\( \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \DeclareMathOperator{\RP}{\mathbf{RP}} \DeclareMathOperator{\C}{C_*} \DeclareMathOperator{\U}{\mathcal{U}} \)

Section3.1Vector Bundles

Definition3.1.1Vector bundle
An \(n\)-dimensional real vector bundle is a triple \((E,B,\pi)\) such that
  1. \(\pi \colon E \to B\).
  2. Each fibre \(\pi^{-1}(p)\) has the structure of an \(n\)-dimensional real vector space.
  3. (Local triviality) For each \(p \in B\) there is \(U \subset B\) open containing \(p\) and a homeomorphism \[f_U\colon \pi^{-1}(U) \to U \times \RR^n\] such that
    1. \(\pi_1\circ f_U = \pi\),
    2. \(\pi_2 \circ f_U\colon \pi^{-1}(p) \to \RR^n\) is a linear map (it must be an isomorphism of vector spaces).
A complex vector bundle is defined similarly with \(\CC\) replacing \(\RR\).

Subsection3.1.1Transition functions

If \(f_1\colon \pi^{-1}(U_1) \to U_1\times \RR^n\) and \(f_2\colon \pi^{-1}(U_2) \to U_2\times \RR^n\) are local trivialisations of \(E\) then the map \(f_1\circ F_2^{-1}\colon (U_1\cap U_2)\times \RR^n \to (U_1 \cap U_2)\) is linear on fibres. So there is a map \(g_{12}\colon U_1\cap U_2 \to \GL_n(\RR)\) such that \(f_1\circ f_2^{-1}(p,v) = (p,g_{12}(p) v)\), \(g_{12}\) is called a transition function.

Exercise 3.1.2
If \(f_3\colon U_3 \to U_3\times \RR^n \) is another local trivialisation then \(g_{32}(p)\cdot g_{21}(p) = g_{31}(p)\), this is called the cocycle condition.
Example3.1.3
  1. \(E = B\times \RR^n\) is the \(n\)-dimensional trivial bundle over \(B\).
  2. The Mobius band \(M = [0,1] \times \RR/\sim\) where \((0,x)\sim (1,-x)\) is a bundle over \(S^1\) via \(\pi\colon M \to [0,1]/\sim\). We have local trivialisations given by \(U_1 = [0-\epsilon, 1/2 + \epsilon]\times \RR\) and \(U_2 = [1/2-\epsilon, 1 + \epsilon]\times \RR\). The transition function is \(g_{12} \colon \{0,1/2\} \to \GL_1(\RR)\) mapping \(1/2\to 1\) and \(0\to -1\).
  3. If \(M\) is a smooth manifold with charts \(\phi_i \colon U_i \to V_i \subset \RR^n\) and transition functions \(\psi_{ij} = \phi_i \circ \phi_j ^{-1}\) where defined. Then the tangent bundle \(TM\) has local trivialisations \(f_i = d\phi_i\colon TM|_{U_i}\to TV_i = V_i\times \RR^n\) and transition functions \[g_{ij} = d\phi_i\circ d\phi_j^{-1} = d(\phi_i\circ \phi_j^{-1}) = d\psi_{ij}.\]