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I have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here
I have spelled out further elementary detail at topological vector bundle.
In (what is now) the section Transition functions I have added a detailed argument that the thing which is glued from the transition functions of a vector bundle is indeed isomorphic to that vector bundle.
Then in (what is now) the section Basic properties I have spelled out a detailed proof that a homomorphism of topological vector bundles is an isomorphism as soon as it is a fiberwise linear isomorphism.
(I was trying to be really explicit, maybe in contrast to what Hatcher offers. The only thing I should still add for completeness is at general linear group the statement that the inclusion $GL(n,k) \hookrightarrow Maps(k^n, k^n)$ into the mapping space with its compact-open topology is continuous.)
The only thing I should still add for completeness is at general linear group the statement that the inclusion GL(n,k)↪Maps(kn,kn) GL(n,k) \hookrightarrow Maps(k^n, k^n) into the mapping space with its compact-open topology is continuous.)
I wonder if one can see this using the fact $GL(n,k)$ is an open subspace of $End(k^n)$, $End(k^n) \simeq k\otimes k*$, and the resulting linear map $k \to k\otimes Maps(k^n,k^n)$. Here we’d need $Maps(k^n,k^n)$ as a topological vector space. But, hmm, what sort of fields $k$ are you allowing? Just $\mathbb{R}$ and $\mathbb{C}$? (and perhaps $\mathbb{H}$…)
Ah, I see you did this on the other thread!
I have spelled out in some detail the proof that topological vector bundles are classified by the relevant Cech cohomology: here.
I have spelled out more statements and proofs in the section Over closed subspaces
Sorry for not reacting earlier. I’d rather we fix an explanation than just removing it. So I have made it this:
(like every finite dimensional vector space, by the Heine-Borel theorem)
you had added this sentence:
Constructions in $Vect(X)$ can be acheived by means of smooth functors, which represent the constructions on vector spaces that can be applied fiberwise to vector bundles.
I am not sure that “smooth functor” is a good term here in the page on topological vector bundles.
Probably you want to refer to “natural operations” (to be created) as in Kolar-Michor-Slovak?
I have moved the sentence to a Remark with that title (now here).
Added:
This result is apparently due to Steenrod, see Theorem 11.4 in \cite{Steenrod}.
Added:
The original reference for many results about bundles, including the theorem that concordance implies isomorphism, is
12: In traditional literature on (topological) vector bundles over paracompact Hausdorff spaces there is a discussion of “continuous functors” (on several covariant and several contravariant vector space variables) on vector spaces (definition in terms of graph of the functor) like exterior product etc. which automatically induce functors on products of categories of vector bundles on the space. This is not the same (I think) as a (newer definition of a) natural operation in the sense of Kolar-Michor-Slovak as the tangent bundle does not make sense in that generality (no manifolds in the game!). I guess some considered also smooth functor in the same sense. Most examples are the same, but I believe “continuous” allows some more possibilities.
12, 14 Milnor, Stasheff, Characteristic classes, page 32, the definition (which is not fully expanded there) and theorem 3.6.
BTW, you can easily grab the formatted bibitem from places like here.
But thanks for insisting. So I have removed the previous remark and instead added one now titled Fiberwise Operations.
Currently it reads as follows:
The category $FinVect$ of finite dimensional vector spaces over a topological ground field is canonically a Top-enriched category, and so are hence its product categories $FinVect^{n}$, for $n \in \mathbb{N}$. Any Top-enriched functor
$F \;\colon\; FinVect^n \longrightarrow FinVect$induces a functorial construction of new topological vector bundles $\widehat{F}(\mathcal{V}_1,, \cdots, \mathcal{V}_n)$ from any n-tuple $(\mathcal{V}_1, \mathcal{V}_2 , \cdots, \mathcal{V}_n)$ of vector bundles over the same base space $B$, by taking the new fiber over a point $b \in B$ to be (e.g. Milnor & Stasheff 1974, p. 32):
$F \big( \mathcal{V}_1, \cdots \mathcal{V}_n \big) _b \;\coloneqq\; F \big( (\mathcal{V}_1)_b, \cdots, (\mathcal{V}_n)_b \big) \,.$For example:
if $F \,\coloneqq\, (-)^\ast \,\colon\, FinVect \longrightarrow FinVect$ is the operation of forming dual vector spaces, then $\widehat{F}$ constructs the fiberwise dual vector bundle;
if $F \,\coloneqq\, det \,\colon\, FinVect \longrightarrow FinVect$ is the operation of forming determinants, then $\widehat{F}$ is the construction of fiberwise determinant line bundles;
if $F \,\coloneqq\, \oplus \,\colon\, FinVect^2 \longrightarrow FinVect$ is the direct sum of vector space, then $\widehat{F}$ constructs the fiberwise direct sum of vector bundles (“Whitney sum”);
if $F \,\coloneqq\, \otimes \,\colon\, FinVect^2 \longrightarrow FinVect$ is the tensor product of vector spaces, then $\widehat{F}$ constructs the fiberwise tensor product of vector bundles.
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