# Where is the compatibility condition in the definition of a vector bundle?

The following definition is given for a vector bundle in Milnor and Stasheff's Characteristic Classes.

A real vector bundle $$\xi$$ over $$B$$ consists of the following:

1. a topological space $$E=E(\xi)$$ called the total space,
2. a map $$\pi:E \rightarrow B$$ called the projection map
3. for each $$b\in B$$ the structure of a vector space over the real numbers in the set $$\pi^{-1}(b)$$/

These must satisfy the following restriction: For each point $$b$$ of $$B$$ there should exist a neighborhood $$U\subset B$$, an integer $$n\geq 0$$, and a homeomoprhism $$h:U\times \mathbb{R}^n \rightarrow \pi^{-1}(U)$$ so that, for each $$b\in U$$, the correspondence $$x\mapsto h(b,x)$$ defines an isomorphism between the vector space $$\mathbb{R}^n$$ and the vector space $$\pi^{-1}(b)$$.

I am confused. It seems that there is no compatibility condition imposed on overlaps. I would have expected the definition to include something like the following:

and if $$h_1:U_1\times \mathbb{R}^n \rightarrow \pi^{-1}(U_1)$$ and $$h_2:U_2\times \mathbb{R}^n \rightarrow \pi^{-1}(U_2)$$ are two such homeomorphisms then we have that $$h_1 \mid_{U_1\cap U_2} = h_2 \mid_{U_1\cap U_2}$$

I do not feel comfortable with the definition. I should also try make this post into more of a question.

1. Is the definition given in Milnor and Stasheff complete?
2. If the definition is complete, have I simply overlooked where the compatibility is stated?
3. How does compatibility and transition maps arise from the definition given?

The definition you quote is correct. There is no compatibility requirement of the sort you are asking for. Indeed, that is kind of the whole point of the notion of a vector bundle: it is a bundle over $$B$$ that locally looks like just the product with $$\mathbb{R}^n$$, but these local fiberwise linear homeomorphisms to $$U\times\mathbb{R}^n$$ are not necessarily compatible and so cannot necessarily be glued together to give a global fiberwise linear homeomorphism $$E\cong B\times\mathbb{R}^n$$.

It may be helpful to see where transition maps arise from this definition. The transition map between two of these local trivializations $$h_1:U_1\times\mathbb{R}^n\to\pi^{-1}(U_1)$$ and $$h_2:U_2\times\mathbb{R}^n\to\pi^{-1}(U_2)$$ arises simply from the composition $$h_1^{-1}\circ h_2$$, which maps $$(U_1\cap U_2)\times\mathbb{R}^n$$ to itself. Since $$h_1$$ and $$h_2$$ are linear isomorphisms on each fiber, so is $$h_1^{-1}\circ h_2$$, so it can be written in the form $$(x,v)\mapsto (x,f_{12}(x)v)$$ where $$f_{12}$$ is some function $$U_1\cap U_2\to GL_n(\mathbb{R})$$ (and this function is what is usually called the "transition map"). Now these transition maps will satisfy the compatibility condition that $$f_{12}(x)f_{23}(x)=f_{13}(x)$$ for all $$x\in U_1\cap U_2\cap U_3$$, simply because $$(h_1^{-1}\circ h_2)\circ(h_2^{-1}\circ h_3)=h_1^{-1}\circ h_3$$. And conversely, it turns out that if you start with a family of functions $$f_{ij}:U_i\cap U_j\to GL_n(\mathbb{R})$$ satisfying this compatibility condition (for some open cover $$(U_i)$$ of $$B$$) then you can use them to build a vector bundle on $$B$$ for which they are transition maps. But this is very different from asking for a compatibility between the local trivializations $$h_i$$ themselves.

It would counterproductive to require compatibility conditions as in your question. The requirement

If $$h_1:U_1\times \mathbb{R}^n \rightarrow \pi^{-1}(U_1)$$ and $$h_2:U_2\times \mathbb{R}^n \rightarrow \pi^{-1}(U_2)$$ are any two trivializing homeomorphisms, then $$h_1 \mid_{U_1\cap U_2} = h_2\mid_{U_1\cap U_2}$$

would imply that

1. For a given open $$U \subset B$$ we are allowed to choose at most one trivializing homeomorphism $$h_U$$. For some $$U$$ there may be no such $$h_U$$, for other $$U$$ exactly one.
This would be unproblematic, we would get a very special "bundle atlas".

2. The open $$U$$ which carry a trivializing homeomorphism $$h_U$$ form an open cover of $$B$$. By your compatibility condition they can be pasted to a global trivializing homeomorphism $$h_B : B \times \mathbb R^n \to \pi^{-1}(B) = E$$.
This means that we would consider only trivial vector bundles.

Milnor and Stasheff's definition is the standard one. Its benefit is that it allows non-trivial bundles. These are needed to get a non-trivial theory. In fact the set of isomorphism classes of vector bundles over $$B$$ contains essential information about the homotopy type of $$B$$.

Update:

Eric Wofsey rightly points out that there plenty of naturally occurring vector bundles (like the tangent bundle of a smooth manifold) which are locally trivial, but globally non-trivial. A general theory of vector bundles should of course cover all these objects.

• I would add that they are needed not just to get a nontrivial theory (e.g., to make K-theory useful for something) but because many very important naturally occurring bundles (e.g., tangent bundles) are nontrivial and we want to study them within a general framework. Commented Mar 8, 2022 at 0:19