ON CALCULATION OF THE BETTI NUMBERS OF A DIFFERENTIABLE MANIFOLD WHICH IS KNOWN ONLY BY ONE OF ITS ATLASSES,

This work provides a method for the calculation of the Betti numbers of a differentiable manifold which is compact and orientable, and which is known only by means of one of its atlasses. A system of maps and of passage homeomorphisms from one map to another is given. It is shown how the atlas uniquely determines the manifold. The solution of the problem uses only procedures which can always be carried out numerically, in spite of being somewhat complicated analytically. Moreover, in calculation of the quantities which occur in the solution of the problem, we shall use only elements provided by our knowledge of the atlas. The calculation of the numbers involves the determination of a numerical sequence whose terms are constructed using the above criteria. It is shown that this sequence converges. Author