Cohomotopy invariants and the universal cohomotopy invariant jump formula

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S 1 -equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphisms of 4-manifolds. Our invariants and the Bauer-Furuta classes are directly comparable for 4-manifolds with b1 =0 ;they are equivalent when b1 =0 and b+ > 1, but are finer in the case b1 =0 ,b+ =1 (they detect the wall-crossing phenomena). We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants.