Atiyah-Floer conjecture

Intuition Beginner

The Atiyah-Floer conjecture says that two different machines should produce the same Floer homology for a three-manifold.

One machine is gauge-theoretic. It studies connections on the three-manifold and counts instanton trajectories between flat connections.

The other machine cuts the three-manifold along a surface. Each half gives a Lagrangian shape inside a symplectic space attached to the cut surface. Then Lagrangian Floer homology counts strips between those two shapes.

The conjecture says these counts are two views of the same invariant.

Visual Beginner

The left side is the gauge picture on the whole three-manifold. The right side is the symplectic picture after cutting along a surface.

Worked example Beginner

Think of a three-manifold made by gluing two handlebodies along the same boundary surface. A flat connection on the whole glued manifold restricts to compatible flat connections on both handlebodies.

In the surface representation space, the first handlebody gives one Lagrangian and the second handlebody gives another. An intersection point of the two Lagrangians is a compatible pair of flat boundary data, so it corresponds to a flat connection on the glued three-manifold.

What this tells us: the generators on both sides are meant to match. The hard part is showing that the differentials, gradings, and continuation maps also match.

Check your understanding Beginner

Formal definition Intermediate+

Let $$ Y=H_0\cup_\Sigma H_1 $$ be a Heegaard splitting of a closed oriented three-manifold along a closed oriented surface $\Sigma$. The surface character variety $$ \mathcal R(\Sigma)=\operatorname{Hom}(\pi_1\Sigma,\mathrm{SU}(2))/\mathrm{SU}(2) $$ has a natural symplectic structure on its smooth irreducible stratum, due to Goldman ^[Goldman].

Restriction of flat connections from the handlebodies to the boundary surface gives maps $$ \mathcal R(H_i)\longrightarrow \mathcal R(\Sigma), $$ and the images are Lagrangian subspaces on the smooth part under suitable irreducibility and transversality hypotheses.

The Atiyah-Floer conjecture predicts an isomorphism $$ HF^{\mathrm{inst}}*(Y)\cong HF^{\mathrm{Lag}}*(\mathcal R(H_0),\mathcal R(H_1);\mathcal R(\Sigma)), $$ with the correct bundle, perturbation, orientation, and grading conventions included. In the instanton setting, the grading is usually mod eight; on the symplectic side, Maslov-index data must reproduce the same grading.

Counterexamples to common slips

• The surface character variety is usually singular. The conjecture is not a statement about a smooth compact symplectic manifold without corrections.
• The Lagrangians are not chosen freely. They come from restriction of flat connections on the two handlebodies.
• Matching generators is only the first step. The conjecture also requires matching differentials, continuation maps, orientations, and gradings.

Key theorem with proof Intermediate+

Proposition (restriction images are isotropic). Let $H$ be a handlebody with boundary $\Sigma$. On the smooth irreducible stratum of the character variety, the image of $\mathcal{R}(H)$ in $\mathcal{R}(\Sigma )$ is isotropic for the Goldman symplectic form. Under the expected dimension condition, it is Lagrangian.

Proof. At a flat connection $\rho$, the tangent space to the surface character variety is modeled by $$ H^1(\Sigma;\operatorname{ad}\rho). $$ Goldman's symplectic form is the cup product paired with the invariant inner product on $\mathfrak{su}(2)$ and evaluated on the fundamental class of $\Sigma$.

A tangent vector coming from the handlebody lies in the image of $$ H^1(H;\operatorname{ad}\rho)\to H^1(\Sigma;\operatorname{ad}\rho). $$ If two such tangent vectors both extend over $H$, their cup product on $\Sigma =\partial H$ is the boundary pairing of a class extending over the three-manifold $H$. The relevant boundary evaluation vanishes by the long exact sequence of the pair $(H,\Sigma )$ and Poincare-Lefschetz duality. Hence the restriction image is isotropic.

For a handlebody of genus $g$, the expected dimension of the handlebody representation space is half the expected dimension of the surface representation space on the irreducible stratum. Isotropic plus half-dimensional gives Lagrangian. $\square$

Bridge. The Atiyah-Floer conjecture builds from 03.07.23 by replacing instanton trajectories on $\mathbb{R}\times Y$ with pseudoholomorphic strips in a symplectic character variety. The symplectic inputs are 05.05.01 and 05.08.02, where Lagrangian submanifolds and Floer homology are developed independently. The foundational reason is that a Heegaard splitting turns flat connections on $Y$ into intersections of two boundary restriction images. The central insight is that the analytic equation should also degenerate: ASD instantons on a stretched three-manifold should limit to holomorphic curves in the surface moduli space.

Exercises Intermediate+

Advanced results Master

The conjecture is a dictionary between two infinite-dimensional Morse theories. The instanton side uses the Chern-Simons functional on the space of connections over $Y$; its gradient trajectories are ASD connections on $\mathbb{R}\times Y$. The symplectic side uses the action functional for paths between two Lagrangians inside a symplectic character variety; its gradient trajectories are pseudoholomorphic strips.

Atiyah's proposal was that a Heegaard splitting should make these two pictures equivalent ^[Atiyah]. The generator-level match is geometric: an intersection point of the two Lagrangians is a flat surface connection extending over both handlebodies, hence a flat connection on the glued three-manifold. The differential-level match is analytic: instanton trajectories should converge, after stretching the metric along $\Sigma$, to holomorphic strips with boundary on the two restriction Lagrangians.

The surface character variety needs care. On the irreducible locus, Goldman's construction gives a symplectic form using cup product in twisted cohomology ^[Goldman]. But reducibles produce singular strata, and low-genus cases can require special handling. Many modern approaches replace the naive character variety by an extended moduli space, impose holonomy constraints, use admissible bundles, or work with quilted Floer theory and symplectic correspondences.

Dostoglou and Salamon proved a related adiabatic-limit theorem for mapping tori: self-dual instantons on certain four-manifolds converge to pseudoholomorphic curves in moduli spaces of flat connections ^{[Dostoglou-Salamon]}. This result is not the full Heegaard-splitting conjecture, but it is a central piece of evidence because it establishes the kind of analytic bridge Atiyah predicted.

Synthesis. The foundational reason the Atiyah-Floer conjecture is plausible is that both sides are Morse theories for the same flat-connection data viewed at different scales. Cutting $Y$ along $\Sigma$ turns flat connections into intersections of two Lagrangian restriction images; stretching the metric should turn ASD flow lines into holomorphic strips. The central insight is not only an isomorphism of groups but a proposed equivalence of field-theoretic descriptions: gauge theory on a three-manifold and symplectic topology of a surface character variety should encode the same Floer invariant.

Full proof set Master

Proposition 1 (generator correspondence under transversality). Suppose the restriction images $\mathcal{R}(H_0)$ and $\mathcal{R}(H_1)$ intersect transversely in the smooth irreducible part of $\mathcal{R}(\Sigma )$. Then their intersection points correspond to irreducible flat connections on $Y=H_0{\cup }_{\Sigma }{H}_1$ with the same boundary convention.

Proof. An intersection point is a representation of ${\pi }_1\Sigma$ that extends to both ${\pi }_1{H}_0$ and ${\pi }_1{H}_1$. Since $Y$ is obtained by gluing the handlebodies along $\Sigma$, the van Kampen presentation of ${\pi }_{1}Y$ is the amalgamated product of the two handlebody groups over the surface group. The two extensions that agree on ${\pi }_1\Sigma$ therefore define a representation of ${\pi }_{1}Y$. Conversely, any representation of ${\pi }_{1}Y$ restricts to compatible representations on the two handlebodies. Passing to conjugacy classes gives the stated correspondence on the irreducible smooth locus. $\square$

Proposition 2 (formal matching of expected dimensions). Under the smooth irreducible assumptions, the expected dimension of the Lagrangian intersection problem agrees with the expected dimension of the flat-connection problem on $Y$.

Proof. The tangent complex for the flat-connection problem on $Y$ is controlled by twisted cohomology of $Y$. The tangent spaces to the two handlebody restriction images are the images of twisted cohomology from $H_0$ and $H_1$ in the twisted cohomology of $\Sigma$. The Mayer-Vietoris sequence for $Y=H_0{\cup }_{\Sigma }{H}_1$ identifies the failure of the two tangent images to intersect transversely with the same cohomological obstruction that appears in the deformation theory of the glued flat connection. Thus the Fredholm index on the instanton side and the Maslov-type index on the Lagrangian side have the same formal source. $\square$

Proposition 3 (adiabatic limit as the analytic bridge). A chain-level Atiyah-Floer comparison would follow from compactness, gluing, orientation, and perturbation results identifying zero- and one-dimensional ASD trajectory moduli spaces with the corresponding pseudoholomorphic-strip moduli spaces.

Proof. The instanton differential counts isolated ASD trajectories, while the Lagrangian Floer differential counts isolated pseudoholomorphic strips. If stretching the metric along $\Sigma$ gives a compactness theorem from ASD trajectories to strips, a gluing theorem from strips back to ASD trajectories, and an orientation-preserving bijection in dimensions zero and one, then the two differentials count the same signed objects after identifying generators. Compatibility in dimension one gives equality of boundary degenerations, so the resulting chain maps commute with differentials and induce an isomorphism on homology. $\square$

Connections Master

• Instanton Floer homology 03.07.23. This is the gauge-theoretic left side of the conjecture.

• Donaldson-Floer surgery exact triangle 03.07.25. A complete comparison should preserve structural features such as exact triangles and cobordism maps.

• Lagrangian submanifold 05.05.01. The two handlebody restriction images are expected to be Lagrangians in the surface character variety.

• Floer homology in symplectic geometry 05.08.02. The right side is Lagrangian Floer homology for those two Lagrangians.

• Polyfolds 03.07.27. Modern transversality frameworks are part of the broader landscape for making such comparison statements precise.

Historical & philosophical context Master

Atiyah formulated the conjectural bridge as part of his 1988 vision for new invariants of three- and four-manifolds ^[Atiyah]. The proposal was striking because it related Yang-Mills gauge theory to symplectic topology before Floer theory had settled into its modern family of parallel theories.

Goldman's symplectic structure on surface character varieties supplied the symplectic target needed by the conjecture ^[Goldman]. Donaldson's exposition later placed the conjecture beside instanton Floer homology and relative invariants as a guiding structural problem ^[Donaldson].

Dostoglou and Salamon's adiabatic-limit theorem for mapping tori gave major evidence that ASD equations can limit to holomorphic-curve equations in moduli spaces of flat connections ^{[Dostoglou-Salamon]}. Later programs, including quilted Floer and symplectic-correspondence approaches, refined the dictionary but did not turn the naive statement into a universally packaged theorem.

Bibliography Master

@incollection{Atiyah1988NewInvariants,
  author = {Atiyah, Michael F.},
  title = {New invariants of 3- and 4-dimensional manifolds},
  booktitle = {The Mathematical Heritage of Hermann Weyl},
  series = {Proceedings of Symposia in Pure Mathematics},
  volume = {48},
  pages = {285--299},
  year = {1988}
}

@article{Goldman1984Symplectic,
  author = {Goldman, William M.},
  title = {The symplectic nature of fundamental groups of surfaces},
  journal = {Advances in Mathematics},
  volume = {54},
  pages = {200--225},
  year = {1984}
}

@article{DostoglouSalamon1994,
  author = {Dostoglou, Stamatis and Salamon, Dietmar A.},
  title = {Self-dual instantons and holomorphic curves},
  journal = {Annals of Mathematics},
  volume = {139},
  pages = {581--640},
  year = {1994}
}

@book{Donaldson2002Floer,
  author = {Donaldson, Simon K.},
  title = {Floer Homology Groups in Yang-Mills Theory},
  series = {Cambridge Tracts in Mathematics},
  volume = {147},
  publisher = {Cambridge University Press},
  year = {2002}
}