# Least number of charts to describe a given manifold

Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.

E.g. a circle requires at least two charts, and so on (I couldn't manage to get anything relevant neither on wikipedia nor on google, so I guess I'm lacking the correct terminology).

Edit: in the case of an open covering of a topological space by n+1 contractible sets (in that space) then n is called the Lusternik-Schnirelman Category of the space, see Andy Putman's answer. The following book seems to be the standard reference http://books.google.fr/books?id=vMREfNN-L4gC&pg=PP1

Great, now I'm still interested by the initial question: does anybody know of another theory without this contractibility assumption (hoping that it allows more freedom)? e.g. would it lead to different numbers say for genus-g surfaces?

Final edit: yes different numbers for genus-g surfaces (see answers below), but not sure there is a theory without contractibility. Right, really lots of interesting literature on the LS category nevertheless, hence the accepted answer. For example there are estimates for non-simply connected compact simple Lie groups like PU(n) and SO(n) in Topology and its Applications, Volume 150, Issues 1-3, 14 May 2005, Pages 111-123.

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It's not quite the same thing, but a related object is the Lyusternik–Schnirelmann category of a topological space. See

http://en.wikipedia.org/wiki/Lyusternik-Schnirelmann_category

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It's pretty close, thanks! –  Thomas Sauvaget Oct 26 '09 at 14:25
Is there some reason this has the same name as the "category" that has objects and morphisms, or is this purely a case of the same English word getting used for two mathematical objects? –  Michael Lugo Oct 26 '09 at 14:34
The concepts are unrelated. The notion of the L-S category predated the "category-theoretic" notion of a category. I suspect that its inspiration comes from the Baire category theorem. –  Andy Putman Oct 26 '09 at 14:47

To answer your last question, the least number of charts needed to cover any orientable 2-manifold is 2. Consider the usual embedding of an orientable surface Σ in R3 which is symmetric across the plane z = 0 (as shown here), and let ε > 0 be sufficiently small. The open subsets Σ ∩ {z > -ε}, Σ ∩ {z < ε} form a covering of Σ by charts: by Morse theory Σ ∩ {z > -ε} is diffeomorphic to Σ ∩ {z > ε}, which is diffeomorphic to an open subset of R2 by projecting onto the xy-plane.

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Note that the two open sets in Reid's answer may not be made contractible in general. Indeed, for orientable 2-manifolds, 3 contractible sets are required, except for S^2, where 2 are required. –  j.c. Oct 26 '09 at 17:53
Yeah, I figured out after asking, so really the LS category measures something different, thanks! –  Thomas Sauvaget Oct 26 '09 at 18:06

I have found on the second page of Michor "Topics in Differential Geometry": "Note finally that any manifold $M$ admits a finite atlas consisting of $\dim{M}+1$ not connected charts. This is a consequence of topological dimension theory [cf. Nagata, Modern Dimension Theory]; a proof for manifolds may be found in [cf. Greub, Halperin, Vanstone, Connections, curvature and cohomology.I]."

I hope to have been useful.

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Yes, sounds like interesting references, and the result is intuitively satisfying, thanks! I'll try to find a library that has them... –  Thomas Sauvaget Feb 5 '11 at 21:19

After "dimension" this is the most basic numerical invariant of a manifold and the least explored. I found this reference some years ago: I. Bernstein, "On Imbedding Numbers of Differentiable Manifolds", Topology, Vol. 7, pp. 95-109.

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Wow, thanks for pointing out this reference! Mike Hopkins has also written on this concept: springerlink.com.ezproxy.webfeat.lib.ed.ac.uk/content/… In particular I think he answers the OP's question for (almost) all projective spaces. –  Mark Grant Jun 17 '11 at 12:52

Orthogonal question: Does the (minimum) number of charts needed to describe a manifold tell you anything about the manifold?

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Is this appropriate as an answer? I would have commented on the question if I had the rep. –  Sonia Balagopalan Oct 26 '09 at 16:15
For the L-S category, having an upper bound on the category can bound the complexity of your manifold (e.g. in terms of cohomology) from above –  j.c. Oct 26 '09 at 17:31
The L-S category gives a lower bound on the number of critical points of a functional on your manifold. That's the big abstract theorem that Lusternik and Schnirelmann proved. –  Jeff Strom Jul 29 '10 at 13:38