In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degener...In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.展开更多
Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,...Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,receive a single arrow from one of its facets,or conversely,send a single arrow into a cell of which it is a facet.By following arrows,one can then construct a natural Floer-type boundary operator.Here,we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows(but not both),again in the absence of closed orbits.Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.展开更多
This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^c structure. Gluing formulae for certain 4-dimensional manifolds splitting along an em...This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^c structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.展开更多
In this paper,we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring.We will also prove that GW-invariants in Floer homology and quantum ho...In this paper,we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring.We will also prove that GW-invariants in Floer homology and quantum homology are equivalent.展开更多
Using a kind of Mayer-Vietoris principle for the symplectic Floer homology of knots,we compute the symplectic Floer homology of the square knot and granny knots.
This article studies the Floer theory of Landau-Ginzburg （LG） model on C^n. We perturb the Kahler form within a fixed Kahler class to guarantee the transversal intersection of Lefschetz thimbles. The C^0 estimate fo...This article studies the Floer theory of Landau-Ginzburg （LG） model on C^n. We perturb the Kahler form within a fixed Kahler class to guarantee the transversal intersection of Lefschetz thimbles. The C^0 estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.展开更多
Using an argument of Baldwin-Hu-Sivek,we prove that if K is a hyperbolic fibered knot with fiber F in a closed,oriented 3-manifold Y,andHFK(Y,K,[F],g(F)−1)has rank 1,then the monodromy of K is freely isotopic to a pse...Using an argument of Baldwin-Hu-Sivek,we prove that if K is a hyperbolic fibered knot with fiber F in a closed,oriented 3-manifold Y,andHFK(Y,K,[F],g(F)−1)has rank 1,then the monodromy of K is freely isotopic to a pseudo-Anosov map with no fixed points.In particular,this shows that the monodromy of a hyperbolic L-space knot is freely isotopic to a map with no fixed points.展开更多
Given a compact,oriented,connected surface F,we show that the set of connected sutured manifolds(M,γ)with R±(γ)■F is generated by the product sutured manifold(F,∂F)×[0,1]through surgery triads.This result...Given a compact,oriented,connected surface F,we show that the set of connected sutured manifolds(M,γ)with R±(γ)■F is generated by the product sutured manifold(F,∂F)×[0,1]through surgery triads.This result has applications in Floer theories of 3-manifolds.The special case where F=D^(2) or F=S^(2) has been a folklore theorem,which has already been used by experts before.展开更多
We prove the Arnold conjecture for a product of finitely many monotone symplectic manifolds and Calabi-Yau manifolds.The key point of our proof is realized by suitably choosing perturbations of the almost complex stru...We prove the Arnold conjecture for a product of finitely many monotone symplectic manifolds and Calabi-Yau manifolds.The key point of our proof is realized by suitably choosing perturbations of the almost complex structures and Hamiltonian functions for the product case.展开更多
文摘In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.
基金funding provided by Max Planck Societysupported by a stipend from the InternationalMax Planck Research School(IMPRS)“Mathematics in the Sciences.”。
文摘Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,receive a single arrow from one of its facets,or conversely,send a single arrow into a cell of which it is a facet.By following arrows,one can then construct a natural Floer-type boundary operator.Here,we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows(but not both),again in the absence of closed orbits.Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.
文摘This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^c structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
文摘In this paper,we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring.We will also prove that GW-invariants in Floer homology and quantum homology are equivalent.
文摘Using a kind of Mayer-Vietoris principle for the symplectic Floer homology of knots,we compute the symplectic Floer homology of the square knot and granny knots.
基金Acknowledgements The author would like to thank his advisor, Gang Tian, for his great help and long-lasting support to carry on this research. Many thanks to Huijun Fan and Yongbin Ruan for sharing many of their insights into the study of Floer theory of LG model. Many thanks to Yong-Guen Oh for rich knowledge of Floer theory of Lagrangian intersection as well as his suggestions. Thanks also goes to Yefeng Shen, Yalong Shi, Dongning Wang, and Ke Zhu for helpful discussion. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11171143）.
文摘This article studies the Floer theory of Landau-Ginzburg （LG） model on C^n. We perturb the Kahler form within a fixed Kahler class to guarantee the transversal intersection of Lefschetz thimbles. The C^0 estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.
基金The author was partially supported by NSF Grant Number DMS-1811900.
文摘Using an argument of Baldwin-Hu-Sivek,we prove that if K is a hyperbolic fibered knot with fiber F in a closed,oriented 3-manifold Y,andHFK(Y,K,[F],g(F)−1)has rank 1,then the monodromy of K is freely isotopic to a pseudo-Anosov map with no fixed points.In particular,this shows that the monodromy of a hyperbolic L-space knot is freely isotopic to a map with no fixed points.
基金supported by U.S.National Science Foundation(Grant No.DMS-1811900).
文摘Given a compact,oriented,connected surface F,we show that the set of connected sutured manifolds(M,γ)with R±(γ)■F is generated by the product sutured manifold(F,∂F)×[0,1]through surgery triads.This result has applications in Floer theories of 3-manifolds.The special case where F=D^(2) or F=S^(2) has been a folklore theorem,which has already been used by experts before.
基金Supported by the National Natural Science Foundation of China
文摘We prove the Arnold conjecture for a product of finitely many monotone symplectic manifolds and Calabi-Yau manifolds.The key point of our proof is realized by suitably choosing perturbations of the almost complex structures and Hamiltonian functions for the product case.
