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. 



Least number of charts to describe a given manifold
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>


