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**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.

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.

Reviewer: Saibal Ganguli (Chennai)

### Citations:

Zbl 1127.57303
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\textit{T. Khandhawit}, Geom. Topol. 19, No. 3, 1631--1655 (2015; Zbl 1316.57022)

### References:

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