I'm currently studying with the Book "From Calculus to Cohomology" by Madsen & Tornehave(free PDF here). Unfortunately I am really struggling to understand the Example 18.14 where Chern classes are used to show that $\mathbb{CP}^{2n}$ is not the boundary of any $4n+1$ dimensional Manifold $R$.
(Here $\tau_{M}$ denotes the tangent bundle for a given Manifold $M$.)
Unfortunately, I'm struggling with almost every point of the calculation where sums are used, which I will list below:
Let $\partial R$ be the boundary of $R$, then we have, that: $$ \tau_{\partial R}\oplus \epsilon^1_{\mathbb{R}} = i^{\ast}(\tau_R) $$ where $\epsilon^1_{\mathbb{R}}$ denotes the trivial real line bundle. (I have no idea how to even approach this, but have the feeling that it's extremely trivial..)
Now, if we assume $\partial R = \mathbb{CP}^{2n}$, and complexify, we get $$ (\tau_{\mathbb{CP}^{2n}})_{\mathbb{RC}} \oplus \epsilon^1_{\mathbb{C}} = i^\ast(\tau_R \otimes \mathbb{C}) $$ (i have no idea how to see this.)
The complexification of a real vector bundle $\eta$ is given by $$ \eta_{\mathbb{C}} = \eta \otimes_{\mathbb{R}} \epsilon^1_{\mathbb{C}},$$ where $\epsilon^1_{\mathbb{C}}$ is the trivial complex line bundle.
Unfortunately I have almost no experience with tensor products and sums, which leaves me kind of confused here. The rest of the argument, I also don't understand.. but I hope that if this part becomes clear, I can piece together the rest..