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nLab > Latest Changes: embedding of smooth manifolds into formal duals of R-algebras

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    • 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

1 to 11 of 11

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