Intersection theory for o-minimal manifolds

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