You are looking for the smallest number of open contractible sets needed to cover $M.$ This is so well studied, that it has a name: *the Lyusternik-Shnirelman category of $M$.$ For references, you can look at the nice paper by Gomez-Larranaga, Heil, and Gonzalez-Acuna, or just look at the Wikipedia article.

You are looking for the smallest number of open contractible sets needed to cover $M.$ This is so well studied, that it has a name: *the Lyusternik-Shnirelman category of $M$." For references, you can look at the nice paper by Gomez-Larranaga, Heil, and Gonzalez-Acuna, or just look at the Wikipedia article.