Ordered abelian groups over a CW complex

Abstract

If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X). In the present work we establish a topological classification of such bundles in terms of the first cohomology group of X with coefficients in the ring Z_2. This result has an amazing application in the theory of characteristic classes of foliations on compact manifolds.