Atiyah-Floer conjecture

A conjecture relating the instanton Floer homology of suitable three-dimensional manifolds with the symplectic Floer homology of moduli spaces of flat connections over surfaces, and hence with the quantum cohomology of such moduli spaces. It was originally stated by M.F. Atiyah for homology $3$-spheres in [a1]. The extension of the conjecture to the case of mapping cylinders was prompted by A. Floer and solved in this case by S. Dostoglou and D. Salamon in [a3].

Instanton Floer homology for three-dimensional manifolds was introduced by Floer in [a10]. Let $( Y , P _ { Y } )$ be a pair consisting of a closed oriented $3$-dimensional manifold $Y$ and an $ \operatorname {SO} ( 3 )$-bundle $P_Y \rightarrow Y$. If either $Y$ is a homology $3$-sphere or $b _ { 1 } ( Y ) > 0$ and the second Stiefel–Whitney class $w _ { 2 } ( P _ { Y } ) \neq 0$, then the instanton Floer homology $\operatorname{HF} _ { * } ^ { \operatorname{inst} } ( Y , P _ { Y } )$ is defined as the homology of the Morse-type complex constructed out of the Chern–Simons functional. The critical points are flat connections and the connecting orbits are anti-self-dual connections on $P _ { Y } \times \mathbf{R} \rightarrow Y \times \mathbf{R}$ decaying exponentially to flat connections $A ^ { \pm }$ when $t \rightarrow \pm \infty$.

The symplectic Floer homology for Lagrangian intersections was introduced by Floer in [a11]. Let $( M , \omega )$ be a symplectic manifold which is monotone and simply connected. Let $L_0$ and $L_1$ be Lagrangian submanifolds of $M$. Then there are Floer homology groups $\operatorname{HF} _ { * } ^ { \text{symp} } ( M , L _ { 0 } , L _ { 1 } )$. Now the critical points are the intersection points $x \in L _ { 0 } \cap L _ { 1 }$ and the connecting orbits are $J$-holomorphic strips $u: [ 0,1 ] \times \mathbf{R} \rightarrow M$ with $u ( 0 , t ) \in L _ { 0 }$, $u ( 1 , t ) \in L _ { 1 }$ and $\operatorname { lim } _ { t \rightarrow \pm \infty } u ( s , t ) = x ^ { \pm }$, where $x ^ { \pm } \in L _ { 0 } \cap L _ { 1 }$ and $J$ is an almost-complex structure compatible with the symplectic form.

Let $\Sigma$ be a closed oriented surface of genus $g \geq 1$ and let $P \rightarrow \Sigma$ be the trivial $ \operatorname {SO} ( 3 )$-bundle. Then the moduli space $\mathcal{M} ( P )$ of flat connections on $P$ is symplectic and smooth except at the trivial connection. Now, let $Y = Y _ { 0 } \cup _ { \Sigma } Y _ { 1 }$ be a Heegaard splitting of a homology $3$-sphere and consider the trivial $ \operatorname {SO} ( 3 )$-bundle $P_{ Y}$ on $Y$. Then the flat connections on $\Sigma$ which extend to $Y _ { 0 }$ define a Lagrangian subspace $\mathcal{L} _ { 0 } \subset \mathcal{M} ( P )$, and analogously $\mathcal{L} _ { 1 } \subset \mathcal{M} ( P )$. Taking care of the singularity one may define $\operatorname {HF} _ { * } ^ { \text{symp} } ( \mathcal{M} ( P ) , \mathcal{L} _ { 0 } , \mathcal{L}_ { 1 } )$. The Atiyah–Floer conjecture reads

\begin{equation} \tag{a1} \operatorname{HF} _ { * } ^ { \text { inst } } ( Y , P _ { Y } ) \overset{\simeq}{\rightarrow} \operatorname{HF} _ { * } ^ { \text { symp } } ( {\cal M} ( P ) , {\cal L} _ { 0 } , {\cal L} _ { 1 } ). \end{equation}

This was originally conjectured by Atiyah in [a1]. An overview of the problem appears in [a8]. The problem is still open (as of 2000).

The symplectic Floer homology for a symplectic mapping was introduced by Floer in [a12]. Let $( M , \Sigma )$ be a symplectic manifold which is monotone and simply connected. Let $\phi : M \rightarrow M$ be a symplectomorphism. Then the symplectic Floer homology $\operatorname{HF} _ { * } ^ { \operatorname{symp} } ( M , \phi )$ can be defined as the Morse-type theory where the critical points are the fixed points of $\phi$ and the connecting orbits are $J$-holomorphic strips $u: [ 0,1 ] \times \mathbf{R} \rightarrow M$ with $u ( 1 , t ) = \phi ( u ( 0 , t ) )$ which converge to fixed points $x ^ { \pm }$ of $\phi$ as $t \rightarrow \pm \infty$. For $\phi = id$, Floer proved [a12] that $\operatorname{HF} _ { * } ^ { \text { symp } } ( M , \text { id } ) \cong H ^ { * } ( M )$. Moreover, there is a natural ring structure for the symplectic Floer homology [a8], and in [a7] it is proved that there is an isomorphism of rings $\operatorname{HF} _ { * } ^ { \text{symp} } ( M , \text { id } ) \cong \operatorname{QH} ^ { * } ( M )$, where $ \operatorname{QH} ^ { * } ( M )$ is the quantum cohomology of $M$.

Let $\Sigma$ be a closed oriented surface of genus $g \geq 1$ and let $Q \rightarrow \Sigma$ be the non-trivial $ \operatorname {SO} ( 3 )$-bundle. The moduli space of flat connections ${\cal M} ( Q )$ is a smooth symplectic manifold. Consider the mapping cylinder $Y_f$ of a diffeomorphism $f : \Sigma \rightarrow \Sigma$. This $Y_f$ fibres over the circle $S ^ { 1 }$ with fibre $\Sigma$. Lift $f$ to a bundle mapping $\tilde { f } : Q \rightarrow Q$. This gives an $ \operatorname {SO} ( 3 )$-bundle $Q _ { \widetilde{f} } \rightarrow Y _ { f }$. On the other hand, $\widetilde { f }$ induces a mapping $\phi _ { \tilde{f} } : \mathcal{M} ( Q ) \rightarrow \mathcal{M} ( Q )$. The Atiyah–Floer conjecture for mapping cylinders was proposed by Floer [a4] and reads:

\begin{equation} \tag{a2} \operatorname{HF}^{\text{inst}}_*(Y_f, Q_{\tilde f}) \xrightarrow{\ \simeq\ } \operatorname{HF}^{\text{symp}}_*( \mathcal{M}(Q), \phi_{\tilde f}). \end{equation}

In [a3], Dostoglou and Salamon prove the existence of an isomorphism between these two Floer homologies by constructing an isomorphism at the chain level and identifying the boundary operators. The idea is named adiabatic limit and consists of stretching $Y_f$ in the direction orthogonal to $\Sigma$.

A very important case is that of $\tilde { f } = \operatorname { id}$. Then $Y _ { \operatorname{id} } = \Sigma \times S ^ { 1 }$ and $Q _ { \operatorname{id} } = Q \times S ^ { 1 } \rightarrow \Sigma \times S ^ { 1 }$ is the $ \operatorname {SO} ( 3 )$-bundle with $w _ { 2 } ( Q _ { \operatorname {id} } ) = \operatorname {PD} [ S ^ { 1 } ]$. Therefore,

\begin{equation} \tag{a3} \operatorname{HF}^{\text{inst}}_*(\Sigma \times S^1, Q\times S^1) \xrightarrow{\ \simeq\ } \operatorname{HF}^{\text{symp}}_*( \mathcal{M}(Q), \operatorname{id}) \cong QH ^ { * } ( \mathcal{M} ( Q ) ). \end{equation}

Both Floer homologies have natural product structures, introduced by S.K. Donaldson (see [a8]). A stronger version of the Atiyah–Floer conjecture establishes that (a3) is an isomorphism of rings.

The existence of such an isomorphism has been proved by V. Muñoz in [a5], [a6] by giving an explicit presentation of both rings in terms of the natural generators of the cohomology of ${\cal M} ( Q )$ and using the relationship of instanton Floer homology of $3$-manifolds with Donaldson invariants of $4$-manifolds [a2]. Also, in [a9] Salamon proves that the adiabatic limit isomorphism is indeed a ring isomorphism.

References

[a1]	M.F. Atiyah, "New invariants of three and four dimensional manifolds" Proc. Symp. Pure Math. , 48 (1988)
[a2]	S.K. Donaldson, "On the work of Andreas Floer" Jahresber. Deutsch. Math. Verein. , 95 (1993) pp. 103–120
[a3]	S. Dostoglou, D. Salamon, "Self-dual instantons and holomorphic curves" Ann. of Math. , 139 (1994) pp. 581–640
[a4]	S. Dostoglou, D. Salamon, "Instanton homology and symplectic fixed points" D. Salamon (ed.) , Symplectic Geometry: Proc. Conf. , London Math. Soc. Lecture Notes , 192 , Cambridge Univ. Press (1993) pp. 57–94
[a5]	V. Muñoz, "Ring structure of the Floer cohomology of $Y$" Topology , 38 (1999) pp. 517–528
[a6]	V. Muñoz, "Quantum cohomology of the moduli space of stable bundles over a Riemann surface" Duke Math. J. , 98 (1999) pp. 525–540
[a7]	S. Piunikhin, D. Salamon, M. Schwarz, "Symplectic Floer–Donaldson theory and quantum cohomology" C.B. Thomas (ed.) , Contact and Symplectic Geometry , Publ. Newton Inst. , 8 , Cambridge Univ. Press (1996) pp. 171–200
[a8]	D. Salamon, "Lagrangian intersections, $3$-manifolds with boundary and the Atiyah–Floer conjecture" , Proc. Internat. Congress Math. , 1 , Birkhäuser (1994) pp. 526–536
[a9]	D. Salamon, "Quantum products for mapping tori and the Atiyah–Floer conjecture" Preprint ETH-Zürich (1999)
[a10]	A. Floer, "An instanton invariant for $3$-manifolds" Comm. Math. Phys. , 118 (1988) pp. 215–240
[a11]	A. Floer, "Symplectic fixed points and holomorphic spheres" Comm. Math. Phys. , 120 (1989) pp. 575–611
[a12]	A. Floer, "Morse theory for the symplectic action" J. Diff. Geom. , 28 (1988) pp. 513–547