Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian ndimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to Z n ; for each k > 0, the k-torsion subgroup of G is isomorphic to (Z/kZ) n , and the o-minimal cohomology algebra over Q of G is isomorphic to the exterior algebra over Q with n generators of degree one.

We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with the study of semialgebraic homotopy done by H. Delfs and M. Knebusch allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem. PreliminariesFor the rest of the paper we fix an o-minimal expansion R of a real closed field R. We always take 'definable' to mean 'definable in R with parameters'. We take the order topology on R and the product topology on

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

LetMbe an o-minimal expansion of an ordered field. Letφbe a formula in the language of ordered domains. In this note we establish some topological properties which are transferred fromφMtoφRand vice versa. Then, we apply these transfer results to give a new proof of a result ofM. Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.

In [2] o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in W -definable groups -which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.