  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 $b_1=0$; they are equivalent when
$b_1=0$ and $b_+>1$, but are finer in the case $b_1=0$, $b_+=1$ (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.
