Monopole-instanton Floer equivalence (Kronheimer-Mrowka)

Advanced results Master

The phrase "monopole-instanton Floer equivalence" should be read carefully. Kronheimer and Mrowka built a detailed foundation for monopole Floer homology in Monopoles and Three-Manifolds ^{[Kronheimer-Mrowka Monopoles]}. They also developed instanton Floer variants, especially in sutured and knot settings, and proved excision theorems that make those theories usable in low-dimensional topology ^{[Kronheimer-Mrowka Sutures]}. This is a strong structural parallel between monopole and instanton packages, but it is not a single blanket theorem saying every instanton Floer group is canonically isomorphic to every monopole Floer group.

The monopole side starts from the Seiberg-Witten equations rather than the ASD equation. Its critical points are solutions of three-dimensional Seiberg-Witten equations, and its flow lines are four-dimensional Seiberg-Witten trajectories on cylinders. Kronheimer and Mrowka organize the theory into several flavors, with exact triangles, cobordism maps, gradings, and contact-topological applications.

The instanton side starts from flat connections and ASD trajectories, as in 03.07.23. Kronheimer and Mrowka's later instanton work constructs versions adapted to knots, links, sutured manifolds, and singular bundles. In these settings, instanton Floer theory can interact with Khovanov homology, contact geometry, and decompositions of three-manifolds.

The common message is that the major three-manifold Floer theories form a comparison landscape. Monopole Floer homology is known to match Heegaard Floer homology and embedded contact homology through major comparison theorems. Instanton Floer homology has parallel formal properties and many analogous constructions, but it also has distinct features, especially over integral coefficients and in singular bundle variants.

Kronheimer and Mrowka's 2010 paper Knots, sutures, and excision develops monopole and instanton Floer homology groups for balanced sutured manifolds and proves excision results ^{[Kronheimer-Mrowka Sutures]}. Excision is central because it allows cut-and-paste arguments, supports knot-complement constructions, and makes sutured decompositions compatible with Floer groups. It is a bridge theorem, not a universal identification theorem.

Their work on Khovanov homology and instantons showed that singular instanton knot homology is powerful enough to detect the unknot ^{[Kronheimer-Mrowka Khovanov]}. This placed instanton Floer theory in the same modern ecosystem as Heegaard and monopole knot invariants, where exact triangles, spectral sequences, sutured decompositions, and contact invariants are compared across theories.

For the present curriculum, this unit is a modern landscape pointer after instanton Floer homology 03.07.23, Atiyah-Floer 03.07.26, and polyfolds 03.07.27. The main takeaway is not a formula to use in computations. It is the recognition that instanton, monopole, Heegaard, and embedded-contact Floer theories are linked by shared structures, comparison theorems, and conjectural or partial correspondences.

Synthesis. The foundational reason monopole and instanton Floer theories are compared is that both arise from gauge-theoretic elliptic equations on three- and four-manifolds. They produce functorial packages with cobordism maps, exact triangles, gradings, and gluing laws. The central insight of the Kronheimer-Mrowka program is that monopole and instanton methods can be developed in parallel for sutured manifolds and knots, giving powerful cut-and-paste tools. The corrected modern statement is therefore a comparison landscape: monopole Floer theory is part of the established equivalence web with Heegaard Floer and ECH, while instanton Floer theory is a closely related gauge-theoretic package with parallel structures and important bridges, rather than a fully interchangeable copy in every coefficient system and variant.

Connections Master

• Instanton Floer homology 03.07.23. This is the instanton-side baseline for the comparison.

• Relative Donaldson invariants 03.07.24. Cobordism maps are the shared formal language for gauge-theoretic Floer theories.

• Donaldson-Floer surgery exact triangle 03.07.25. Exact triangles are one of the formal structures that make cross-theory comparison meaningful.

• Atiyah-Floer conjecture 03.07.26. Atiyah-Floer compares instanton Floer theory with symplectic Floer theory; monopole comparison places instantons in the broader low-dimensional Floer web.

• Polyfolds 03.07.27. Abstract transversality frameworks address analytic issues that recur across instanton, monopole, and symplectic Floer theories.

Historical & philosophical context Master

Donaldson theory and instanton Floer homology came first historically in this strand, using ASD equations and flat connections. Seiberg-Witten theory then changed four-manifold topology by replacing ASD connections with monopole equations that are often more computable.

Kronheimer and Mrowka's Monopoles and Three-Manifolds gave a comprehensive construction of monopole Floer homology ^{[Kronheimer-Mrowka Monopoles]}. It became one of the core foundations for Seiberg-Witten Floer theory and its three-manifold applications.

Their 2010 sutured paper developed monopole and instanton Floer theories for balanced sutured manifolds and proved excision theorems ^{[Kronheimer-Mrowka Sutures]}. This work made both theories more flexible for knot complements and decomposition arguments. It also clarified how closely the two gauge-theoretic packages can run in parallel.

The broader 2000s and 2010s comparison story includes Heegaard Floer homology, monopole Floer homology, and embedded contact homology. Ozsvath and Szabo introduced Heegaard Floer homology as a holomorphic-disk invariant of three-manifolds ^{[Ozsvath-Szabo]}. Subsequent comparison programs related Heegaard Floer, monopole Floer, and ECH in deep ways ^{[Kutluhan-Lee-Taubes / Colin-Ghiggini-Honda / Taubes]}.

Philosophically, this landscape says that low-dimensional topology has several analytic languages for similar topological information. The differences matter: coefficients, gradings, functoriality, reducibles, and singular-bundle choices can change what the theory sees. The value of the comparison viewpoint is that it separates shared structure from theory-specific strength.

Bibliography Master

@book{KronheimerMrowka2007Monopoles,
  author = {Kronheimer, Peter B. and Mrowka, Tomasz S.},
  title = {Monopoles and Three-Manifolds},
  series = {New Mathematical Monographs},
  volume = {10},
  publisher = {Cambridge University Press},
  year = {2007}
}

@article{KronheimerMrowka2010Sutures,
  author = {Kronheimer, Peter B. and Mrowka, Tomasz S.},
  title = {Knots, sutures, and excision},
  journal = {Journal of Differential Geometry},
  volume = {84},
  pages = {301--364},
  year = {2010}
}

@article{KronheimerMrowka2011Khovanov,
  author = {Kronheimer, Peter B. and Mrowka, Tomasz S.},
  title = {Khovanov homology is an unknot-detector},
  journal = {Publications mathematiques de l'IHES},
  volume = {113},
  pages = {97--208},
  year = {2011}
}

@article{OzsvathSzabo2004Heegaard,
  author = {Ozsvath, Peter and Szabo, Zoltan},
  title = {Holomorphic disks and topological invariants for closed three-manifolds},
  journal = {Annals of Mathematics},
  volume = {159},
  pages = {1027--1158},
  year = {2004}
}