Least number of charts to describe a given manifold - MathOverflow

Least number of charts to describe a given manifold

2009-10-26T14:00:44Z

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.

Answer by Andy Putman for Least number of charts to describe a given manifold

2009-10-26T14:07:12Z

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

Answer by Sonia Balagopalan for Least number of charts to describe a given manifold

2009-10-26T16:05:17Z

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

Answer by Reid Barton for Least number of charts to describe a given manifold

2009-10-26T16:54:28Z

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.

Answer by Aaron Mazel-Gee for Least number of charts to describe a given manifold

2009-10-26T19:18:18Z

I believe that Cech cohomology could yield the sort of answer you're looking for (at least in the contractible case). The general idea is that it computes cohomology based on nothing but the so-called "incidence data" of a good cover (that is, which n-fold intersections of open sets in the cover are nonempty) -- in fact, a n-chain is nothing more than a formal sum of (nonempty) (n+1)-fold intersections, with R coefficients. Of course, this cohomology theory is usually isomorphic to singular cohomology, de Rham cohomology, et al. (in particular, they agree in the case of manifolds). So if your manifold has a high h¹=rank(H¹), for example, then there must be lots of different 2-fold intersections to generate H¹, and also if your manifold has nonzero H^k then there must exist a (k+1)-fold intersection of sets in the open cover, which means that you must have at least that many sets in any good cover. (Note that if M is a k-dimensional orientable manifold, then H^k(M)=R by Poincare duality.)

Answer by Giuseppe for Least number of charts to describe a given manifold

2011-02-04T19:25:40Z

I do not know if the following exactly answers your question.

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.

Answer by Al-burcas for Least number of charts to describe a given manifold

2011-06-17T12:19:56Z

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.