of the cup product, j∗(u) ∪ u = u ∪ u since j is just an inclusion. Since ϕ is an isomorphism, it follows that ϕ−1(u ∪ u) = (π∗)−1j∗(u) = e(ξ).

Corollary 3.5.4. For any n-bundle ξ, ωn(ξ) = e(ξ) mod 2. Corollary 3.5.5. For any bundle ξ with Euler class e(ξ), (1) (Naturality) If f : B0 → B is a map, then e(f ∗ξ) = f ∗e(ξ). (2) If η is another bundle, then e(ξ ⊕ η) = e(ξ) ∪ e(η). (3) If ξ has a nonvanishing section then e(ξ) = 0.

Remark. Suppose ϕ is an oriented n-bundle over an oriented n-manifold B. Let s0 : B → E be the zero section and s : B → E be any other smooth section of ξ whose image in E lies in general position relative to s0(B). If B is a closed manifold, then s0(B) ∪ s(B) can only be a finite collection of points for transversality to hold. Thus s0(B) • s(B) = e(ξ) ∩ [B], where • is the algebraic intersection number and [B] ∈ Hn(B; Z) is the fundamental class. A special case of this is (3) of Corollary 3.5.5, since a nonvanishing section has trivial intersection with s0(B) and thus e(ξ) = 0. A related fact is given in the theorem below. Theorem 3.5.6. Let M be an oriented n-manifold. Then e(TM) ∩ [M] = χ(M). Proof. A smooth section s : M → TM is just a smooth tangent vector field on M and the points of s0(M) ∩ s(M) are just the zeroes of s. The sign of each zero is called the index of the zero, written I(x). It is well-known that one can compute the Euler characteristic using indices as follows: X χ(M) = I(x). s(x)=0 Thus we can apply the Remark to obtain the desired equality.

54 3.6 Chern Classes 3 Characteristic Classes

3.6 Chern Classes

Recall (Section 2.4) that a structure group for ξ with fibres V is a subgroup of GL(V ) containing the images of the transition maps of ξ. We saw in Section 3.1 that, after endowing a bundle with a metric, we may assume the orthogonal group On(R) ⊂ GLn(R) is a structure group for ξ. This observation is called a “reduction of the structure group”. In this section we define complex vector bundles. Parallel to the linear algebra, we will see that the structure group of such a bundle may be chosen to be Un(C) ⊂ GLn(C), the group of unitary matrices.

Definition. A complex vector bundle of dimension n is a vector bundle ω : E −→π B such that each fibre π−1(x) has the structure of a complex vector space of dimension n, and the local trivializations U × Cn → π−1(U) reduce to a complex vector space isomorphism {x} × Cn → π−1(x) on each fibre. Notice that every complex vector bundle is also a real vector bundle if we “forget” the complex structure on the fibres. In particular, a complex n-bundle ω is a real 2n-bundle which we denote by ωR. Example 3.6.1. For a complex n-manifold M, the complex tangent bundle TM is a complex vector bundle of dimension n, and therefore a real vector bundle of dimension 2n. That is, TM is a real manifold of dimension 4n.

For a complex vector space V , let J : V → V be the C-linear operator J(~v) = i~v. 2 Then J = −idV . In fact, there is an equivalence between complex n-vector spaces and real 2 2n-vector spaces with a linear operator J : V → V such that J = −idV . Given any complex vector space V , there is an induced orientation on the underlying real vector space VR: if {x1, . . . , xn} is a C-basis of V then {x1, Jx1, . . . , xn, Jxn} is an R-basis of VR. The fact that this is well-defined on VR (i.e. a difference choice of xi determines the same orientation on VR) is a consequence of GLn(C) being connected. Similarly, we have:

Lemma 3.6.2. For a complex bundle ω, there is an induced orientation on ωR.

2n As a consequence, the Euler class e(ωR) ∈ H (B; Z) is defined. This coincides with the nth Chern class of the bundle ω.

Definition. For a complex n-bundle ω over B, a set of Chern classes for ω is a choice 2i of elements ci = ci(ω) ∈ H (B; Z) for each i ≥ 0 and a total Chern class c(ω) = c0(ω) + c1(ω) + c2(ω) + ... which satisfy:

(0) ci(ω) = 0 for i > n.

0 (1) c0(ω) = 1 in H (B; Z). (2) For any other complex bundle ζ, c(ω ⊕ ζ) = c(ω) · c(ζ).

0 ∗ ∗ (3) For any map f : B → B and each i ≥ 0, ci(f ω) = f ci(ω).

1 1 1 2 1 (4) For the tautological line bundle γ1 over CP , c1(γ1 ) is a generator of H (CP ; Z).

55 3.6 Chern Classes 3 Characteristic Classes

In analogy to the construction of Stiefel-Whitney classes, we can consider complex Grass- mannian manifolds n+k k→∞ ∞ Grn(C ) −−−→ Grn = Grn(C ). Each of these has a canonical bundle γn(Cn+k) which fit together to define a universal complex n bundle γ over Grn.

• ∼ ∞ Theorem 3.6.3. H (Grn; Z) = Z[c1, . . . , cn] for the complex Grassmannian Grn = Grn(C ), 2i for generators ci ∈ H (Grn; Z) satisfying the axioms for Chern classes on Grn. Theorem 3.6.4. Any complex bundle ω of dimension n over B is induced by a map f : ∞ ∼ ∗ n n B → Grn(C ) such that ω = f γ , where γ is the universal bundle. Moreover, there is a one-to-one correspondence between isomorphism classes of complex bundles over B and ∞ homotopy classes of maps B → Grn(C ). These theorems imply that Chern classes, if they exist, must be unique. Now the construc- tion of ci(ω) proceeds as in Section 3.4, using the oriented version of the Thom isomorphism theorem (3.5.2). π Alternatively, for a complex bundle ω : E −→ B, one can define cn(ω) := e(ωR) ∈ H2n(B; Z) using the existence of the Euler class from Section 3.5.2. Then for the set of π0 −1 nonzero vectors E0 ⊂ E, one constructs a bundle ω0 : E(ω0) −→ E0 by taking π0 (~v) := (Span(~v))⊥, the orthogonal complement of the vector subspace of π−1(x) spanned by ~v 6= 0, where the orthogonal complement is taken with respect to a Hermitian inner product on ω. (The existence of such a Hermitian inner product is constructed in exactly the same manner as Euclidean metrics in Proposition 3.1.4.) Then ω0 is a complex (n−1)-bundle. To proceed, recall the Gysin sequence from algebraic topology:

Theorem 3.6.5 (Gysin Sequence). For a real, oriented vector bundle ξ over B with Euler class e = e(ξ), there is an exact sequence

∗ i −∪e i+n ρ i+n i+1 · · · → H (B) −−→ H (B) −→ H (E0) → H (B) → · · ·

where ρ = π|E0 . Now by induction, suppose the Chern classes are constructed for any (n − 1)-bundle. • In particular, c1(ω0), . . . , cn−1(ω0) ∈ H (E0) exist for the complex bundle ω0 defined above. Notice that when 1 ≤ k ≤ n − 1, the Gysin sequence gives us an isomorphism

∗ 2k ρ 2k H (B) −→ H (E0)

∗ −1 so we can define ck(ω) := (ρ ) ck(ω0). Together with cn(ω) = e(ωR), this constructs the Chern classes for ω.

Corollary 3.6.6. If ω has k sections which are linearly independent over C, then ci(ω) = 0 for each n − k + 1 ≤ i ≤ n.

56 3.7 Pontrjagin Classes 3 Characteristic Classes

1 1 ∼ 2 Example 3.6.7. Let γ1 be the tautological line bundle over CP = S , the Riemann sphere. Then H2(CP 1; Z) has a canonical choice of generator agreeing with the orientation on S2 1 induced by its complex structure. By the axioms for Chern classes, we must have c1(γ1 ) = 2 1 1 n 1 a ∈ H (CP ). Likewise, for the canonical line bundle γn over CP , we have c1(γn) = a ∈ H2(CP n). A similar proof as the one in Example 3.2.5 for RP n shows that the total Chern class of the complex projective space is

n n n+1 c(CP ) := c(T CP ) = (1 + a) . Example 3.6.8. Consider a complex line bundle ω over an oriented 4-manifold M. Let s0, s : M → E = E(ω) be sections intersecting in general position in E, with s0 the 0- 0 section. Set S = s0(M) ∩ s(M) ⊆ E. Since ωR is a 2-dimensional real bundle, we must have dim S0 = 2 by transversality. Thus, since π is a local diffeomorphism, S = π(S0) ⊆ M is a 2-dimensional (oriented) submanifold. Let [S] ∈ H2(S) be the fundamental class of S. For i : S,→ M, we get the following duality between the fundamental classes of M and S, given by the first Chern class: ∗ c1(ω) ∩ [M] = i [S]. This can be interpreted similarly for oriented bundles of arbitrary dimension.

3.7 Pontrjagin Classes

For any real vector bundle ξ : E → X, we obtain an associated complex bundle ξC : EC → X by setting (EC)x = Ex ⊗ C for every x ∈ X. If ξ is a bundle of real dimension n, then ξC is a bundle of complex dimension n. Further, every fibre Fx of EC has a decomposition ∼ Fx = Ex ⊕ iEx, so as a real vector bundle of dimension 2n, ξC = ξ ⊕ ξ.

Lemma 3.7.1. For any real vector bundle, ξC is isomorphic to the complex conjugate ξC. For a real bundle ξ : E → X, consider the total Chern class of the associated complex bundle:

c(ξC) = 1 + c1(ξC) + ... + cn(ξC). j It is clear that for each 1 ≤ j ≤ n, cj(ξC) = (−1) cj(ξC), so we have

n c(ξC) = 1 − c1(ξC) + ... + (−1) cn(ξC).

• Thus each c2k+1(ξC) is an element of order two in H (X; Z). Definition. For a real vector bundle ξ : E → X, define the jth Pontrjagin class of ξ to be j 4j pj(ξ) := (−1) c2j(ξC) ∈ H (X; Z) and the total Pontrjagin class to be

p(ξ) := 1 + p1(ξ) + ... + p n (ξ). b 2 c We don’t quite have a Whitney sum formula for Pontrjagin classes, but we do have the following sum formula.

57 3.8 General Characteristic Classes 3 Characteristic Classes

Proposition 3.7.2. Let ξ and η be two real vector bundles over X. Then

2(p(ξ ⊕ η) − p(ξ)p(η)) = 0,

or in other words, p(ξ ⊕ η) ∼= p(ξ)p(η) mod 2. Example 3.7.3. Let X = Sn and E = TSn. Then by Example 3.2.2, c(TSn) = c(TSn ⊕ ε1) = c(εn+1) = 1, so p(TSn) = 1.

Example 3.7.4. Let X = CP n and E = T CP n, so that by Example 3.6.7, c(T CP n) = (1 + x)n+1 where x is a generator in H2(CP n; Z). Then by definition of Pontrjagin classes n pj = pj(T CP ), we have

n n (1 − p1 + ... (−1) pn) = (1 − c1 + ... (−1) cn)(1 + c1 + ... + cn) = (1 − x)n+1(1 + x)n+1 = (1 − x2)n+1.

Therefore p(T CP n) = (1 + x2)n+1.

3.8 General Characteristic Classes

We adopt a more abstract perspective in this section to describe all possible characteristic classes of vector bundles. This affords an algebraic description of the Stiefel-Whitney, Chern and Euler classes we constructed in previous sections and also allows us access to other types of classes. Let K be a field (usually K = R or C) and let Vectn(K) be the category of vector n bundles with fibre K . For a base X, Vectn(K,X) will denote the subcategory of Vectn(K) consisting of bundles with base X.

Definition. For a triple (n, p, R) consisting of n, p ∈ N0 and R an abelian group, a char- acteristic class for this data is a natural transformation

p c : Vectn(K, −) −→ H (−; R).

That is, for each bundle E → X, there is a class c(E) ∈ Hp(X; R) which is natural with respect to pullback.

Example 3.8.1. If E → B is a trivial vector bundle, then it is the pullback of a vector bundle over a point, so it follows that c(E) = 0 for any characteristic class c.

Lemma 3.8.2. The set of characteristic classes for (n, p, R) forms an abelian group under addition. Moreover, if R is a ring then this group acquires the structure of a ring.

Proof. Pullback is natural. When R is a ring, the multiplication operation on characteristic classes is given by cup product.

Lemma 3.8.3. For all n, p ∈ N0 and abelian groups R, there is a bijection between charac- p teristic classes for (n, p, R) and the set H (BGLn(K); R).

58 3.8 General Characteristic Classes 3 Characteristic Classes

p Proof. If c0 ∈ H (BGLn(K); R), then for any vector n-bundle E classified by a map f, ∗ c(E) := f c0 defines a characteristic class for (n, p, R). Conversely, any characteristic class p p c : Vectn(K,X) → H (X; R) determines a class c0 := c(EGLn(K)) ∈ H (BGLn(K); R), ∗ where f c(EGLn(K)) = c(E) for f any classifying map of E → X. These associations are clearly inverses.

Example 3.8.4. When n = 1, we have

∞ ∞ • ∞ ∼ BO(1) = B(Z/2Z) = R /(Z/2Z) = RP and H (RP ; Z/2Z) = Z/2Z[ω] 1 ∞ 1 ∞ • ∞ ∼ BU(1) = BS = S /S = CP and H (CP ; Z) = Z[c] for generators ω ∈ H1(RP ∞; Z/2Z) and c ∈ H2(CP ∞; Z). So in some sense, the only “interesting” characteristic class of a line bundle is for p = 1. By Proposition 3.3.6 and The- 1 orem 3.6.4, these are the first Stiefel-Whitney class ω1(L) ∈ H (X; Z/2Z) and the first Chern 2 class c1(L) ∈ H (X; Z). Altogether, the classification of line bundles can be summarized by the isomorphism

∞ ∼ 1 Vect1(R,X) = [X, RP ] = [X,K(Z/2Z, 1)] = H (X; Z/2Z) ∞ ∼ 2 Vect1(C,X) = [X, CP ] = [X,K(Z, 2)] = H (X; Z).

These are a rather special case, due to the fact that RP ∞ = K(Z/2Z, 1) and CP ∞ = K(Z, 2). Definition. Let p : E → X be a K-vector bundle of dimension n. The associated projec- tive bundle is a fibre bundle p¯ : PE → X with fibre KP n−1, the projective (n − 1)-space over K, and total space

n−1 −1 PE = {(x, `) ∈ B × KP : ` is a line in Ex = p (x)}.

n−1 Equivalently, PE = F (E) ×GLn(K) KP , the associated bundle for the projective action of n−1 GLn(K) on KP .

Each fibre of PE admits a tautological line bundle which induces a line bundle L → PE given by L = {(x, `, v) ∈ PE × Kn | v ∈ `}. 1 Since L is a line bundle, it has a distinguished characteristic class q = ω1(L) ∈ H (PE; Z/2Z) 2 when K = R or q = c1(L) ∈ H (PE; Z) when K = C.

Lemma 3.8.5. Let R = Z/2Z when K = R and R = Z when K = C. The classes 1, q, q2, . . . , qn−1 ∈ H•(PE; R) restrict to a generating set for H•(KP n−1; R) under the fibre inclusion KP n−1 ,→ PE.

Proof. We give the proof for K = R, but the proof for K = C is identical, with appropriate dimension changes. n−1 ∗ If i : RP ,→ PE is the inclusion of the fibre over x ∈ X, then i (L) = L0 is the the tautological line bundle on RP n−1. Thus by naturality of characteristic classes,

∗ ∗ 1 n−1 i q = i ω1(L) = ω1(L0) ∈ H (RP ; Z/2Z).

59 3.8 General Characteristic Classes 3 Characteristic Classes

Notice that the inclusion RP n−1 ,→ RP ∞ is a classifying map for the tautological bundle over RP n−1 (this follows from the proof of Theorem 3.3.9). Then the induced map on cohomology is • ∞ • n−1 n Z/2Z[ω] = H (RP ; Z/2Z) −→ H (RP ; Z/2Z) = Z/2Z[ω]/(ω ), so ω maps to a generatorω ¯ of H•(RP n−1; Z/2Z). This provesω ¯ is the Stiefel-Whitney class of the tautological line bundle over RP n−1 and i∗qj =ω ¯j for each 0 ≤ j ≤ n − 1 by construction.

Theorem 3.8.6 (Leray-Hirsch). Let p : E → X be a general fibre bundle with fibre F , let R be a commutative ring and suppose

(i) Hk(F ; R) is a finitely generated, free R-module for all k ≥ 0.

• ∗ (ii) There exist classes q1, q2, . . . , qn ∈ H (E; R) whose restrictions i qj along the inclu- sion i : F,→ E form a basis for the cohomology ring H•(F ; R).

• • Then H (E; R) is a free module over H (X; R) with basis {q1, q2, . . . , qn}. Proof. It is equivalent to prove that the map

Φ: H•(X; R) ⊗ H•(F ; R) −→ H•(E; R) ∗ ∗ x ⊗ i qj 7−→ p x ∪ qj

is an isomorphism of R-modules. First assume X is a finite dimensional CW-complex. In this case, we induct on its dimension d, with the case d = 0 being trivial. For d > 0, let X0 be the complement of the centers of all the d-cells of X (viewing them as disks) and let E0 be the pullback of E along the inclusion X0 ,→ X. We will drop the coefficients in H•(−; R) for convenience. Consider the diagram ··· Hk(X,X0) ⊗ H•(F ) Hk(X) ⊗ H•(F ) Hk(X0) ⊗ H•(F ) ···

ΦX,X0 ΦX ΦX0

··· Hk(E,E0) Hk(E) Hk(E0) ···

The top row is obtained by tensoring the long exact sequence in cohomology for the pair (X,X0) with H•(F ), which remains exact since H•(F ) is free. The bottom row is the long 0 exact sequence for the pair (E,E ). The vertical arrow ΦX,X0 can be defined in analogy to the Φ in the theorem using relative cup products. Since pullbacks and cup products are natural, the entire diagram commutes so by the Five Lemma from commutative algebra, if two out of the three Φ are isomorphisms, so is the third. Now notice that ΦX0 is an isomorphism because X0 deformation retracts to the (d − 1)-skeleton X(d−1) by construction, meaning −1 (d−1) 0 p (X ) → E is a weak homotopy equivalence. One can prove that ΦX,X0 is also an isomorphism by excision, so it follows that ΦX is an isomorphism by induction. For an arbitrary CW-complex X, the pair (X,X(d)) is d-connected by homotopy theory, and one can show this implies (E, p−1(X(d))) is also d-connected. As a result, the diagram

60 3.8 General Characteristic Classes 3 Characteristic Classes

Hk(X) ⊗ H•(F ) Hk(X(d)) ⊗ H•(F )

ΦX ΦX(d)

Hk(E) Hk(p−1(X(d)))

commutes and the horizontal maps are isomorphisms for k < n. By the finite dimensional

case above, ΦX(d) is an isomorphism, which means ΦX is as well. For arbitrary X, the result follows by cellular approximation. Remark. Theorem 3.8.6 is a special case of the Leray-Serre spectral sequence, which, for any p,q p q fibration E → X with homotopy fibre F , has an E2-page E2 = H (X; R) ⊗ H (F ; R) and p,q p+2,q−1 differential d2 : E2 → E2 . The inclusion F,→ E and the corresponding restriction map H•(E) → H•(F ) induce a map

• •,• 0 q H (E) → E∞ → H (X) ⊗ H (F )

∗ explicitly given by c 7→ 1⊗i c, whose image lies in the kernels of all of the dr. When E → X is a fibre bundle satisfying the conditions of Theorem 3.8.6, restriction H•(E) → H•(F ) is • surjective, so H (F ) ⊆ ker dr for all r and therefore there are no nonzero differentials on the • • • E2-page of the spectral sequence. Hence H (X) ⊗ H (F ) → H (E) is an isomorphism.

Let E → X be a real, dimension n vector bundle and let PE → X be the associated projective bundle with tautological line bundle L → PE. As we saw, there is a Stiefel- 1 2 n−1 Whitney class q = ω1(L) ∈ H (PE; Z/2Z) with the property that 1, q, q , . . . , q generate H•(RP n−1; Z/2Z). That is, PE satisfies the hypotheses of the Leray-Hirsch theorem, so we have proven:

Corollary 3.8.7. For all real vector bundles E → X, the cohomology ring H•(PE; Z/2Z) is a free module over H•(X; Z/2Z) with basis {1, q, q2, . . . , qn−1}. Corollary 3.8.8. The Stiefel-Whitney classes of the bundle E → X are the unique classes • ω1(E), . . . , ωn(E) ∈ H (X; Z/2Z) such that n n−1 n−2 q + ω1(E)q + ω2(E)q + ... + ωn−1(E)q + ωn(E) = 0

in H•(PE; Z/2Z). Similarly, when E → X is a complex vector bundle with distinguished Chern class 2 q = c1(L) ∈ H (PE; Z), the same argument proves the following corollaries. Corollary 3.8.9. For all complex vector bundles E → X, H•(PE; Z) is a free module over H•(X; Z) with basis {1, q, q2, . . . , qn−1}.

Corollary 3.8.10. The Chern classes of E → X are the unique classes c1(E), . . . , cn(E) ∈ H•(X; Z) such that n n−1 n−2 q + c1(E)q + c2(E)q + ... + cn−1(E)q + cn(E) = 0

in H•(PE; Z).

61 3.8 General Characteristic Classes 3 Characteristic Classes

Example 3.8.11. These results give an explicit proof of the statement in Example 3.8.1: E → X is trivial if and only if H•(X) ⊗ H•(F ) ∼= H•(E) as rings, in which case qn = 0. So E is a trivial bundle if and only if ωi(E) = 0 (for real bundles) or ci(E) = 0 (for complex bundles) for all 1 ≤ i ≤ n. Theorem 3.8.12. Suppose p : E → X is a vector bundle. Then there exists a space Y and a map f : Y → X so that for any coefficient ring R, (1) f ∗ : H•(X; R) → H•(Y ; R) is injective. (2) The induced bundle f ∗E → Y is isomorphic to a Whitney sum of line bundles.

Proof. Letp ¯ : PE → X be the projective bundle associated to E. Then by the Leray- Hirsch theorem (3.8.6),p ¯∗ : H•(X; R) → H•(PE; R) is injective. Now the induced bundle p¯∗E → PE contains the tautological line bundle L → PE as a subbundle, so there is an exact sequence of vector bundles 0 → L → p¯∗E → Q → 0

∗ ∗ ∼ where Q is the quotient bundle with fibres Qx = (¯p E)x/Lx. This shows thatp ¯ E = L ⊗ Q. Since Q is then a bundle of strictly smaller dimension than E, we may repeat this process on q : PQ → PE to obtain isomorphisms q∗(¯p∗E) = q∗(L ⊗ Q) ∼= q∗L ⊗ q∗Q ∼= q∗L ⊗ (q∗L0 ⊗ q∗Q0) for some line bundle L0 and quotient bundle Q0. Continue this process until Q0 is a line bundle. Remark. Let V be a real n-dimensional vector space and recall that a (complete) flag in V is a sequence of subspaces

F• : 0 = F0 ( F1 ( F2 ( F3 ( ··· ( Fn−1 ( Fn = V. A flag is completely described by a maximal orthogonal collection of 1-dimensional subspaces (once an inner product is chosen) V1,...,Vn such that V1 ⊕ · · · ⊕ Vj = Fj for each 1 ≤ j ≤ n. Let F(V ) be the flag manifold of V , i.e. the space of all flags F• in V with the topology inherited from the identification F(V ) ∼= O(n)/O(1)⊕n. Then there are tautological line bundles L1,...,Ln over F(V ), explicitly given by

Lj = {((V1,...,Vn), w) ∈ F(V ) × V | w ∈ Vj}. For any vector bundle p : E → X, let F(p): F(E) → X be the associated flag bundle with fibres F(E)x = F(Ex), the flag manifold on the fibre Ex. Let L1,...,Ln be the tautological line bundles over F(E) as above. Then there is an isomorphism of bundles ∼ = ∗ L1 ⊕ · · · ⊕ Ln F(p) E

F(E)

62 3.8 General Characteristic Classes 3 Characteristic Classes

Taking Y = F(E) gives an alternate proof of Theorem 3.8.12. In the complex case, we can take F(V ) ∼= U(n)/U(1)⊕n and the same construction works. By Example 3.8.4, we know the characteristic class for any line bundle: it is the pullback along a classifying map of the generator ω ∈ H•(RP ∞; Z/2Z) in the real case and c ∈ H•(CP ∞; Z) in the complex case. Now Theorem 3.8.12 says that the characteristic classes of a bundle of any dimension can (up to pullback) be written as products of characteristic classes for line bundles. Using classifying spaces, we can give another interpretation of orientability (see Sec- tion 3.5). Proposition 3.8.13. Let p : E → X of dimension n be a real vector bundle. Then the following are equivalent: (1) p is orientable. (2) The structure group of p reduces to SO(n), that is, the classifying map f : X → BO(n) lifts to BSO(n):

BSO(n) f˜ π

X BO(n) f

(3) ω1(E) = 0. Proof. (1) ⇐⇒ (2) follows from Proposition 3.5.1. (2) ⇐⇒ (3) We know O(n) has two connected components, with SO(n) identified as the connected component of the identity. Consider the short exact sequence of groups

1 → SO(n) → O(n) → Z/2Z → 1. By the construction in Corollary 1.3.9, BSO(n) = EO(n)/SO(n) and there exists a principal ∼ bundle BSO(n) → BO(n) with fibre O(n)/SO(n) = Z/2Z. Note that π : BSO(n) → BO(n) is a double covering, so by covering theory a classifying map f : X → BO(n) ∼ lifts to BSO(n) if and only if f∗(π1(X)) ⊆ π∗(π1(BSO(n))). Since SO(n) = ΩBSO(n), ∼ πi(SO(n)) = πi+1(BSO(n)) for all i. In particular, ∼ ∼ π1(BO(n)) = π0(O(n)) = Z/2Z and π1(BSO(n)) = π0(SO(n)) = 1 so we see that π : BSO(n) → BO(n) is a universal cover. Thus f : X → BO(n) lifts if and only if f∗ : π1(X) → π1(BO(n)) is trivial. This in turn is equivalent to f∗ : H1(X; Z) → π1(BO(n)) being trivial, since π1(BO(n)) = Z/2Z is abelian. Notice that ∼ f∗ ∈ Hom(H1(X; Z), Z/2Z) = Hom(H1(X; Z) ⊗ Z/2Z, Z/2Z) ∼ = Hom(H1(X; Z/2Z), Z/2Z) by universal coefficients.

63 3.8 General Characteristic Classes 3 Characteristic Classes

∗ So we see that f∗ = 0 in Hom(H1(X; Z/2Z), Z/2Z) is equivalent to having f = 0 in 1 1 ∗ Hom(H (BO(n); Z/2Z),H (X; Z/2Z)), which in turn is equivalent to f ω1 = ω1(E), where 1 ω1 is a generator in H (BO(n); Z/2Z). Definition. A relative fibre bundle (or bundle pair) is a pair of spaces (E,E0) with 0 0 0 E ⊆ E and a fibre bundle p : E → X such that p := p|E0 : E → X is a fibre bundle and, if F (resp. F 0) is the fibre of p (resp. p0), then the structure group G of p preserves F 0 as a subset of F .

Example 3.8.14. Let p : E → X be a vector bundle and σ : X → E the zero section, i.e. σ(x) = ~0 ∈ p−1(x) for all x ∈ X. Then E0 = E r σ(X) forms a bundle pair (E,E0) with fibre pair (Rn, Rn r {0}) which retracts to the pair (Dn,Dn r {0}). Given a bundle pair (E,E0) → X with structure group G, given by a homomorphism ρ : G → Homeo(F,F 0) – the subgroup of Homeo(F ) preserving F 0 – there is an induced action n 0 ρ∗ : G −→ Aut(H (F,F ; R)) for any coefficient ring R and any n ≥ 0. Let Hn(F,F 0; R) → X be the associated bundle

n 0 n 0 H (F,F ; R) := PG ×G H (F,F ; R) → X, where PG is the principal G-bundle associated to E.

Definition. Let p : E → X be a real vector bundle of dimension n and set E0 = E r σ(x). The orientation bundle for E with coefficients in R is the bundle

n n n ΘR(E) := H (R , R r {0}; R) → X.

We say E → X is R-orientable if ΘR(E) is a trivial bundle.

n n n ∼ Since H (R , R r {0}; R) = R, a bundle E → X is R-orientable if and only if ΘR(E) admits a section σ : X → ΘR(E) such that σ(x) is a unit in R for all x ∈ X.

Example 3.8.15. For R = Z/2Z and any bundle E → X,ΘZ/2Z(E) always has a section n 0 ∼ given by x 7→ σ(x) where σ(x) is the unique nonzero element of H (Ex,Ex; Z/2Z) = Z/2Z. That is, every vector bundle is Z/2Z-orientable.

Proposition 3.8.16. A vector bundle E → X is orientable if and only if it is Z-orientable, that is, if and only if ΘZ(E) is trivial. ∼ Vn Proof. We claim that ΘZ(E) ⊗ R = E, where n is the dimension of E as a bundle. Given transition functions gαβ : Uα ∩ Uβ −→ O(n) Vn for E, the transition functions for E are given by hαβ : det ◦gαβ : Uαβ → O(n) →

{±1}. On the other hand, the transition maps for ΘZ(E) are given by the associated bundle construction, n n n (gαβ)∗ : H (R , R r {0}; Z) = Z −→ Z.

64 3.8 General Characteristic Classes 3 Characteristic Classes

That is, (gαβ)∗ ∈ Aut(Z) = {±1} and (gαβ)∗ = 1 if and only if gαβ is orientation-preserving, i.e. det(gαβ) = 1. Tensoring with R, we see that ΘZ(E)⊗R has precisely the same transition Vn ∼ Vn maps as E, so ΘZ(E) ⊗ R = E as claimed. Vn Now, it’s easy to see that E being trivial is equivalent to ω1(E) = 0, so by Proposi-

tion 3.8.13, E → X is orientable if and only if ΘZ(E) is trivial.

Corollary 3.8.17. For any vector bundle E → X, there is a double covering q : Xe → X such that q∗E → Xe is orientable, and Xe is connected if and only if E → X is not orientable.

By the results above, a bundle E → X is orientable if there exists a class u ∈ Hn(E,E0; Z) n 0 with the property that u|Ex ∈ H (Ex,Ex; Z) is a generator for all x ∈ X. This generalizes to any coefficient ring R as follows.

Definition. A class u ∈ Hn(E,E0; R) is called an R-Thom class for the bundle E → X n 0 if for every x ∈ X, ux := u|Ex is a generator in H (Ex,Ex; R). When R = Z, such a u is simply called a Thom class.

Note that a Thom class u ∈ Hn(E,E0; Z) is precisely the class prescribed by the Thom isomorphism theorem (3.4.1). To relate this class to orientability, we need a relative version of Theorem 3.8.6.

Theorem 3.8.18 (Relative Leray-Hirsch). Suppose (E,E0) → X is a relative fibre bun- dle such that H•(F,F 0; R) is a finitely generated, free R-module and there exist classes • 0 • 0 q1, . . . , qn ∈ H (E,E ; R) whose restrictions are free generators of each H (Ex,Ex; R), x ∈ • 0 • X. Then H (E,E ; R) is free over H (X; R) with basis {q1, . . . , qn}.

Proof. Letp ˆ : Eb → X be the relative mapping cylinder of p defined by Eb = E ∪ Mp0 , where 0 0 p = p|E : E → X and

0 0 Mp0 = (E × [0, 1] ∪ X)/(e, 1) ∼ p (e).

Then (E,Mb p0 ) is an excisive pair with

• ∼ • 0 H (E,Mb p0 ; R) = H (E,E ; R).

Observe that the natural inclusion i0 : X,→ Mp0 has the property thatp ˆ ◦ i0 = id, i.e.p ˆ is ∗ ∗ • ∗ a retraction. Thus i0p = 0 on H (X; R), so i0 is surjective. This shows that (E,Mb p0 ) and (E,Xb ) have the same cohomology, so we get along exact sequence

i∗ 0 0 · · · → Hk(Eb; R) −→0 Hk(X; R) −→ Hk(E,Xb ; R) → Hk+1(Eb; R) → Hk+1(X; R) −→· · ·

By exactness, H•(Eb; R) ∼= H•(E,Xb ; R) ⊕ H•(X; R) and result follows from the ordinary Leray-Hirsch theorem (3.8.6) applied to the bundle Eb. Theorem 3.8.19. A dimension n vector bundle E → X is R-orientable if and only if there exists an R-Thom class u ∈ Hn(E,E0; R).

65 3.8 General Characteristic Classes 3 Characteristic Classes

Proof. ( ⇒ = ) follows from the definition of R-orientability. ( =⇒ ) Suppose E → X is R-orientable. We will prove the more general statement that, when X is connected, the restriction k 0 k 0 H (E,E ; R) −→ H (Ex,Ex; R) is an isomorphism for all k ≤ n; the result we are after is the special case k = n. Suppose X is a CW-complex of finite dimension d and induct on d. Let U be the complement in X of the centers of all the d-cells in X, so that U deformation retracts to X(d−1). Let V = X rX(d−1). ` d−1 Then X = U ∪ V and U ∩ V deformation retracts to a disjoint union σ S of a copy of Sd−1 for each d-cell σ. Consider the Mayer-Vietoris sequence (with coefficients suppressed) 0 0 for (E|U ,E |U ), (E|V ,E |V ): n−1 0 n 0 · · · → H (E|U∩V ,E |U∩V ) → H (E|U∪V ,E |U∪V ) → n 0 n 0 ψ n 0 → H (E|U ,E |U ) ⊕ H (E|V ,E |V ) −→ H (E|U∩V ,E |U∩V ) → · · · n−1 0 By induction, H (E|U∩V ,E |U∩V ) = 0 and we also have n 0 ∼ H (E|U ,E |U ) = R n 0 ∼ M H (E|V ,E |V ) = R σ n 0 ∼ M H (E|U∩V ,E |U∩V ) = R. σ n 0 ∼ For each d-cell σ, H (E|U∩V ∩σ,E |U∩V ∩σ) = R and the maps ∼ n 0 R = H (E|U∩σ,E |U∩σ)

n 0 ∼ H (E|U∩V ∩σ,E |U∩V ∩σ) = R

∼ n 0 R = H (V |V ∩σ,E |V ∩σ)

∼ 0 are both isomorphisms R = R since they factor through the restrictions to (Ex,Ex) for any n 0 x ∈ U ∩ V ∩ σ. Let ηx ∈ H (Ex,Ex) be the generator determined by the trivialization of n 0 n 0 ΘR(E). This determines units ηU ∈ H (E|U∩σ,E |U∩σ) and ηV ∈ H (E|V ∩σ,E |V ∩σ), but n 0 both of these map to the same element of H (E|U∩V ∩σ,E |U∩V ∩σ) by the above maps and they depend continuously on x ∈ U ∩ V ∩ σ. Thus the ηU , ηV are independent of x. This ∼ shows that the ηU , ηV for all d-cells σ generate ker ψ, so ker ψ = R. Thus by exactness of the n 0 ∼ n 0 ∼ ∼ long sequence above, H (E|U∪V ,E |U∪V ) = H (E,E ) = ker ψ = R, and ΘR(E) is therefore trivial. This proves the theorem when X is a finite-dimensional CW-complex. The general case follows by a similar argument to that in the proof of the Leray-Hirsch theorem (3.8.6). Remark. The Thom class is natural: if E → X is an oriented bundle, i.e. an orientable bundle with choice of Thom class u ∈ Hn(E,E0; R), then for any map f : Y → X, f˜ : f ∗E → E is an orientation-preserving isomorphism on fibres, so f˜∗ : Hn(E,E0; R) −→ Hn(f ∗E, f ∗E0; R) determines a Thom class f ∗u := f˜∗u for the induced bundle.

66 3.8 General Characteristic Classes 3 Characteristic Classes

Let p : E → X be an oriented bundle (for R = Z) and let σ : X → E be the zero section. Then p ◦ σ = id and σ ◦ p is homotopic to the identity. Consider the diagram

i∗ j∗ δ ··· Hk(E,E0) Hk(E) Hk(E0) Hk+1(E,E0) ···

Φ p∗ id Φ ψ ρ ··· Hk−n(X) Hk(X) Hk(E0) Hk−n+1(X) ···

where Φ(α) = p∗α ∪ u is the Thom isomorphism (Theorem 3.4.1), the top row is the long exact sequence for (E,E0) and the bottom row is the Gysin sequence (Theorem 3.6.5). Since p and σ are homotopy inverses, p∗ is an isomorphism. This diagram shows that the Thom isomorphism theorem and the Gysin sequence are logically equivalent: either Φ is an isomorphism by exactness of the bottom row and the Five Lemma, or the bottom row is defined so that the entire diagram commutes, in which case exactness follows immediately. Note that for a class α ∈ H•(X), commutativity gives us

ψ(α) = σ∗i∗Φ(α) = σ∗i∗(p∗α ∪ u) = α ∪ σ∗i∗u.

Definition. The Euler class of an oriented bundle E → X is the class e(E) := σ∗i∗u ∈ Hn(X; Z), where σ : X → E is the zero section, i : E,→ (E,E0) is the natural inclusion and u ∈ Hn(E,E0; Z) is a Thom class. The following properties of the Euler class from Section 3.5 are now immediate from the commutative diagram above: ˆ e(E) = Φ−1(u2) (Lemma 3.5.3).

ˆ Euler class is natural (Corollary 3.5.5(1)).

ˆ If E → X is trivial, then e(E) = 0.

ˆ e(E1 ⊕ E2) = e(E1) ∪ e(E2) (Corollary 3.5.5(2)). ˆ If E admits a nonvanishing section, then e(E) = 0 (Corollary 3.5.5(3)).

Corollary 3.8.20. Euler class is a characteristic class of type (n, n, Z) for oriented vector bundles. Remark. Note that the converse to Corollary 3.5.5(3) does not hold in general, that is, there are vector bundles with trivial Euler class that do not admit nonvanishing global sections. However, the converse does hold when n ≤ 2, or when n = d and X is a CW-complex of finite dimension d. Example 3.8.21. Consider an oriented n-bundle E → Sk. Then E is trivial over the open k k hemispheres S+ and S− (they are each contractible) so E is determined by a the homotopy class of a single transition map

k k ∼ k−1 g : S+ ∩ S− = S −→ SO(n),

67 3.8 General Characteristic Classes 3 Characteristic Classes

k−1 or equivalently, by an element of [S ,SO(n)] = πk−1(SO(n))/π1(SO(n)). For k = 4 and SO k k small values of n for example, the category Vectn (S ) of oriented n-bundles over S can be computed:

SO 4 ∼ Vect1 (S ) = π3(SO(1))/π1(SO(1)) = π3(∗)/π1(∗) = 0 SO 4 ∼ Vect2 (S ) = π3(SO(2))/π1(SO(2)) = 0 SO 4 ∼ Vect3 (S ) = π3(SO(3))/π1(SO(3)) = Z/(Z/2Z). In particular, there is some nontrivial oriented vector bundle of dimension 3 over S4. Such a bundle E → S4 has e(E) ∈ H3(S4) = 0 but no nonvanishing sections, since otherwise it would be a Whitney sum of lower rank, nontrivial bundles, but by the above computation, none exist.

Recall that for a complex vector bundle p : E → X of complex dimension n, with associated tautological line bundle L → CPE (over its complex projective bundle) classified 2 2i by q ∈ H (CPE; Z), the Chern classes ci(E) ∈ H (X; Z) are the unique classes satisfying the relation n n−1 q + c1(E)q + ... + cn−1(E)q + cn(E) = 0.

Every complex bundle E can be regarded as a real vector bundle of dimension 2n, say ER, so we may consider the Stiefel-Whitney classes ωi(ER). Let RPE → X be the associated n−1 real projective bundle for ER. Then RPE → X has fibres RP and tautological (real) line bundle L → RPE which is classified by some h : RPE → RP ∞. Likewise, CPE → X has fibres CP n−1 and tautological (complex) line bundle Q → CPE classified by g : CPE → CP ∞. Define a map ϕ : RPE → CPE by sending a real line ` ⊂ E to the unique complex ∞ ∞ line ϕ(`) containing `. There is a similarly defined map ϕe : RP → CP .

Lemma 3.8.22. For a complex line bundle E → X with g, h, ϕ, ϕe as above, the diagram ϕ RPE CPE

h g ϕe RP ∞ CP ∞ commutes up to homotopy.

Proof. A classifying map for a vector bundle is equivalent to a map from the total space to R∞ (in the real case) or C∞ (in the complex case) that is a linear injection on fibres. For g, h above, fix such mapsg ˜ : Q → C∞ and h˜ : L → R∞. One can see that h˜ is equal to the composition of L → Q withg ˜, proving the lemma. This allows us to make an explicit comparison between the Chern classes for E and the

Stiefel-Whitney classes for ER. Theorem 3.8.23. For a complex vector bundle E → X of dimension n and for all i ≥ 0, 2i+1 2i ω2i+1(ER) = 0 in H (X; Z/2Z) and ω2i(ER) ≡ ci(E) mod 2 in H (X; Z/2Z).

68 3.8 General Characteristic Classes 3 Characteristic Classes

Proof. By Lemma 3.8.22, the diagram L Q ϕ RPE CPE

h g Le Qe

RP ∞ CP ∞ ϕe

∞ 1 1 commutes up to homotopy. Observe that ϕe is a fibre bundle over CP with fibre RP = S . 1 ∞ 1 1 ∼ The generator x ∈ H (RP ; Z/2Z) restricts to the generator of H (S ; Z/2Z) = Z/2Z, so by Leray-Hirsch (Theorem 3.8.6), H•(RP ∞; Z/2Z) is a free module over H•(CP ∞; Z/2Z) 2 i ∞ 2 with basis {1, x}. Hence x = d1x + d2 for some classes di ∈ H (CP ; Z/2Z), but x 2 ∞ 1 ∞ generates H (RP ; Z/2Z) and d1 ∈ H (CP ; Z/2Z) = 0, so we must have d2 6= 0 in 2 2 ∼ 2 ∗ H (CP ; Z/2Z) = Z/2Z. In particular, d2 generates this group, so x = d2 = ϕe c for some class c ∈ H2(CP ∞; Z/2Z). Set q = h∗x and r = q∗c. Then by commutativity of the diagram,

2 ∗ 2 ∗ 2 ∗ ∗ ∗ ∗ ∗ q = (h x) = h (x ) = h ϕe c = ϕ g c ≡ ϕ r mod 2.

By Corollary 3.8.10, the Chern classes ci(E) satisfy

n n−1 r + c1(E)r + ... + cn−1(E)r + cn(E) = 0 ∗ n ∗ n−1 ∗ =⇒ (ϕ r) + c1(E)(ϕ r) + ... + cn−1(E)(ϕ r) + cn(E) ≡ 0 (mod 2) by naturality 2n 2n−2 2 =⇒ q + c1(E)q + ... + cn−1(E)q + cn(E) ≡ 0 (mod 2).

Therefore the odd Stiefel-Whitney classes of ER are trivial and by Corollary 3.8.8, ω2i(ER) ≡ ci(E) mod 2.

69 4 K-Theory

4 K-Theory

In this chapter we follow Hatcher’s notes on K-theory to study the following main topics:

ˆ Complex K-theory K(X) and reduced K-theory Ke(X)

ˆ Real (ordinary) K-theory KO(X) and its reduced form KOg(X) ˆ Bott periodicity for complex K-theory.

For the most part we will assume all vector bundles are complex bundles over a fixed compact base X unless otherwise specified. To first give some abstract nonsense, we record that any abelian semigroup (S, ⊕) induces a canonical group in one of two (equivalent) ways:

(1) Let F (S) be the free abelian group generated by the elements of S and let E(S) be the subgroup of F (S) generated by all elements of the form (x ⊕ y) − (x + y). The quotient group K(S) = K0(S) = F (S)/E(S) is called the Grothendieck group of S. Note that i : S → K(S), s 7→ [s] is an embedding and that K(S) is universal with respect to all abelian groups in the following sense: for every morphism of semigroups S → G where G is an abelian group, there is a unique homomorphism of abelian groups K(S) → G making the diagram commute:

K(S)

i

S G

(2) The product S × S is a semigroup and the image of the diagonal map ∆ : S,→ S × S is a sub-semigroup. We claim the quotient (S × S)/∆(S) is a group. An element of this quotient is (x, y)∆ = {(x, y) + (a, a) | a ∈ S}, and addition and inverses are given by

(x, y)∆ + (x0, y0)∆ = (x + x0, y + y0)∆ and − (x, y)∆ = (y, x)∆.

The map i : S → (S × S)/∆(S), s 7→ (s, 0)∆ in fact satisfies the same universal property as in (1), so (S × S)/∆(S) ∼= K(S) as abelian groups. The notation in construction (2) suggests that we view (x, y)∆ ∈ K(S) as a formal difference x − y.

When S = Vect(X), the semigroup (category) of vector bundles over a compact Hausdorff space X under Whitney sum, K(X) := K(Vect(X)) is called the K-theory of X. In the next section, we give an explicit construction of this abelian group.

70 4.1 K-Theory and Reduced K-Theory 4 K-Theory

4.1 K-Theory and Reduced K-Theory

Informally, the idea of K-theory is to construct an abelian group out of vector bundles over a fixed base, with the addition operation given by Whitney sum ξ ⊕ η. We first make note of the following types of equivalence relations on vector bundles over X. Let εn denote the trivial n-bundle X × Cn → X (or X × Rn → X in the real case). ˆ A natural isomorphism of bundles is denoted ξ ∼= η. Isomorphism is of course an equivalence relation.

ˆ We say ξ and η are stably equivalent, written ξ ≈ η if ξ ⊕ εn ∼= η ⊕ εn for some n. Note that for ξ ≈ η to make sense, ξ and η must have the have fibres of the same dimension. It is easy to check stable equivalence is an equivalence relation on the set of n-bundles over X for each n ≥ 1.

ˆ Define a third equivalence relation by ξ ∼ η if ξ ⊕ εn ∼= η ⊕ εm for some m, n. In this case, ξ and η may have fibres of different dimensions.

ξ η ˆ If E1 −→ X and E2 −→ X are the total spaces of the vector bundles over X, we will ∼ usually abuse notation in each of three cases above by writing E1 = E2,E1 ≈ E2 or E1 ∼ E2. Example 4.1.1. For any sphere Sk, the tangent bundle TSk is stably equivalent to εn (for k+1 k+1 R ∼ 1 k R ∼ k+1 any n) since VSk = ε and TS ⊕ VSk = ε by Corollary 3.1.6.

Let ξ1 : E1 → X, ξ2 : E2 → X be two finite-dimensional complex bundles over X. To define an abelian group structure on bundles, we introduce the formal difference E1 − E2. 0 0 0 0 Two formal differences E1 − E2 and E1 − E2 are said to be equal if E1 ⊕ E2 ≈ E1 ⊕ E2. Definition. The K-theory of X is the abelian group

K(X) = {E1 − E2 | E1,E2 are bundles over X}

(with equivalence of formal differences as defined above), equipped with the addition operation 0 0 0 0 (E1 − E2) + (E1 − E2) = E1 ⊕ E1 − E2 ⊕ E2. Proposition 4.1.2. For any compact space X, K(X) is an abelian group with identity class 0 0 [E − E] = [ε − ε ] and inverses given by −(E1 − E2) = E2 − E1. Proof. For any bundle E over X, notice that

(E1 − E2) + (E − E) = E1 ⊕ E − E2 ⊕ E = E1 − E2, ∼ since E1 ⊕ E2 ⊕ E = E1 ⊕ E ⊕ E2. Therefore E − E represents the identity class (for any E; 0 0 in particular, [ε − ε ] = 0). Similarly, (E1 − E2) + (E2 − E1) = 0. Corollary 4.1.3. K(X) coincides with the Grothendieck group of the semigroup Vect(X).

71 4.1 K-Theory and Reduced K-Theory 4 K-Theory

Definition. The reduced K-theory of X is the set of ∼-equivalence classes of complex vector bundles over X, Ke(X) = {E → X}/ ∼ with addition given by [E1] + [E2] = [E1 ⊕ E2].

Proposition 4.1.4. Ke(X) is an abelian group with identity class [ε0] and inverses given by −[E] = [E⊥], where E is viewed as a subbundle of X × RN for some large N and E⊥ is the orthogonal complement of E in X × RN with respect to some fixed metric on E. Proof. It is clear that E ⊕ ε0 ∼= E and E ⊕ E⊥ ∼= εN ∼ ε0 for any E. ∼ Lemma 4.1.5. For any space X, there is an isomorphism K(X) = Ke(X) ⊕ Z.

n Proof. Note that any class in K(X) may be represented by E−ε for some n, since E1 −E2 = ⊥ ⊥ ⊥ n n E1 ⊕ E2 − E2 ⊕ E2 = E1 ⊕ E2 − ε . Define ϕ : K(X) → Ke(X) by ϕ(E − ε ) = [E]. For E,E0, suppose E − εn = E0 − εm in K(X). Then E ⊕ εm ≈ E0 ⊕ εn but this implies E ∼ E0. Thus ϕ is well-defined. Clearly it is also surjective and an abelian group homomorphism. To finish, note that for any bundle E,

E − εn ∈ ker ϕ ⇐⇒ E ∼ ε0 ⇐⇒ E ⊕ εk ∼= ε0 ⊕ ε` for some k, ` ⇐⇒ E − εn = E ⊕ εk − εn ⊕ εk ∼= ε` − εn+k for some k, `.

Therefore ker ϕ = {εa − εb | a, b ∈ Z} and there is an obvious map ker ϕ → Z given by εa − εb 7→ a − b which is an isomorphism. Define a splitting K(X) → ker ϕ by E 7→ n m E|x0 = ε − ε for a fixed basepoint x0 ∈ X. This gives us the desired isomorphism ∼ K(X) = Ke(X) ⊕ Z which depends on the choice of basepoint.

Proposition 4.1.6. K(X) and Ke(X) are rings with multiplication given by ⊗ and unit represented by the trivial line bundle ε1.

Proof. Similar to the proofs that K(X) and Ke(X) are abelian groups. We will commonly replace εn with n and write nE = E + ... + E = E ⊗ εn. | {z } n

Theorem 4.1.7. K and Ke are both contravariant functors Top → Rings. In particular, for any f : X → Y there are induced maps

f ∗ : K(Y ) −→ K(X) ∗ ∗ E1 − E2 7−→ f E1 − f E2 and f ∗ : Ke(Y ) −→ Ke(X) E 7−→ f ∗E

∗ ∼ ∗ ∗ ∗ ∼ ∗ ∗ which both satisfy f (E1 ⊕ E2) = f E1 ⊕ f E2 and f (E1 ⊗ E2) = f E1 ⊗ f E2.

72 4.2 External Products in K-theory 4 K-Theory

Definition. For a pair of spaces (X,A), the relative K-theory of the pair is

K(X,A) := Ke(X/A). ` By convention X/∅ = X ∗, the disjoint union of X with a point, so that K(X, ∅) = Ke(X ` ∗) = K(X). There is an alternative construction that yields relative K-theory. For a pair (X,A), let Ln(X,A) be the set of equivalence classes of all sequences

(E0,E1,...,En; α1, . . . , αn)

where Ei ∈ Vect(X), αi : Ei−1|A → Ei|A are bundle morphisms and the sequence of bundles

α1 α2 αn 0 → E0|A −→ E1|A −→· · · −→ En|A → 0 is exact. The equivalence relation on sequences is defined as follows. Define an elementary sequence in Ln(X,A) to be a sequence of the form

(0, 0,...,Ei−1,Ei, 0,..., 0; 0, . . . , αi,..., 0)

where Ei−1 = Ei and αi = id. Then two elements of Ln(X,A) are called equivalent if they become isomorphic after a direct sum with some finite number of elementary sequences.

Example 4.1.8. For n = 1, L1(X,A) consists of triples (E0,E1; α) where α : E0 → E1 is a bundle isomorphism over A. Lemma 4.1.9. For each n ≥ 1, the map

Ln(X,A) −→ Ln+1(X,A)

(E0,...,En; α1, . . . , αn) 7−→ (E0,...,En, 0; α1, . . . , αn, 0) is an isomorphism. Therefore,

L(X,A) = lim Ln(X,A) −→

is defined and equals Ln(X,A) for any n. Proposition 4.1.10. There is a unique natural isomorphism of functors χ : L(X,A) −→ K(X,A)

n such that when A = ∅, χ(E0,E1,...,En) = E0 − E1 + ... + (−1) En in K(X).

4.2 External Products in K-theory

Recall the algebraic cross product on cohomology, H•(X) ⊗ H•(Y ) → H•(X × Y ). There is an analogous external product in K-theory, defined by: K(X) ⊗ K(Y ) −→ K(X × Y ) ∗ ∗ α ⊗ β 7−→ (π1α) ⊗ (π2β)

where π1 : X × Y → X and π2 : X × Y → Y are the natural projections.

73 4.2 External Products in K-theory 4 K-Theory

Proposition 4.2.1. The external product on K-groups is a ring homomorphism.

Definition. A functor H : Top×Top → GradedAbGps is called a generalized (or extraor- dinary) cohomology theory if it satisfies the following four out of five Eilenberg-Steenrod Axioms:

(1) (Homotopy) If f ' g then f ∗ = g∗.

(2) (Excision) For any pair (X,A) and any open set U ⊆ X such that U ⊆ A, the inclusion i :(X r U, A r U) ,→ (X,A) induces an isomorphism i∗. (3) (Additivity) If P = {x} is a point space, then Hn(P ) = 0 unless n = 0.

(4) (Exactness) For each pair (X,A), there is an exact sequence

· · · → Hn−1(A) → Hn(X,A) → Hn(X) → Hn(A) → · · ·

Theorem 4.2.2. K and Ke are generalized cohomology theories. There is an external product for reduced K-theory,

Ke(X) ⊗ Ke(Y ) −→ Ke(X ∧ Y )

where X ∧ Y = (X × Y )/X ∨ Y is the smash product of X and Y .

Proposition 4.2.3. For a compact, Hausdorff space X and a closed subset A ⊆ X, the p sequence A −→i X −→ X/A induces an exact sequence

p∗ i∗ Ke(X/A) −→ Ke(X) −→ Ke(A).

Proof. The sequence A → X → X/A factors through a point A/A = ∗:

A X X/A

∗

Therefore the induced sequence of reduced K-groups factors through 0:

Ke(X/A) Ke(X) Ke(A)

0

so i∗p∗ = 0. In other words im p∗ ⊆ ker i∗. ∗ On the other hand, an element of ker i is a bundle E → X whose restriction E|A is stably n ∼ m n m trivial, i.e. E|A ⊕εA = εA for some trivial bundles εA, εA over A. Consider the trivial bundle n n n εX over X. Then (E ⊕ εX )|A is trivial since [E ⊕ εX ] = [E]. Thus there is some trivialization −1 n −1 h : π (A) × C → E|A. We can use this to identify all the fibres π (a) for a ∈ A. Define

74 4.3 Bott Periodicity 4 K-Theory

−1 −1 the bundle E/h → X/A by E/h = E/π (a1) ∼ π (a2) for any a1, a2 ∈ A. One now checks that E/h is locally trivial – this reduces to checking local triviality near A/A = ∗ which follows from Tietze’s extension theorem: if E is trivial over A, there exists an open set A ⊆ U ⊆ X such that E is trivial over U. Alternatively, we can use a deformation retract ∗ ∗ r : U → A in X and pull back the trivial bundle E|A to U. This proves ker i ⊆ im p . Theorem 4.2.4 (Long Exact Sequence in Reduced K-theory). For a closed subset A ⊆ X, there is a long exact sequence of abelian groups

· · · → Ke(Σ2A) → Ke(Σ(X/A)) → Ke(ΣX) → Ke(ΣA) → Ke(X/A) → Ke(X) → Ke(A).

Definition. For a space X, the nth reduced K-group is defined to be Ke n(X) := Ke(ΣnX). Theorem 4.2.5. For any spaces X and Y , there is an isomorphism

Ke(X × Y ) ∼= Ke(X ∧ Y ) ⊕ Ke(X) ⊕ Ke(Y ).

Proof. In the exact sequence

Ke(Σ(X × Y )) → Ke(Σ(X ∨ Y )) → Ke(X ∧ Y ) → Ke(X × Y ) → Ke(X ∨ Y ) there are splittings Ke(Σ(X ∨ Y )) → Ke(Σ(X × Y )) and Ke(X ∨ Y ) → Ke(X × Y ). Then use the fact that Ke(X ∨ Y ) ∼= Ke(X) ⊕ Ke(Y ).

Now fix a point (x0, y0) ∈ X × Y and suppose E ∈ ker(K(X) → K({x0)}) = Ke(X) and 0 ∗ ∗ 0 E ∈ ker(K(Y ) → K({y0)}) = Ke(Y ). Then p1E is trivial over X × {y0}, p2E is trivial over ∗ ∗ 0 {x0} × Y and therefore p1E ⊗ p2E is trivial over X ∨ Y . This defines the reduced external product Ke(X) ⊗ Ke(Y ) → Ke(X ∧ Y ).

4.3 Bott Periodicity

One version of Bott periodicity says that for any space X, there is an isomorphism Ke(X) ∼= Ke(Σ2X), where Σ is the reduced suspension functor. In the course of proving this, we will establish that the external product

K(X) ⊗ K(S2) −→ K(X × S2)

is an isomorphism. We will also compute the K-theory of the 2-sphere to be the polynomial ring K(S2) = Z[H]/(H − 1)2. 1 2 To this end, let H = [γ1 ] be the equivalence class in K(S ) of the (complex) canonical 2 ∼ 1 line bundle over S = CP . Concretely, we will view the complex projective line as the space of homogeneous coordinates

1 ∗ CP = {[z0, z1]: z0, z1 ∈ C, [λz0, λz1] = [z0, z1] for all λ ∈ C }.

1 −1 z0 The fibres of γ are given by π ([z0, z1]) = {λ(z0, z1) | λ ∈ }. Identifying [z0, z1] with , 1 C z1 we get a bijection between CP 1 and the Riemann sphere S2 = C ∪ {∞}. We may cover

75 4.3 Bott Periodicity 4 K-Theory

2 2 the sphere with two charts, D0 containing 0 ∈ C and D∞ containing the point at infinity. 1 2 2 Then explicit trivializations of the bundle γ1 over D0 and D∞ are given by the following nonvanishing sections:

2 1 D0 −→ E(γ1 )   z0 [z0, z1] 7−→ , 1 z1 2 1 and D∞ −→ E(γ1 )   z1 [z0, z1] 7−→ 1, . z0

−1 2 −1 2 The map ϕ : π (D0) → π (D∞) is defined on the boundary by

1 1 S × C −→ S × C (z, 1) 7−→ (1, z−1),

1 ∗ so that the transition map S → GL1(C) = C is given by z 7→ multiplication by z. Let 1 = [ε1] ∈ K(S2) be the unit class. Lemma 4.3.1. (H − 1)2 = 0 in K(S2). Proof. Note that (H − 1)2 = 0 is equivalent to H2 + 1 = 2H, so it’s enough to show (H ⊗ H) ⊕ ε1 ∼= H ⊕ H. We prove this equivalence by analyzing both sides as a 2-bundle 2 2 over S . The two transition maps S → GL2(C) are given by matrices z2 0 z 0 and . 0 1 0 z It’s enough to show these are homotopic, since then the two bundles are isomorphic. Using the fact that GL2(C) is connected, we may choose a path γ : [0, 1] → GL2(C) with γ0 = 1 0 0 1 and γ = . Then the homotopy 0 1 1 1 0

z 0 1 0 Φ = γ γ t 0 1 t 0 z t

z 0 z2 0 has Φ = and Φ = so the maps are homotopic. 0 0 z 1 0 1

The lemma shows there exists an injective map Z[H]/(H − 1)2 ,→ K(S2). We claim the composition 2 2 2 K(X) ⊗ Z[H]/(H − 1) → K(X) ⊗ K(S ) → K(X × S ) is an isomorphism. To show this, we must further analyze bundles over X × S2. Consider a bundle E −→π X. Taking the Cartesian product bundle (Section 2.4) with the trivial bundle D2 → D2 gives a bundle 0 E × D2 −→π X × D2 0 −1 −1 2 2 2 where (π ) (x, z) = π (x). This gives two bundles E × D0 → X × D0 and E × D∞ → 2 1 X × D∞ which glue together along S by an automorphism

76 4.3 Bott Periodicity 4 K-Theory

f D × S1 E × S1 π0 π0 X × S1 called a generalized clutching function. Given such a clutching function f, we get a bundle 2 2 ` 2 over X × S by gluing E × D0 f E × D∞. Denote this bundle by [E, f]. We will show that any bundle over X × S2 is of this form. Example 4.3.2. The identity map is a clutching function, and for any bundle E → X, the resulting bundle [E, id] → X × S2 is equal to E × S2. On the level of K-groups, [E, id] = E × 1 ∈ K(X × S2). Example 4.3.3. Define a clutching function f : E × S1 → E × S1 by (~v, z) 7→ (z~v, z). On a fibre π−1(x), this acts by

−1 −1 π (x) ⊗ C −→ π (x) × C ~v ⊗ 1 7−→ ~v ⊗ z.

Here, we get [E, f] = E × H ∈ K(X × S2). More generally, for each n define

1 1 fn : E × S −→ E × S (~v, z) 7−→ (zn~v, z).

n 2 n ⊗n Then [E, fn] = E × H ∈ K(X × S ), where H = H is the n-fold tensor power of the canonical class in K(S2). Lemma 4.3.4. For any generalized clutching function f : E × S1 → E × S1, [E, znf] = ∗ n 2 2 [E, f] ⊗ p2H , where p2 : X × S → S is the canonical projection. 0 ∗ n ∼ n 0 n Proof. Set E = [E, f] and He = p2H. Then [E, z f] = [[E, f], z ] = [E , z ] so it’s enough to prove the statement when f = id. In this case we have

n 2 2 [E, z ] = E × D0 ∪zn E × D∞ ∼ 2 2 = (E × D0) ⊗ He ∪1⊗zn (E × D∞) ⊗ He ∼ 2 2 = (E × D0) ⊗ He ∪zn⊗1 (E × D∞) ⊗ He ∼ 2 2 n = (E × D0 ∪1 E × D∞) ⊗ He = [E, 1] ⊗ He n.

Lemma 4.3.5. Any bundle over X × S2 is isomorphic to [E, f] for some bundle E → X and some generalized clutching function f.

2 2 Proof. Given a bundle E1 → X × S , consider the restrictions E0 → X × D0 and E∞ → 2 1 2 2 X ×D∞. Set E = E1|E1×{1} → X ×{1} where we view 1 ∈ S ⊂ S . Since D0 is contractible, the maps id and i ◦ p in the diagram

77 4.3 Bott Periodicity 4 K-Theory

2 id 2 X × D0 X × D0 p i X × {1} are homotopic. Since homotopic maps induce isomorphic bundles, we get

∗ ∼ ∗ ∗ ∗ ∗ 2 E0 = id E0 = (i ◦ p) E0 = p i E0 = p E = E × D0.

∼ 2 1 Similarly, E∞ = E × D∞, so a clutching function can be defined by restricting to X × S on each piece. The hardest piece of the proof that K(X) ⊗ K(S2) → K(X × S2) is an isomorphism is contained in the following lemma.

Lemma 4.3.6. µ : K(X) ⊗ K(S2) → K(X × S2) is surjective.

Proof. By Lemma 4.3.5, any bundle over X × S2 is isomorphic to [E, f] for some bundle E → X and clutching map f : E × S1 → E × S1. We now change f by homotopy through generalized clutching maps to simplify the scenario. Ultimately, we will demonstrate that [E, f] is the image of some linear combination of elements of K(X) ⊗ K(S2). The following steps will lead towards this goal:

(1) Given a generalized clutching map f : E ×S1 → E ×S1, there is a Laurent polynomial generalized clutching map N X n q(x, z) = bn(x)z , n=−N

with bn(x) continuous functions on X, to which f is homotopic through generalized clutching maps.

(2)[ E, q] = [E, p] for a polynomial clutching map p(x, z).

Pk n (3) For any polynomial clutching map p(x, z) = n=0 an(x)z , where an(x) are continuous on X, there exists a linear clutching function L(x, z) such that

[E, p] ⊗ [kE, 1] ∼= [(k + 1)E,L].

(4) For any bundle Ee over X and any linear clutching function L,[E,Le ] is in the image of µ : K(X) ⊗ K(S2) → K(X × S2).

78 4.3 Bott Periodicity 4 K-Theory

Once we have (1) – (4), surjectivity of µ will follow from:

[E, f] = [E, q] = [E, z−N p] by (1) – (2) ∗ −N −1 ∗ = [E, p] ⊗ p2H where H = H is the dual bundle to H ∗ −N = ([(k + 1)E,L] − [kE, 1]) ⊗ p2H by (3) ∗ −N ∗ −N = [(k + 1)E,L] ⊗ p2H − [kE, 1] ⊗ p2H ∗ −N −N = [(k + 1)E,L] ⊗ p2H − µ(kE ⊗ H ) ∗ −N ∗ −N −N = [Ee+, 1] ⊗ p2H + [Ee−, z] ⊗ p2H − µ(kE ⊗ H ) for (k + 1)E = Ee+ ⊕ Ee− −N −N+1 = µ(Ee+ ⊗ H ) + µ(Ee− ⊗ H ) by (4). (1) In local trivializations of E → X, the clutching function f(x, z): E × S1 → E × S1 is just a matrix with entries depending on x and z. Each entry is of the form h : X × S1 → C. By Fourier analysis, as long as X is compact, any such function X × S1 → C can be approximated uniformly by Laurent polynomial functions in z:

∞ X  1 Z 2π  h(x, z) = f(x, eiθ)e−inθ dθ zn. 2π n=−∞ 0 Next, any endomorphism E × S1 → E × S1 can be approximated by Laurent polynomial n endomorphisms by defining the approximation on each local trivialization hi : Ui × C → −1 1 π (Ui) and using a partition of unity to define on the entire space E × S . Finally, since Aut(E×S1) is an open subset of End(E×S1) (with respect to the operator norm), this implies that any f ∈ Aut(E × S1) can be approximated by Laurent polynomial clutching functions. Let q(x, z) be the Laurent polynomial clutching function approximating f(x, z) within some ε-neighborhood of f(x, z) in Aut(E × S1). Then the linear homotopy tq + (1 − t)f induces a bundle isomorphism [E, f] ∼= [E, q] since each level of the homotopy is also a clutching function. (2) Define the polynomial clutching map p(x, z) = zN q(x, z). Then by Lemma 4.3.4,

N ∗ N ∗ N [E, p] = [E, z q] = [E, q] ⊗ p2H = [E, f] ⊗ p2H . (3) Consider two (k + 1) × (k + 1) matrices whose entries are endomorphisms of bundles:

 1 −z 0 0 ··· 0  1 0 0 ··· 0   0 1 −z 0 ··· 0    0 1 0 ··· 0   0 0 1 −z ··· 0    A =  ......  and B = 0 0 1 ··· 0   ......  . . . . .    ......   0 0 0 ··· 1 −z      0 0 0 ··· p(x, z) ak(x) ak−1(x) ······ a1(x) a0(x) Then A, B ∈ End((k + 1)E) and in fact B :(k + 1)E → (k + 1)E is an automorphism, that is, a generalized clutching map. Notice that A is similar to B (by elementary row/column operations), so A ∈ Aut((k + 1)E) as well. Even better, A and B are homotopic through clutching functions, and B represents [E, p] ⊗ [kE, 1] while A represents [(k + 1)E,L] for some linear L, since all the entries of A are linear in z.

79 4.3 Bott Periodicity 4 K-Theory

(4) Let L(x, z) = a1(x)z + a0(x) be a linear clutching function. Define Ht(x, z) = z+t a1(x) 1+tz + a0(x) and consider the homotopy

Ht(x, z)(1 + tz) = a1(x)(z + t) + a0(x)(1 + tz)

= (a1(x) + ta0(x))z + (ta1(x) + a0(x)).

1 z+t 1 Note that if z ∈ S ⊂ C, then 1+tz ∈ S as well:

z + t z + t zz¯ + tz¯ 1 + tz¯ ||1 + tz¯|| = · z¯ = = = = 1. 1 + tz 1 + tz 1 + tz 1 + tz ||1 + tz|| So this implies  z + t  z + t L x, = a (x) + a (x) 1 + tz 1 1 + tz 0 is a clutching function as well (i.e. is an automorphism on fibres). Further, multiplication  z+t  by the nonzero 1 + tz ∈ C is also an automorphism, so a1(x) 1+tz + a0(x) (1 + tz) is an automorphism on each fibre for any 0 ≤ t < 1. At t = 0, we get a1(x)z + a0(x) = L(x, z). −1 Plugging in t = 1, we do get an automorphism a1(x) + a0(x) of the fibre π (x), but since X −1 is compact, this means a1(x) + ta0(x) is still an automorphism of π (x) for any 0 ≤ t ≤ 1. Therefore we can invert this coefficient to obtain

−1 ta1(x) + a0(x) (a1(x) + ta0(x)) Ht(x, z) = z + , 0 ≤ t ≤ 1. a1(x) + ta0(x)

In other words, the linear clutching function L(x, z) = a1(x)z + a0(x) is homotopic through clutching functions to a linear clutching function of the form z +a ˜0(x). Assume L(x, z) = −1 z+a0(x) to begin with. Since this is an automorphism on π (x) for each x, z, notice that the −1 −1 function a0(x): π (x) → π cannot have any eigenvalues on the unit circle, for a0(x)v = λv implies (z + a0(x))v = zv + λv = 0 which is only possible if z = −λ, contradicting z + a0(x) being an automorphism. −1 Now fix a fibre V = π (x). Then a0(x): V → V has no eigenvalues on the unit circle, so we can decompose V = V+ ⊕ V− where V+ is the sum of the eigenspaces for eigenvalues of a0(x) outside the unit circle and V− is the same for eigenvalues inside the unit circle. Consider the characteristic polynomial of a0(x), P (t) = det(tI − a0(x)). This admits a decomposition P (t) = P+(t)P−(t) corresponding to the eigenvalues inside/outside the unit circle. By the Cayley-Hamilton theorem, P (a0(x)) = 0, but notice that V+ = ker(p+(a0(x)) : V → V ), V− = ker(p−(a0(x)) : V → V ) and V+ ∩ V− = 0, so V = V+ ∩ V−. In fact, this decomposition depends continuously on x so that we get a bundle decomposition E = E+ ⊕ E−, where E+ has fibres V+ and E− has fibres V−. Set a+(x) = a0(x)|E+ and a−(x) = a0(x)|E− . Then for 0 ≤ t ≤ 1, there exists a homotopy through clutching functions exhibiting the isomorphisms ∼ ∼ [E+, z + a+(x)] = [E+, tz + a+(x)] = [E+, a+(x)],

1 since each tz + a+(x) is an isomorphism (there are no eigenvalues on S ). Likewise, [E−, z + ∼ ∼ a−(x)] = [E−, z + ta−(x)] = [E−, z]. Finally, a+(x) is just a bundle isomorphism E → E, so as a clutching function it is the same as a+(x) × idS2 . In other words, [E+, a+(x)] = [E+, id], ∼ so we have the desired decomposition [E,L] = [E+, id] ⊕ [E, z].

80 4.3 Bott Periodicity 4 K-Theory

Theorem 4.3.7. K(X)⊗Z[H]/(H−1)2 → K(X)⊗K(S2) → K(X ×S2) is an isomorphism. Proof. To check the composition is one-to-one, we need only return to each step of the previous proof and check that there is a unique choice of generalized clutching functions up to homotopy. This is done in Hatcher’s notes.

2 ∼ 2 Corollary 4.3.8. K(S ) = Z[H]/(H − 1) . Proof. Apply Theorem 4.3.7 to X = {x}.

Corollary 4.3.9. The external product map K(X)⊗K(S2) → K(X×S2) is an isomorphism.

As abelian groups, Z[H]/(H − 1)2 = Z ⊕ ZhHi. Since Ke(S2) = ker(K(S2) → K(∗)), this implies Ke(S2) = ZhH − 1i is infinite cyclic. We will use Theorem 4.2.5 to analyze the reduced K-theory external product. Note that when Y = S2, the smash product X ∧ Y becomes the reduced suspension Σ2X.

Theorem 4.3.10 (Bott Periodicity). Let X be a compact space and Σ the reduced suspension functor. Then there is an isomorphism Ke(X) ∼= Ke(Σ2X). Proof. Consider the diagram ∼= K(X) ⊗ K(S2) K(X × S2)

∼= ∼=

(Ke(X) ⊕ Z) ⊗ (Ke(S2) ⊕ Z) Ke(X × S2) ⊕ Z

The top row is an isomorphism by Corollary 4.3.9 and the vertical arrows are isomorphisms by Lemma 4.1.5. Moreover, the diagram commutes so the bottom row is also an isomorphism. Now

2 2 2 (Ke(X) ⊕ Z) ⊗ (Ke(S ) ⊕ Z) = (Ke(X) ⊗ Ke(S )) ⊕ Ke(X) ⊕ Ke(S ) ⊕ Z ∼ 2 = (Ke(X) ⊗ ZhH − 1i) ⊕ Ke(X) ⊕ Ke(S ) ⊕ Z ∼ 2 = Ke(X) ⊕ Ke(X) ⊕ Ke(S ) ⊕ Z. On the other hand, Theorem 4.2.5 implies that

2 ∼ 2 2 Ke(X × S ) ⊕ Z = Ke(X ∧ S ) ⊕ Ke(X) ⊕ Ke(S ) ⊕ Z ∼ 2 2 = Ke(Σ X) ⊕ Ke(X) ⊕ Ke(S ) ⊕ Z.

Therefore the isomorphism on the bottom row of the diagram implies Ke(X) ∼= Ke(Σ2X).

Corollary 4.3.11. For any n ≥ 1, Ke(Sn) ∼= Ke(Sn+2).

81 4.4 Characteristic Classes in K-Theory 4 K-Theory

Bott’s original incarnation of periodicity went as follows. Let

U = lim Un( ) −→ R be the infinite unitary group, the direct limit of the finite (real) unitary groups Un(R) for n ≥ 1. Then πk+2(U) = πk(U) for all k ≥ 0. It is a fact that the infinite Grassmannian

∞ Gr∞ = lim Grn( ) −→ R (the limit of the Grassmannians constructed in Section 3.3) is the classifying space of U, i.e. BU = Gr∞. By the results in Section 3.3, this means BU is the base of the universal bundle γ. The version of periodicity in Theorem 4.3.10 then follows from the original version of n periodicity by comparing πn(U) to Ke (BU). Some vital consequences of Bott periodicity include:

n ˆ For any CW-complex X and any fixed k ≥ 0, the groups πn+k(Σ X) are eventually constant as n gets large. This is the beginning of stable homotopy theory. ˆ S n The above allows one to define the kth stable homotopy group, πk = lim πn+k(S ), and −→ n the J-homomorphism J : πk(SO(n)) → πn+k(S ) for each n ≥ 1. ˆ Real Bott periodicity, i.e. for real vector bundles, says that real reduced K-theory KOg(X) is 8-periodic: KOg(X) ∼= KOg(Σ8X).

n ˆ This further implies the image of the J-homomorphism in πn+k(S ) is 8-periodic.

4.4 Characteristic Classes in K-Theory

Let X be any topological space and consider the total Chern class c : Vect(X) → H•(X; Z). Lemma 4.4.1. The total Chern class extends naturally to a functor

• c : K(X) −→ H (X; Z) E − F 7−→ c(E)c(F )−1.

More generally, if R is a commutative ring and f(t) ∈ R[[t]] is a formal power series over R which is a unit (the constant term of f is 1), then f induces a characteristic class in K-theory • as follows. First, if L → X is a complex line bundle, set cf (L) = f(c1(L)) ∈ H (X; R).

Example 4.4.2. For the polynomial f(t) = 1 + t, cf (L) = 1 + c1(L) is the ordinary Chern class of L.

If E → X is a bundle which splits as a Whitney sum of line bundles E = L1 ⊕ · · · ⊕ Ln, then define cf (E) = cf (L1) ··· cf (Ln) = f(c1(L1)) ··· f(c1(Ln)). In general, by Theorem 3.8.12 any E → X pulls back along some q : Y → X to a sum of ∗ ∼ line bundles over Y , say q E = L1 ⊕ · · · ⊕ Ln.

82 4.4 Characteristic Classes in K-Theory 4 K-Theory

• ∗ Definition. The elements x1, . . . , xn ∈ H (X; Z) such that q xi = c1(Li) for each i are called the Chern roots of E. Therefore for an arbitrary vector bundle E → X, we can define a characteristic class

cf (E) = f(x1) ··· f(xn)

where xi are the Chern roots of E. Since cf is symmetric in the xi, it can be written as a polynomial in σ1, . . . , σn, the elementary symmetric polynomials in the xi, but since • σj = cj(E) for all 1 ≤ j ≤ n, cf gives a value in H (X; Z) which is a sum of Chern classes of X. To make this construction compatible with the tensor product of bundles, i.e. the ring structure of K(X), we define: Definition. For a space X, the Chern character of X is the function

• ch : K(X) −→ H (X; Q) defined uniquely by the following:

n c1(L) t P∞ t (1) For a line bundle L → X, ch(L) = e , where e = n=0 n! is the formal exponen- tial.

(2) For a Whitney sum of line bundles E = L1 ⊕ · · · ⊕ Ln,

n X ch(E) = ec1(Li). i=1

(3) For an arbitrary vector bundle E → X,

n X ch(E) = exi i=1

where x1, . . . , xn are the Chern roots of E.

Example 4.4.3. If E = L1 ⊕ L2 is a sum of line bundles, then  x2 x3   x2 x3  ch(E) = 1 + x + 1 + 1 + ... 1 + x + 2 + 2 + ... 1 2 6 2 2 6 1 = 2 + σ (x , x ) + (σ2 − 2σ )(x , x ) + ... 1 1 2 2 1 2 1 2 1 = rank(E) + c (E) + (c (E)2 − 2c (E)) + ... 1 2 1 2 Such formulas are easily obtained for vector bundles of arbitrary rank.

Lemma 4.4.4. The Chern character ch : K(X) → H•(X; Q) is a ring homomorphism. Moreover, there is an induced ring homomorphism ch : K(X,A) → H•(X,A; Q) for any pair (X,A).

83 4.4 Characteristic Classes in K-Theory 4 K-Theory

Example 4.4.5. Let π : E → X be a complex vector bundle of rank n and write Ee = π∗E → E. Then there is a sequence of vector bundles

^ α1 ^ α2 ^ α3 ^ 0 → 0Ee −→ 1Ee −→ 2Ee −→· · · → nEe → 0

Vj−1 where over v ∈ Ex, αj(v)(ω) = (v, v ∧ ω) for (v, ω) ∈ Ee. In fact, this sequence is exact when restricted to E0 = E r s0(X), so it defines an element λE ∈ K(E,E0) by Proposition 4.1.10. Taking the pullback along the zero section gives

n ∗ X j ^j s0λE = (−1) E ∈ K(X). j=0

This allows us to calculate the Chern character of E as follows. Assume E = L1 ⊕ · · · ⊕ Ln, V• Nn V• Nn or pullback to PE to be in this situation. Then E = i=1 ( Li) = i=1(C ⊕ Li) so we have n ! n n   ^• O Y Y xi ch E = ch (1 + Li) = ch(1 + Li) = (1 + e ) i=1 i=1 i=1 where xi = c1(Li). On the other hand, notice that

n ! n Y Y xi ch(λE) = ch (1 − Li) = (1 − e ). i=1 i=1 Theorem 4.4.6 (Thom Isomorphism for K-Theory). Let E → X be a complex vector bundle. Then

(1) The ring Ke(E) = K(E,E0) is a module over K(X) via pullback and tensor product.

(2) K(E,E0) is a free K(X)-module generated by λE. In particular, the map

ΦK : K(X) −→ K(E,E0), α 7−→ αλE

is an isomorphism.

Definition. λE ∈ K(E,E0) is called the Thom class in K-theory of E → X.

84 5 Chern-Weil Theory

5 Chern-Weil Theory

Let M be a smooth manifold and E → M a smooth vector bundle. Denote by Γ(M,E) the space of smooth sections of E → M. Definition. A differential k-form on a smooth manifold M is a smooth section ω ∈ Ωk(M) := Γ M, Vk T ∗M
, where T ∗M = (TM)∗ is the cotangent bundle of M.

Example 5.0.1. A differential 0-form is just a smooth function f : M → R. A differential 1 1-form ω ∈ Ω (M) is a smooth assignment x 7→ ωx ∈ Hom(TM, R).

If x1, . . . , xn are local coordinates in a chart U of a manifold M, there is a corresponding n ∂ ∂ o basis ,..., of TM|U . Let {dx1, . . . , xn} denote the corresponding dual basis of ∂x1 ∂xn ∗ k T M|U . Then any differential form ω ∈ Ω (M) can be written locally on U as X ω = ωi1,...,ik dxi1 ∧ · · · ∧ dxik

i1<···

for a smooth function ωi1,...,ik on U defined by  ∂ d  ωi1,...,ik = ω ,..., . ∂xi1 dxik Recall that for any smooth manifold M, there is a unique operator d :Ωk(M) → Ωk+1(M) called the exterior derivative which satisfies (a) For a smooth function f ∈ Ω0(M) and a vector field X ∈ Γ(M,TM), df(X) = X(f). (b) If ω ∈ Ωk(M) and η ∈ Ω`(M) then d(ω ∧ η) = dω ∧ η + (−1)kω ∧ dη. (c) For all ω ∈ Ωk(M), d2ω = 0. (d) For a smooth function f : M → N, there is a unique linear map f ∗ :Ωk(N) → Ωk(M) satisfying: (i) If g ∈ Ω0(N), then f ∗g = g ◦ f. (ii) f ∗(ω ∧ η) = f ∗ω ∧ f ∗η for all ω, η ∈ Ω•(N). (iii) f ∗(dω) = d(f ∗ω) for all ω ∈ Ω•(N). Since d2 = 0, the sequence · · · → Ωk(M) → Ωk+1(M) → · · · forms a chain complex with differential d. The cohomology of this complex is called the de Rham cohomology of M, • written HdR(M). De Rham cohomology has the following properties: ˆ • ∧ descends to a signed-commutative operation on HdR(M), also denoted ∧, and this gives the de Rham cohomology the structure of a ring.

ˆ ∗ k k Every smooth function f : M → N induces linear maps f : HdR(N) → HdR(M) for all k. In fact, f ∗ commutes with ∧ and so preserves the ring structure of the de Rham cohomology.

85 5.1 Connections on a Vector Bundle 5 Chern-Weil Theory

ˆ • ∼ • (De Rham’s Theorem) There is a natural isomorphism HdR(M) = H (M; R) given by integration along a smooth singular chain. Our goal is to generalize this construction by considering “differential forms with values in vector bundles”, i.e. sections of VkT ∗M ⊗ E for various bundles E → M.

5.1 Connections on a Vector Bundle

Our goal is to generalize de Rham theory on a smooth manifold by considering “vector bundle-valued differential forms”, i.e. sections of Vk T ∗M ⊗ E for a vector bundle E → M. Given a (real or complex) vector bundle E → M, any section of Vk T ∗M ⊗E may be thought of as an alternating linear map T ∗M × · · · × T ∗M → E. Set Ωk(M; E) = Γ Vk T ∗M ⊗ E
. | {z } k k Vk ∗

In general, for U ⊆ M we will write Ω (U; E) = Γ T M ⊗ E |U . Definition. A (smooth) connection on a real (or complex) vector bundle E → M is an R-linear (or C-linear) map ∇ :Ω0(M; E) −→ Ω1(M; E),

written (∇σ)(X) = ∇X σ, which satisfies:

∞ (1) ∇fX+gY σ = f∇X σ + g∇Y σ for all f, g ∈ C (M) and X,Y ∈ Γ(TM).

∞ 0 (2) ∇X (f · σ) = df(X) · σ + f∇X σ for all f ∈ C (M) and σ ∈ Ω (M; E).

Example 5.1.1. If E = M × R is a trivial line bundle, then the exterior derivative d : Ω0(M) → Ω1(M) is a connection, called the trivial connection. Each a ∈ Ω1(M) gives another connection on M × R by defining da :Ω0(M) −→ Ω1(M) f 7−→ daf := df + a · f.

Note that daf ∈ Ω1(M) since daf(gX) = gdaf(X) for any f, g ∈ Ω0(M). Also,

da(gf) = d(gf) + a · gf = (dg)f + g(df) + g · af = (dg)f + gdaf.

So da is a connection. In particular, there is an infinite family of connections on M × R 1 ∼ ∼ parametrized by Ω (M) = R (or = C in the complex case). Proposition 5.1.2. If ∇1 and ∇2 are connections on a bundle E → M, then ∇1 − ∇2 satisfies (∇1 − ∇2)(fσ) = f(∇1 − ∇2)σ for all f ∈ C∞(M) and σ ∈ Ω0(E). That is, ∇1 − ∇2 is a section of End(E) = E∗ ⊗ E.

Remark. For two real bundles E1,E2 → M, an R-linear F : Γ(E1) → Γ(E2) corresponds to ∗ ∞ a section of Hom(E1,E2) = E1 ⊗ E2 if and only if F (fσ) = fF (σ) for all f ∈ C (M) and σ ∈ Γ(E1). Such an F is called tensorial.

86 5.1 Connections on a Vector Bundle 5 Chern-Weil Theory

Example 5.1.3. Proposition 5.1.2 shows that every connection on M × R is of the form a 1 ∼ k d for some a ∈ Ω (M). More generally, let E = M × R be a trivial k-bundle. Then any 0 σ ∈ Γ(E) is of the form σ = (σ1, . . . , σk) for σi ∈ Ω (M). There is a trivial connection d on M × Rk given by the componentwise differential operator: 0 k 1 k d :Ω (M; M × R ) −→ Ω (M; M × R ) (σ1, . . . , σk) 7−→ (dσ1, . . . , dσk).

Any connection on M × Rk is then of the form ∇ = d + A for some 1 k 1 A ∈ Ω (M; End(M × R )) = Ω (M; glk(R))

where glk(R) = Mk(R) is the space of k × k matrices with real entries (the Lie algebra of GLk(R) in fact). One can regard A as a matrix of 1-forms on M, and for a section σ ∈ Ω0(M; M × Rk), ∇σ is explicitly given by

k X ∇σ = dσ + Aijσj. j=1 Proposition 5.1.4. Let E → M be a vector bundle. Then (1) E → M admits a connection. (2) Any convex combination of connections is again a connection. (3) The space Conn(E) of all connections on E is an affine space modelled on Ω1(M; End(E)). Proposition 5.1.5. Let E → M be a vector bundle with connection ∇. Then there is a unique linear differential operator d∇ :Ωk(M; E) → Ωk+1(M; E) satisfying: (1) d∇σ = ∇σ for all σ ∈ Ω0(M; E). (2) If α ∈ Ωp(M) and ω ∈ Ωq(M; E), then d∇(α ∧ ω) = dα ∧ ω + (−1)pα ∧ d∇ω.

Pr p 0 Proof. Locally, ω can be written ω = i=1 ωi ⊗ σi for some ωi ∈ Ω (M) and σ ∈ Ω (M; E). Then if d∇ exists, it must satisfy

r r ∇ X p ∇ X p d ω = (dωi ⊗ σi + (−1) ωi ⊗ d σi) = (dωi ⊗ σi + (−1) ωi ⊗ ∇σi). i=1 i=1 So we may take this as the definition of d∇, or we may define d∇ globally by

p ∇ X i (d ω)(X0,...,Xp) = (−1) ∇Xi ω(X0,..., Xbi,...,Xp) i=1 X i+j + (−1) ω([Xi,Xj],X0,..., Xbi,..., Xbj,...,Xp), i

where Xb` denotes the deletion of X` from the given tuple. Now one can check that either definition gives an operator satisfying (1) and (2).

87 5.1 Connections on a Vector Bundle 5 Chern-Weil Theory

Definition. For a connection ∇, the operator d∇ :Ω•(M; E) → Ω•(M; E) is called the covariant exterior derivative with respect to ∇. Compare the properties of d∇ in Proposition 5.1.5 to those of d in the previous section. In particular, notice that d2 = 0, but this is not required for d∇. In general, (d∇)2 6= 0 but we can measure how far a given connection is from defining a de Rham cohomology theory by studying the value of (d∇)2. First, note that the composition ^  ^  ^ ^  p T ∗M ⊗ End(E) ⊗ q T ∗M ⊗ E ∼= p T ∗M ⊗ q T ∗M ⊗ (End(E) ⊗ E) ↓ ∧ ⊗ ev ^ p+q T ∗M ⊗ E defines a wedge operation ∧ :Ωp(M; End(E)) ⊗ Ωq(M; E) → Ωp+q(M; E). Proposition 5.1.6. For a connection ∇ on a vector bundle E → M, there is a unique 2 ∇ 2 p 2-form R∇ ∈ Ω (M; End(E)) satisfying (d ) ω = R∇ ∧ ω for all ω ∈ Ω (M; E). Proof. For p = 0, note that the assignment

Ω0(M; E) −→ Ω2(M; E) σ 7−→ (d∇)2σ is tensorial (in the sense of the remark above) since for all smooth functions f ∈ C∞(M),

d∇(d∇(fσ)) = d∇(df ∧ σ + f ∧ d∇σ) = d2f ∧ σ − df ∧ d∇σ + df ∧ d∇σ + f ∧ (d∇)2σ = f ∧ (d∇)2σ since d2f = 0. Therefore by that earlier remark, σ 7→ (d∇)2σ corresponds to a section 2 ∇ 2 R∇ ∈ Ω (M; End(E)). So we have (d ) σ = R∇ ∧ σ for all 0-forms σ. Now we show that p Pr this extends to all p-forms. Any ω ∈ Ω (M; E) may be written locally as ω = i=1 αi ∧ σi p 0 for some αi ∈ Ω (M) and σi ∈ Ω (M; E), so it suffices to prove the case when ω = α ∧ σ identically. In this case, we have

d∇(d∇ω) = d∇(d∇(α ∧ σ)) = d∇(dα ∧ σ + (−1)pα ∧ d∇σ) = d2α ∧ σ + (−1)p+1dα ∧ d∇σ + (−1)pdα ∧ d∇σ + α ∧ (d∇)2σ = α ∧ (d∇)2σ.

2 Definition. The 2-form R∇ ∈ Ω (M; End(E)) is called the curvature of the connection ∇. Example 5.1.7. When E = TM and ∇ is a particular connection called the Levi-Civita con- nection (or Riemannian connection; see Section 5.3), R∇ recovers the Riemannian curvature of the manifold M. (In the case of a surface, this is the Gaussian curvature.)

88 5.1 Connections on a Vector Bundle 5 Chern-Weil Theory

Example 5.1.8. Take E = M × Rk to be the trivial bundle and ∇ = d to be the trivial connection. Then a section of E may be viewed as a column vector   σ1  .  k σ =  .  : M −→ R σk and dσ is just the corresponding column vector of 1-forms   dσ1  .  dσ =  .  . dσk

∇ 2 Thus d σ = dσ and d σ is just the zero vector, so R∇ = 0. In general, R∇ may be regarded as an obstruction in Ω2(M; End(E)) to the connection ∇ being locally trivial on E.

Locally, any bundle looks like U ×Rk for U ⊆ M open, so any connection ∇ on the bundle k 1 looks like ∇ = d+A on U, where d is the trivial connection on U ×R and A ∈ Ω (U; glk(R)) is a matrix of 1-forms. If {e1, . . . , ek} is a basis of sections of E, then for each 1 ≤ i ≤ k, dei = 0 so k X ∇ei = Ajiej. j=1 Pk Here, A is called the connection matrix for ∇ on U. More generally, for σ = i=1 σi ∧ ei, we have

k X ∇σ = (d + A)σ = (d(σi ∧ ei) + A(σi ∧ ei)) i=1 k k ! X X = dσi ∧ ei + Aji(σi ∧ ej) i=1 j=1 k k ! X X = dσi + Ajiσj ∧ ei. i=1 j=1

Pk For curvature, R∇ locally looks like a matrix of 2-forms satisfying R∇ ∧ei = j=1(R∇)ji ∧ej. Thus for σ ∈ Ω0(M; E),

R∇ ∧ σ = (d + A)(d + A)σ = (d + A)(dσ + A ∧ σ) = d2σ + d(A ∧ σ) + A ∧ dσ + A ∧ (A ∧ σ) = dA ∧ σ − A ∧ dσ + A ∧ dσ + (A ∧ A) ∧ σ = (dA + A ∧ A)σ

where A∧A is the ‘wedge product of matrices’ done by standard matrix multiplication using wedge product for each product of entries. This shows that R∇ = dA + A ∧ A.

89 5.1 Connections on a Vector Bundle 5 Chern-Weil Theory

Example 5.1.9. Let M = H2 = {(x, y) ∈ R2 | y > 0} be the upper half-plane and E = T H2 its tangent bundle. There is a nontrivial connection ∇ = d + A on T H2 given by the matrix

 dy dx  − y − y A =   . dx dy y − y

Consider the vector fields ∂x and ∂y, the partial differential operators in the x and y direc- 1 0 tions, respectively, which can be viewed as the column vectors ∂ = and ∂ = . x 0 y 1 Then σ = ∂x is a section whose value under the connection ∇ is:     − dy − dx − dy 1 1 y y 1 y ∇∂x = (d + A) = d +   =   . 0 0 dx dy 0 dx y − y y

Applying this to the vector fields ∂x, ∂y gives:

 dy  ! − y 0 ∇∂ σ = ∂x = x  dx  1 y y  dy  1 ! − y − y and ∇∂ σ = ∂y = . y  dx  y 0 To compute curvature, it’s easy to show that A ∧ A = 0 and

 dx∧dy  0 − y2 dA = .  dx∧dy  y2 0

dx∧dy 2 It turns out that y2 is the standard volume form on H , so the curvature of the connection ∇ encodes some geometry of H2. Globally, we defined d∇ explicitly by

p ∇ X i (d ω)(X0,...,Xp) = (−1) ∇Xi ω(X0,..., Xbi,...,Xp) i=1 X i+j + (−1) ω([Xi,Xj],X0,..., Xbi,..., Xbj,...,Xp) i

p 0 ∇ for ω ∈ Ω (M; E). As a special case, if σ ∈ Ω (M; E) = Γ(E), then (d σ)(X) = ∇X σ for all vector fields X. Applying d∇ twice gives

∇ ∇ ∇ ∇ ∇ d (d σ)(X,Y ) = ∇X (d σ)(Y ) − ∇Y (d σ)(X) − (d σ)([X,Y ])

= ∇X ∇Y σ − ∇Y ∇X σ − ∇[X,Y ]σ. This proves:

90 5.1 Connections on a Vector Bundle 5 Chern-Weil Theory

Proposition 5.1.10. For any connection ∇ on E → M and any two vector fields X,Y ∈ Γ(TM), the curvature of ∇ along (X,Y ) equals

R∇(X,Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ]. Remark. The proof that d2 = 0 really relies on the fact that mixed partial derivatives commute. Therefore, in some sense R∇ is a measurement of the failure of ∇X and ∇Y to commute for general vector fields X and Y . This can be seen by the formula R∇ = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ].

Above, we showed that locally, R∇ = dA + A ∧ A where A is the connection matrix on a locally trivial coordinate chart U ⊆ M. We now investigate what happens under change of coordinates. Suppose {e1, . . . , en} and {f1, . . . , fn} are two local bases of E → M. Then there is a matrix of smooth functions g = (gij): U → GLn(R) such that each fi can be written n n X X −1 fi = gjiej or equivalently ej = (g )kjfk. j=1 k=1 If ∇ is a connection on E, let A and B be the two connection matrices with respect to these local bases, i.e. the matrices satisfying

n n X X ∇ei = Ajiej and ∇fi = Bjifj j=1 j=1 for each 1 ≤ i ≤ n. Then we have

n n ! X X Bjifj = ∇fi = ∇ gjiej j=1 j=1 n X = ((dgji)ej + gji∇ej) by axiom (2) for connections j=1 n n X X = (dgji)ej + gjiAkjek j=1 j,k=1 X −1 X −1 = (dgji(g )kjfk + gjiAki(g )`kf` j,k j,k,` X −1 X −1 = (dgki)(g )jkfj + gkiA`k(g )j`fj after reindexing j,k j,k,` X −1 −1 = ((g )jkdgki + (g )j`A`kgki)fj. j,k,`

More succinctly, B = g−1dg + g−1Ag. For curvature, we have the following proposition:

A B Proposition 5.1.11. Let R∇ and R∇ be the curvature forms for ∇ associated to each local basis and let g : M → GLn(R) be the change of basis function between the local bases. Then

B −1 A R∇ = g R∇g.

91 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

A B −1 −1 Proof. By the calculations above, R∇ = dA+A∧A, R∇ = dB+B∧B and B = g dg+g Ag. Putting these together,

B R∇ = dB + B ∧ B = d(g−1dg + g−1Ag) + (g−1dg + g−1Ag) ∧ (g−1dg + g−1Ag) = (−g−1dg g−1 ∧ dg + g−1 ∧ d2g − g−1dg g−1Ag + g−1dA ∧ g − g−1A ∧ dg) + (g−1dg ∧ g−1dg + g−1dg ∧ g−1Ag + g−1Ag ∧ g−1dg + g−1Ag ∧ g−1Ag) = g−1dAg + g−1A ∧ Ag −1 −1 A = g (dA + A ∧ A)g = g R∇g.

Remark. Globally, any vector bundle E → M is identified with the principal GLn(R)- bundle PE = F (E) → M (the frame bundle of Example 1.2.2). Further, the Lie group GLn(R) acts on its Lie algebra gln(R) = Mn(R) by the adjoint representation

GLn(R) × gln(R) −→ gln(R) −1 (g, A) 7−→ Adg(A) := gAg .

For each pair of vector fields X,Y on M, the curvature R∇(X,Y ) is a matrix, so R∇(−, −) may really be thought of as a gln(R)-valued function R∇ : PE → gln(R). Moreover, Propo- sition 5.1.11 shows that R∇ satisfies

R∇(X,Y )(p · g) = Adg−1 (R∇(X,Y )(p))

for all X,Y ∈ Γ(TM), p ∈ PE, g ∈ GLn(R). Therefore R∇ is a section of the associated bundle PE × gln(R).

5.2 Characteristic Classes from Curvature

In this section, we use the curvature of a connection on a vector bundle to produce some de Rham cohomology classes that encode the obstruction to d∇ being a differential (i.e. −1 squaring to zero). Suppose f : gln(R) → R is a function satisfying f(g Ag) = f(A) for all A ∈ gln(R) and g ∈ GLn(R). Let In(R) be the R-algebra of all such functions f which are polynomials.

Example 5.2.1. Examples of functions in In(R) are tr(A), det(A) and in general, any coefficient of the characteristic polynomial χ(A) = det(A − tI).

Fix a connection ∇ on a vector bundle E → M. For each f ∈ In(R), f(R∇) is a differential form on M which is independent of the choice of local bases by Proposition 5.1.11. 2k 2k Notice that if f is homogeneous of degree k, then f(R∇) ∈ Ω (M) = Ω (M; R). We can 2k similarly construct f(R∇) ∈ Ω (M; C) for a connection on any complex vector bundle. Recall the elementary symmetric functions from linear algebra, X σk(x1, . . . , xn) = xi1 ··· xik .

i1<...

92 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

Lemma 5.2.2. In(R) is a polynomial algebra generated by the functions σk : gln(R) → R, where σk(A) is the kth elementary symmetric function on the eigenvalues of A. The same things holds for In(C).

Proof. The proof is identical for R and C, so we will prove the real case only. Let Sn(R) ⊂ R[x1, . . . , xn] be the subalgebra of symmetric polynomials, i.e. polynomials invariant under permutation of the variables x1, . . . , xn. To define a map In(R) → Sn(R), take f ∈ In(R) and

consider its restriction to the subspace Tn ⊂ gln(R) of diagonal matrices. Then f|Tn may be thought of as a polynomial in the n entries along the diagonal of such a matrix. Moreover, f

is invariant under conjugation, so f|Tn is a symmetric polynomial in x1, . . . , xn, the variables

representing diagonal entries in Tn. Thus we have a map α : In → Sn(R), f 7→ f|Tn . ∼ From linear algebra, Sn(R) = R[σ1, . . . , σn], so because the coefficients of the characteristic polynomial χ = det((xij) − tI) are elementary symmetric functions, it follows that α is surjective. On the other hand, the same proof shows that α : In(C) → Sn(C) is well-defined and surjective. If f is in the kernel of α, it vanishes on all matrices similar to a diagonal matrix. In particular, f vanishes on all upper triangular matrices with distinct, nonzero diagonal entries, and by continuity, all upper triangular matrices. But since f is invariant under the symmetric group action, f vanishes on all matrices in GLn(R), so f = 0. This shows ∼ In(C) = Sn(C). Finally, since In(C) = In(R) ⊗ C and Sn(C) = Sn(R) ⊗ C, we get the same isomorphism in the real case.

j Lemma 5.2.3. For each j ≥ 1, let sj ∈ In(R) be defined by sj(A) = tr(A ). Then sj(A) = j j α1 + ... + αn where αi are the eigenvalues of A, and In(R) = R[s1, . . . , sn]. The same thing holds for In(C). Proof. Standard linear algebra.

Let ∇ be a connection on a vector bundle E → M. Then the connection R∇, viewed as a matrix of differential 2-forms, can be turned into a single differential form by applying any function in In(R) (or In(C)). Example 5.2.4. Suppose L → M is a complex line bundle with connection ∇. Then 2 2 R∇ ∈ Ω (M; End(L)) = Ω (M) since End(L) is a trivial line bundle. Thus R∇ is a 1 × 1 matrix, i.e. a single 2-form R∇ = (ω). Locally, R∇ = dA + A ∧ A = dA since A is a local 2 1-form, so we see that dR∇ = d A = 0. So here, R∇ = (ω) is a closed 2-form. The de Rham 2 class [R∇] = [ω] ∈ HdR(M) corresponds to the trace (or the determinant, since they are the same thing in the one-dimensional case). Moreover, any other connection ∇0 on L is of the form ∇0 = ∇ + a for some a ∈ Ω1(M; End(L)) = Ω1(M). If A0 is its local connection matrix, then A0 = A + a, so 0 R∇0 = dA = dA + da = R∇ + da

and as a consequence [R∇] = [R∇0 ]. This generalizes as follows. Lemma 5.2.5 (Bianchi’s Identity). Let ∇ be a connection with local connection matrix A A = (Aij) and curvature matrix R = R∇ = (Rij). Then dR = [R,A] = R ∧ A − A ∧ R.

93 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

Pn Proof. Write Rij = dAij + k=1 Aik ∧ Akj. Then

n 2 X dRij = d Aij + (dAik ∧ Akj − Aik ∧ dAkj) k=1 n X = [(Rik − Ai` ∧ A`k) ∧ Akj − Aik ∧ (Rkj − Ak` ∧ A`j)] k,`=1 n " n # X X = (Rik ∧ Akj − Aik ∧ Rkj) − (Ai` ∧ A`k ∧ Akj − Aik ∧ Ak` ∧ A`j) k=1 `=1 3 3 = [R,A]ij − ((A )ij − (A )ij) = [R,A]ij.

Since they agree in the ijth component for all 1 ≤ i, j ≤ n, dR = [R,A].

Theorem 5.2.6. Let E → M be a vector bundle of rank n and ∇ a connection on E. Then for any f ∈ In(R) of degree k,

2k (1) f(R∇) ∈ Ω (M) is a closed 2k-form, i.e. d(f(R∇)) = 0.

2k (2) The de Rham cohomology class [f(R∇)] ∈ HdR(M) is independent of ∇.

j Proof. (1) By Lemma 5.2.3, it suffices to show the case when f = sj = tr(A ) is the jth elementary symmetric power function. In this case, we have

j j d(sj(R∇)) = d(tr(R∇)) = tr(d(R∇)) by linearity of d j−1 j−2 j−1 = tr(dR∇ ∧ R∇ + R∇ ∧ dR∇ ∧ R∇ + ... + R∇ ∧ dR∇) j−1 j−2 j−1 = tr([R∇,A] ∧ R∇ + R∇ ∧ [R∇,A] ∧ R∇ + ... + R∇ ∧ [R∇,A]) by Lemma 5.2.5 j−1 j−1 = tr((R∇ ∧ A − A ∧ R∇) ∧ R∇ + ... + R∇ ∧ (R∇ ∧ A − A ∧ R∇)) j j = tr(R∇ ∧ A − A ∧ R∇) after cancelling terms j = tr([R∇,A]) = 0 by linearity again. Therefore [sj(R∇)] is closed. (2) If ∇, ∇0 are two connections on E with curvatures R,R0, respectively, then consider the product bundle Ee = E × [0, 1] → M × [0, 1]. A sections ˜ ∈ Γ(M × [0, 1], Ee) is of the form (x, t) 7→ s˜(x, t) ∈ Ex × {t} = Ex, so in facts ˜(·, t) is a section of E for each fixed t ∈ [0, 1]. Write T (M × [0, 1]) = TM ⊕ Rh∂ti. One can prove that there is a unique connection ∇e on Ee satisfying the following conditions: ˆ Ifs ˜ ∈ Γ(M × [0, 1], Ee) is independent of t, then ∇e ∂t s˜ = 0. ˆ 0 If X ∈ TM ⊆ T (M × [0, 1]), then ∇e X s˜ = (1 − t)∇X (˜s|M×{t}) + t∇X (˜s|M×{t}).

Observe that if f ∈ In(R), then

f(R )| = f(R ) and f(R )| = f(R 0 ). ∇e M×{0} ∇ ∇e M×{1} ∇

94 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

Let i0, i1 : M,→ M × [0, 1] be the inclusions of M × {0} and M × {1}, respectively. Then

[f(R )] = i∗[f(R )| ] = i∗[f(R )] ∇ 0 ∇e M×{0} 0 ∇e = i∗[f(R )] since i , i are homotopic 1 ∇e 0 1 ∗ = i1[f(R∇0 )|M×{1}] = [f(R∇0 )].

Therefore [f(R∇)] is independent of ∇.

Corollary 5.2.7. For any f ∈ In(R) (or In(C) in the complex case), the assignment

• Vectn(M) −→ HdR(M)

E 7−→ [f(R∇)],

for any connection ∇ on E, is a characteristic class for the category of smooth real (or complex) manifolds.

Proof. We need only check [f(R∇)] is natural. Suppose h : M → N is a smooth map, E → N is a vector bundle and Ee = h∗E is the pullback bundle, so that we have a diagram h˜ Ee E

h M N

If {e1, . . . , en} is a local basis of E on an open neighborhood U ⊆ N, let {e˜1,..., e˜n} be the induced local basis of Ee on h−1(U) ⊆ M. Then for a connection ∇ on E, define the induced connection ∇e = h∗∇ locally on h−1(U) by

∇e X e˜j = ∇h∗(X)ej

−1 for any X ∈ Γ(h (U),TM), where h∗ denotes the pushforward: (h∗(X))x(v) = (d∗h(X))h(x)(v). One now checks that ∇e is a connection on Ee, and it clear that for any f ∈ In(R) or In(C), we have h∗[f(R )] = [h∗f(R )] = [f(R )]. ∇ ∇ ∇e Remark. There is a more general notion of a connection on a principal G-bundle, with the special cases G = O(n) and G = U(n) recovering the theory described in this chapter. The replacement for In(R) and In(C) in that setting is the space of polynomial functions on the Lie algebra g = Lie(G) which are invariant under Ad(G).

2 ∼ 1 2 Example 5.2.8. Let M = S = CP = C ∪ {∞} and let E = TS be the tangent bundle, which is a complex line bundle. Viewing S2 in stereographic coordinates, we can write down a connection ∇ on TS2 by specifying a 1 × 1 connection matrix A = (ω), where −2 ω = (x dx + y dy + i(x dy − y dx)). 1 + x2 + y2

95 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

2 ∼ This defines ∇ locally on the coordinate chart S r{∞} = C, but under the natural transition   2 ∼ x y function from this chart to S r {0} = C given by (x, y) 7→ x2+y2 , x2+y2 , the definition of A is preserved. Hence A defines a global connection ∇ on TS2. By Example 5.2.4, R∇ = dA = (dω), and we have  −2  dω = d ∧ (x dx + y dy + i(x dy − y dx)) 1 + x2 + y2  −2  + ∧ d(x dx + y dy + i(x dy − y dx)) 1 + x2 + y2  4x 4y  = dx + dy ∧ (x dx + y dy + i(x dy − y dx)) (1 + x2 + y2)2 (1 + x2 + y2)2  −2  + ∧ (dx ∧ dx + dy ∧ dy + i(dx ∧ dy − dy ∧ dx)) 1 + x2 + y2 4 = (x2 dx ∧ dx + xy dx ∧ dy + i(x2 dx ∧ dy − xy dx ∧ dx)) (1 + x2 + y2)2 4 + (xy dy ∧ dx + y2 dy ∧ dy + i(xy dy ∧ dy − y2 dy ∧ dx)) (1 + x2 + y2)2 2i − (dx ∧ dy − dy ∧ dx) since dx ∧ dx = dy ∧ dy = 0 1 + x2 + y2 4 4i(1 + x2 + y2) = (xy + ix2 − xy + iy2) dx ∧ dy − dx ∧ dy (1 + x2 + y2)2 (1 + x2 + y2)2 −4i = dx ∧ dy. (1 + x2 + y2)2

2 2 Now to compute the cohomology class [R∇] ∈ HdR(S ; C), recall that by de Rham’s theorem, 2 2 ∼ 2 R HdR(S ; C) = C via integration over the fundamental class: α 7→ hα, [S ]i = S2 α. Therefore for α = dω, we have Z Z Z −4i dω = dω = 2 2 2 dx dy S2 CP 1r{∞} C (1 + x + y ) Z 2π Z ∞ −4ir Z 2π  2i r=∞ = 2 2 dr dθ = 2 dθ 0 0 (1 + r ) 0 1 + r r=0 Z 2π = −2i dθ = −4πi. 0

So [R∇] = −4πi. Compare this to the Chern class (which is equal to the Euler class in this case) using Example 3.6.7:

1 1 2 2 c(CP ) = c(T CP ) = (1 + x) = 1 + 2x + x ,

1  i  1 so c1(CP ) = 2. This shows that 2π R∇ = c1(CP ). Definition. For a complex vector bundle E → M over a complex manifold M of dimension dR 2j n, the de Rham-Chern classes of E are the classes cj (E) ∈ HdR(M; C) appearing as the

96 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory jth coefficient of the total de Rham-Chern class  i  cdR(E) := det 1 + R 2π ∇ where ∇ is any connection on E. Explicitly, for 0 ≤ j ≤ n, " #  i j cdR(E) = σ (R ) j 2π j ∇ where σj ∈ In(C) is the jth elementary symmetric function. dR 1 1 2 1 ∼ Example 5.2.9. Example 5.2.8 shows that c1 (T CP ) = c1(T CP ), viewed in HdR(CP ; C) = H2(CP 1; Z) ⊗ C. To extend this identification of Chern classes to any manifold and any vector bundle, it will suffice to check the axioms for Chern classes hold for cdR(E). Note that naturality is guaranteed by Corollary 5.2.7. Given any complex vector bundle E → M with connection ∇, we can define associated connections on various related bundles such as E∗,E⊗n, Vk E, etc. For example, a connection ∇∗ on the dual bundle E∗ should satisfy the following product rule for all σ ∈ Γ(M,E), α ∈ Γ(M,E∗): d(α(σ)) = (∇∗α)(σ) + α(∇σ). In fact, we can take this as a definition of ∇∗, so that for X ∈ Γ(TM),

∗ (∇X α)(σ) = d(α(σ))(X) − α(∇X σ). Similarly, for E,F → M with connections ∇E, ∇F , we can define ∇ on E ⊗ F as follows, for σ1 ∈ Γ(M,E), σ2 ∈ Γ(M,F ):

∇(σ1 ⊗ σ2) = ∇σ1 ⊗ σ2 + σ1 ⊗ ∇σ2.

1 2 In the case of two complex line bundles L1,L2 → M with connections ∇ , ∇ , let e1 be a local parameter for L1, e2 a local parameter for L2 and let

1 2 ∇ e1 = A1e1 and ∇ e2 = A2e2 define the local connection matrices. Then the connection ∇ on L1 ⊗ L2 satisfies

1 2 ∇(e1 ⊗ e2) = ∇ e1 ⊗ e2 + e1 ⊗ ∇ e2 = A1e1 ⊗ e2 + e1 ⊗ A2e2 = (A1 + A2)(e1 ⊗ e2).

That is, ∇ has local connection matrix A = A1 + A2. For curvature, we compute:

R∇ = dA + A ∧ A = dA = d(A1 + A2) = dA1 + dA2 = R∇1 + R∇2 .

This shows [R∇] = [R∇1 ] + [R∇2 ], which implies:     dR i i c (L ⊗ L ) = R = (R 1 + R 2 ) 1 1 2 2π ∇ 2π ∇ ∇     i i dR dR = R 1 + R 2 = c (L ) + c (L ). 2π ∇ 2π ∇ 1 1 1 2

97 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

1 1 dR Proposition 5.2.10. For the tautological line bundle γ1 = γ1 on CP , c1 (γ1) = ±c1(γ1) 2 1 in HdR(CP ; C).

Proof. By Lemma 3.8.3, complex line bundles over CP 1 are classified by H2(CP 1; Z) via L 7→ c1(L), and this assignment is additive with respect to ⊗. We know from Example 3.6.7 2 1 1 that x = c1(γ1) is a generator in HdR(CP ; C) and from Example 5.2.8, c1(T CP ) = ±2x. 1 ∼ ∗ Thus T CP = ±(γ1 ⊗ γ1), where for a line bundle L, −L = L . Moreover, by the work above, dR dR dR 1 2c1 (γ1) = c1 (γ1 ⊗ γ1) = ±c1 (T CP ) = ±2x dR so c1 (γ1) = ±x.

2 1 By convention, we choose the “positive generator” in HdR(CP ; C) to be the x such that 1 1 ∼ ∗ c1(T CP ) = 2x. In this case, T CP = (γ1 ⊗ γ1) , so c1(γ1) = −x. It remains to establish the Whitney formula for de Rham-Chern classes.

Proposition 5.2.11. Let E,F → M be complex vector bundles. Then cdR(E ⊕ F ) = cdR(E)cdR(F ).

E F Proof. Let ∇ and ∇ be connections on E and F , respectively. For σ1 ∈ Γ(M,E) and σ2 ∈ Γ(M,F ), the formula E F ∇(σ1, σ2) = (∇ σ1, ∇ σ2) defines a connection ∇ on E ⊕ F , and the corresponding covariant exterior derivative d∇ is ∇ ∇E ∇F p q of the form d (ω1, ω2) = (d ω1, d ω2) for ω1 ∈ Ω (M; E), ω2 ∈ Ω (M; F ). In particular, the curvature of ∇ is given by R∇(σ1, σ2) = (R∇E σ1,R∇F σ2). On the level of matrices, R∇ = R∇E ⊕ R∇F , so       dR i i i dR dR c (E ⊕ F ) = det 1 + R = det 1 + R E det 1 + R F = c (E)c (F ). 2π ∇ 2π ∇ 2π ∇

Theorem 5.2.12. If E → M is a complex vector bundle, then cdR(E) is equal to the image • • ∼ • of c(E) in HdR(M; C) via the de Rham isomorphism H (M; Z) ⊗ C = HdR(M; C).

Recall that any real vector bundle E → M has an associated complex bundle EC → M given by tensoring every fibre with C. The Pontrjagin class of E was defined to be

n p(E) = 1 − c2(EC) + ... + (−1) c2n(EC). Corollary 5.2.13. If E → M is a real vector bundle, then the total Pontrjagin class of E is  1  p(E) = det 1 + R ∈ H• (M) 2π ∇ dR

where ∇ is any connection on E, and the jth Pontrjagin class pj(E) is the jth homogeneous component of this determinant.

98 5.2 Characteristic Classes from Curvature 5 Chern-Weil Theory

Example 5.2.14. Viewing M = S2 in Example 5.2.8 as a real 2-manifold, one can compute

−4  0 −dx ∧ dy Rreal = . ∇ (1 + x2 + y2)2 dx ∧ dy 0

Therefore p(TS2) = 1 so the top Pontrjagin class is trivial.

Let E → M be a real vector bundle equipped with a metric. Recall that this is equivalent to E having structure group O(n), where n is the dimension of the fibre, and since GLn(R) retracts to O(n), every real bundle admits a metric.

Definition. A connection ∇ on E → M is compatible with the metric h·, ·i on E if for any sections v, w ∈ Γ(M,E),

dhv, wi = h∇v, wi + hv, ∇wi.

Remark. A choice of metric on a bundle E → M is equivalent to a choice of section σ ∈ Γ(M,E∗ ⊗ E∗) satisfying

ˆ h(v, w) = h(w, v) for all v, w ∈ Γ(M,E);

ˆ h(v, v) ≥ 0 for all v ∈ Γ(M,E), with h(v, v) = 0 if and only if v = 0.

Then a connection ∇ is compatible with the metric if and only if ∇∗h = 0, where ∇∗ is the connection induced by ∇ on E∗ ⊗ E∗.

Lemma 5.2.15. Every real vector bundle with a metric admits a compatible connection.

Proposition 5.2.16. Suppose E → M is a vector bundle with a metric and a compatible connection ∇ and let {e1, . . . , en} be a local orthonormal basis for E. Then the connection matrix A for ∇ with respect to the ej is skew-symmetric, i.e. Aij = −Aji.

Proof. An orthonormal basis satisfies hei, eji = δij for all i, j, so the compatibility condition gives us

0 = dδij = dhei, eji = h∇ei, eji + hei, ∇eji n X = (hAikek, eji + hei,Ajkeji) k=1 n X = (Aikδkj + Ajkδik) k=1

= Aij + Aji.

Therefore A is skew.

Remark. The set o(n) of skew-symmetric n × n skew-symmetric matrices is precisely the Lie algebra of O(n). A compatible connection is sometimes called an O(n)-connection, and there is a more general theory of G-connections as hinted at earlier.

99 5.3 The Levi-Civita Connection 5 Chern-Weil Theory

Corollary 5.2.17. The curvature of a compatible connection ∇ on a vector bundle E → M 2 is an element R∇ of Ω (M; Endskew(E)), where Endskew(E) denotes the subbundle of End(E) given by skew-symmetric operators on each fibre.

Proof. Locally, R∇ = dA + A ∧ A by Example 5.1.8, so for each 1 ≤ i, j ≤ n,

n n X X (R∇)ij = dAij + Aki ∧ Ajk = −dAji − Akj ∧ Aik = −(R∇)ji. k=1 k=1

Corollary 5.2.18. For any symmetric polynomial f ∈ In(R) of odd degree k and any con- 2k nection ∇ on E → M, [f(R∇)] = 0 in HdR(M; R).

Proof. By Theorem 5.2.6, [f(R∇)] is independent of ∇ so we may choose ∇ compatible with k the metric, so that by Corollary 5.2.17, R∇ is skew. In the case f = sk, sk(A) = tr(A ) and k if A is skew, so is A . Therefore sk(R∇) = 0 and by Lemma 5.2.3, this proves it for any f ∈ In(R).

1
2j 4j Corollary 5.2.19. The Pontrjagin classes pj(E) = − 2π σ2j(R∇) ∈ HdR(M; R) are triv- ial for j odd.

Remark. The Euler class also has an interpretation in terms of curvature. Let so(n) be the space of n × n skew-symmetric matrices, which is equivalently the Lie algebra of SO(n). (In fact, since SO(n) is a connected component of O(n), so(n) = o(n).) The Pfaffian is a skew-symmetric polynomial PF ∈ I2n(so(2n)) defined by

1 X P f(x ) = x x ··· x . ij 2nn! σ(1)σ(2) σ(3)σ(4) σ(2n−1)σ(2n) σ∈S2n

Then for any compatible connection ∇, the matrix R∇ is skew, so P f(R∇) defines a de Rham cohomology class which is independent of ∇. One can prove that: 1 e(E) = [P f(R )] in H2n(M; ). (2π)n ∇ dR R

dR 2 It’s easy to see that this class satisfies e(E) = cn (E) and e(E) = pn(ER) as expected.

5.3 The Levi-Civita Connection

Let (M, g) be a Riemannian manifold.

Lemma 5.3.1. If {e1, . . . , en} is a local orthonormal basis for TM with respect to g and ∗ {θ1, . . . , θn} is the dual basis for T M, then there is a unique matrix A = (Aij) of differential 1-forms satisfying:

(i) Aij = −Aji (that is, A is skew-symmetric);

100 5.3 The Levi-Civita Connection 5 Chern-Weil Theory

n X (ii) dθi = − Aji ∧ θj. j=1 Proof. In general, we can write X X dθi = aijkθj ∧ θk = 2 aijkθj ∧ θk j,k j

for functions aijk satisfying aijk = −aikj. Being 1-forms, the Aij should be of the form Aij = Pn k=1 bijkθk for some coefficients bijk. Notice that (i) is equivalent to having bijk = −bikj, while (ii) is equivalent to any of the following: X X X dθi = − Aji ∧ θj ⇐⇒ dθi = − bijkθk ∧ θj = (bijk − bikj)θj ∧ θk j,k j,k j

⇐⇒ 2aijk = bijk − bikj

⇐⇒ 2ajik = bjik − bjki

⇐⇒ 2akji = bkji − bkij.

One can combine the latter three equations to obtain bijk = aijk − ajik + akji, so the bijk are all defined and A satisfies (i) and (ii).

Definition. The Levi-Civita connection on a Riemann manifold M is the unique con- nection ∇ on TM such that for any orthonormal basis {e1, . . . , en} with matrix of 1-forms A = (Aij) satisfying the conditions of Lemma 5.3.1,

n X ∇ei = Aijej j=1

for all 1 ≤ i ≤ n.

∗ ∗ ∗ Pn Remark. By construction, the dual connection ∇ on T M satisfies ∇ θi = − j=1 Aji ∧θj if {θi} is the dual basis to {ei}. Theorem 5.3.2. Let ∇ be the Levi-Civita connection on TM. Then the composition

∇∗  ^  Γ(M,T ∗M) −→ Γ(M,T ∗M ⊗ T ∗M) → Γ M, 2 T ∗M

is equal to the exterior derivative.

∗ Pn Proof. Locally, if α ∈ Γ(M,T M) can be written α = i=1 αiθi, then

n n ∗ X X ∇ α = ∇(αiθi) = (dαi ⊗ θi + αi ⊗ ∇θi) i=1 i=1 n n ! X X = dαi ⊗ θi − αi Aji ⊗ θj i=1 j=1

101 5.3 The Levi-Civita Connection 5 Chern-Weil Theory and the image of this element in Γ (M, V2 T ∗M) is

n n ! n X X X dαi ∧ θi − αi Aji ∧ θj = (dαi ∧ θi + αi ∧ dθi) i=1 j=1 i=1 n X = d(αiθi) = dα. i=1

Remark. The Levi-Civita connection is uniquely characterized by the following two prop- erties:

(1) ∇ is compatible with the inner product h·, ·i = g(·, ·) on M, i.e. for all X,Y ∈ Γ(M,TM), dhX,Y i = h∇X,Y i + hX, ∇Y i.

(2) ∇ is torsion-free, i.e. ∇X Y − ∇Y X − [X,Y ] = 0 for any vector fields X,Y .

102 6 Index Theory

6 Index Theory

6.1 Differential Operators

Suppose M is a Riemannian manifold, E → M is a vector bundle with connection ∇ and σ : T ∗M ⊗ E → F is any morphism of vector bundles over M. Then the composition D = σ ◦ ∇ : Γ(M,E) −→∇ Γ(M,T ∗M ⊗ E) −→σ Γ(M,F ) is an example of a first order linear differential operator E → F . Example 6.1.1. The Levi-Civita connection ∇ on TM (see Section 5.3) determines the exterior derivative ∇∗  ^  d : Γ(M,T ∗M) −→ Γ(M,T ∗M ⊗ T ∗M) → Γ M, 2 T ∗M by Theorem 5.3.2. Suppose ∇0 is another connection on E given by ∇ = ∇0 + A for some A ∈ Γ(M,T ∗M ⊗ End(E)). Then for any bundle map σ : T ∗M ⊗ E → F , the composition σ ◦ A is also a bundle map, so D = σ ◦ ∇ = σ ◦ (∇0 + A) = σ ◦ ∇0 + σ ◦ A. We can view σ ◦ A as a 0th order linear differential operator E → F . Thus we make the following definition. Definition. Let E,F → M be vector bundles and ∇ a connection on E.A first order linear differential operator E → F is any operator D : Γ(M,E) → Γ(M,F ) of the form D = σ◦∇+K where σ : T ∗M ⊗E → F and K : E → F are bundle maps. The corresponding ∗ map σ = σD : Γ(M,T M ⊗ E) → Γ(M,F ) is called the symbol of D. Example 6.1.2. Extending the above, the exterior derivative d :Ωk(M) → Ωk+1(M) is a first order linear differential operator given by d = σ ◦ ∇ where ∇ is the extension of the Levi-Civita connection to Γ M, Vk T ∗M
→ Γ M,T ∗M ⊗ Vk T ∗M
and σ is the bundle map  ^  ^  σ :Γ M,T ∗M ⊗ k T ∗M −→ Γ k+1 T ∗M α ⊗ β 7−→ α ∧ β.

∞ For a point p ∈ M, let Ip ⊂ C (M) be the ideal of smooth functions vanishing at p. Definition. A kth order linear differential operator D from E → F is a map D : Γ(M,E) → Γ(M,F ) such that k is the smallest integer with the property that D(fσ)(p) = 0 k+1 for all p ∈ M, σ ∈ Γ(E) and f ∈ Ip . Example 6.1.3. A 0th order linear differential operator is just a bundle map E → F . ∞ Indeed, take any g ∈ C (M) and p ∈ M, so that g = g(p) + f for some f ∈ Ip. Then a 0th order operator D satisfies D(gσ)(p) = D(g(p)σ + fσ)(p) = g(p)D(σ)(p) + D(fσ)(p) = g(p)D(σ)(p). Thus D is linear over C∞(M), so it corresponds to a bundle map E → F .

103 6.1 Differential Operators 6 Index Theory

Definition. Let D : Γ(M,E) → Γ(M,F ) be a kth order linear differential operator. The symbol of D is the bundle map

k ∗ σD : Sym (T M) −→ Hom(E,F )

∗ defined as follows. For p ∈ M and ω1, . . . , ωk ∈ Tp M, choose functions f1, . . . , fk ∈ Ip such that dpfi = ωi for 1 ≤ i ≤ k. For a section ψ ∈ Γ(M,E), set 1 σ (ω ⊗ · · · ⊗ ω )(ψ)(p) = D(f ··· f ψ)(p). D 1 k k! 1 k (Technically, this defines a map on sections Γ(M, Symk(T ∗M)) → Γ(M, Hom(E,F )) which ∞ k ∗ is linear over C (M), thus corresponding to σD : Sym (T M) → Hom(E,F ).)

Lemma 6.1.4. The definition of σD is independent of the choice of f1, . . . , fk.

Proof. Clearly the order of the f1, . . . , fk does not matter, so suppose we have these as in 2 the definition above and we also have g1 ∈ Ip with dpg1 = ω1 = dpf1. Then f1 − g1 ∈ Ip , so

D(f1 ··· fkψ)(p) = D(g1f2 ··· fkψ)(p) + D((f1 − g1)f2 ··· fkψ)(p)

= D(g1f2 ··· fkψ)(p) + 0

k+1 since D is a kth order operator and (f1 − g1)f2 ··· fk ∈ Ip .

Remark. The symbol σD is uniquely determined by the values of σD(ω ⊗ · · · ⊗ ω) for each d ω ∈ T ∗M, since Sym (Rn) may be identified as the space of homogeneous polynomials of degree d on Rn. Definition. A kth order linear differential operator D : Γ(M,E) → Γ(M,F ) is called ellip- ∗ tic if its symbol is an isomorphism σD(ω ⊗ · · · ⊗ ω): Ep → Fp for all p ∈ M, ω ∈ Tp M. Example 6.1.5. Take k = 1 and let ∇ : Γ(M,E) → Γ(M,T ∗M ⊗ E) be a connection on a vector bundle E (which is a first order operator . For p ∈ M, f ∈ Ip and ψ ∈ Γ(M,E), we have ∇(fψ)(p) = dpf ⊗ ψ(p) + f(p)∇ψ(p) = dpf ⊗ ψ(p).

This shows that for the first order operator D = σ◦∇, the symbol σD is given by σD(ω)(ψ) = ω ⊗ ψ.

Example 6.1.6. More generally, if d :Γ M, Vk T ∗M
→ Γ M, Vk+1 T ∗M
is the exterior derivative, we know d = ∧ ◦ ∇ where ∧ : α ⊗ β 7→ α ∧ β. For f ∈ C∞(M) and α ∈ Vk ∗
Γ M, T M , d(fα) = df ∧ α + f dα, so if f ∈ Ip and ω = dpf, the symbol of d is given by σd(ω)(α) = ω ∧ α. Vk−1 ∗ Notice that if β ∈ T M such that α = ω ∧ β, then σd(ω)(α) = ω ∧ ω ∧ β = 0, so d is not elliptic (unless for example dim M = 1). In general, a linear differential operator D : Γ(M,E) → Γ(M,F ) can’t be elliptic unless E and F are vector bundles of the same dimension, so the exterior derivative d :Ωk(M) → Ωk+1(M) is certainly not elliptic in general.

104 6.1 Differential Operators 6 Index Theory

Example 6.1.7. Suppose D is a kth order linear differential operator acting on functions on Rn (every linear differential operator is locally so). Then X ∂j D = ai1,...,ij (p) . ∂xi ··· ∂xi j≤k 1 j i1<...

n n ∗ n So if p ∈ R and f1, . . . , fk vanish at p, then for ψ : R → R and 1-forms ω1, . . . , ωk ∈ Tp R , we have 1 σ (ω ⊗ · · · ⊗ ω )(ψ)(p) = D(f ··· f ψ)(p) D 1 k k! 1 k j 1 X ∂ (f1 ··· fkψ) = ai1,...,ij (p) k! ∂xi ··· ∂xi j≤k 1 j p i1<...

1 X ∂fσ(1) ∂fσ(k) = ai1,...,ik (p) ··· ψ(p) k! ∂xi1 ∂xik i1<...

i1,...,ik

∂2 ∂2 ∂2 ∂2 2 For example, if D = a11 ∂x2 + a12 ∂x∂y + a21 ∂y∂x + a22 ∂y2 is a second order operator on R , 2 2 then σD(ω ⊗ η) = a11ω + (a12 + a21)ω ∧ η + a22η . Substituting x = ω and y = η, we may 2 view σD as a function on R :

2 2 σD(x, y) = a11x + (a12 + a21)xy + a22y .

2 2 For D to be an elliptic operator, σD must be an isomorphism, so a11x +(a12+a21)xy+a22y 6= 2 0 for all (x, y) 6= (0, 0) in R . In the special case that the aij are constant functions, the 2 2 above holds if and only if a11x + (a12 + a21)xy + a22y = c describes an ellipse for all c 6= 0. (Hence the name “elliptic”.)

n Pn ∂2 Example 6.1.8. On M = R , let ∆ = − i=1 2 be the Laplacian operator. Then ∆ is a ∂xi second order linear differential operator with symbol

n X 2 2 σD(ω) = − ωi = −|ω| · id i=1

if ω = ω1 ⊗ · · · ⊗ ωn. Thus σ∆(ω) 6= 0 whenever ω 6= 0, so ∆ is elliptic. Definition. Let (M, g) be an oriented Riemannian manifold of dimension n with induced inner product h·, ·i on each Vk T ∗M, k ≥ 1. Then the volume form of (M, g) is the unique n n-form volg ∈ Ω (M) such that | volg | = 1 pointwise on M and volg induces the given orientation on M as a section of the orientation bundle.

105 6.1 Differential Operators 6 Index Theory

∗ Remark. Locally, volg = ε1 ∧ · · · ∧ εn where {εi} is an oriented, orthonormal basis for T M. Equivalently, if x1, . . . , xn are local coordinates on M, then q volg = det(g(xi, xj)) dx1 ∧ · · · ∧ xn.

Definition. For a Riemannian manifold (M, g), the Hodge star operator on M is the bundle map ? : Vk T ∗M → Vn−k T ∗M defined uniquely by

α ∧ ?β = hα, βi volg

Vk ∗ ∗ for all α, β ∈ T M. Equivalently, for {ε1, . . . , εn} an oriented normal basis for T M, ? may be defined by σ ?(εi1 ∧ · · · ∧ εik ) = (−1) εj1 ∧ · · · ∧ εjn−k where {i1, . . . , ik} ∪ {j1, . . . , jn−k} = {1, . . . , n} and σ denotes the sign of the permutation (i1, . . . , ik, j1, . . . , jn−k) of this set.

n Example 6.1.9. When M = R with dual basis {ε1, ε2, ε3, ε4},

?(ε1 ∧ ε3) = −ε2 ∧ ε4,?(ε2) = −ε1 ∧ ε3 ∧ ε4, etc.

Remark. On any compact, oriented, Riemannian n-manifold (M, g), the integral of an n- form can be computed locally on compatible oriented coordinate patches by ordinary iterated integration. For instance, the integral of a function f on M may be taken with respect to the volume form volg on M and we have Z Z Z f := f volg = ?f volg M M M Definition. Let E,F → M be vector bundles with inner product on a closed, oriented, Riemannian manifold (M, g) of dimension n and let D : Γ(M,E) → Γ(M,F ) be a linear differential operator. The formal adjoint of D is the linear differential operator D∗ : Γ(M,F ) → Γ(M,E) defined uniquely by Z Z hDϕ, ψi = hϕ, D∗ψi M M for all compactly supported sections ϕ ∈ Γ(M,E), ψ ∈ Γ(M,F ).

Lemma 6.1.10. The formal adjoint of a linear differential operator exists. Moreover, if D is a kth order operator then so is D∗.

Proof. Existence follows from the fact that the spaces Γcpt(M,E), Γcpt(M,F ) of compactly supported sections are Hilbert spaces. The second statement is obvious.

106 6.1 Differential Operators 6 Index Theory

Example 6.1.11. Let d : Vk−1 T ∗M → Vk T ∗M be the exterior derivative on an n-manifold M. Then for α ∈ Ωk−1(M) and β ∈ Ωk(M), we have ?β ∈ Ωn−k(M) so by Stokes’ theorem, Z Z 0 = d(α ∧ ?β) = (dα ∧ ?β + (−1)k−1α ∧ d(?β)) M M Z Z = hdα, βi + (−1)k−1 (−1)(k−1)(n−k+1)α ∧ ? ? d(?β) M M Z = hdα, βi + (−1)(k−1)(n−k)hα, ?d(?β)i
M Z = (−1)n(k−1)+1 hα, ?d(?β)i. M This shows that the formal adjoint of d is d∗ = (−1)n(k−1)+1 ? d?. Definition. Let X ∈ TM be a vector field. The interior product (or contraction) of X Vk ∗ Vk−1 ∗ is the bundle map iX : T M → T M defined by

(iX ω)(Y1,...,Yk−1) = ω(X,Y1,...,Yk−1). Lemma 6.1.12. Let X ∈ TM be a vector field. Then ∼ ∗ (a) iX is formally adjoint to ∧ with respect to the identification TM = T M; explicitly, ∗ If αX ∈ T M is the functional αX (Y ) = hX,Y i, then

hiX ω, ηi = hω, αX ∧ ηi for all ω ∈ Ωk(M), η ∈ Ωk−1(M).

k k (b) iX is a derivation: iX (ω ∧ η) = iX ω ∧ η + (−1) ω ∧ iX η if ω ∈ Ω (M).

∗ (c) With respect to an orthonormal basis {e1, . . . , en} for TM, the adjoint d of the exterior derivative can be written n ∗ X d = − iej ◦ ∇ej j=1 where ∇ is the Levi-Civita connection on TM. Proof. Straightforward from the definitions. Proposition 6.1.13. The operator D = d + d∗ :Ω•(M) → Ω•(M) is elliptic. Proof. The symbol of D = d + d∗ is defined for each ψ ∈ Γ(M,E) by

σD(ω)(ψ) = ω ∧ ψ − iω# ψ # where ω is the tangent vector dual to ω, i.e. αω# = ω. Now for all ψ, we have 2 [σD(ω)] (ψ) = ω ∧ (ω ∧ ψ − iω# ψ) − iω# (ω ∧ ψ − iω# ψ) 2 = 0 − ω ∧ iω# ψ − iω# (ω ∧ ψ) + 0 since (iω# ) = 0 2 = −iω# ω ∧ ψ = −|ω| ∧ ψ. 2 2 2 So [σD(ω)] = −|ω| ∧id which is an isomorphism unless ω = 0. Hence σD is an isomorphism, so σD is as well.

107 6.2 Hodge Decomposition 6 Index Theory

Recall from Example 6.1.8 that the Laplacian operator ∆ has symbol −|ω|2 ∧ id. For this reason, a second order linear differential operator D : Γ(M,E) → Γ(M,F ) with symbol 2 σD(ω) = −|ω| ∧ id is called a generalized Laplacian operator. Definition. A first order linear differential operator D : Γ(M,E) → Γ(M,F ) is a Dirac- type operator if D2 = D ◦ D is a generalized Laplacian operator. Example 6.1.14. By Proposition 6.1.13, D = d + d∗ is a Dirac-type operator. The gener- alized Laplacian operator ∆ = (d + d∗)2 = dd∗ + d∗d is called the Laplace-Beltrami operator on M. (Notice that it depends on the choice of metric on M.) Definition. A k-form ω ∈ Ωk(M) is harmonic if ∆ω = 0. The vector space of all harmonic k-forms is written Hk(M). Lemma 6.1.15. Suppose M is a closed, Riemannian manifold. Then a differential form ω ∈ Ωk(M) is harmonic if and only if it is both closed (dω = 0) and co-closed (d∗ω = 0). Proof. ( ⇒ = ) is trivial. ( =⇒ ) Suppose ∆ω = 0. Then Z 0 = h∆ω, ωi M Z Z = hdd∗ω, ωi + hd∗dω, ωi M M Z Z = hd∗ω, d∗ωi + hdω, dωi since M is closed M M = ||d∗ω||2 + ||dω||2 where || · || is the operator norm. This is only possible if ||d∗ω||2 = ||dω||2 = 0, so both dω and d∗ω are zero.

6.2 Hodge Decomposition

The elliptic condition on a linear differential operator D is really an algebraic condition: the symbol σD is a linear map on vector spaces, so isomorphism has an algebraic meaning. The goal of index theory is to interpret this information topologically. Definition. A linear differential operator D : Γ(M,E) → Γ(M,F ) is Fredholm if im D ⊆ Γ(M,F ) is closed (with respect to the norm) and ker D and coker D are both finite di- mensional vector spaces. The index of a Fredholm operator is defined to be ind(D) = dim ker D − dim coker D. There is also an analytic component to index theory, as the vector spaces in question, ker D = {f ∈ Γ(M,E) | Df = 0} and coker D = ker(D∗) = {g ∈ Γ(M,F ) | D∗g = 0}, are spaces of homogeneous solutions to some differential equations coming from de Rham theory.

108 6.2 Hodge Decomposition 6 Index Theory

Theorem 6.2.1. Every elliptic operator is Fredholm. Remark. A property called elliptic regularity shows that ker D and coker D consist of smooth functions, so we may interpret these spaces in terms of de Rham theory. Theorem 6.2.2 (Hodge Decomposition). For a closed Riemannian manifold M, there is an orthogonal decomposition

Ωk(M) = Hk(M) ⊕ dΩk−1(M) ⊕ d∗Ωk+1(M)

for all k ≥ 1. Proof. Observe that ∆ = dd∗ + d∗d is self-adjoint, so we have a decomposition

Ω•(M) = ker ∆ ⊕ (ker ∆)⊥ = ker ∆ ⊕ (ker ∆∗)⊥ = ker ∆ ⊕ im ∆.

• L k Note that ker ∆ = H (M) = k≥0 H (M). On the other hand, for ω ∈ im ∆, write ω = (dd∗ + d∗d)γ for γ ∈ Ω•(M). Set α = d∗γ and β = dγ. Then Z Z hdγ, d∗ηi = hd2γ, ηi = 0 M M for all η implies im d ⊥ im(d∗), so it follows that α and β are unique.

k Corollary 6.2.3. Every class in HdR(M) has a unique harmonic representative. k ∗ Proof. By Theorem 6.2.2, any class [ω] ∈ HdR(M) is represented by ω = h + dα + d β for some h ∈ Hk(M), α ∈ Ωk−1(M) and β ∈ Ωk+1(M). Thus

0 = dω = dh + d2α + dd∗β = 0 + 0 + dd∗β

so d∗β is closed and co-closed, meaning d∗β ∈ Hk(M) ∩ im(d∗) = 0. Hence ω = h + dα, so we see that [ω] = [h]. For uniqueness, suppose [h] = [h0] for some other harmonic form h0 ∈ Hk(M). Then h = h0 + dα for some α ∈ Ωk−1(M), but dα = h − h0 is harmonic, so dα = 0. Corollary 6.2.4. For any Riemannian manifold M, there is an isomorphism of graded rings • ∼ k H (M) = HdR(M). From this, we get a primitive form of the index theorem: Corollary 6.2.5. Let D = d + d∗ :Ωeven(M) → Ωodd(M). Then ind(D) = χ(M), the Euler characteristic of M. Proof. By definition,

ind(D) = dim ker D − dim coker D and χ(M) = dim Heven(M) − dim Hodd(M),

so Corollary 6.2.4 gives the result. Corollary 6.2.6. If ∆ = ∆∗ is a self-adjoint, Fredholm, linear differential operator, then ind(D) = 0. In particular, the Laplacian operator ∆ = d∗d + dd∗ has index 0.

109 6.3 The Index Theorem 6 Index Theory

6.3 The Index Theorem

Suppose D : Γ(X,E) → Γ(X,F ) is an elliptic linear differential operator over a Riemannian manifold X. If π : TX → X is the tangent bundle, write Ee = π∗E and Fe = π∗F . Denote the frame bundle associated to TX by TX0, or equivalently, TX0 = TX r s0(X) where s0 is the zero section.

Lemma 6.3.1. There is a bundle isomorphism σ˜D : Ee|TX0 → Fe|TX0 such that for each ∗ ∼ x ∈ X and ω ∈ T X = TX, σ˜|(x,ω) = σD(ω): Ex → Fx is the symbol of D. ∼ Using the notation of Section 4.1, this defines an element of K-theory σD ∈ L1(TX,TX0) = K(TX,TX0) by Proposition 4.1.10. The pair (TX,TX0) is homotopy equivalent to the Thom space DX/SX, where DX and SX are, respectively, the unit disk and unit sphere bundle on X. Thus σD ∈ Ke(DX/SX). Using the notation in Section 4.4, let ch : K(X) → H•(X; Q) be the Chern character of X. Then by Theorem 4.4.6, for any complex vector bundle E → X, there is an isomorphism ΦK : K(X) → K(E,E0) given by α 7→ αλE, where λE is the Thom class in K-theory of E → X. Observe that we have a diagram

ΦK K(X) K(E,E0)

ch ch

• • H (X; Q) H (E,E0; Q) Φ

where Φ is the ordinary Thom isomorphism. However, the diagram does not commute! In • ∗ • fact, we have Φ(ch(α)) = t ch(ΦK (α)) for some t ∈ H (X; Q). Applying i : H (E,E0; Q) → • ∗ ∗ H (X; Q) gives i Φ(ch(α)) = t i ch(ΦK (α)). In particular, for α = 1, our computations in Example 4.4.5 give us n ∗ ∗ Y xi i uE = t i ch(λE) = t (1 − e ) i=1 where uE is the ordinary Thom class and x1, . . . , xn are the Chern roots of E → X. On ∗ the other hand, i uE = e(E) is the Euler class of E, and by the construction in Section 3.6, e(E) = cn(E) = x1 ··· xn is the top Chern class. This shows that

n Y xi t = . 1 − exi i=1 This is sometimes called the Thom defect of the bundle E.

Definition. For a complex vector bundle E → X, the Todd class of E is the characteristic • class Td(E) = cf (E) ∈ H (X; Q) determined by the formal power series t 1 1 1 f(t) = = 1 + t + t2 − t4 + ... 1 − e−t 2 12 720

110 6.3 The Index Theorem 6 Index Theory

Explicitly, the Todd class is equal to

n Y xi Td(E) = 1 − e−xi i=1 where x1, . . . , xn are the Chern roots of E.

Lemma 6.3.2. Let t ∈ H•(X; Q) be the Thom defect of a vector bundle E → X of rank n. Then t = (−1)n Td(E) where E is the complex conjugate bundle to E. Proof. Obvious from our computation of t above. Proposition 6.3.3. The Todd class is a characteristic class which is uniquely characterized by the property that Z n n n hTd(T CP ), [CP ]i = Td(T CP ) = 1 CP n for all n ≥ 1, where [CP n] is the fundamental class. We now come to the crowning achievement in index theory, the Atiyah-Singer index theorem. Theorem 6.3.4 (Atiyah-Singer Index Theorem). Let X be a compact, closed n-manifold with complex vector bundles E,F → X and elliptic linear differential operator D : Γ(X,E) → Γ(X,F ). Then n ind(D) = (−1) ch(σD) Td(TX ⊗ C)[TX]

where σD ∈ K(TX,TX0) is the K-theory symbol associated to D and [TX] is the fundamental class of the total space of the tangent bundle TX → X.

• • Remark. If π! : H (TX,TX0; Q) → H (X; Q) is the pushforward map, then the equality in the Atiyah-Singer index theorem can instead be written

n(n+1)/2 ind(D) = (−1) π! ch(σD) Td(TX ⊗ X)[X].

We will not prove the index theorem, but simply marvel at all the information it contains: ind(D) is an algebraic invariant that really comes out of the linear algebra of some differential equations describing the elliptic operator D; likewise, σD is an algebraic expression from which we obtain a characteristic class ch(σD); the Todd class is some measurement of the size of the Chern roots of the (complexified) bundle, which themselves encode the topological data of Chern classes of the bundle; and finally, [TX] is a purely topological gadget. Rather than giving a proof, we will conclude this chapter with several applications of the index theorem. First, we need the following analogue of Theorem 3.8.12 for oriented bundles: Proposition 6.3.5. Let V → X be an oriented real vector bundle of dimension 2n. Then there is a space Y and a map π : Y → X such that

(1) π∗ : H•(X; Z) → H•(Y ; Z) is injective.

111 6.3 The Index Theorem 6 Index Theory

∗ ∼ (2) π V = E1 ⊕ · · · ⊕ En for some oriented, real, dimension 2 bundles Ei → Y . In particular, n n ∗ ∼ M M π (V ⊗ C) = (Ei ⊗ C) = (Li ⊕ Li) i=1 i=1 ∼ where Li is a complex line bundle such that Ei = (Li)R.

Proof. Same as Theorem 3.8.12, replacing PE with Grf2(E), the (fibrewise) Grassmannian of oriented dimension 2 subspaces of E.

Definition. Let Dj : Γ(X,Ej) → Γ(X,Ej+1) be a sequence of linear differential operators on complex line bundles Ej → X such that Dj+1 ◦ Dj = 0 for all j ≥ 0. Then (Dj) is called a Fredholm sequence if ker Dj/ im Dj+1 is finite dimensional for all j ≥ 0.

Definition. An elliptic sequence is a sequence of linear differential operators (Dj) such that for all ω ∈ T ∗X, the sequence

σD0 (ω) σD1 (ω) E0 −−−−→ E1 −−−−→ E2 → · · · is exact. Theorem 6.3.6. Every elliptic sequence is Fredholm. Moreover, if one defines

∞ ∞ ∞ ∞ 0 M 1 M 0 M 1 M E = E2j,E = E2j+1,D = D2j and D = D2j+1, j=0 j=0 j=0 j=0

0 1 ∗ then the sequence (Dj) is elliptic if and only if the linear differential operator D + (D ) : E0 → E1 is elliptic. In this case,

∞ 0 1 ∗ X j ind(D + (D ) ) = (−1) (dim ker Dj − dim im Dj+1). j=0 Example 6.3.7. Consider the elliptic operator D = d + d∗ on a Riemannian manifold X. Vi ∗ Vi+1 ∗ ∗ The symbol σD : T X ⊗ C → T X ⊗ C acts as − ∧ ω for some nonzero ω ∈ T X ⊗ C and the sequence

^i−1 ∗ σD ^i ∗ σD ^i+1 ∗ · · · → T X ⊗ C −→ T X ⊗ C −→ T X ⊗ C → · · ·

is exact. Thus on the level of K-theory, σD = λT ∗X⊗C, the Thom class. Notice that the Chern roots of TX ⊗ C are x1, −x1, x2, −x2, . . . , xn, −xn where xi = c1(Li) for the line bundles Li in the decomposition n ∼ M TX ⊗ C = (Li ⊕ Li) i=0

∗ Qn xi −xi from Proposition 6.3.5. Hence i ch(σD) = i=1(1 − e )(1 − e ). On the other hand,

n 2 n 2 Y −xi n Y xi Td(TX ⊗ C) = = (−1) . (1 − e−xi )(1 − exi ) (1 − exi )(1 − e−xi ) i=1 i=1

112 6.4 Dolbeault Cohomology 6 Index Theory

Qn Also recall that e(TX) = i=1 xi. Putting this all together, we have by the index theorem:

n ind(D) = (−1) ch(σD) Td(TX ⊗ C)[TX] n ! n ! Y Y x2 = (−1)n (1 − exi )(1 − e−xi ) (−1)n i [TX] (1 − exi )(1 − e−xi ) i=1 i=1 n ! Y 2 = xi [TX] after cancelling terms i=1 n ! Y = xi [X] by the Thom isomorphism theorem (3.4.1) i=1 = e(TX)[X] = χ(X), the Euler characteristic of X, by Theorem 3.5.6. 2n Going further, Theorem 3.4.1 says that every β ∈ H (TX,TX0) is of the form β = ∗ α ∪ uTX where uTX is the Thom class, so if s0 : X → TX is the zero section, then s0β = ∗ ∗ ∗ s0β s0α ∪ s0uTX = α ∪ e(TX). Thus if e(TX) 6= 0, we can say that α = e(TX) and

α ∩ [TX] = (−1)n(n−1)/2β ∩ [TX].

Proposition 6.3.8. If D is an elliptic linear differential operator on a closed manifold X of odd dimension, then ind(D) = 0.

Proof. The map a : ω 7→ −ω on TX changes the sign of the fundamental class [TX], but ∗ k k a σD = σD in K(TX,TX0). Indeed, for ω ∈ Ω (X) we have σD(−ω) = (−1) σD(ω) but iπt multiplication by e gives a homotopy σD(−ω) → σD(ω). Feeding this information into the index theorem, we get ind(D) = − ind(D), so the index of D must be zero.

6.4 Dolbeault Cohomology

In this section we give an application of the Atiyah-Singer index theorem to Dolbeault cohomology, which produces an analogue of the Riemann-Roch theorem in complex algebraic geometry known as the Hirzebruch-Riemann-Roch theorem. If V is a complex vector space, then the map J : V → V, v 7→ iv is linear with J 2 = −1, so it has eigenvalues i and −i. Write V 1,0 for the i-eigenspace of J acting on V ⊗ C; likewise, write V 0,1 for the −i-eigenspace.

Example 6.4.1. If VR = SpanR{e1, e2} as a real vector space, with Je1 = e2, then

1,0 0,1 V = Span{e1 − ie2} and V = Span{e1 + ie2}.

Now suppose X is a compact, complex manifold of (complex) dimension n. Then TX is a complex vector bundle, so its complexification TX ⊗ C can be decomposed into

1,0 0,1 TX ⊗ C = TX ⊕ TX

113 6.4 Dolbeault Cohomology 6 Index Theory

where TX1,0 ∼= TX and TX0,1 ∼= TX, using the decomposition described above on each ∗ ∗ 1,0 ∗ 0,1 ∗ fibre. Similarly, T X ⊗ C = T X ⊕ T X . If dz1, . . . , dzn is a local basis for T X with respect to zj = xj + iyj for each 1 ≤ j ≤ n, then

∗ 1,0 ∗ 0,1 T X = Span{dz1, . . . , dzn} and T X = Span{dz¯1, . . . , dz¯n}.

This extends to a decomposition

^• ∗ M ^p,q ∗ (T X ⊗ C) = T X p,q≥0

where Vp,qT ∗X = Vp(T ∗X1,0) ⊗ Vq(T ∗X0,1). Set Ωp,q(X) = Vp,qT ∗X.

Definition. The Dolbeault operator is the linear differential operator

0,q 0,q+1 ð :Ω (X) −→ Ω (X) ω 7−→ π0,q+1 ◦ dω

r 0,r where π0,r :Ω (X) → Ω (X) is the natural projection map.

P 0,q Locally, if α = αi1,...,ik dz¯i1 ∧ · · · ∧ dz¯ik ∈ Ω (X), then X ðα = ðαi1,...,ik ∧ dz¯i1 ∧ · · · ∧ dz¯ik

Pn ∂f ∂ 1  ∂ ∂  where f = dz¯j and = − i . Notice that f = 0 if and only if f ð j=1 ∂zj ∂z¯j 2 ∂x ∂y ð satisfies the Cauchy-Riemann equations, i.e. is holomorphic.

Proposition 6.4.2. ð2 = 0 on Ω0,•(X). Definition. The Dolbeault complex on a complex manifold X is the complex

· · · → Ω0,q−1(X) −→ð Ω0,q(X) −→ð Ω0,q+1(X) → · · ·

The qth Dolbeault cohomology of X, written H0,q(X), is the qth cohomology of this complex.

Theorem 6.4.3 (Dolbeault). Let X be a complex manifold with structure sheaf X. Then for all q, there is an isomorphism

0,q ∼ q H (X) = H (X, OX ),

q where H (X, OX ) denotes the qth sheaf cohomology.

Remark. One can define the Dolbeault operator ð :Ωp,q(X) → Ωp,q+1(X) for any fixed p, giving a similar cohomology theory Hp,q(X). Then the isomorphism in Theorem 6.4.3 becomes Hp,q(X) ∼= Hq(X, Ωp(X)).

Proposition 6.4.4. The Dolbeault operator ð is elliptic.

114 6.4 Dolbeault Cohomology 6 Index Theory

0,q 0,q Definition. For a complex manifold X, set h (X) = dimC H (X) and define the holo- morphic Euler characteristic of X to be

∞ X q 0,q χh(X) = (−1) h (X). q=0

Corollary 6.4.5. For any complex manifold X, ind(ð) = χh(X).

Corollary 6.4.6. χh(X) = Td(TX)[X].

Proof. We have χh(X) = ind(ð). The right side of the equation in the index theorem (6.3.4) n is (−1) ch(σð) Td(TX ⊗ C)[TX]. Note that if s0 : X → TX is the zero section, then we have ∞ ∞ ∗ X q ^q ∗ 0,1 X q ^q s0σð = (−1) (T X ) = (−1) TX q=0 q=0 in K-theory, so n ∗ Y xi s0 ch(σð) = (1 − e ) i=1 where xi are the Chern roots of TX. Meanwhile, the Todd class can be computed as

Td(TX ⊗ C) = Td(TX ⊕ TX) = Td(TX) Td(TX) n ! n ! Y xi Y −xi = 1 − e−xi 1 − e−xi i=1 i=1 n Y x2 = (−1)n i . (1 − exi )(1 − e−xi ) i=1 Since TX is a real vector bundle of dimension 2n, the index theorem reads (by the subsequent remark): 1 ind( ) = (−1)2n(2n+1)/2s∗ ch(σ ) Td(TX ⊗ ) [X] ð 0 ð C e(X) n ! n ! n ! Y Y x2 Y 1 = (−1)n (1 − exi ) (−1)n i [X] (1 − exi )(1 − e−xi ) x i=1 i=1 i=1 i n ! Y xi = [X] = Td(TX)[X]. 1 − e−xi i=1

Definition. The Todd genus of a complex manifold is Td(TX)[X].

Thus Corollary 6.4.6 says that the Todd genus equals the holomorphic Euler character- istic.

115 6.4 Dolbeault Cohomology 6 Index Theory

If E → X is any holomorphic vector bundle, then one can define an operator 0,q 0,q+1 ðE :Ω (X; E) −→ Ω (X : E) 2 in a similar fashion to the constructions in Section 5.1. Then ðE = 0 and ðE is still elliptic, but we have σðE = σð[E] where [E] is the fundamental class of the total space of E. This implies:

Corollary 6.4.7. For any holomorphic vector bundle E → X, χh(X; E) = ch(E) Td(TX)[X]. Suppose now that X is a Riemann surface, i.e. a closed, connected complex manifold of dimension 1. Then as a real manifold, X is an oriented surface of genus g = g(X). For any holomorphic vector bundle E → X, Example 4.4.3 and the definition of the Todd class show that 1 1 Td(E) = 1 + c (E) + (c (E)2 + c (E)) + ... 2 1 12 1 2 1 and ch(E) = rank(E) + c (E) + (c (E)2 − 2c (E)) + ... 1 2 1 2 0,0 0,1 Also note that χh(X) = h (X) − h (X) and 1 1 Td(X)[X] = hc (TX), [TX]i = (2 − 2g) = 1 − g. 2 1 2 This proves: Corollary 6.4.8. For a Riemann surface X, h0,0(X) − h0,1(X) = 1 − g. Corollary 6.4.9. For a Riemann surface X, h0,1(X) = g. Proof. By Liouville’s theorem, every global holomorphic differential 1-form on a closed Rie- 0,0 ∼ mann surface is trivial, so H (X) = C. If E → X is a holomorphic vector bundle on a Riemann surface, we have a similar result q 0,q which is famously known as the Riemann-Roch theorem. Set h (E) = dimC H (X,E). Corollary 6.4.10 (Riemann-Roch Theorem). Let X be a Riemann surface of genus g and let E → X be a holomorphic vector bundle. Then h0(E) − h1(E) = deg(D) + 1 − g where deg(E) = hc1(E), [X]i is the degree of E. Proof. By Corollary 6.4.7, the holomorphic Euler characteristic of E can be written h0(E) − h1(E) = ch(E) Td(TX)[X]  1  = 1 + c (X) (1 + c (E))[X] 2 1 1  1  = c (E) + c (X), [X] 1 2 1 1 = hc (E), [X]i + hc (X), [X]i 1 2 1 = deg(E) + 1 − g by Corollary 6.4.8.

116 6.4 Dolbeault Cohomology 6 Index Theory

Corollary 6.4.11. If deg(E) + 1 − g > 0, then E admits a holomorphic section.

Now suppose X is a complex surface, that is, a complex manifold of dimension 2.

Corollary 6.4.12 (Noether’s Formula). If X is a complex surface, then 1 χ (X) = (c (X)2 + c (X))[X]. h 12 1 2

2 In particular, (c1(X) + c2(X))[X] is an integer multiple of 12. Proof. Apply Corollary 6.4.6.

Example 6.4.13. For X = CP 2, by Example 3.6.7 we have

2 3 2 c(CP ) = (1 + x) = 1 + 3x + 3x

2 2 2 2 2 2 2 for a generator x ∈ H (CP ). Thus c1(CP ) = 9x , c2(CP ) = 3x and we see that 2 2 (c1(X) + c2(X))[X] = 9 + 3 = 12. In particular, χh(CP ) = 1.

1 1 1 1 Example 6.4.14. For X = CP × CP , let c1(CP × ∗) = 2x and c1(∗ × CP ) = 2y, 2 2 2 2 so that c1(X) = 2x + 2y. Then c1(X) = (2x + 2y) = 4x + 8xy + 4y = 8xy, while c2(X) = e(X) = 4xy, so

2 (c1(X) + c2(X))[X] = (8xy + 4xy)[X] = 12

1 1 and once again χh(CP × CP ) = 1. Finally, when L → X is a line bundle, we obtain the following analogue of the Riemann- Roch theorem.

Corollary 6.4.15 (Hirzebruch-Riemann-Roch Theorem). Let X be a complex surface, L → X a complex line bundle and let D denote the Poincar´edual of c1(L) in H2(X; Z) and −K denote the Poincar´edual of c1(X) in H2(X; Z). Then 1 χ (X,L) = χ (X) + (D • D − K • D) h h 2 where • denotes the intersection product.

Proof. By Corollary 6.4.7, χh(X,L) = ch(L) Td(X)[X], but the right side of this expression evaluates as:  1   1 1  ch(L) Td(X)[X] = 1 + c (L) + c (L)2 1 + c (X) + (c (X)2 + c (X)) [X] 1 2 1 2 1 12 1 2 1 = χ (X) + c (L)c (X) + c (L)2
[X]. h 2 1 1 1

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