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I am trying to understand why a fiber bundle over a fiber bundle is a fiber bundle (called composite). This appears in the book "Differential geometric structures" of Poor, p. 9.

There, it is said that if $ E_1 $ is a fiber bundle over $E$ with a bundle chart $(\pi_1,\psi)$ (where $\psi:E_1\to F_1$ is the map "on the fiber", $F_1$ the typical fiber) on $V\subset E$ and $E$ is a fiber bundle over $M$ with a bundle chart $(\pi,\phi)$ over $U\subset M$ with $U\cap \pi(V)\neq \emptyset$, then the map $(\pi\circ \pi_1,\phi\circ \pi_1,\psi)$ is a bundle chart over $W$.

My problem is that I don't even see why, in the last map, $\psi$ is well defined on the inverse image of $W$ in $E_1$.

Also, I haven't found this kind of statement in any more standard mathematical reference (eg from Husemoller or Steenrod); I would be interested if anyone has seen this there too.

Thank you

Amd
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1 Answers1

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After looking at long, I believe the answer is yes, but relies on the notion of connections. This approach is in the books of Sardanashvily (a physicist but the books are readable). I would still be curious to see an answer without using connections.

Amd
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