# Download Math 8306, Algebraic Topology Homework 12 Due in-class on Wednesday, December 3

Survey
Was this document useful for you?
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Covering space wikipedia, lookup

General topology wikipedia, lookup

Fundamental group wikipedia, lookup

Affine connection wikipedia, lookup

Grothendieck topology wikipedia, lookup

Riemannian connection on a surface wikipedia, lookup

Orientability wikipedia, lookup

Transcript
```Math 8306, Algebraic Topology
Homework 12
Due in-class on Wednesday, December 3
1. Show that if a principal bundle P → B has a section, then there is a
homeomorphism to the trivial principal bundle: P ∼
= B × G as right
G-spaces.
2. Let G and H be topological groups. Suppose P1 → B is a principal Gbundle and P2 → B is a principal H-bundle. Show P1 ×B P2 (pullback!)
is a principal G × H-bundle.
3. Suppose G → H is a homomorphism of topological groups, and P → B
is a principal G-bundle. Show that the mixing construction gives a
principal H-bundle P ×G H → B.
4. We identified CP1 with the space of lines in C2 . Associated to this,
there is a vector bundle ξ → CP1 :
ξ = {(L, v)|L ∈ CP1 , v ∈ L}.
Find an open cover {Uα } together with transition functions {hα,β :
Uα ∩ Uβ → GL1 (C)} to reconstruct the associated principal GL1 (C)bundle.
```