Cohomotopy invariants and the universal cohomotopy invariant jump formula

J. Math. Sci. Univ. Tokyo
Vol. 15 (2008), No. 3, Page 325--409.

Okonek, Christian ; Teleman, Andrei
Cohomotopy invariants and the universal cohomotopy invariant jump formula
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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.

Mathematics Subject Classification (2000): 57R57, 55Q55.
Mathematical Reviews Number: MR2510627