Cohomotopy invariants and the universal cohomotopy invariant jump formula

Abstract

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.

LaTeX, 51 pages. v2: References added. More details in the introduction. v3: New comments about the functorial properties of the groups to which the new invariants belong (and about the functoriality of the Bauer-Furuta classes) have been added. To appear in Journal of Mathematical Sciences the University of Tokyo