# Problem Sheet H

- 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 introduces the notion of a
*pullback bundle*. This problem is important, since we will use pullback bundles throughout the remainder of Differential Geometry I and II. - Problem 5 introduces the
*external**product*of two bundles. - Problem 6 relates the external product to the direct sum (of vector bundles) and the product (of prinicpal bundles).
- Problem 7 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 8 investigates kernels and cokernels of vector bundle homomorphisms.
- Problem 9 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 10 is used in gauge theory. Time permitting, we hope to discuss this in Differential Geometry II.

Comments and questions?