Not signed in (Sign In)

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).

nLab > Latest Changes: topological vector bundle

Bottom of Page

1 to 20 of 20

  1.  
    • 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

    • CommentRowNumber19.
    • CommentAuthorTokarak
    • CommentTimeJan 13th 2025

    There is currently a problem with the Slice category construction. I will list the problems in order of importance:

    1. At the “local triviallity” section:
    U× Topℝ n⟶≃U× XE, U \times_{Top} \mathbb{R}^n \overset{\simeq}{\longrightarrow} U \times_X E \,,

    “for some n”. Supposedly this reads as “there exists an open cover U≔⊔i∈IU iU \coloneqq \underset{i \in I}{\sqcup} U_i (equipped with disjoint union of inclusion maps because we are in Top /XTop_{/X}) such that there exists a non-negative integer n such that there exists a linear isomorphism”. However, this is equivalent to saying that the rank of the vector bundle is constant across the module, which isn’t necessarily true.

    1. Since we are not working in Top /XTop_{/X}, but in the category Vect kVect_{k} internal to Top/XTop/X, which I will call VT XVT_{X}, we should say that the above isomorphism is in (VT X) /U(VT_{X})_{/U} with U→XU \rightarrow X being a zero-dimensional vector space. Right now, the concept that we have a “linear isomorphism” is ambiguous because we have to go through the same ordeal of defining the canonical vector space structure on objects in Top/UTop/U as we already did for Top/XTop/X and then having to somehow prove or visualise that this is the same structure, when we could just carry everything over from VT XVT_X by composing an object in VT X/U→XVT_{X}_{/ U \rightarrow X} with U→XU \rightarrow X itself to form an object of VT XVT_X. This is using that we have an equivalence of categories (Top /X) /(U→X)=Top /U(Top_{/X})_{/(U \rightarrow X)}=Top_{/U}.

    2. Here is how I would construct the category of topological vector bundles over X. I’m still learning and, apart from potential beginner mistakes, I want to resolve before editing the page: defining an open cover categorically.

    Let VT XVT_X be the category of k-vector spaces (Vect kVect_k) internal to the ambient category of Top/XTop / X for some topological space X∈TopX \in Top. Vect(X)Vect(X) is defined to be a full subcategory of VT XVT_X, where an object E∈VT XE \in VT_X is also in Vect(X)Vect(X) if there exists an open cover of XX as the family {U i⊆X} i∈I\{U_i \subseteq X\}_{i \in I} regarded in Top/XTop/X as the family of inclusion maps {U i↪inc iX} i∈I\{U_i \xhookrightarrow{inc_i} X\}_{i \in I} and also a family of non-negative integers indexed by the same set {n i} i∈I\{n_i\}_{i \in I} such that there exists an isomorphism (of vector spaces) in VT XVT_X: U i× Topk n i≃U i× XEU_i \times_Top k^{n_i} \simeq U_i \times_X E. The LHS has vector space structure from K n iK^{n_i} and has the map to X given by projection to U iU_i composed with incl iincl_i; the RHS has vector space structure transfered from EE and is the pullback of functions incl iincl_i and EE.

    I believe this to be a full construction of Vect(X)Vect(X). One of the advantages of this construction is that some properties of Vector bundles are quite easily derived from properties of VectVect, for example, once you prove that Vect(X)Vect(X) is a cartesian category with the cartesian product given by the cartesian product of Top/XTop/X, then, since VectVect is a cartesian closed category, we have the Vect(X)Vect(X) is also cartesian closed! We should also be able to generalise pretty easily to the global category of F.D topological vector bundles over all topological spaces by using the arrow category of TopTop rather than a slice of TopTop.

    Please implement some or all of these suggestions! Thanks.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2025
    • (edited Jan 13th 2025)

    Thanks for looking into this.

    Regarding the “problem”:

    That UU denotes the disjoint union of patches in an open cover is stated in the item just before, so I am not sure on this point what issue you see. (?) But I admit I have only glanced over the entry and your message for now. (It’s late here, need to call it a day.)

    Regarding the definition of “nn”: True, this was a little mix-up. I have now edited to declare n:I→ℕn \colon I \to \mathbb{N} and made the product with ℝ n\mathbb{R}^n a fiber product over II.

    Apart from this I don’t see a substantial issue?

    I guess your motivation is to give a more streamlined account, for which there is probably room.

    You could go ahead and add your version below the existing one, introduced by a line like like:

    Another way to say this is the following, which has the advantage that…

    diff, v48, current

1 to 20 of 20

  1.