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

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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.