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Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and rarely if ever seems to play any role in differential geometry/topology.

There is a much better definition of differentiable manifolds, which I don't know a good textbook reference for, via sheaves of local rings. This definition does not involve any strange arbitrary choices, and is coordinate free. Paragraph 3 in Wikipedia (which is the actual definition) states:

A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is a sheaf of local $R$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.

This confuses me, because I don't see why such a sheaf should be acyclic, or where conditions like "paracompact" or "complete metric space" or "second countable Hausdorff" are implicit. So either:

  1. The wikipedia entry has a mistake (I would want to sanity-check this before editing the entry, because this is such a fundamental definition which thousands must have read).
  2. Somewhere in that definition, the condition that $M$ be paracompact is implicit.
Question: Should the definition above indeed require that $M$ be second-countable Hausdorff or paracompact or whatever? Or is it somehow implicit somewhere, and if so, where?

Also, is this definition given carefully in any textbook?

Update: I have editted the Wikipedia article to require that $M$ be second-countable Hausdorff. But I'm still wondering if there is a textbook covering this stuff, and whether requiring the sheaf to be acyclic might have worked instead as an alternative.

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Does being locally isomorphic to $(\mathbb{R}^n,\mathcal{O})$ require the topology to be paracompact, second-countable, and Hausdorff? –  Neal Feb 9 '12 at 3:57
    
What about "when O_M is an acyclic sheaf of local R-algebras". Would that also do the job? –  Daniel Moskovich Feb 9 '12 at 5:43
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@Qiaochu: being locally isomorphic to $(\mathbb{R}^n,\mathcal{O})$ certainly does not imply that the space is Hausdorff. There are standard examples of "non-Hausdorff differentiable manifolds". For a slightly non-standard example, see p. 4 of math.uga.edu/~pete/modularcurves.pdf -- these are notes for lectures I gave last week in a course on modular curves. The better part of a week was spent nailing down conditions for the quotient under a group action to be Hausdorff! –  Pete L. Clark Feb 9 '12 at 6:09
    
@Pete: whoops. Of course. –  Qiaochu Yuan Feb 9 '12 at 6:30
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3 Answers 3

up vote 19 down vote accepted

To expand a bit on my comment above:

Being isomorphic as a locally ringed space to $(\mathbb{R}^n,\mathcal{O})$ doesn't impose additional conditions on the underlying topological space of a locally ringed space beyond requiring it to be locally homeomorphic to $\mathbb{R}^n$. (Well, that's a lie: a differentiable structure does of course place limitations on the topology of a manifold, but only very subtle ones: it doesn't impose either of the limitations you are asking about. See below!)

Thus, if you want your definition of a manifold to include Hausdorff and second countable and/or paracompact, you had better put that in explicitly. (And, although it's a matter of taste and terminology, in my opinion you do want this.)

I think you will find these lecture notes enlightening on these points. In particular, on page 4 I give an example (taken from Thurston's book on 3-manifolds!) of a Galois covering map where the total space is a manifold but the quotient space is not Hausdorff. (When I gave this example I mentioned that I wish someone had told me that covering maps could destroy the Hausdorff property! And indeed the audience looked suitably shaken.)

With regard to your other question ("Also, is this definition given carefully in any textbook?")...I completely sympathize. When I was giving these lectures I found that I really wanted to speak in terms of locally ringed spaces! See in particular Theorem 9 in my notes, which contains the unpleasantly anemic statement: "If $X$ has extra local structure, then $\Gamma \backslash X$ canonically inherits this structure." What I really wanted to say is that if $\pi: X \rightarrow \Gamma \backslash X$, then $\mathcal{O}_{\Gamma \backslash X} = \pi_* \mathcal{O}_X$! (I am actually not the kind of arithmetic geometer who has to express everything in sheaf-theoretic language, but come on -- this is clearly the way to go in this instance: that one little equation is worth a thousand words and a lot of hand waving about "local structure".)

What is even more ironic is that my course is being taken by students almost all of whom have taken a full course on sheaves in the context of algebraic geometry. But whatever differential / complex geometry / topology they know, they know in the classical language of coordinate charts and matrices of partial derivatives. It's really kind of a strange situation.

I fantasize about teaching a year long graduate course called "modern geometry" where we start off with locally ringed spaces and use them in the topological / smooth / complex analytic / Riemannian categories as well as just for technical, foundational things in a third course in algebraic geometry. (As for most graduate courses I want to teach, improving my own understanding is a not-so-secret ulterior motive.) In recent years many similar fantasies have come true, but this time there are two additional hurdles: (i) this course cuts transversally across several disciplines so implicitly "competes" with other graduate courses we offer and (ii) this should be a course for early career students, and at a less than completely fancy place like UGA such a highbrow approach would, um, raise many eyebrows.

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I wish I had taken a class like your proposed "modern geometry" class. Off hand are there any books you would recommend (aside from your forthcoming lecture notes :-)) for this unified treatment? –  Willie Wong Feb 9 '12 at 9:22
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Thank you for this answer. Some points of confusion: 1) Which was my other question? 2) Would requiring the sheaf O_M to be acyclic solve the problem? The course sounds interesting- I wish I could listen in :). Maybe you could upload video lectures if you give it? Also, indeed sheaf theory language somehow seems more natural and enlightening for fundamentals of differential geometry, for me at the moment anyway. –  Daniel Moskovich Feb 9 '12 at 10:05
    
@Willie: I have no (existing!) texts to recommend for this, but other people have recommended some texts in their answers. (Wait, I thought of something: Wells's Differential analysis on complex manifolds has at least some of the desired material.) They are worth looking into. –  Pete L. Clark Feb 9 '12 at 14:00
    
@Daniel: I edited my answer to clarify which question I was referring to. Also, off the top of my head I don't see why requiring acyclicity of the structure sheaf should help, but I don't really know or have any particular insight there. If this is a serious question, you may want to ask again (maybe on MO: it sounds "research level" to me). –  Pete L. Clark Feb 9 '12 at 14:03
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I asked the sheaf question on MO: mathoverflow.net/questions/88056/… –  Daniel Moskovich Feb 10 '12 at 1:35
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1) Godement has written a book on Lie groups where he defines and uses manifolds through sheaves.
This is not surprising since he wrote a treatise on sheaf theory nearly sixty years ago, which surprisingly is still the standard reference on the subject.
The sheaf theory is very easy, since the structural sheaf is a ring of functions and thus automatically separated (= satisfies first axiom for a presheaf to be a sheaf).

The book has no English translation (to my knowledge), but if you can overcome that hurdle you will be able to savour Godelment's inimitably idiosyncratic style, as well as the expertise of this great mathematician .

2) Since you mention acyclicity, let me remark that it does not follow from either definition but is a theorem.
It is a consequence of the existence of partitions of unity, which implies that the structural sheaf $\mathcal C^k_M$ is fine, hence acyclic.
However partitions of unity require $M$ to be paracompact, which might be an argument for including paracompactness (or equivalent conditions) in the definition.

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Thanks!! Glancing briefly through Godement's book, isn't this the "structure sheaf" definition? en.wikipedia.org/wiki/Differentiable_manifold#Structure_sheaf –  Daniel Moskovich Feb 9 '12 at 12:56
    
If I demand that the sheaf of local R-algebras be fine, and local isomorphism to $(\mathbb{R}^n,O)$, does that imply that M must be paracompact Hausdorff? (at least for M connected?) If so, that looks like a nice alternative formulation. –  Daniel Moskovich Feb 9 '12 at 13:04
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Dear @Daniel: 1) yes, Godement's definition is the structure sheaf definition. I mentioned his book because you asked "is this definition given carefully in any textbook ?" and Godement is a very careful mathematician, as you can expect from a collaborator of Bourbaki. 2) Whether you use the atlas definition or the sheaf definition, it's up to you to demand that a manifold be Hausdorff or paracompact: it doesn't follow from either definition. 3) I don't know if it is sufficient to assume that the structure sheaf be acyclic to ensure paracompactness. It would indeed be nice if it were the case –  Georges Elencwajg Feb 9 '12 at 15:22
    
Thanks you very much for this answer! I'll ask about whether sheaf conditions suffice on MO. –  Daniel Moskovich Feb 9 '12 at 23:29
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I'm afraid I don't know the answer to your main question, but I would like to mention a textbook that approaches manifolds from the sheaf-theoretic perspective: Ramanan's Global Calculus.

He explicitly includes the Hausdorff + second-countable conditions, defining a manifold as follows:

Definition. A differential manifold $M$ (of dimension $n$) consists of

a) a topological space which is Hausdorff and admits a countable base for open sets, and

b) a sheaf $\mathcal{A}^M=\mathcal{A}$ of subalgebras of the sheaf of continuous functions on $M$.

These are required to satisfy the following local condition. For any $x\in M$, there is an open neighborhood $U$ of $x$ and a homeomorphism of $U$ with an open set $V$ in $\mathbb{R}^n$ such that the restriction of $\mathcal{A}$ to $U$ is the inverse image of the sheaf of differentiable functions on $V$.

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Thanks! This looks like a version of the "structure sheaf" definition. en.wikipedia.org/wiki/Differentiable_manifold#Structure_sheaf –  Daniel Moskovich Feb 9 '12 at 5:23
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