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nLab > Latest Changes: topological vector bundle

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2017
    • (edited May 26th 2017)

    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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2017
    • (edited May 29th 2017)

    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)↪Maps(k n,k n)GL(n,k) \hookrightarrow Maps(k^n, k^n) into the mapping space with its compact-open topology is continuous.)

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 30th 2017

    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)GL(n,k) is an open subspace of End(k n)End(k^n), End(k n)≃k⊗k*End(k^n) \simeq k\otimes k*, and the resulting linear map k→k⊗Maps(k n,k n)k \to k\otimes Maps(k^n,k^n). Here we’d need Maps(k n,k n)Maps(k^n,k^n) as a topological vector space. But, hmm, what sort of fields kk are you allowing? Just ℝ\mathbb{R} and ℂ\mathbb{C}? (and perhaps ℍ\mathbb{H}…)

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 30th 2017

    Ah, I see you did this on the other thread!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2017

    I have spelled out in some detail the proof that topological vector bundles are classified by the relevant Cech cohomology: here.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2017

    I have spelled out more statements and proofs in the section Over closed subspaces

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeFeb 13th 2019
    It is said that k^n is locally compact (as every metric space). Athough it certainly is true that k^n is compact, the argument given here is wrong since metric does not imply local compactness (see e.g. infinite dimensional Hilbert space as a counter-example)
    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 14th 2019

    Removed “as every metric space” (a mistake pointed out in a recent nForum comment).

    diff, v33, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2019

    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)

    diff, v34, current

  10. Corrected two typos in the proof of Lemma ’CoverForProductSpaceWithIntrval’.

    Pierre PC

    diff, v36, current

  11. Added link to smooth functors page, in order to add smooth functors page.

    Owen Lynch

    diff, v39, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2022

    you had added this sentence:

    Constructions in Vect(X)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).

    diff, v40, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 3rd 2022

    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

    • Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951, 1957, 1960.

    diff, v41, current

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2022
    • (edited Jul 20th 2022)

    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.

    • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2022
    • (edited Jul 20th 2022)

    12, 14 Milnor, Stasheff, Characteristic classes, page 32, the definition (which is not fully expanded there) and theorem 3.6.

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2022
    • John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press 1974

    diff, v43, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2022
    • (edited Jul 20th 2022)

    BTW, you can easily grab the formatted bibitem from places like here.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2022
    • (edited Jul 20th 2022)

    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 FinVectFinVect of finite dimensional vector spaces over a topological ground field is canonically a Top-enriched category, and so are hence its product categories FinVect nFinVect^{n}, for n∈ℕn \in \mathbb{N}. Any Top-enriched functor

    F:FinVect n⟶FinVect F \;\colon\; FinVect^n \longrightarrow FinVect

    induces a functorial construction of new topological vector bundles F^(𝒱 1,,⋯,𝒱 n)\widehat{F}(\mathcal{V}_1,, \cdots, \mathcal{V}_n) from any n-tuple (𝒱 1,𝒱 2,⋯,𝒱 n)(\mathcal{V}_1, \mathcal{V}_2 , \cdots, \mathcal{V}_n) of vector bundles over the same base space BB, by taking the new fiber over a point b∈Bb \in B to be (e.g. Milnor & Stasheff 1974, p. 32):

    F(𝒱 1,⋯𝒱 n) b≔F((𝒱 1) b,⋯,(𝒱 n) b). 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:

    diff, v44, current

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