CHARACTERISTIC CLASSES
CONTENTS
1. Introduction
CHAPTER 1.
2. Linear algebra
3.Vector bundles.
4.Cohomology
5. Grassmannian
CHAPTER 2.
1.Characteristic classes
Axioms for:
a. Euler class
a. Stiefel- Whitney
b. Chern- Weil
c. Pontrjagin classes
d. Construction
- topological
- differential geometrical
CHAPTER 3.
1. Characteristic numbers
2. Chern character
4. Genus
A- genus, L=genus, Todd -genus,
CHAPTER 4.
Gauss Bonnett,
Hirtzebruch signature,
Riemann Roch theorem
Atiyah-Singer index theorem
References:
M. F. Atiyah, KTheory, W. A. Benjamin, 1967.
A. Hatcher, Vector bundles an K-theory (available to download from
http://www.math.cornell.edu/ hatcher/ VBKT).
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton
Univ. Press, 1974.
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2
INTRODUCTION.
The theory of characteristic classes can be developed in two
contexts:
1. Principal G- bundles , G a Lie group (usually G= GL(n:R),
GL(n:C), O(n), U(n), Sp(n)
2. Vector bundles possibly with additional structures ( Euclidean, Hermitian, Symplectic .
When restricted to the Lie groups mentioned above the two
contexts are equivalent.
In this course we will consider the second context only VECTOR BUNDLES.
Since a vector bundle can be regarded as a parametrized
vector space and the vector bundle theory as parametrized
linear algebra, it is expected that characteristic classes are
more or less obvious extensions of the familiar invariants:
• dimension of vector space,
• trace, determinant of a matrix,
• Pfaffian of a skew symmetric matrix,
• Characteristic polynomial, Heat trace of a matrix.
So will begin be reviewing a few more or less familiar facts
in linear algebra.
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LINEAR ALGEBRA.
VECTOR SPACES over a field.
• Sum: V, W ⇒ V ⊕ W
• Tensor product V, W ⇒ V ⊗ W
- k−Symmetric product V ⇒ S k V
∴ Symmetric algebra S ∗ V ; Graded algebra (commutative as
an algebra)
If dim V = n then S ∗ V = κ[z1 , · · · zn ] the polynomial algebra
in n variable
- k− Exterior product V ⇒ Λk V
∴ Exterior algebra Λ∗ V ; Commutative graded algebra.
- Λeven V (commutaive algebra)
• Hom : V, W ⇒ V ⊕ W
Dual: V ⇒ V ∗ t := Hom(V ; κ)
• Determinant Line: V ⇒ det V := Λdim V V
SQUARE MATRICES= M (n)
• A ∈ M (n), B ∈ M (m) ⇒ A ⊕ B ∈ M (m + n), A ⊗ B ∈ M (nm)
• A, B ∈ M (n)L
⇒ A · B ∈ M (n)
exp(A) = n≥0 n!1 An
• A ∈ M (n), B ∈ M (m) ⇒ A ⊗ B ∈ M (nm), A ∗ B ∈ M (mn)
A ∗ B = A ⊗ Im + In ⊗ B
INVARIANTS
(1) Eigenvalues, λ1 , · · · , λn
(2) Jordan blocks (λ, k)

λ 1

0 λ

T (λ, k) = 
0 0
. .
..
 ..
0 ···
(3) Trace: T r(A)
0
1
...
..
.
0

0
.. 
.

...
0


λ 1
0 λ .
···
...
4
(4)
(5)
(6)
(7)
Determinant det(A)
Characteristic Polynomial PA (λ) = det λ(I − A)
Heat trace T r(exp(λA))
Pfaffian of a skew symmetric matrix A= square root of the
determinant for a skew symmetric matrix. (see the attached
pdf file)
Remarks :
Suppose the eigenvalues of A are λ1 , · · · λn and the eigenvalues of B
are µ1 , · · · µk
then the eigenvalues of
(1) A ⊕ B are λ1 , · · · λn , µ1 , · · · µk
(2) A ⊗ B are λi µ̇j fot all pairs (i, j)
(3) A ∗ B are λi + µj fot all pairs (i, j)
(4) det exp(A) = exp(tr(A)
Properties:
• T r exp(tA ⊕ tB) = T r exp(tA) + T r exp(tB)
• T r exp(tA ∗ tB)P= T r exp(tA) · T r exp(tB)
• det(I + zA) = ni=0 z k T r(|Lambdak A)
Note :
P
If ||A||1 := 1≤i,j≤n |ai,j | then
||Λk (A)||1 ≤ ||A||k1 /k! and
P
k+1
T rAk /kz k
log det(I + zA) = T r(log(I + zA)) = ∞
i=0 (−1)
Exercises:
(1) E1
Let α : V → W ∴ Show that:
1. dim(ker α) = T rexp(A∗ A)
2. dim(co ker α) = exp(T r(A · A∗ ))
3. IndexA = T r(expT r(A∗ · A)) − T r(exp(A · A∗ ))
4. Derive
exp(tA ⊕ tB) = exp(tA) ⊕ exp(tB)
exp(tA ∗ tB) = exp(tA) ⊗ exp(tB)
(2) E2: Show that invariants (3), (4), (5) (7) can be etended to matrices with entries in any commutative unital ring (algebra)A.
(3) E3: Show that invariants (3), (4), (5) (7) can be etended to
endomorphisms of any endomorphism of a finitely generated
projective module over A as in E2.
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VECTOR BUNDLES
Reference A Hatcher,
Problems Hatcher and J Milnor
Continuous family of vector spaces p : E → X
Homomorphism, Isomorphism, trivial family, sections
Vector bundles
Local trivialization , cocycle construction.
Examples
Properties of trivial bundles, interpretation of sections.
Constructions
1. Continuous functors extension to vector bundles:
2. Pull backs
Category of vector bundles over a base X.
1. sum, tensor product, Hom- bundle and the dual bundle, symmetric and exterior power
2. subbundle and quotient bundle
The main lemmas
1. Given two bundles E1 → X and E2 → X a continuous map ϕ :
E1 → E2 with the property that for any x ∈ X ϕ sends E1x → E + 2x
isomorphically is an isomorphism .
2. If ϕ : E1 → E2 is an homomorphism then {x ∈ X} with ϕx an
isomorphism / monomorphism / epimorphism is an open set.
The main elementary theorems
1. Any vector bundle over a compact space has a complement,
2. If E → X × I is a vector bundle then the restriction to X × 0 and
X × 1 are isomorphic.
Classification
1. Grasmannian and the universal bundle.
2. Classification theorem.
3. Topology of the Grassmannian manifold(s)
K-Theory
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COHOMOLOGY
Singular cohomology algebra,
Chech Cohomology algebra
DeRham cohomology algebra.
Cohomology with Z2 coefficients, Steenrod operations.