Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 29th 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)kk*End(k^n) \simeq k\otimes k*, and the resulting linear map kkMaps(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 \mathbb{R} and \mathbb{C}? (and perhaps \mathbb{H}…)

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 29th 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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)