View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302902.png" />-spheres in [[#References|[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 [[#References|[a3]]].

Instanton Floer homology for three-dimensional manifolds was introduced by Floer in [[#References|[a10]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302903.png" /> be a pair consisting of a closed oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302904.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302905.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302906.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302907.png" />. If either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302908.png" /> is a homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a1302909.png" />-sphere or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029010.png" /> and the second [[Stiefel–Whitney class|Stiefel–Whitney class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029011.png" />, then the instanton Floer homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029012.png" /> is defined as the homology of the Morse-type complex constructed out of the [[Chern–Simons functional|Chern–Simons functional]]. The critical points are flat connections and the connecting orbits are anti-self-dual connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029013.png" /> decaying exponentially to flat connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029014.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029015.png" />.

The symplectic Floer homology for Lagrangian intersections was introduced by Floer in [[#References|[a11]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029016.png" /> be a [[Symplectic manifold|symplectic manifold]] which is monotone and simply connected. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029018.png" /> be Lagrangian submanifolds of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029019.png" />. Then there are Floer homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029020.png" />. Now the critical points are the intersection points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029021.png" /> and the connecting orbits are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029022.png" />-holomorphic strips <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029028.png" /> is an [[Almost-complex structure|almost-complex structure]] compatible with the symplectic form.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029029.png" /> be a closed oriented surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029030.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029031.png" /> be the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029032.png" />-bundle. Then the moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029033.png" /> of flat connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029034.png" /> is symplectic and smooth except at the trivial connection. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029035.png" /> be a Heegaard splitting of a homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029036.png" />-sphere and consider the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029037.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029039.png" />. Then the flat connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029040.png" /> which extend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029041.png" /> define a Lagrangian subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029042.png" />, and analogously <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029043.png" />. Taking care of the singularity one may define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029044.png" />. The Atiyah–Floer conjecture reads

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

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

The symplectic Floer homology for a symplectic mapping was introduced by Floer in [[#References|[a12]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029046.png" /> be a symplectic manifold which is monotone and simply connected. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029047.png" /> be a symplectomorphism. Then the symplectic Floer homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029048.png" /> can be defined as the Morse-type theory where the critical points are the fixed points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029049.png" /> and the connecting orbits are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029050.png" />-holomorphic strips <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029051.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029052.png" /> which converge to fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029054.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029055.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029056.png" />, Floer proved [[#References|[a12]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029057.png" />. Moreover, there is a natural ring structure for the symplectic Floer homology [[#References|[a8]]], and in [[#References|[a7]]] it is proved that there is an isomorphism of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029059.png" /> is the quantum cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029060.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029061.png" /> be a closed oriented surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029062.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029063.png" /> be the non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029064.png" />-bundle. The moduli space of flat connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029065.png" /> is a smooth symplectic manifold. Consider the [[Mapping cylinder|mapping cylinder]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029066.png" /> of a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029067.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029068.png" /> fibres over the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029069.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029070.png" />. Lift <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029071.png" /> to a bundle mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029072.png" />. This gives an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029073.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029074.png" />. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029075.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029076.png" />. The Atiyah–Floer conjecture for mapping cylinders was proposed by Floer [[#References|[a4]]] and reads:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

In [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029078.png" /> in the direction orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029079.png" />.

A very important case is that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029080.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029082.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029083.png" />-bundle with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029084.png" />. Therefore,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029085.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029086.png" /></td> </tr></table>

Both Floer homologies have natural product structures, introduced by S.K. Donaldson (see [[#References|[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 [[#References|[a5]]], [[#References|[a6]]] by giving an explicit presentation of both rings in terms of the natural generators of the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029087.png" /> and using the relationship of instanton Floer homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029088.png" />-manifolds with Donaldson invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029089.png" />-manifolds [[#References|[a2]]]. Also, in [[#References|[a9]]] Salamon proves that the adiabatic limit isomorphism is indeed a ring isomorphism.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.F. Atiyah,   "New invariants of three and four dimensional manifolds"  ''Proc. Symp. Pure Math.'' , '''48'''  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.K. Donaldson,   "On the work of Andreas Floer"  ''Jahresber. Deutsch. Math. Verein.'' , '''95'''  (1993)  pp. 103–120</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Dostoglou,   D. Salamon,   "Self-dual instantons and holomorphic curves"  ''Ann. of Math.'' , '''139'''  (1994)  pp. 581–640</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V. Muñoz,   "Ring structure of the Floer cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029090.png" />"  ''Topology'' , '''38'''  (1999)  pp. 517–528</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V. Muñoz,   "Quantum cohomology of the moduli space of stable bundles over a Riemann surface"  ''Duke Math. J.'' , '''98'''  (1999)  pp. 525–540</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Salamon,   "Lagrangian intersections, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029091.png" />-manifolds with boundary and the Atiyah–Floer conjecture" , ''Proc. Internat. Congress Math.'' , '''1''' , Birkhäuser  (1994)  pp. 526–536</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Salamon,   "Quantum products for mapping tori and the Atiyah–Floer conjecture"  ''Preprint ETH-Zürich''  (1999)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Floer,   "An instanton invariant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130290/a13029092.png" />-manifolds"  ''Comm. Math. Phys.'' , '''118'''  (1988)  pp. 215–240</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Floer,   "Symplectic fixed points and holomorphic spheres"  ''Comm. Math. Phys.'' , '''120'''  (1989)  pp. 575–611</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A. Floer,   "Morse theory for the symplectic action"  ''J. Diff. Geom.'' , '''28'''  (1988)  pp. 513–547</TD></TR></table>