[vect.bund]

This is a scratchpad for my summer school course in vector bundles and K-theory.

# Clutching functions

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 .