## A new gauge slice for the relative Bauer-Furuta invariants.(English)Zbl 1316.57022

The author of the paper under review studies Manolescu’s construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on $$4$$-manifolds with boundary. He introduces a new gauge fixing condition called the double Coulomb condition for studying these invariants.
Bauer and Furuta constructed their invariant for closed 4-manifolds as an element in the stable cohomotopy group. “The basic idea of the constuction is to consider the Seiberg-Witten map and then to consider approximated maps between finite dimensional vector spaces to obtain a stable map between spheres. In [Geom. Topol. 7, 889–932 (2003; Zbl 1127.57303)], C. Manolescu constructed a Floer spectrum for a rational homology 3-sphere. It comes from a finite dimensional approximation of the Seiberg-Witten flow on an infinite dimensional space. This allows one to extend these invariants to 4-manifolds with a rational homology 3-sphere as a boundary.”
The author tries to provide a framework to generalize these invariants to general 4-manifolds $$X$$ with (arbitrary) boundary $$Y$$. The generalization is for configuration spaces satisfying a certain gauge fixing condition called double Coulomb condition. An advantage of this condition is that the restriction map of the corresponding slice on $$X$$ to the Coulomb slice on $$Y$$ is linear in contrast to the previously used Coulomb-Neumann slice on $$X$$.
The main result (Theorem 1.1) is that when $$b_1(Y)=0$$, the Seiberg Witten map with boundary term on the double Coulomb slice gives an $$S^{1}$$-equivariant stable homotopy class of maps $SWF(X):\Sigma^{-b^{+}(X)}Th_{Dir}(X) \rightarrow SWF(Y)$ where $$SWF(Y)$$ is the Floer spectrum associated to $$Y$$ and $$Th_{Dir}(X)$$ is the Thom spectrum associated to a family of Dirac operators on $$X$$ parametrized by the Picard torus on $$X$$. In the special case $$b_1(X)=0$$ it is shown that $SWF(X):S^{-b^{+}(X)\mathbb R -(\sigma(X)/8) C} \rightarrow SWF(Y)$ where $$S$$ is the sphere spectrum.
The proof depends on finding a map to some order reduced suspension of the Conley Index of a certain set, induced from a map depending on the Seiberg Witten map and the restriction map defined above. Then the result follows with some work and desuspending arguments.

### MSC:

 57R57 Applications of global analysis to structures on manifolds 57R58 Floer homology

Zbl 1127.57303
Full Text:

### References:

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