Cohomotopy invariants and the universal cohomotopy invariant jump formula

2007/04/19 by Okonek, Christian, Teleman, Andrei
#55Q10 #55Q50 #55Q55 #57R57 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0704.2615

Abstract

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S¹-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.

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