Intersection theory for o-minimal manifolds

✍ Scribed by Alessandro Berarducci; Margarita Otero

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 233 KB
• Volume: 107
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(00)00027-0

✦ Synopsis

We develop an intersection theory for deÿnable C p -manifolds in an o-minimal expansion of a real closed ÿeld and we prove the invariance of the intersection numbers under deÿnable C p -homotopies (p ¿ 2). In particular we deÿne the intersection number of two deÿnable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer ÿxed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under deÿnable C p -homotopies. A. Pillay has shown that any deÿnable group admits an abstract manifold structure. We apply the intersection theory to deÿnable groups after proving an embedding theorem for abstract deÿnably compact C p -manifolds. In particular using the Lefschetz ÿxed point theorem we show that the Lefschetz number of the identity map on a deÿnably compact group, which in the classical case coincides with the Euler characteristic, is zero.