80-min Lesson Plan
Math 533: Fiber bundles and Characteristic classes
Lesson A1: Introduction
Spring 2016
Unit A: Bundles
Objective. At the end of this lesson, students will be able to:
• Define fiber bundle, isomorphism, pull-back, & characteristic class
• Classify bundles as cocycles/coboundaries
• State the goal of the course
Assignments:
• Read
– MS section 2*,
– VB 4-9, and
– BT 47-48 & 53-55.
• Survey
Materials and Equipment needed:
• Survey
Introduction.
My goal for today is to introduce the objects that we will be working with in this
course: vector bundles and characteristic classes. I will also let you know what we
will accomplish in this course. I will start with the basic course mechanics.
Outline.
Course Mechanics: 20 min lecture.
Read introduction
Basic Info:
Name: Susan Tolman
E-mail: [email protected] NOT [email protected]
Course: Math 533, Fiber Bundles and Characteristic Classes
webpage: www.math.uiuc.edu/~stolman/m533
Survey:
Hand out surveys.
I want to schedule office hours when people are free.
I also hope to figure out what background material I need to cover. Depending one what students
know, I may set aside some classes to cover background.
Prerequisites:
• Good grasp of cohomology, preferably with arbitrary coefficients.
• Manifolds a plus.
Books:
We will mostly be following:
• Characteristic Classes, by Milnor and Stasheff (MS)
We will also take material from:
• Vector Bundles and K-theory by Hatcher (VB) – online
• Differential Forms in Algebraic Topology by Bott and Tu (BT)
Math 533 Lesson A1
p. 2
Grade:
15% Homework. Due every other week. Grade based on discussions.
85% Either
(1) a take-home final exam, or
(2) a project tying characteristic classes to your interests.
Give reading
Give assignment
Fiber bundles: 20 min lecture.
A fiber bundle is a space that is locally a product space.
Let F be a space. Let a topological group G act effectively on F (on the left), i.e. G ⊂
Homeomorphisms(F ).
A surjection π : E → B is a fiber bundle with fiber F and structure group G if there
exists an open cover {Uα } of B and homeomorphisms
∼
=
φα : E|Uα := π −1 (Uα ) → Uα × F
such that
• proj1 ◦φα = π (draw diagram) &
• ∃ continuous transition functions gαβ : Uαβ := Uα ∩ Uβ → G s.t.
φα ◦ φ−1
β : Uαβ × F → Uαβ × F
is given by
φα ◦ φ−1
β (x, ξ) = (x, gαβ (x) · ξ) ∀α, β
Terminology:
•
•
•
•
E is the total space
B is the base
π is the projection map
Given b ∈ B, Fb := π −1 (b) is the fiber over b
Example. The trivial bundle with E = B × F and π(b, f ) = b.
What is the structure group?
G = {1} or any subgroup of the homeomorphism group of F .
Example. The Möbius strip M = [0, 1] × R/ ∼, where (0, x) ∼ (1, −x) is a R bundle over S 1 .
What is the structure group now? G = Z/(2) = O(1) or R× = Gl(1) or any subgroup of homeos
of R that contains Z/(2)
When we want G to be a proper subgroup of Homeomorphisms(F ), we need to include these
trivializations as part of our data, like an atlas for a manifold.
We focus on three cases:
• n-dimensional real vector bundles, i.e., F = Rn and G = Gl(n, R)
• n-dimensional oriented real vector bundles, i.e., F = Rn and G = Gl(n, R)+
• n-dimensional complex vector bundles, i.e., F = Cn and G = Gl(n, C)
Here, Gl(n, R)+ := {A ∈ Gl(n, R) | det A > 0}.
In these cases, each fiber Fb is naturally a vector space.
π
π0
An isomorphism between bundles F → E → B and F → E 0 → B is a homeomorphism
ψ : E → E 0 such that
Math 533 Lesson A1
p. 3
• π0 ◦ ψ = π &
• ∃ continuous maps hα : Uα → G s.t.
φ0α ◦ ψ ◦ φ−1
α (x, ξ) = (x, hα (x) · ξ).
Draw diagrams.
Given an isomorphism, write E ∼
= E0.
Example. For vector bundles, an isomorphism E → E 0 is a homeomorphism that induces a linear
isomorphism Eb → Eb0 ∀ b.
Cocycles and coboundaries: 10 min lecture.
Claim: The transition functions gαβ satisfy the cocycle condition
gαβ · gβγ = gαγ
on Uα ∩ Uβ ∩ Uγ .
This is clear from the definition.
Given a cocycle {gαβ }, construct a fiber bundle with transition functions gαβ by setting
a
Uα × F/ ∼,
E=
α
where
(x, ξ) ∈ Uα × F ∼ (x, gαβ (x) · ξ) ∈ Uβ × F.
0 are isomorphic exactly if there
Lemma. Two bundles E and E 0 w/ transition function gαβ and gαβ
exists hα : Uα → G such that
0
h−1
α · gαβ · hβ = gαβ ;
0
and say that g and g 0 differ by a coboundary
we write gαβ ∼ gαβ
Proof: HW # 1.
Bundles E → B with structure group G (trivializable over {Uα }) are classified by
H 1 ({Uα }, G)) := {cocycles gαβ }/ ∼ .
pull-backs: 20 min small group/discussion.
Given a bundle π : E → B and map f : B 0 → B, define the pull-back bundle
f ∗ (E) = {(b0 , e) ∈ B 0 × E | f (b0 ) = π(e)},
with projection (b0 , e) 7→ b0 .
Lemma. It is a bundle with the same fiber and structure group.
Prove this claim. What are the transition functions? Give students time to think. Ask for
volunteer.
Proof. Trivializations for f ∗ (E):
f ∗ (E)|f −1 (Uα ) = {(b0 , e) ∈ f −1 (Uα ) × E|Uα | f (b0 ) = π(e)}
' {(b0 , b, ξ) ∈ f −1 (Uα ) × Uα × F | b = f (b0 )} ' f −1 (Uα ) × F,
where (b0 , e) 7→ (b0 , φα (e)).
On Uαβ :
0
0
0
(b0 , b, ξ) 7→ (b0 , φβ ◦ φ−1
α (b, ξ)) = (b , b, gαβ (b) · ξ) = (b , b, gαβ (f (b )) · ξ)
Math 533 Lesson A1
So the transition functions for f ∗ (E) are f ∗ (gαβ ).
p. 4
Claim:
• Given g : B 00 → B 0 , g ∗ (f ∗ (E)) = (f ◦ g)∗ (E).
• Id∗ (E) = E.
So we have defined a contravariant functor from the category of spaces to the category of sets.
It associates the set of isomorphism classes of (real, oriented, or complex) vector bundles to each
space.
characteristic classes: 10 min lecture.
Goal of this course: Classify vector bundles up to isomorphism.
Spoiler alert: We will fail.
In general, it is very hard – too hard – to classify bundles up to isomorphism. So we will focus
on the next best thing.
Instead...
A characterstic class is a way of associating to each bundle π : E → B a cohomology class
c(E) ∈ H ∗ (B) such that:
• If E 0 ∼
= E then c(E 0 ) = c(E).
• Given f : B 0 → B and π : E → B, c(f ∗ (E)) = f ∗ (c(E)).
Cohomology is also a contravariant functor – from spaces to graded rings. A characteristic class
is natural transformation from the functor of isomorphism classes of bundles to the cohomology
functor.
π
Example. Let E → B be a vector bundle of rank n.
•
•
•
•
If E
If E
If E
if E
is a real vector bundle, have the Stiefel-Whitney Class w(E) ∈ H ∗ (B; Z/(2)).
is a real vector bundle, have the Pontrjagin Class p(E) ∈ H ∗ (B; Z).
is an oriented real vector bundle, have the Euler class e(E) ∈ H n (B; Z).
is a complex vector bundle, have the Chern class c(E) ∈ H ∗ (B; Z).
In each case, every possible characteristic class is a polynomial in the given characteristic
classes.
Overall, the timing was very accurate. Consider adding a few more words about Characteris
classes.