This is a scratchpad for my summer school course in vector bundles and K-theory.
Given local trivializations of a vector bundle on open sets whose intersection is non-empty, these local trivializations define a clutching function (or ), since on the intersection the composite restricted to each is a linear isomorphism.
Direct sum of vector bundles
Given vector bundles , define .
- Fibers. The obvious thing we want for the direct sum is to let the fiber over each be . So for in the total space the projection map is .
- We need local triviality. Define an open cover and local maps on each that satisfy the properties of local trivializations (except the homeomorphism bit, because we haven’t put a topology on the total space yet).
- Topologize. Finally, topologize the total space so that the local trivializations on are in fact local homeomorphisms from trivial bundles on to the preimage .