2 1 Ja n 20 09 Description of the vector G - bundles over G - spaces with quasi - free proper action of discrete group

We give a description of the vector G-bundles over G-spaces with qusi-free proper action of discrete group G in terms of the classifying space. 1 The setting of the problem This problem naturally arises from the Conner-Floyd's description ([2]) of the bordisms with the action of a group G using the so-called fix-point construction. This construction reduces the problem of describing the bordisms to two simpler problems: a) description of the fixed-point set (or, more generally, the stationary point set), which happens to be a submanifold attached with the structure of its normal bundle and the action of the same group G, however, this action could have stationary points of lower rank; b) description of the bordisms of lower rank with an action of the group G. We assume that the group G is discrete.