On the Betti numbers of real varieties
John Milnor · Proceedings of the American Mathematical Society · 1964
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