I know that $E(\gamma^1_n)$ is non-trivial, but when I try to find local trivializations, I am finding that the transition functions are the identity, rendering the bundle trivial. I am obviously doing something wrong here.
Can anyone point out the wrong step for me. The reasoning goes:
Let $\tilde{U}_i$ be the subspace of $\mathbb{R}^3$ with $x_i>0$.
Let $U_i$ be its image in $\mathbb{RP^2}$.
Let $p:E(\gamma_2^1)\rightarrow \mathbb{RP}^2$ be the canonical vector bundle.
Define $h_i: p^{-1}(U_i)\rightarrow U_i \times \mathbb{R}$ as $h_i([x],v) = ([x],v \cdot x)$.
This is obviously a homeomorphism with continuous inverse: $h_i^{-1} ([x],t) = ([x],tx)$.
Then the transition functions $t_{ij}$ on $U_i \cap U_j$ are the identity since $h_i$ does not depend on $U_i$.
This implies the bundle is trivial. Since it is not, what did I do wrong?