Isomorphism between Frøyshov’s limiting monopole Floer homology and Kronheimer–Mrowka’s monopole Floer homology

Establish a natural isomorphism between the limiting monopole Floer homology group constructed by Frøyshov for a rational homology three-sphere Y—obtained by letting the chamber parameter m_s tend to infinity so that the resulting homology is an invariant of Y in all degrees—and one of the Kronheimer–Mrowka flavors of monopole Floer homology for the same three-manifold.

Background

In the paper the author reviews Frøyshov’s construction of monopole Floer homology for rational homology three-spheres and explains that, in an allowable range of degrees, the homology groups HM_q are invariants of the underlying 3-manifold. Frøyshov proved a stronger result: by letting the chamber parameter m_s go to infinity, the resulting homology group becomes an invariant of the manifold independent of homological degree.

The author notes that it is conjectured that this chamber-stabilized group agrees with one of the standard flavors of monopole Floer homology developed by Kronheimer–Mrowka. Establishing this identification would clarify the relationship between Frøyshov’s chamber-stabilized construction and the established Kronheimer–Mrowka framework.