# Cohomotopy invariants and the universal cohomotopy invariant jump formula

Okonek, C; Teleman, A (2008). Cohomotopy invariants and the universal cohomotopy invariant jump formula. Journal of Mathematical Sciences (Tokyo), 15(3):325-409.

## 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.

## 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.

## Statistics

### Citations

1 citation in Web of Science®
1 citation in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Airy differential equation, hyperbolic Schwarz map, flat front, swallowtail singularity English 2008 09 Nov 2009 02:24 18 Feb 2018 11:16 University of Tokyo 1340-5705 Green http://journal.ms.u-tokyo.ac.jp/abstract/jms150301.html http://arxiv.org/abs/0704.2615

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