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w 1. IntroductionOn a compact complex manifold X it is an interesting problem to compare the continuous and holomorphic vector bundles. The case of line-bundles is classical and is well understood in the framework of sheaf theory. On the other hand for bundles E with dimE>dimX we are in the stable topological range and one can use K-theory. Much is known in this direction, for example the topological and holomorphic K-groups of all complex projective spaces are isomorphic. This paper deals with what is perhaps the simplest case not covered by the methods indicated above. We shall consider 2-dimensional complex vector bundles over the 3-dimensional complex projective space P3. Our aim is to prove (1.1) Theorem. Every continuous 2-dimensional vector bundle over P3 admits a holomorphic structure.The corresponding result for P2 was proved by Schwarzenberger [13], but this falls into the class of stable problems. In particular 2-dimensional vector bundles over P2 are determined by their Chern classes c 1 , c 2 . This is no longer true on P3 and therein lies the main difficulty and also the interest of this paper. In fact Horrocks in [-10] has already constructed holomorphic (actually algebraic) bundles with arbitrary cl, c 2 subject only to the topologically necessary condition that c~c 2 be even [8; p. 166].It is not hard to see that, topologically, there are at most two bundles on P3 with given cl, c 2 . The two possibilities arise because the homotopy group n 5 (U(2)) ~ n 5 (S 3) which classifies 2-dimensional bundles over S 6, and acts on the bundles over P3, has order 2. It turns out that there are two sharply different cases depending on the parity of c~.In w 2 we study the case of even c 1 . By tensoring with line-bundles one reduces to the case of cl =0 in which case the structure group is SU(2)~-Sp(1). We view our 2-dimensional complex vector bundle as a quaternion line-bundle and this simplifies the classification because quaternion line-bundles over P3 are already

A constructive proof of the classical theorem of Gel'fand and Kolmogorov (1939) characterising the image of the evaluation map from a compact Hausdorff space X into the linear space C(X) * dual to the ring C(X) of the continuous functions on X is given. Our approach to the proof enabled us to obtain a more general result characterising the image of an evaluation map from the symmetric products Sym n (X) into C(X) * . A similar result holds if X = C m and leads to explicit equations for symmetric products of affine algebraic varieties as algebraic sub-varieties in the linear space dual to the polynomial ring. This leads to a better understanding of the algebra of multi-symmetric polynomials.The proof of all these results is based on a formula used by Frobenius in 1896 when defining higher characters of finite groups. This formula had no further applications for a long time; however, it occurred in several independent contexts during the last fifteen years. The formula was used by A. Wiles and R. L. Taylor when studying representations and by H.-J. Hoehnke and K. W. Johnson and later by J. McKay when studying finite groups. It plays an important role in our work concerning multi-valued groups. We describe several properties of this remarkable formula. We also use it to prove a theorem on the structure constants of Frobenius algebras, which have recently attracted attention due to constructions taken from topological field theory and singularity theory. This theorem develops a result of H.-J. Hoehnke published in 1958. As a corollary, we obtain a direct self-contained proof of the fact that the 1-, 2-, and 3-characters of the * The present paper is based on the talk given by E. G. Rees at the conference "Kolmogorov and contemporary mathematics," Moscow, 2003.

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