# Scratchpad for Vector bundles & K-theory

[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 $U, V$ whose intersection is non-empty, these local trivializations define a clutching function $\tilde{h}_{UV} : U \cap V \rightarrow GL_n(\mathbb{R})$ (or $GL_n(\mathbb{C})$), since on the intersection the composite $h_{UV} = h_V^{-1} \circ h_U : (U \cap V) \times \mathbb{R}^n \rightarrow (U \cap V) \times \mathbb{R}^n$ restricted to each $x \in U \cap V$ is a linear isomorphism.

# Direct sum of vector bundles

Given vector bundles $p_1 : E_1 \rightarrow X, p_2 : E_2 \rightarrow X$, define $E_1 \oplus E_2 = \bigcup_{x\in X} (E_1)_x \oplus (E_2)_x$.

1. Fibers. The obvious thing we want for the direct sum is to let the fiber over each $x$ be $(E_1)_x \oplus (E_2)_x$. So for $(e_1, e_2)$ in the total space the projection map is $(p_1 \oplus p_2)(e_1,e_2) = p_1(e_1) = p_2(e_2)$.
2. We need local triviality. Define an open cover $\{U_\alpha\}$ and local maps $h_\alpha$ on each $U_\alpha$ that satisfy the properties of local trivializations (except the homeomorphism bit, because we haven’t put a topology on the total space yet).
3. Topologize. Finally, topologize the total space so that the local trivializations on $U_\alpha$ are in fact local homeomorphisms from trivial bundles on $U_\alpha$ to the preimage $(p_1\oplus p_2)^{-1}(U_\alpha)$.