I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological manifold and $E$ is another topological manifold with a surjective continuous map $\pi:E\to M$.Then we say $E$ is a vector bundle of rank $k$ over $M$ if the following properties are satisfied:
$1.$ For each $p\in M$ the fiber $E_p=\pi^{-1}(p)$ is a $k$-dimensional vector space.
$2.$(Local Trivialization)For each $p\in M$ there exists an open nbhd $U$ of $p$ in $M$ such that there is a homeomorphism $\varphi:\pi^{-1}(U)\to U\times \mathbb R^k$ such that $\pi_1\circ \varphi=\pi$ ( $\pi_1:U\times \mathbb R^k\to U$ being the projection map onto $U$.)
$3.$ For each $q\in U$,the map $\varphi|_{E_q}:\pi^{-1}(q)\to \{q\}\times \mathbb R^k\cong\mathbb R^k$ is an isomorphism of vector spaces.
Then spaces $E$ and $M$ are referred to as the total space and the base space and $\pi:E\to M$ is called the bundle map/projection.
I am not quite satisfied with the conditions in this definition.Being a beginner I am curious about whether all of these conditions are needed or we can drop some of them as they might be redundant.My doubt centres around the first and the third conditions.In the first condition we are already saying that each fiber lying over some point is a copy of $\mathbb R^k$ but in the third condition,we are again saying that the map $\varphi|_{E_q}:E_q\to \{q\}\times\mathbb R^k$ is a vector space isomorphism.So,given the third condition,the first one should be automatically implied.I do not understand why at all we are putting the first condition then?
This is a source of dissatisfaction in my understanding of vector bundles.Can someone give me examples to show that both $(1)$ and $(3)$ are essential by giving examples where one of them holds and the other doesn't.Looking eagerly to my MathStackExchange folks for answers and suggestions.
First of all, the assumption that $M$ and $E$ are manifolds is too strong: There are many reasons to consider vector bundles over topological spaces which are not manifolds. Second, once you assume that $M$ is a topological manifold, Property 2 will imply that $E$ is a manifold as well. Thus, in what follows, I will not assume that $M$ is a manifold, it is totally irrelevant for the discussion.
Now, let's start with Property 2 instead of Property 1: It says that $E\to M$ is a (locally trivial) fiber bundle with fibers homeomorphic to ${\mathbb R}^k$. However, if we consider "transition maps" for pairs $(U, \varphi), (V,\psi)$, namely, maps $$ \varphi \circ \psi^{-1}: (U\cap V)\times {\mathbb R}^k\to (U\cap V)\times {\mathbb R}^k, $$ Property 2 only ensures that for all $x\in U\cap V$, $$ \theta_{x}=\varphi \circ \psi^{-1}: \{x\}\times {\mathbb R}^k \to \{x\}\times {\mathbb R}^k $$ is a homeomorphism that does not preserve, a priori, any further structure on the vector space $\{x\}\times {\mathbb R}^k$.
The point of Properties 1 and 3 is to ensure that the "transition maps" $\theta_x$ are linear. My personal preference is to replace Properties 1 and 3 by a single condition:
Property 13: For all $(U, \varphi), (V, \psi)$ and $x\in U\cap V$, the maps $\theta_x$ as above are linear.
This allows one to equip each fiber $E_x$ with the structure of a vector space via transfer of the structure using the homeomorphism $\varphi^{-1}: \{x\}\times {\mathbb R}^k\to E_x$. Property 13 implies that this structure of a vector space is independent of a "chart" $(U, \varphi)$. With this approach, the collection of "charts" $(U, \varphi)$ becomes a part of the data of a vector bundle (just like an atlas with smooth maps is a datum defining a differentiable structure on a topological manifold).
Now, back to Properties 1 and 3. Property 1 is badly stated, it should be formulated as
"Each $E_x$ comes equipped with the structure of a vector space."
Note that the dimension $k$ assumption is irrelevant here (it is a consequence of Property 2). What's important is the fact that such a vector space structure on the fibers is a part of the data of a vector bundle (just like the map $\pi$ is a part of the bundle data). If one omits Property 1, then Property 3 simply does not make sense since you cannot talk about a linear isomorphism between two spaces, one of which is just a topological space homeomorphic to ${\mathbb R}^k$ (in view of Property 2) and the other is ${\mathbb R}^k$. Now, if you omit Property 3, then Property 1 contains no information (besides one which is already in Property 2); if one uses Property 1 as a part of a vector bundle data, the vector space structures on the fibers $E_x$, a priori, vary discontinuously and, hence, this datum is useless for topological purposes. The point of Properties 1 and 3 combined is to ensure that fibers $E_x$ have structure of vector spaces (Property 1) which "depends continuously on $x$" (this is what Property 3 formalizes).
One last thing: One can ask if Property 2 alone implies existence of a vector bundle structure on a topological bundle satisfying Property 2. This turns out not to be the case, as follows, for instance, from
Browder, William, Open and closed disk bundles, Ann. Math. (2) 83, 218-230 (1966). ZBL0148.17503.