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: embedding of smooth manifolds into formal duals of R-algebras

Bottom of Page

1 to 16 of 16

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2015

    I finally gave this statement its own entry, in order to be able to conveniently point to it:

    embedding of smooth manifolds into formal duals of R-algebras

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 10th 2015

    Such a statement is true for manifolds with finitely many or countably many connected components.

    As Theo Johnson-Freyd once pointed out to me, and later expanded in this answer on MO: http://mathoverflow.net/a/91445 it is false for arbitrary paracompact Hausdorff manifolds, in particular, given two uncountable (discrete) sets S and T one can find a morphism of real algebras C^∞(T)→C^∞(S) that is not induced by a function S→T.

    However, the construction is a very subtle set-theoretical argument that uses measurable cardinals.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2015

    Thanks. I have made the standard regularity assumptions explicit in the entry now and added pointer to this MO discussioon.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 11th 2015
    • (edited Mar 11th 2015)

    Ah, I was just wondering what sort of things break for uncountable disjoint unions of second countable manifolds.

    EDIT: I was thinking continuum-many summands, which is still better behaved than for general uncountable coproducts.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020
    • (edited Jul 18th 2020)

    added doi to

    and

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020

    added pointer to Milnor’s original statement

    and to these proofs for the case of isomorphisms:

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 18th 2021

    Added redirect: Milnor duality. To satisfy a link at duality between geometry and algebra.

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 7th 2022

    Added:

    The case of the category of smooth manifolds and diffeomorphisms is proved in

    • Lyle Eugene Pursell, Algebraic structures associated with smooth manifolds, PhD dissertation, Purdue University, 1952. 93 pp. ISBN: 978-1392-88143-9. PDF.

    diff, v16, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2022
    • (edited Dec 8th 2022)

    Interesting that such an early reference exists.

    In trying to check it out on my phone, I only get to see the first 19 pages. Do you mean to say Pursell’s proof covers only diffeos/ring-isos, but not non-invertible maps?

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 8th 2022

    Yes. There is also this announcement by Pursell’s PhD advisor M. E. Shanks: https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf, see page 295.

    They never published any of this, except that they have a similar paper about Lie algebras: https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2022

    Thanks. I have added pointer also to Shanks’s announcement.

    Also, I adjusted the wording of the attribution paragraph (here).

    diff, v18, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2023
    • (edited Oct 21st 2023)

    added pointer to:

    diff, v22, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2024
    • (edited Sep 12th 2024)

    Made the statement of Milnor’s exercise an actual numbered Lemma (now here)

    and then spelled out (here) the proof, from this lemma, of the full embedding statement (following Cor. 35.10 in Kolář, Michor & Slovák 1993)

    diff, v24, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2024

    Seeing that the entry still didn’t have any Idea-section, I added some paragraphs (here).

    diff, v24, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTime4 days ago
    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTime4 days ago

1 to 16 of 16

  1.