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Prove that $\tau(\Bbb{C}P^n)\oplus1=\eta\otimes\ldots\otimes\eta$ (n+1 times), where $1$ denotes the trivial complex line bundle over $\Bbb{C}P^n$, $\tau(\Bbb{C}P^n)$ is the complex tangent bundle of $\Bbb{C}P^n$ and $\eta$ denotes the Hopf bundle $p:S^{2n+1}\rightarrow\Bbb{C}P^n$ with fiber $F=S^1$.

I think I have to show that there is a fiber bundle equivalence, so I have to show that there is a common fiber. The fiber of the LHS is $\Bbb{C}^n\oplus\Bbb{C}=\Bbb{C}^{n+1}$ and the fiber of the RHS is ${S^1}\otimes\ldots\otimes{S^1}$. I don't understand how they can be equal since ${S^1}\otimes\ldots\otimes{S^1}$ seems to be bounded whereas $\Bbb{C}^{n+1}$ is not bounded. Also I don't understand how to take the tensor product of $S^1$ as it is not a vector space under normal addition and multiplication. Can someone help me understand?

Austin
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    You will find people somewhat interchangeable talk about a vector bundle and its associated principal bundle. Instead of the whole fiber $\Bbb C$, we only need to know the unit circle $S^1$ inside of $\Bbb C$. This is analogous to how a real line bundle is determined by its associated $O(1)={\pm1}$-bundle. Once you know where $1$ and $-1$ are, you can fill in $\Bbb R$ by linearity, and vice versa. We sort of do this intuitively when we draw a the Mobius line bundle with arrows rotating around a circle. The arrows only show where $+1$ is. – pancini Jan 22 '22 at 05:36

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