Let $H \to \mathbb{CP}^1$ be the canonical line bundle over $\mathbb{CP}^1 =S^2$.
Then from the text Vector Bundles and K-theory by Hatcher, given the chern character, $ch$, and first chern class $c_1(H)$, we get:
$ch(H)=e^{c_1(H)}=1+c_1(H)$
I don't understand where all the higher order $c_1(H)$ terms go when expanding the exponential. Could anyone explain this please?
In the proof of Corollary 4.4, on page 101, Hatcher says that for the canonical line bundle on $\textbf{CP}^n$ and $c$ the first Chern class, the Chern character is given by $$ 1+ c +\frac{c^2}{2}+\frac{c^3}{3!}+\cdots + \frac{c^n}{n!}. $$ The reason why this sum stops at $n$ is because the cohomology groups of $\textbf{CP}^n$ are trivial past $H^n$ (and the Chern classes are elements in these groups). If we continue to apply the cup product to an element in $H^n$ (such as $c^n$) and an element in $H^1$ (such as $c$), then we should get an element of $H^{n+1}$, but that group is trivial. Hence we get zero, so all products $c^k$ for $k>n$ are zero, so they disappear in the expansion of the exponential.
Your question is when $n=1$, so everything vanishes for degree 2 and above.