812:). It may be viewed as the Morse homology of the ChernâSimonsâDirac functional on U(1) connections on the three-manifold. The associated gradient flow equation corresponds to the SeibergâWitten equations on the 3-manifold crossed with the real line. Equivalently, the generators of the chain complex are translation-invariant solutions to SeibergâWitten equations (known as monopoles) on the product of a 3-manifold and the real line, and the differential counts solutions to the SeibergâWitten equations on the product of a three-manifold and the real line, which are asymptotic to invariant solutions at infinity and negative infinity.
731:. Kronheimer and Mrowka first introduced the contact element in the SeibergâWitten case. Ozsvath and Szabo constructed it for Heegaard Floer homology using Giroux's relation between contact manifolds and open book decompositions, and it comes for free, as the homology class of the empty set, in embedded contact homology. (Which, unlike the other three, requires a contact structure for its definition. For embedded contact homology see
1213:
the graph of the symplectomorphism. The construction of
Heegaard Floer homology is based on a variant of Lagrangian Floer homology for totally real submanifolds defined using a Heegaard splitting of a three-manifold. SeidelâSmith and Manolescu constructed a link invariant as a certain case of Lagrangian Floer homology, which conjecturally agrees with
701:. The 3-manifold Floer homologies should also be the targets of relative invariants for four-manifolds with boundary, related by gluing constructions to the invariants of a closed 4-manifold obtained by gluing together bounded 3-manifolds along their boundaries. (This is closely related to the notion of a
1657:
Floer homologies are generally difficult to compute explicitly. For instance, the symplectic Floer homology for all surface symplectomorphisms was completed only in 2007. The
Heegaard Floer homology has been a success story in this regard: researchers have exploited its algebraic structure to compute
1576:
over closed Reeb orbits. It further includes a linear homology theory, called cylindrical or linearized contact homology (sometimes, by abuse of notation, just contact homology), whose chain groups are vector spaces generated by closed orbits and whose differentials count only holomorphic cylinders.
738:
These theories all come equipped with a priori relative gradings; these have been lifted to absolute gradings (by homotopy classes of oriented 2-plane fields) by
Kronheimer and Mrowka (for SWF), Gripp and Huang (for HF), and Hutchings (for ECH). Cristofaro-Gardiner has shown that Taubes' isomorphism
1012:
An analog of embedded contact homology may be defined for mapping tori of symplectomorphisms of a surface (possibly with boundary) and is known as periodic Floer homology, generalizing the symplectic Floer homology of surface symplectomorphisms. More generally, it may be defined with respect to any
1212:
Several kinds of Floer homology are special cases of
Lagrangian Floer homology. The symplectic Floer homology of a symplectomorphism of M can be thought of as a case of Lagrangian Floer homology in which the ambient manifold is M crossed with M and the Lagrangian submanifolds are the diagonal and
1017:
on the 3-manifold; like contact structures, stable
Hamiltonian structures define a nonvanishing vector field (the Reeb vector field), and Hutchings and Taubes have proven an analogue of the Weinstein conjecture for them, namely that they always have closed orbits (unless they are mapping tori of a
684:
These homologies are closely related to the
Donaldson and Seiberg invariants of 4-manifolds, as well as to Taubes's Gromov invariant of symplectic 4-manifolds; the differentials of the corresponding three-manifold homologies to these theories are studied by considering solutions to the relevant
1001:
that a contact 3-manifold has a closed Reeb orbit for any contact form holds on any manifold whose ECH is nontrivial, and was proved by Taubes using techniques closely related to ECH; extensions of this work yielded the isomorphism between ECH and SWF. Many constructions in ECH (including its
120:
The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the
1603:
of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is well-defined and given by the symplectic Floer homologies of powers of the symplectomorphism, (rational) symplectic field theory and contact homology can be considered as generalized symplectic Floer
133:
is then used to show that the counts of flow lines defining the differential are finite, so that the differential is well-defined and squares to zero. Thus the Floer homology is defined. For instanton Floer homology, the gradient flow equation is exactly the YangâMills equation on the
1628:
and for the symplectic Floer homology of cotangent bundles by Cohen. This approach was the basis of
Manolescu's 2013 construction of Pin (2)-equivariant SeibergâWitten Floer homology, with which he disproved the Triangulation Conjecture for manifolds of dimension 5 and higher.
1577:
However, cylindrical contact homology is not always defined due to the presence of holomorphic discs and a lack of regularity and transversality results. In situations where cylindrical contact homology makes sense, it may be seen as the (slightly modified)
237:
of the free loop space of M (proofs of various versions of this statement are due to
Viterbo, SalamonâWeber, AbbondandoloâSchwarz, and Cohen). There are more complicated operations on the Floer homology of a cotangent bundle that correspond to the
1386:
as a
Lagrangian submanifold. One can consider the Lagrangian intersection Floer homology. Alternately, we can consider the Instanton Floer homology of the 3-manifold Y. The AtiyahâFloer conjecture asserts that these two invariants are isomorphic.
985:
and its differential counts certain holomorphic curves with ends at certain collections of Reeb orbits. It differs from SFT in technical conditions on the collections of Reeb orbits that generate itâand in not counting all holomorphic curves with
1658:
it for various classes of 3-manifolds and have found combinatorial algorithms for computation of much of the theory. It is also connected to existing invariants and structures and many insights into 3-manifold topology have resulted.
1434:
on the mirror CalabiâYau manifold. In this situation, one should not focus on the Floer homology groups but on the Floer chain groups. Similar to the pair-of-pants product, one can construct multi-compositions using pseudo-holomorphic
1205:" of Lalonde and Cornea offer a different approach to it. The Floer homology of a pair of Lagrangian submanifolds may not always exist; when it does, it provides an obstruction to isotoping one Lagrangian away from the other using a
1193:
1637:
Many of these Floer homologies have not been completely and rigorously constructed, and many conjectural equivalences have not been proved. Technical difficulties come up in the analysis involved, especially in constructing
677:. A knot in a three-manifold induces a filtration on the chain complex of each theory, whose chain homotopy type is a knot invariant. (Their homologies satisfy similar formal properties to the combinatorially-defined
173:
Here, nondegeneracy means that 1 is not an eigenvalue of the derivative of the symplectomorphism at any of its fixed points. This condition implies that the fixed points are isolated. SFH is the homology of the
202:
corresponding to fixed points of the symplectomorphism. A relative index may be defined between pairs of fixed points, and the differential counts the number of holomorphic cylinders with relative index 1.
1568:. The symplectic field theory as well as its subcomplexes, rational symplectic field theory and contact homology, are defined as homologies of differential algebras, which are generated by closed orbits of the
384:
66:. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.
980:
to certain non-compact symplectic 4-manifolds (namely, a contact three-manifold cross R). Its construction is analogous to symplectic field theory, in that it is generated by certain collections of closed
542:
1604:
homologies. In the important case when the symplectomorphism is the time-one map of a time-dependent
Hamiltonian, it was however shown that these higher invariants do not contain any further information.
829:, where it is known as monopole Floer homology. Taubes has shown that it is isomorphic to embedded contact homology. Alternate constructions of SWF for rational homology 3-spheres have been given by
858:
716:). These are related to the invariants for closed 3-manifolds by gluing formulas for the Floer homology of a 3-manifold described as the union along the boundary of two 3-manifolds with boundary.
631:
1198:
which is defined by counting holomorphic triangles (that is, holomorphic maps of a triangle whose vertices and edges map to the appropriate intersection points and Lagrangian submanifolds).
69:
Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free
1572:
of a chosen contact form. The differential counts certain holomorphic curves in the cylinder over the contact manifold, where the trivial examples are the branched coverings of (trivial)
451:
1592:. Its generators are Reeb chords, which are trajectories of the Reeb vector field beginning and ending on a Lagrangian, and its differential counts certain holomorphic strips in the
956:, is an invariant of 3-manifolds (with a distinguished second homology class, corresponding to the choice of a spin structure in SeibergâWitten Floer homology) isomorphic (by work of
1581:
of the action functional on the free loop space, which sends a loop to the integral of the contact form alpha over the loop. Reeb orbits are the critical points of this functional.
315:
and for the definition of both Floer homology and quantum cohomology. The semi-positive condition means that one of the following holds (note that the three cases are not disjoint):
233:
of a manifold M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness. For Hamiltonians that are quadratic at infinity, the Floer homology is the
206:
The symplectic Floer homology of a Hamiltonian symplectomorphism of a compact manifold is isomorphic to the singular homology of the underlying manifold. Thus, the sum of the
785:
Soon after Floer's introduction of Floer homology, Donaldson realized that cobordisms induce maps. This was the first instance of the structure that came to be known as a
1491:
1464:
1384:
1303:
1355:
1335:
1274:
1250:
1974:
2028:
1877:
1649:
and a "general Fredholm theory". While the polyfold project is not yet fully completed, in some important cases transversality was shown using simpler methods.
1530:
1424:
937:. The "plus" and "minus" versions of Heegaard Floer homology, and the related OzsvĂĄthâSzabĂł four-manifold invariants, can be described combinatorially as well (
2308:
1067:
130:
1645:
of pseudoholomorphic curves. Hofer, in collaboration with Kris Wysocki and Eduard Zehnder, has developed new analytic foundations via their theory of
1030:
of a symplectic manifold is the homology of a chain complex generated by the intersection points of the two submanifolds and whose differential counts
774:, i.e. anti-self-dual connections on the three-manifold crossed with the real line. Instanton Floer homology may be viewed as a generalization of the
705:.) For Heegaard Floer homology, the 3-manifold homology was defined first, and an invariant for closed 4-manifolds was later defined in terms of it.
190:
of the symplectomorphism. This itself is a symplectic manifold of dimension two greater than the original manifold. For an appropriate choice of
1225:
The AtiyahâFloer conjecture connects the instanton Floer homology with the Lagrangian intersection Floer homology. Consider a 3-manifold Y with a
2054:
2988:
1925:
1858:
1832:
1802:
1771:
1745:
884:
announced a proof that the plus-version of Heegaard Floer homology (with reverse orientation) is isomorphic to embedded contact homology.
323:
1951:
1357:
bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary embeds into
2598:
2396:
Kutluhan, Cagatay; Lee, Yi-Jen; Taubes, Clifford Henry (2020). "HF=HM I: Heegaard Floer homology and SeibergâWitten Floer homology".
1786:
869:
481:
2922:
1719:
953:
887:
A knot in a three-manifold induces a filtration on the Heegaard Floer homology groups, and the filtered homotopy type is a powerful
89:
functional. Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer
2099:
786:
702:
690:
125:
for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as
2261:
1639:
214:
for the number of fixed points for a nondegenerate symplectomorphism. The SFH of a Hamiltonian symplectomorphism also has a
1009:
The contact element of ECH has a particularly nice form: it is the cycle associated to the empty collection of Reeb orbits.
2979:
Piunikhin, Sergey; Salamon, Dietmar; Schwarz, Matthias (1996). "Symplectic FloerâDonaldson theory and quantum cohomology".
3045:
3015:
1943:
1620:
to such a spectrum could yield other interesting invariants. This strategy was proposed by Ralph Cohen, John Jones, and
973:
694:
566:
3070:
969:
98:
122:
3010:
1396:
246:
215:
62:. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the
2649:———; Szabo (2004). "Holomorphic disks and three-manifold invariants: properties and applications".
257:
In 1996 S. Piunikhin, D. Salamon and M. Schwarz summarized the results about the relation between Floer homology and
1824:
1794:
1763:
1737:
1613:
817:
1508:
in homage to the underlying physics. The Homological Mirror Symmetry conjecture states there is a type of derived
1014:
2223:
1589:
179:
2306:; Wysocki, Kris; Zehnder, Eduard (2007). "A General Fredholm Theory I: A Splicing-Based Differential Geometry".
968:) to the plus-version of Heegaard Floer homology (with reverse orientation). It may be seen as an extension of
3065:
2266:
756:
651:
639:
411:
191:
126:
82:
1466:-relations making the category of all (unobstructed) Lagrangian submanifolds in a symplectic manifold into an
1404:
752:
3005:
708:
There are also extensions of the 3-manifold homologies to 3-manifolds with boundary: sutured Floer homology (
3055:
2741:
2702:
1850:
919:
258:
183:
39:
2582:
Manolescu, Ciprian; OzsvĂĄth, Peter; Thurston, Dylan (2009). "Grid diagrams and Heegaard Floer invariants".
2492:
2357:
1585:
1027:
114:
113:
vector field connecting fixed pairs of critical points (or collections thereof). Floer homology is the
59:
55:
50:
introduced the first version of Floer homology, now called symplectic Floer homology, in his proof of the
26:
2192:
2087:
686:
63:
3060:
2603:
2536:
1612:
One conceivable way to construct a Floer homology theory of some object would be to construct a related
1202:
3050:
2902:
2760:
2668:
2621:
2554:
2327:
2275:
2163:
2062:
1983:
998:
896:
779:
2848:
Sarkar, Sucharit; Wang, Jiajun (2010). "An algorithm for computing some Heegaard Floer homologies".
1970:"Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions"
1505:
1314:
1206:
880:
announced a proof that Heegaard Floer homology is isomorphic to SeibergâWitten Floer homology, and
655:
298:
291:
167:
143:
74:
31:
908:
2967:
2949:
2928:
2892:
2875:
2857:
2836:
2818:
2791:
2778:
2750:
2711:
2684:
2658:
2637:
2611:
2583:
2570:
2544:
2518:
2500:
2466:
2448:
2423:
2405:
2384:
2366:
2343:
2317:
2291:
2249:
2231:
2179:
2078:
1561:
1509:
1469:
1442:
1230:
1226:
873:
312:
223:
1699:
1360:
1279:
766:-bundle over the three-manifold (more precisely, homology 3-spheres). Its critical points are
2984:
2918:
2478:
2011:
1947:
1921:
1854:
1828:
1798:
1767:
1741:
1715:
1646:
1617:
1569:
1214:
1031:
923:
728:
678:
234:
211:
195:
155:
151:
147:
102:
51:
1340:
1320:
1259:
1235:
86:
2959:
2910:
2867:
2828:
2768:
2721:
2676:
2629:
2562:
2510:
2458:
2436:
2415:
2376:
2335:
2283:
2241:
2207:
2171:
2138:
2107:
2070:
2036:
2001:
1991:
1886:
1812:
1782:
1729:
1707:
1573:
1557:
1549:
1537:
1431:
1400:
892:
865:
850:
822:
809:
775:
760:
748:
245:
The symplectic version of Floer homology figures in a crucial way in the formulation of the
230:
2121:
1900:
911:
for the Heegaard splittings, knot Floer homology was given a combinatorial construction by
558:
can be defined as the tensor products of the ordinary cohomology with Novikov ring Î, i.e.
2806:
2531:
2117:
1896:
1755:
1593:
1494:
1253:
957:
767:
670:
667:
239:
163:
159:
1061:
of a symplectic manifold, there is a product structure on the Lagrangian Floer homology:
990:
1 with given ends, but only those that also satisfy a topological condition given by the
815:
One version of SeibergâWittenâFloer homology was constructed rigorously in the monograph
2906:
2764:
2672:
2625:
2558:
2331:
2279:
2167:
2066:
1987:
723:
Floer homologies also come equipped with a distinguished element of the homology if the
3028:
2006:
1969:
1695:
1515:
1501:
1409:
987:
888:
724:
720:
674:
43:
2439:; Thurston, Dylan (2008). "Bordered Heegaard Floer homology: Invariance and pairing".
3039:
2971:
2940:
Taubes, Clifford (2007). "The SeibergâWitten equations and the Weistein conjecture".
2840:
2641:
2470:
2427:
2183:
2082:
2023:
1917:
1816:
1600:
1500:
To be more precise, one must add additional data to the Lagrangian â a grading and a
1034:
826:
199:
187:
175:
94:
90:
47:
2879:
2782:
2574:
2388:
2347:
2221:
FrĂžyshov, Kim A. (2010). "Monopole Floer homology for rational homology 3-spheres".
1910:
2932:
2688:
2601:(2004). "Holomorphic disks and topological invariants for closed three-manifolds".
2522:
2303:
2295:
2253:
1642:
1621:
1565:
739:
between ECH and SeibergâWitten Floer cohomology preserves these absolute gradings.
647:
638:
This construction of Floer homology explains the independence on the choice of the
287:
207:
3023:
1891:
1872:
1188:{\displaystyle HF(L_{0},L_{1})\otimes HF(L_{1},L_{2})\rightarrow HF(L_{0},L_{2}),}
301:
structure on the cohomology of M with the pair-of-pants product on Floer homology.
2871:
2680:
2633:
2151:
2129:——— (1989). "Cuplength estimates on Lagrangian intersections".
2049:
1711:
1201:
Papers on this subject are due to Fukaya, Oh, Ono, and Ohta; the recent work on "
2566:
1935:
1868:
1842:
1533:
994:, which in particular implies that the curves considered are (mainly) embedded.
982:
307:
The above condition of semi-positive and the compactness of symplectic manifold
219:
20:
1706:. Proceedings of Symposia in Pure Mathematics. Vol. 48. pp. 285â299.
1403:
predicts an equality between the Lagrangian Floer homology of Lagrangians in a
872:
of a closed 3-manifold equipped with a spin structure. It is computed using a
2773:
2736:
2726:
2697:
2245:
977:
933:
The "hat" version of Heegaard Floer homology was described combinatorially by
805:
279:
198:(of finite energy) in it have cylindrical ends asymptotic to the loops in the
106:
70:
2212:
2112:
2094:
2963:
2914:
2419:
1996:
1553:
1427:
960:) to SeibergâWitten Floer cohomology and consequently (by work announced by
771:
78:
2832:
2481:(2003). "SeibergâWittenâFloer stable homotopy type of three-manifolds with
2380:
2142:
2040:
2015:
101:
of the function (or possibly certain collections of critical points). The
54:
in symplectic geometry. Floer also developed a closely related theory for
2514:
226:. A version of the product also exists for non-exact symplectomorphisms.
242:
operations on the homology of the loop space of the underlying manifold.
110:
35:
2809:(2008). "Instanton Floer homology with Lagrangian boundary conditions".
2755:
2716:
2462:
2322:
1624:, and carried out in certain cases for SeibergâWittenâFloer homology by
1596:
of the contact manifold whose ends are asymptotic to given Reeb chords.
876:
of the space via a construction analogous to Lagrangian Floer homology.
210:
of that manifold yields the lower bound predicted by one version of the
2287:
2175:
2074:
1616:
whose ordinary homology is the desired Floer homology. Applying other
2339:
42:
that arises as an infinite-dimensional analogue of finite-dimensional
2954:
2891:. CRM Proceedings and Lecture Notes. Vol. 49. pp. 263â297.
2862:
2823:
2796:
2663:
2616:
2549:
2505:
2371:
142:
Symplectic Floer Homology (SFH) is a homology theory associated to a
2355:
JuhĂĄsz, AndrĂĄs (2008). "Floer homology and surface decompositions".
2026:(1988). "The unregularized gradient flow of the symplectic action".
2887:
Hutchings (2009). "The embedded contact homology index revisited".
1504:. A Lagrangian with a choice of these structures is often called a
182:
of such a symplectomorphism, where the differential counts certain
2897:
2588:
2453:
2410:
2236:
763:
379:{\displaystyle \langle ,A\rangle =\lambda \langle c_{1},A\rangle }
922:
of S^3 branched over a knot is related by a spectral sequence to
673:. Each yields three types of homology groups, which fit into an
646:
and the isomorphism to Floer homology provided from the ideas of
2790:
Rasmussen, Jacob (2003). "Floer homology and knot complements".
1026:
The Lagrangian Floer homology of two transversely intersecting
537:{\displaystyle \langle c_{1},\pi _{2}(M)\rangle =N\mathbb {Z} }
2534:(2009). "A combinatorial description of knot Floer homology".
2264:(1985). "Pseudo holomorphic curves in symplectic manifolds".
666:
There are several equivalent Floer homologies associated to
2889:
New Perspectives and Challenges in Symplectic Field Theory
2737:"On the Heegaard Floer homology of branched double-covers"
1791:
Floer Homology, Gauge Theory, And Low-dimensional Topology
2193:"Witten's complex and infinite dimensional Morse Theory"
938:
782:
of the Floer homology agrees with the Casson invariant.
81:) version for three-manifolds, it is the space of SU(2)-
751:
introduced by Floer himself. It is obtained using the
713:
262:
912:
1518:
1472:
1445:
1412:
1363:
1343:
1323:
1282:
1262:
1238:
1070:
569:
484:
414:
326:
554:
The quantum cohomology group of symplectic manifold
1968:Colin, Vincent; Ghiggini, Paolo; Honda, Ko (2011).
1540:of coherent sheaves of the mirror, and vice versa.
658:between homology and cohomology as the background.
626:{\displaystyle QH_{*}(M)=H_{*}(M)\otimes \Lambda .}
270:The Floer cohomology groups of the loop space of a
1909:
1762:. Cambridge Tracts in Mathematics. Vol. 147.
1700:"New invariants of 3- and 4-dimensional manifolds"
1524:
1485:
1458:
1418:
1378:
1349:
1329:
1297:
1276:modulo gauge equivalence is a symplectic manifold
1268:
1244:
1187:
965:
881:
625:
536:
445:
378:
166:of a symplectic manifold. SFH is invariant under
2152:"Symplectic fixed points and holomorphic spheres"
961:
877:
2983:. Cambridge University Press. pp. 171â200.
1599:In SFT the contact manifolds can be replaced by
747:This is a three-manifold invariant connected to
105:of the chain complex is defined by counting the
77:with the symplectic action functional. For the (
1584:SFT also associates a relative invariant of a
1512:between the Fukaya category of the CalabiâYau
1002:well-definedness) rely upon this isomorphism (
1940:Fukaya Categories and Picard Lefschetz Theory
1878:Bulletin of the American Mathematical Society
1793:. Clay Mathematics Proceedings. Vol. 5.
697:, respectively) on the 3-manifold cross
278:,Ï) are naturally isomorphic to the ordinary
8:
2441:Memoirs of the American Mathematical Society
2309:Journal of the European Mathematical Society
1217:, a combinatorially-defined link invariant.
927:
900:
520:
485:
434:
415:
373:
354:
345:
327:
2095:"Morse theory for Lagrangian intersections"
1873:"Floer theory and low dimensional topology"
907:. It is known to detect knot genus. Using
134:three-manifold crossed with the real line.
1916:. Progress in Mathematics. Vol. 111.
1760:Floer homology groups in YangâMills theory
934:
2953:
2896:
2861:
2822:
2795:
2772:
2754:
2725:
2715:
2662:
2615:
2587:
2548:
2504:
2452:
2409:
2370:
2321:
2235:
2211:
2111:
2005:
1995:
1890:
1704:The Mathematical Heritage of Hermann Weyl
1625:
1517:
1477:
1471:
1450:
1444:
1411:
1362:
1342:
1322:
1281:
1261:
1237:
1173:
1160:
1135:
1122:
1097:
1084:
1069:
904:
830:
732:
599:
577:
568:
530:
529:
505:
492:
483:
446:{\displaystyle \langle c_{1},A\rangle =0}
422:
413:
361:
325:
85:on a three-dimensional manifold with the
2050:"An instanton-invariant for 3-manifolds"
834:
186:in the product of the real line and the
154:, the homology arises from studying the
2698:"Holomorphic disks and knot invariants"
2530:Manolescu, Ciprian; OzsvĂĄth, Peter S.;
1672:
263:Piunikhin, Salamon & Schwarz (1996)
2735:OzsvĂĄth, Peter; SzabĂł, ZoltĂĄn (2005).
2696:OzsvĂĄth, Peter; SzabĂł, ZoltĂĄn (2004).
1679:
1439:-gons. These compositions satisfy the
1022:Lagrangian intersection Floer homology
1003:
939:Manolescu, OzsvĂĄth & Thurston 2009
913:Manolescu, OzsvĂĄth & Sarkar (2009)
709:
899:. Knot Floer homology was defined by
714:Lipshitz, OzsvĂĄth & Thurston 2008
7:
1040:Given three Lagrangian submanifolds
150:of it. If the symplectomorphism is
1847:Introduction to Symplectic Topology
918:The Heegaard Floer homology of the
1758:; M. Furuta; D. Kotschick (2002).
1478:
1451:
1370:
1344:
1324:
1289:
1263:
1239:
882:Colin, Ghiggini & Honda (2011)
617:
14:
891:, called knot Floer homology. It
878:Kutluhan, Lee & Taubes (2020)
662:Floer homology of three-manifolds
297:This isomorphism intertwines the
1556:between them, originally due to
972:, known to be equivalent to the
966:Colin, Ghiggini & Honda 2011
849:
804:is a homology theory for smooth
787:topological quantum field theory
703:topological quantum field theory
261:and formulated as the following.
2981:Contact and Symplectic Geometry
1588:of a contact manifold known as
962:Kutluhan, Lee & Taubes 2020
712:) and bordered Floer homology (
2191:——— (1989).
2150:——— (1989).
2093:——— (1988).
2048:——— (1988).
1373:
1367:
1292:
1286:
1179:
1153:
1144:
1141:
1115:
1103:
1077:
654:, where we must recognize the
611:
605:
589:
583:
517:
511:
336:
330:
1:
1944:European Mathematical Society
1892:10.1090/S0273-0979-05-01080-3
1821:Monopoles and Three-Manifolds
1544:Symplectic field theory (SFT)
1337:. In the Heegaard splitting,
818:Monopoles and Three-manifolds
798:SeibergâWitten Floer homology
793:SeibergâWitten Floer homology
311:is required for us to obtain
38:. Floer homology is a novel
3024:Heegaard Floer Knot Homology
2872:10.4007/annals.2010.171.1213
2681:10.4007/annals.2004.159.1159
2634:10.4007/annals.2004.159.1027
1391:Relations to mirror symmetry
1015:stable Hamiltonian structure
544:is greater than or equal to
3011:Encyclopedia of Mathematics
2567:10.4007/annals.2009.169.633
1908:Schwarz, Matthias (2012) .
1845:; Salamon, Dietmar (1998).
1486:{\displaystyle A_{\infty }}
1459:{\displaystyle A_{\infty }}
1397:homological mirror symmetry
837:; they are known to agree.
755:functional on the space of
247:homological mirror symmetry
218:product that is a deformed
3087:
1825:Cambridge University Press
1795:Clay Mathematics Institute
1764:Cambridge University Press
1738:Kluwer Academic Publishers
1734:Lectures on Morse Homology
1732:; David Hurtubise (2004).
1379:{\displaystyle M(\Sigma )}
1298:{\displaystyle M(\Sigma )}
901:OzsvĂĄth & SzabĂł (2004)
478: â„ 0 defined by
170:of the symplectomorphism.
131:Gromov compactness theorem
3006:"Atiyah-Floer conjecture"
2774:10.1016/j.aim.2004.05.008
2727:10.1016/j.aim.2003.05.001
2246:10.1215/00127094-2010-060
1590:relative contact homology
976:, from closed symplectic
970:Taubes's Gromov invariant
950:Embedded contact homology
945:Embedded contact homology
286:, tensored by a suitable
138:Symplectic Floer homology
2267:Inventiones Mathematicae
1712:10.1090/pspum/048/974342
1548:This is an invariant of
974:SeibergâWitten invariant
935:Sarkar & Wang (2010)
928:OzsvĂĄth & SzabĂł 2005
743:Instanton Floer homology
685:differential equations (
652:pseudoholomorphic curves
640:almost complex structure
292:covering transformations
290:associated the group of
192:almost complex structure
184:pseudoholomorphic curves
127:pseudoholomorphic curves
16:Symplectic topology tool
2964:10.2140/gt.2007.11.2117
2811:Geometry & Topology
2742:Advances in Mathematics
2703:Advances in Mathematics
2420:10.2140/gt.2020.24.2829
2398:Geometry & Topology
2358:Geometry & Topology
1997:10.1073/pnas.1018734108
1851:Oxford University Press
1785:; Stipsicz, AndrĂĄs I.;
1536:underlying the bounded
1350:{\displaystyle \Sigma }
1330:{\displaystyle \Sigma }
1269:{\displaystyle \Sigma }
1245:{\displaystyle \Sigma }
1221:AtiyahâFloer conjecture
1028:Lagrangian submanifolds
864:is an invariant due to
846:Heegaard Floer homology
841:Heegaard Floer homology
802:monopole Floer homology
770:and its flow lines are
123:CauchyâRiemann equation
117:of this chain complex.
56:Lagrangian submanifolds
30:is a tool for studying
2833:10.2140/gt.2008.12.747
2381:10.2140/gt.2008.12.299
2213:10.4310/jdg/1214443291
2143:10.1002/cpa.3160420402
2113:10.4310/jdg/1214442477
2041:10.1002/cpa.3160410603
2029:Comm. Pure Appl. Math.
1586:Legendrian submanifold
1526:
1493:-category, called the
1487:
1460:
1420:
1380:
1351:
1331:
1309: â 6, where
1299:
1270:
1246:
1189:
627:
538:
447:
380:
2515:10.2140/gt.2003.7.889
2131:Comm. Pure Appl. Math
2100:J. Differential Geom.
1527:
1488:
1461:
1421:
1381:
1352:
1332:
1300:
1271:
1247:
1190:
903:and independently by
628:
539:
448:
381:
274:symplectic manifold (
64:YangâMills functional
3046:Mathematical physics
1633:Analytic foundations
1516:
1470:
1443:
1410:
1361:
1341:
1321:
1280:
1260:
1252:. Then the space of
1236:
1068:
999:Weinstein conjecture
897:Alexander polynomial
780:Euler characteristic
567:
482:
473:minimal Chern Number
412:
324:
146:and a nondegenerate
34:and low-dimensional
3071:Symplectic topology
2915:10.1090/crmp/049/10
2907:2008arXiv0805.1240H
2765:2003math......9170O
2673:2001math......5202O
2626:2001math......1206O
2559:2006math......7691M
2332:2006math.....12604H
2280:1985InMat..82..307G
2168:1988CMaPh.120..575F
2067:1988CMaPh.118..215F
1988:2011PNAS..108.8100C
1781:Ellwood, David A.;
1405:CalabiâYau manifold
1207:Hamiltonian isotopy
727:is equipped with a
299:quantum cup product
168:Hamiltonian isotopy
158:functional on the (
144:symplectic manifold
93:is formed from the
75:symplectic manifold
32:symplectic geometry
2805:Salamon, Dietmar;
2479:Manolescu, Ciprian
2435:Lipshitz, Robert;
2288:10.1007/BF01388806
2176:10.1007/BF01260388
2075:10.1007/BF01218578
1562:Alexander Givental
1522:
1510:Morita equivalence
1483:
1456:
1416:
1376:
1347:
1327:
1295:
1266:
1242:
1227:Heegaard splitting
1185:
623:
534:
443:
376:
259:quantum cohomology
224:quantum cohomology
196:holomorphic curves
109:of the function's
2990:978-0-521-57086-2
2463:10.1090/memo/1216
2055:Comm. Math. Phys.
1982:(20): 8100â8105.
1962:Research articles
1927:978-3-0348-8577-5
1860:978-0-19-850451-1
1834:978-0-521-88022-0
1813:Kronheimer, Peter
1804:978-0-8218-3845-7
1783:OzsvĂĄth, Peter S.
1773:978-0-521-80803-3
1747:978-1-4020-2695-9
1689:Books and surveys
1618:homology theories
1570:Reeb vector field
1550:contact manifolds
1525:{\displaystyle X}
1419:{\displaystyle X}
1215:Khovanov homology
1032:pseudoholomorphic
954:Michael Hutchings
924:Khovanov homology
808:(equipped with a
729:contact structure
679:Khovanov homology
235:singular homology
212:Arnold conjecture
178:generated by the
156:symplectic action
148:symplectomorphism
52:Arnold conjecture
3078:
3019:
2994:
2975:
2957:
2948:(4): 2117â2202.
2936:
2900:
2883:
2865:
2856:(2): 1213â1236.
2844:
2826:
2807:Wehrheim, Katrin
2801:
2799:
2786:
2776:
2758:
2731:
2729:
2719:
2692:
2666:
2657:(3): 1159â1245.
2645:
2619:
2610:(3): 1027â1158.
2597:OzsvĂĄth, Peter;
2593:
2591:
2578:
2552:
2532:Sarkar, Sucharit
2526:
2508:
2474:
2456:
2431:
2413:
2404:(6): 2829â2854.
2392:
2374:
2351:
2325:
2299:
2257:
2239:
2217:
2215:
2197:
2187:
2156:Comm. Math. Phys
2146:
2125:
2115:
2086:
2044:
2019:
2009:
1999:
1957:
1931:
1915:
1904:
1894:
1864:
1838:
1808:
1777:
1751:
1730:Augustin Banyaga
1725:
1682:
1677:
1626:Manolescu (2003)
1558:Yakov Eliashberg
1538:derived category
1531:
1529:
1528:
1523:
1492:
1490:
1489:
1484:
1482:
1481:
1465:
1463:
1462:
1457:
1455:
1454:
1432:coherent sheaves
1425:
1423:
1422:
1417:
1401:Maxim Kontsevich
1385:
1383:
1382:
1377:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1304:
1302:
1301:
1296:
1275:
1273:
1272:
1267:
1254:flat connections
1251:
1249:
1248:
1243:
1203:cluster homology
1194:
1192:
1191:
1186:
1178:
1177:
1165:
1164:
1140:
1139:
1127:
1126:
1102:
1101:
1089:
1088:
905:Rasmussen (2003)
874:Heegaard diagram
863:
862:
861:
860:
853:
831:Manolescu (2003)
823:Peter Kronheimer
776:Casson invariant
768:flat connections
749:Donaldson theory
733:Hutchings (2009)
656:Poincaré duality
632:
630:
629:
624:
604:
603:
582:
581:
543:
541:
540:
535:
533:
510:
509:
497:
496:
460:
452:
450:
449:
444:
427:
426:
385:
383:
382:
377:
366:
365:
231:cotangent bundle
58:of a symplectic
3086:
3085:
3081:
3080:
3079:
3077:
3076:
3075:
3066:Homology theory
3036:
3035:
3004:
3001:
2991:
2978:
2939:
2925:
2886:
2847:
2804:
2789:
2756:math.GT/0209056
2734:
2717:math.GT/0209056
2695:
2648:
2596:
2581:
2529:
2489:
2477:
2434:
2395:
2354:
2340:10.4171/JEMS/99
2323:math.FA/0612604
2302:
2262:Gromov, Mikhail
2260:
2220:
2195:
2190:
2149:
2128:
2092:
2047:
2022:
1967:
1964:
1954:
1934:
1928:
1907:
1867:
1861:
1841:
1835:
1811:
1805:
1789:, eds. (2006).
1780:
1774:
1756:Simon Donaldson
1754:
1748:
1728:
1722:
1696:Atiyah, Michael
1694:
1691:
1686:
1685:
1678:
1674:
1669:
1664:
1655:
1635:
1610:
1594:symplectization
1552:and symplectic
1546:
1514:
1513:
1495:Fukaya category
1473:
1468:
1467:
1446:
1441:
1440:
1408:
1407:
1393:
1359:
1358:
1339:
1338:
1319:
1318:
1317:of the surface
1278:
1277:
1258:
1257:
1234:
1233:
1223:
1169:
1156:
1131:
1118:
1093:
1080:
1066:
1065:
1060:
1053:
1046:
1024:
958:Clifford Taubes
947:
857:
856:
855:
848:
843:
835:FrĂžyshov (2010)
795:
745:
671:three-manifolds
664:
595:
573:
565:
564:
548: â 2.
501:
488:
480:
479:
463:
458:
418:
410:
409:
393:
357:
322:
321:
255:
253:PSS isomorphism
240:string topology
164:free loop space
160:universal cover
140:
99:critical points
97:spanned by the
17:
12:
11:
5:
3084:
3082:
3074:
3073:
3068:
3063:
3058:
3056:Gauge theories
3053:
3048:
3038:
3037:
3034:
3033:
3029:The Knot Atlas
3020:
3000:
2999:External links
2997:
2996:
2995:
2989:
2976:
2937:
2923:
2884:
2845:
2817:(2): 747â918.
2802:
2787:
2732:
2693:
2646:
2594:
2579:
2543:(2): 633â660.
2527:
2499:(2): 889â932.
2485:
2475:
2437:OzsvĂĄth, Peter
2432:
2393:
2365:(1): 299â350.
2352:
2316:(4): 841â876.
2300:
2274:(2): 307â347.
2258:
2230:(3): 519â576.
2218:
2206:(1): 202â221.
2188:
2162:(4): 575â611.
2147:
2137:(4): 335â356.
2126:
2106:(3): 513â547.
2090:
2088:Project Euclid
2061:(2): 215â240.
2045:
2035:(6): 775â813.
2024:Floer, Andreas
2020:
1963:
1960:
1959:
1958:
1953:978-3037190630
1952:
1932:
1926:
1912:Morse Homology
1905:
1865:
1859:
1839:
1833:
1817:Mrowka, Tomasz
1809:
1803:
1778:
1772:
1752:
1746:
1726:
1720:
1690:
1687:
1684:
1683:
1671:
1670:
1668:
1665:
1663:
1660:
1654:
1651:
1634:
1631:
1609:
1608:Floer homotopy
1606:
1579:Morse homology
1545:
1542:
1521:
1502:spin structure
1480:
1476:
1453:
1449:
1415:
1399:conjecture of
1392:
1389:
1375:
1372:
1369:
1366:
1346:
1326:
1305:of dimension 6
1294:
1291:
1288:
1285:
1265:
1241:
1222:
1219:
1196:
1195:
1184:
1181:
1176:
1172:
1168:
1163:
1159:
1155:
1152:
1149:
1146:
1143:
1138:
1134:
1130:
1125:
1121:
1117:
1114:
1111:
1108:
1105:
1100:
1096:
1092:
1087:
1083:
1079:
1076:
1073:
1058:
1051:
1044:
1023:
1020:
988:Fredholm index
946:
943:
889:knot invariant
842:
839:
810:spin structure
794:
791:
744:
741:
725:three-manifold
721:three-manifold
695:CauchyâRiemann
691:SeibergâWitten
675:exact triangle
663:
660:
636:
635:
634:
633:
622:
619:
616:
613:
610:
607:
602:
598:
594:
591:
588:
585:
580:
576:
572:
552:
551:
550:
549:
532:
528:
525:
522:
519:
516:
513:
508:
504:
500:
495:
491:
487:
469:
461:
442:
439:
436:
433:
430:
425:
421:
417:
407:
391:
375:
372:
369:
364:
360:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
305:
304:
303:
302:
295:
254:
251:
222:equivalent to
139:
136:
44:Morse homology
15:
13:
10:
9:
6:
4:
3:
2:
3083:
3072:
3069:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3043:
3041:
3031:
3030:
3025:
3021:
3017:
3013:
3012:
3007:
3003:
3002:
2998:
2992:
2986:
2982:
2977:
2973:
2969:
2965:
2961:
2956:
2951:
2947:
2943:
2938:
2934:
2930:
2926:
2924:9780821843567
2920:
2916:
2912:
2908:
2904:
2899:
2894:
2890:
2885:
2881:
2877:
2873:
2869:
2864:
2859:
2855:
2851:
2846:
2842:
2838:
2834:
2830:
2825:
2820:
2816:
2812:
2808:
2803:
2798:
2793:
2788:
2784:
2780:
2775:
2770:
2766:
2762:
2757:
2752:
2748:
2744:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2710:(1): 58â116.
2709:
2705:
2704:
2699:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2665:
2660:
2656:
2652:
2647:
2643:
2639:
2635:
2631:
2627:
2623:
2618:
2613:
2609:
2606:
2605:
2604:Ann. of Math.
2600:
2599:Szabo, ZoltĂĄn
2595:
2590:
2585:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2551:
2546:
2542:
2539:
2538:
2537:Ann. of Math.
2533:
2528:
2524:
2520:
2516:
2512:
2507:
2502:
2498:
2495:
2494:
2488:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2455:
2450:
2446:
2442:
2438:
2433:
2429:
2425:
2421:
2417:
2412:
2407:
2403:
2399:
2394:
2390:
2386:
2382:
2378:
2373:
2368:
2364:
2360:
2359:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2324:
2319:
2315:
2311:
2310:
2305:
2304:Hofer, Helmut
2301:
2297:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2268:
2263:
2259:
2255:
2251:
2247:
2243:
2238:
2233:
2229:
2226:
2225:
2224:Duke Math. J.
2219:
2214:
2209:
2205:
2201:
2200:J. Diff. Geom
2194:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2148:
2144:
2140:
2136:
2132:
2127:
2123:
2119:
2114:
2109:
2105:
2102:
2101:
2096:
2091:
2089:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2057:
2056:
2051:
2046:
2042:
2038:
2034:
2031:
2030:
2025:
2021:
2017:
2013:
2008:
2003:
1998:
1993:
1989:
1985:
1981:
1977:
1976:
1971:
1966:
1965:
1961:
1955:
1949:
1945:
1941:
1937:
1933:
1929:
1923:
1919:
1914:
1913:
1906:
1902:
1898:
1893:
1888:
1884:
1880:
1879:
1874:
1870:
1866:
1862:
1856:
1852:
1848:
1844:
1840:
1836:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1800:
1796:
1792:
1788:
1787:SzabĂł, ZoltĂĄn
1784:
1779:
1775:
1769:
1765:
1761:
1757:
1753:
1749:
1743:
1739:
1735:
1731:
1727:
1723:
1721:9780821814826
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1692:
1688:
1681:
1676:
1673:
1666:
1661:
1659:
1652:
1650:
1648:
1644:
1643:moduli spaces
1641:
1632:
1630:
1627:
1623:
1619:
1615:
1607:
1605:
1602:
1597:
1595:
1591:
1587:
1582:
1580:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1543:
1541:
1539:
1535:
1519:
1511:
1507:
1503:
1498:
1496:
1474:
1447:
1438:
1433:
1429:
1413:
1406:
1402:
1398:
1390:
1388:
1364:
1316:
1312:
1308:
1283:
1255:
1232:
1228:
1220:
1218:
1216:
1210:
1208:
1204:
1199:
1182:
1174:
1170:
1166:
1161:
1157:
1150:
1147:
1136:
1132:
1128:
1123:
1119:
1112:
1109:
1106:
1098:
1094:
1090:
1085:
1081:
1074:
1071:
1064:
1063:
1062:
1057:
1050:
1043:
1038:
1036:
1035:Whitney discs
1033:
1029:
1021:
1019:
1016:
1010:
1007:
1005:
1000:
995:
993:
989:
984:
979:
975:
971:
967:
963:
959:
955:
951:
944:
942:
940:
936:
931:
929:
925:
921:
916:
914:
910:
909:grid diagrams
906:
902:
898:
894:
890:
885:
883:
879:
875:
871:
867:
866:Peter OzsvĂĄth
859:
852:
847:
840:
838:
836:
832:
828:
827:Tomasz Mrowka
824:
820:
819:
813:
811:
807:
803:
799:
792:
790:
788:
783:
781:
777:
773:
769:
765:
762:
758:
754:
750:
742:
740:
736:
734:
730:
726:
722:
717:
715:
711:
706:
704:
700:
696:
692:
688:
682:
680:
676:
672:
669:
661:
659:
657:
653:
649:
645:
641:
620:
614:
608:
600:
596:
592:
586:
578:
574:
570:
563:
562:
561:
560:
559:
557:
547:
526:
523:
514:
506:
502:
498:
493:
489:
477:
474:
470:
467:
456:
440:
437:
431:
428:
423:
419:
408:
405:
401:
398:) where λâ„0 (
397:
389:
370:
367:
362:
358:
351:
348:
342:
339:
333:
320:
319:
318:
317:
316:
314:
310:
300:
296:
293:
289:
285:
281:
277:
273:
272:semi-positive
269:
268:
267:
266:
265:
264:
260:
252:
250:
248:
243:
241:
236:
232:
227:
225:
221:
217:
216:pair of pants
213:
209:
208:Betti numbers
204:
201:
200:mapping torus
197:
193:
189:
188:mapping torus
185:
181:
177:
176:chain complex
171:
169:
165:
161:
157:
153:
149:
145:
137:
135:
132:
128:
124:
118:
116:
112:
108:
104:
100:
96:
95:abelian group
92:
91:chain complex
88:
84:
80:
76:
72:
67:
65:
61:
57:
53:
49:
48:Andreas Floer
45:
41:
37:
33:
29:
28:
22:
3061:Morse theory
3027:
3009:
2980:
2955:math/0611007
2945:
2941:
2888:
2863:math/0607777
2853:
2850:Ann. of Math
2849:
2824:math/0607318
2814:
2810:
2797:math/0306378
2746:
2740:
2707:
2701:
2664:math/0105202
2654:
2651:Ann. of Math
2650:
2617:math/0101206
2607:
2602:
2550:math/0607691
2540:
2535:
2506:math/0104024
2496:
2493:Geom. Topol.
2491:
2486:
2482:
2444:
2440:
2401:
2397:
2372:math/0609779
2362:
2356:
2313:
2307:
2271:
2265:
2227:
2222:
2203:
2199:
2159:
2155:
2134:
2130:
2103:
2098:
2058:
2053:
2032:
2027:
1979:
1973:
1939:
1936:Seidel, Paul
1911:
1882:
1876:
1869:McDuff, Dusa
1846:
1843:McDuff, Dusa
1820:
1790:
1759:
1733:
1703:
1675:
1656:
1640:compactified
1636:
1622:Graeme Segal
1611:
1601:mapping tori
1598:
1583:
1578:
1566:Helmut Hofer
1547:
1499:
1436:
1394:
1310:
1306:
1224:
1211:
1200:
1197:
1055:
1048:
1041:
1039:
1025:
1011:
1008:
996:
991:
949:
948:
932:
920:double cover
917:
893:categorifies
886:
870:ZoltĂĄn SzabĂł
845:
844:
816:
814:
801:
797:
796:
784:
778:because the
753:ChernâSimons
746:
737:
718:
707:
698:
683:
665:
648:Morse theory
643:
637:
555:
553:
545:
475:
472:
465:
454:
403:
399:
395:
387:
313:Novikov ring
308:
306:
288:Novikov ring
283:
275:
271:
256:
249:conjecture.
244:
228:
205:
194:, punctured
180:fixed points
172:
141:
119:
103:differential
87:ChernâSimons
68:
24:
18:
3051:3-manifolds
2942:Geom. Topol
2749:(1): 1â33.
1680:Atiyah 1988
1653:Computation
1534:dg category
1004:Taubes 2007
983:Reeb orbits
978:4-manifolds
806:3-manifolds
757:connections
710:JuhĂĄsz 2008
220:cup product
152:Hamiltonian
83:connections
21:mathematics
3040:Categories
1918:BirkhÀuser
1662:References
1554:cobordisms
1428:Ext groups
1018:2-torus).
772:instantons
687:YangâMills
453:for every
386:for every
280:cohomology
107:flow lines
71:loop space
3016:EMS Press
2972:119680690
2898:0805.1240
2841:119680541
2642:119143219
2589:0910.0078
2471:115166724
2454:0810.0687
2428:118772589
2411:1007.1979
2237:0809.4842
2184:123345003
2083:122096068
1885:: 25â42.
1667:Footnotes
1647:polyfolds
1574:cylinders
1479:∞
1452:∞
1371:Σ
1345:Σ
1325:Σ
1290:Σ
1264:Σ
1240:Σ
1145:→
1107:⊗
992:ECH index
952:, due to
761:principal
618:Λ
615:⊗
601:∗
579:∗
521:⟩
503:π
486:⟨
435:⟩
416:⟨
374:⟩
355:⟨
352:λ
346:⟩
334:ω
328:⟨
79:instanton
40:invariant
2880:55279928
2783:17245314
2575:15427272
2447:(1216).
2389:56418423
2348:14716262
2016:21525415
1938:(2008).
1871:(2005).
1819:(2007).
1698:(1988).
1614:spectrum
1426:and the
1229:along a
404:monotone
229:For the
162:of the)
115:homology
111:gradient
60:manifold
36:topology
27:homology
3018:, 2001
2933:7751880
2903:Bibcode
2761:Bibcode
2689:8154024
2669:Bibcode
2622:Bibcode
2555:Bibcode
2523:9130339
2328:Bibcode
2296:4983969
2276:Bibcode
2254:8073050
2164:Bibcode
2122:0965228
2063:Bibcode
2007:3100941
1984:Bibcode
1901:2188174
1313:is the
1231:surface
129:. The
2987:
2970:
2931:
2921:
2878:
2839:
2781:
2687:
2640:
2573:
2521:
2490:= 0".
2469:
2426:
2387:
2346:
2294:
2252:
2182:
2120:
2081:
2014:
2004:
1950:
1924:
1899:
1857:
1831:
1801:
1770:
1744:
1718:
1532:and a
1054:, and
693:, and
668:closed
25:Floer
2968:S2CID
2950:arXiv
2929:S2CID
2893:arXiv
2876:S2CID
2858:arXiv
2837:S2CID
2819:arXiv
2792:arXiv
2779:S2CID
2751:arXiv
2712:arXiv
2685:S2CID
2659:arXiv
2638:S2CID
2612:arXiv
2584:arXiv
2571:S2CID
2545:arXiv
2519:S2CID
2501:arXiv
2467:S2CID
2449:arXiv
2424:S2CID
2406:arXiv
2385:S2CID
2367:arXiv
2344:S2CID
2318:arXiv
2292:S2CID
2250:S2CID
2232:arXiv
2196:(PDF)
2180:S2CID
2079:S2CID
1506:brane
1315:genus
854:
764:SU(2)
759:on a
73:of a
2985:ISBN
2919:ISBN
2012:PMID
1975:PNAS
1948:ISBN
1922:ISBN
1855:ISBN
1829:ISBN
1799:ISBN
1768:ISBN
1742:ISBN
1716:ISBN
1564:and
1395:The
997:The
964:and
895:the
868:and
833:and
825:and
719:The
650:and
471:The
390:in Ï
3026:",
2960:doi
2911:doi
2868:doi
2854:171
2829:doi
2769:doi
2747:194
2722:doi
2708:186
2677:doi
2655:159
2630:doi
2608:159
2563:doi
2541:169
2511:doi
2459:doi
2445:254
2416:doi
2377:doi
2336:doi
2284:doi
2242:doi
2228:155
2208:doi
2172:doi
2160:120
2139:doi
2108:doi
2071:doi
2059:118
2037:doi
2002:PMC
1992:doi
1980:108
1887:doi
1708:doi
1430:of
1256:on
1006:).
941:).
930:).
821:by
800:or
681:.)
642:on
457:in
402:is
282:of
46:.
19:In
3042::
3014:,
3008:,
2966:.
2958:.
2946:11
2944:.
2927:.
2917:.
2909:.
2901:.
2874:.
2866:.
2852:.
2835:.
2827:.
2815:12
2813:.
2777:.
2767:.
2759:.
2745:.
2739:.
2720:.
2706:.
2700:.
2683:.
2675:.
2667:.
2653:.
2636:.
2628:.
2620:.
2569:.
2561:.
2553:.
2517:.
2509:.
2465:.
2457:.
2443:.
2422:.
2414:.
2402:24
2400:.
2383:.
2375:.
2363:12
2361:.
2342:.
2334:.
2326:.
2312:.
2290:.
2282:.
2272:82
2270:.
2248:.
2240:.
2204:30
2202:.
2198:.
2178:.
2170:.
2158:.
2154:.
2135:42
2133:.
2118:MR
2116:.
2104:28
2097:.
2077:.
2069:.
2052:.
2033:41
2010:.
2000:.
1990:.
1978:.
1972:.
1946:.
1942:.
1920:.
1897:MR
1895:.
1883:43
1881:.
1875:.
1853:.
1849:.
1827:.
1823:.
1815:;
1797:.
1766:.
1740:.
1736:.
1714:.
1702:.
1560:,
1497:.
1209:.
1047:,
1037:.
915:.
851://
789:.
735:.
689:,
468:).
406:).
23:,
3032:.
3022:"
2993:.
2974:.
2962::
2952::
2935:.
2913::
2905::
2895::
2882:.
2870::
2860::
2843:.
2831::
2821::
2800:.
2794::
2785:.
2771::
2763::
2753::
2730:.
2724::
2714::
2691:.
2679::
2671::
2661::
2644:.
2632::
2624::
2614::
2592:.
2586::
2577:.
2565::
2557::
2547::
2525:.
2513::
2503::
2497:7
2487:1
2483:b
2473:.
2461::
2451::
2430:.
2418::
2408::
2391:.
2379::
2369::
2350:.
2338::
2330::
2320::
2314:9
2298:.
2286::
2278::
2256:.
2244::
2234::
2216:.
2210::
2186:.
2174::
2166::
2145:.
2141::
2124:.
2110::
2085:.
2073::
2065::
2043:.
2039::
2018:.
1994::
1986::
1956:.
1930:.
1903:.
1889::
1863:.
1837:.
1807:.
1776:.
1750:.
1724:.
1710::
1520:X
1475:A
1448:A
1437:n
1414:X
1374:)
1368:(
1365:M
1311:g
1307:g
1293:)
1287:(
1284:M
1183:,
1180:)
1175:2
1171:L
1167:,
1162:0
1158:L
1154:(
1151:F
1148:H
1142:)
1137:2
1133:L
1129:,
1124:1
1120:L
1116:(
1113:F
1110:H
1104:)
1099:1
1095:L
1091:,
1086:0
1082:L
1078:(
1075:F
1072:H
1059:2
1056:L
1052:1
1049:L
1045:0
1042:L
926:(
699:R
644:M
621:.
612:)
609:M
606:(
597:H
593:=
590:)
587:M
584:(
575:H
571:Q
556:M
546:n
531:Z
527:N
524:=
518:)
515:M
512:(
507:2
499:,
494:1
490:c
476:N
466:M
464:(
462:2
459:Ï
455:A
441:0
438:=
432:A
429:,
424:1
420:c
400:M
396:M
394:(
392:2
388:A
371:A
368:,
363:1
359:c
349:=
343:A
340:,
337:]
331:[
309:M
294:.
284:M
276:M
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.