On homology of real algebraic sets
Given a compact, connected, smooth (i.e. \(C^{\infty})\) surface, not diffeomorphic to the two-sphere, real projective plane, or Klein bottle, we construct a nonsingular real algebraic surface diffeomorphic to it whose first homology group with coefficients in the integers modulo 2 is not generated by algebraic cycles. Such algebraic surfaces diffeomorphic to smooth, orientable surfaces of positive genus \(\leq 9\) and of genus 1 have previously been found by Risler and Silhol, respectively [\textit{J. J. Risler}, Géométrie algébrique réelle et formes quadratiques, Journ. S.M.F., Univ. Rennes 1981, Lect. Notes Math. 959, 381-385 (1982; Zbl 0503.14014) and \textit{R. Silhol}, ibid. 443-450 (1982)]. The higher dimensional case is also considered.
scientific article; zbMATH DE number 3871542
scientific article