Algebraische und topologische reelle Zykeln unter birationalen Transformationen. (Algebraic and topological real cycles under birational transformations.) (Q1071078)

Let X be complete smooth variety defined over a real closed field R. If Y is an algebraic k-cycle containing a regular rational point, then \(| Y|\) represents a homology class in \(H_ k(X(R),{\mathbb{Z}}/2{\mathbb{Z}})\). A fundamental problem in real algebraic geometry is to determine which classes are represented in this way - i.e. to determine the graded ring of homology modulo real algebraic homology. Let \(T_*(X)\) be this ring and \(\pi\) : \(\tilde X\to X\) be a blowing-up with smooth irreducible center Z of codimension d. The author uses the work of Borel-Haefliger and H. Delfs, together with the Leray-Hirsch theorem, various characteristic classes, and the observation that the fibre product of \(\tilde X\) with Z over X is a projective bundle over Z to prove that \(T_ k(\tilde X)=T_ k(X)\oplus \oplus^{d-2}_{i=0}T_{k-d+1- i}(Z).\) It follows from Hironaka's work that \(T_*(X)\) is a birational invariant if X is a smooth projective real surface and that \(T_ 1(X)\) is a birational invariant for any smooth projective real variety. For \(k=n-1\), \(T_{n-1}(X)\) is homology modulo divisor classes and the result yields \(T_{n-1}(\tilde X)=T_{n-1}(X)\oplus ({\mathbb{Z}}/2{\mathbb{Z}})^{s- 1}\), where s is the number of connected components of Z(R).