Characteristic Classes and Chern Classes

Bailey Whitbread October 23, 2020

1 What are Characteristic Classes and Chern Classes?

Characteristic classes are the natural analogue of a cohomology class for vector bundles. Chern classes are a type of characteristic class for complex vector bundles.

2 Outline of Talk

(1) Vector Bundles. (a) Definitions. (b) Examples. (2) Characteristic Classes. (a) The Cohomology Ring. (b) Characteristic Classes. (c) Chern Classes. n (3) Is CP Parallelisable? (a) Complex Projective Space, CPn. (b) Parallelisablity. (4) Recovering Steifel-Whitney Classes.

3 What’s a Vector Bundle?

Definition (Complex Vector Bundles) A complex vector bundle over a topological space B is

(i) A topological space E called the total space, and (ii) A continuous surjection p : E → B called the projection map

...together with the conditions...

(a) For each b ∈ B, the fibre p−1(b) is a complex vector space, and...

4 What’s a Vector Bundle?

Definition (Complex Vector Bundles)

(b)( Local Triviality) Each point b0 ∈ B must possess a neighbourhood U with p−1(U) being homeomorphic to r r −1 U × C . This homeomorphism h : U × C → p (U) must −1 r map the fibres p (b) linearly onto {b} × C .

r h −1 U × C p (U)

π1 p B

where π1 is the projection map onto the first coordinate.

5 What’s a Vector Bundle?

Example (Tangent Bundle) Fix a differentiable manifold B.

Each point x ∈ B has a tangent space, Tx B. We take the space of pairs

TB = {(x, y): x ∈ B, y ∈ Tx B}

and form the tangent bundle, with

1. Base space B, 2. Total space TB, and 3. Projection map p : TB → B given by p(x, y) = x.

6 Characteristic Classes

Fix an abelian group A. Recall the cohomology ring of B,

∞ M H∗(B; A) = Hi (B; A) = H0(B; A) + H1(B; A) + ... i=0

A characteristic class ci of vector bundles is an assignment of each bundle E → B to a cohomology class

∞ M i c0(E) + c1(E) + ... ∈ H (B; A). i=0

7 Characteristic Classes

Certain choices of B and A give rise to different classes:

i (1) Stiefel-Whitney classes: wi (E) ∈ H (B; Z2), E → B a real vector bundle. 2i (2) Chern classes: ci (E) ∈ H (B; Z), E → B a complex vector bundle. n (3) The Euler class: e(E) ∈ H (B; Z), E → B an oriented, n-dimensional real vector bundle.

8 What are Chern Classes?

Definition (Chern Classes)

Fix a complex vector bundle E over B. The Chern classes ck (E) are defined to be the unique classes satisfying:

(i) c0(E) = 1. (ii)( Naturality) If f : B0 → B is a homeomorphism then recall f ∗E is the pullback vector bundle over B0. Then

∗ ∗ ck (f E) = f ck (E).

(iii)( Whitney Sum Formula) Let F → B be another complex vector bundle over B. Then

k X ck (E ⊕ F ) = ci (E)ck−i (F ). i=0 9 What are Chern Classes?

Definition (Chern Classes)

(iv)( Normalisation) The first Chern class of the tautological line 1 2 1 ∼ bundle of CP is equal to −1 in H (CP ; Z) = Z. P The total Chern class of E is c(E) = i≥0 ci (E). That is,

∗ c(E) = c0(E) + c1(E) + ... ∈ H (B; Z).

10 Complex Projective Space, CPn

Definition (Complex Projective Space) n+1 Consider the non-zero vectors in C = {(z1, ..., zn+1): zi ∈ C}. n+1 We identify non-zero vectors in C with their scalar multiples.

(z1, ..., zn+1) ∼ (λz1, ..., λzn+1), λ ∈ C.

n n+1 Then CP = (C \{0})/ ∼.

n CP has much more structure than just a topological space, it’s a complex manifold.

1 2 CP is the Riemann sphere, CP is the complex projective plane.

11 Parallelisability

Definition (Parallelisability) Fix a differentiable manifold M of dimension n.

We say M is parallelisable if there exist a family of smooth vector

fields on the manifold, V1, ..., Vn, such that every point p in M provides a basis {V1(p), ..., Vn(p)} of TpM.

Example (Hairy Ball Theorem)

12 Parallelisability

Theorem (Milnor, Stasheff) As before, M is a differentiable manifold. Then

M is parallelisable ⇐⇒ M has trivial tangent bundle.

13 Is CPn Parallelisable?

n Recall the cohomology ring of CP over Z,

∗ n ∼ n+1 H (CP ; Z) = Z[x]/hx i.

n n Consider the tangent bundle, T CP . If T CP were trivial, we n would have c(T CP ) = 1. This has total Chern class

n n+1 ∗ n ∼ n+1 c(T CP ) = (1 + x) ∈ H (CP ; Z) = Z[x]/hx i.

n Then x 6= 0 and n ≥ 0, so c(T CP ) 6= 1.

14 Recovering Stiefel-Whitney Classes

Proposition Regard an n-dimensional complex vector bundle E → B as a 2n-dimensional real vector bundle. Then

• w2i+1(E) = 0, and

• w2i (E) is the image of ci (E) under the coefficient 2i 2i homomorphism H (B; Z) → H (B; Z2).

15 References

• Characteristic Classes, • Algebraic Topology, John W. Milnor, Allen Hatcher, James D. Stasheff, pi.math.cornell.edu/ Princeton University Press, ∼hatcher/AT/AT.pdf 1974. 2001. • Differential Forms in • Vector Bundles and K-Theory, Algebraic Topology, Allen Hatcher, Graduate Texts in Mathematics, pi.math.cornell.edu/ Raoul Bott, Loring W. Tu, ∼hatcher/VBKT/VB.pdf Springer, 2017. 1995.

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