The aim of the present paper is to study definably compact abelian groups. Its structure is as follows: In Section 2 the authors introduce definable covering maps and their groups of definable covering transformations, and prove their basic properties. In Section 3 they introduce the concept of o-minimal cohomology of a definable set. It is proved that the o-minimal cohomology with rational coefficients of a definably connected definable group can be equipped with the structure of a Hopf-algebra which is isomorphic to a finitely generated exterior algebra over the field of rationals. The notion of degree of a definable map and its properties and applications are the subject of Section 4. In Section 5 an o-minimal version of the Poincaré-Hurewitz theorem (Theorem 5.1) is proved and the above results are applied to establish the main result of the paper:
Let \(G\) be a definably compact definably connected abelian definable group of dimension \(n\). Then, a) the o-minimal fundamental group of \(G\) is isomorphic to \(Z^{n}\); b) for each \(k\succ 0,\) the \(k\)-torsion subgroup of \(G\) is isomorphic to \( (Z/kZ)^{n}\), and c) the o-minimal cohomology rational algebra of \(G\) is isomorphic to the exterior rational algebra with \(n\) generators of degree one.