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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
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.
Thanks. I have made the standard regularity assumptions explicit in the entry now and added pointer to this MO discussioon.
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.
added doi to
and
added pointer to Milnor’s original statement
and to these proofs for the case of isomorphisms:
Janusz Grabowski, Isomorphisms and ideals of the Lie algebras of vector fields, Inventiones mathematicae volume 50, pages 13–33 (1978) (doi:10.1007/BF01406466)
Jerrold Marsden, Ratiu, Abraham, Theorem 4.2.36 in: Manifolds, tensor analysis, and applications, Springer 2003 (ISBN:978-1-4612-1029-0)
Janusz Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. 85 (2005), 190-196 (arXiv:math/0310295)
Added redirect: Milnor duality. To satisfy a link at duality between geometry and algebra.
Added:
The case of the category of smooth manifolds and diffeomorphisms is proved in
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?
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.
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