# Problem Sheet H

This Problem Sheet is based on Lectures 14 and 15.

- Problem 1 asks you to prove that there are natural isomorphisms $V \otimes W \cong W \otimes V$ and $U \otimes ( V \otimes W) \cong (U \otimes V) \otimes W)$.
*Hint:*Use the universal property of tensor products. - Problem 2 asks you to prove the universal property for $ \bigwedge (V)$.
- Problem 3 covers more properties of $ \bigwedge(V)$.
- Problem 4 asks you identify when two vector bundles are isomorphic in terms of their transition functions.
- Problem 5 looks insane, but it is actually very easy. It is another application of the “univeral property” idea. It shows that pullback bundles are pullbacks (in the sense of category theory) in the category of vector bundles.
*Hint:*Mimic the proof of the universal property for tensor products (or Problem 2). - Problem 6 shows that any smooth map between the total spaces of two vector bundles that is linear on the fibres can be factored as the composition of a vector bundle homomorphism and a vector bundle morphism along a smooth map.
- Problem 7 investigates kernels and cokernels of vector bundle homomorphisms.

(This sheet probably looks much harder than most of them, but the three “horrible” looking problems (4,5,6) have very short proofs. 😃 )

Comments and questions?