Key points are not available for this paper at this time.
The object of this note will be to give an upper bound for the sum of the Betti numbers of a real affine algebraic variety. (Added in proof. Similar results have been obtained by R. Thom lO. ) Let F be a variety in the real Cartesian space Rm, defined by polynomial equations /i (xi, ■ ■ ■, xm) = 0, ■ • •, fP (xx, ■ ■ ■, xm) = 0. The qth Betti number of V will mean the rank of the Cech cohomology group Hq (V), using coefficients in some fixed field F. Theorem 2. If each polynomial f, has degree S k, then the sum of the Betti numbers of V is ᵏ (2k — l) m-1. Analogous statements for complex and/or projective varieties will be given at the end. I wish to thank W. May for suggesting this problem to me. Remark A. This is certainly not a best possible estimate. (Compare Remark B. ) In the examples which I know, the sum of the Betti
John Milnor (Wed,) studied this question.