Localisation of The

This article is a rst step in establishing a link between the Donaldson polynomials and Seiberg-Witten invariants of a smooth 4-manifold. As the result of analysis of the structure of the Donaldson's invariant of a smooth simply connected 4 manifold X Kronheimer and Mrowka K-M1] (see also F-S]) proved the existence of k X classes such that the Donaldson's invariant of X is deened uniquely by these pairs. These classes are called the basic classes of X and the set fK i g; for i = 1; : : :; k X ; in H 2 (X; Z) is a diieomorphism invariant. So the diieomorphism group DiffX admits a representation to the symmetric group on k X letters. A little bit later Seiberg and Witten introduced another set of classes fSWg and polynomials and Witten predicted the shape of the Donaldson's invariant in terms of these data. S-W classes are labelling the moduli spaces fM SW g of the solutions of the Seiberg-Witten system of equations (see x1) and the dimensions of these moduli spaces are the degrees of the polynomials. Using other type of system of equations we will prove that the classes of Kronheimer and Mrowka and Seiberg-Witten coincide. This paper is the rst step of this proof and contains those facts from gauge theory which we need for the proof. At various stages of this project useful were conversations with Ed Witten, S.Donaldson, S.Bauer. We are grateful to S.Donaldson who invited us to the conference at the Newton Institute in December 1994 where we described these results and to I. Hambleton for his kind invitation to the MPI, Bonn. The rst author acknowledges with gratitude nancial support of the AMS (fSU grant) and the VW-Stiftung during the work over this project as well as the kind invitation of S.Bauer and the hospitality of Bielefeld University.