Atjaunināt sīkdatņu piekrišanu

Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Pt. 1 [Hardback]

4.00/5 (2 ratings by Goodreads)
, , ,
  • Formāts: Hardback, 396 pages
  • Sērija : AMS/IP Studies in Advanced Mathematics No. 46.1
  • Izdošanas datums: 01-Oct-2009
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 0821848364
  • ISBN-13: 9780821848364
Citas grāmatas par šo tēmu:
Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Pt. 1Palielināt
  • Formāts: Hardback, 396 pages
  • Sērija : AMS/IP Studies in Advanced Mathematics No. 46.1
  • Izdošanas datums: 01-Oct-2009
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 0821848364
  • ISBN-13: 9780821848364
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780821848364.html
Keywords:
This two-volume research monograph on the general Lagrangian Floer theory and the accompanying homological algebra of filtered $A_
Preface xi
Introduction
1(38)
What is Floer (co)homology
1(4)
General theory of Lagrangian Floer cohomology
5(8)
Applications to symplectic geometry
13(3)
Relation to mirror symmetry
16(9)
Chapter-wise outline of the main results
25(10)
Acknowledgments
35(1)
Conventions
36(3)
Review: Floer cohomology
39(38)
Bordered stable maps and the Maslov index
39(10)
The Maslov index: the relative first Chern number
39(4)
The moduli space of bordered stable maps
43(6)
The Novikov covering and the action functional
49(4)
The Γ-equivalence
50(1)
The action functional and the Maslov-Morse index
51(2)
Review of Floer cohomology I: without anomaly
53(7)
The L2-gradient equation
53(4)
Floer's definition: 2-coefficients
57(2)
Bott-Morse Floer cohomology
59(1)
Review of Floer cohomology II: anomaly appearance
60(17)
The Floer cochain module
61(1)
The Floer moduli space
62(4)
The Novikov ring ΛR (L)
66(3)
Monotone Lagrangian submanifolds
69(2)
Appearance of the primary obstruction
71(6)
The A∞ algebra associated to a Lagrangian submanifold
77(114)
Outline of
Chapter 3
77(9)
Algebraic framework on filtered A∞ algebras
86(8)
A∞ algebras and homomorphisms
89(1)
Filtered A∞ algebras and homomorphisms
89(5)
Algebraic framework on the homotopy unit
94(3)
Definition of the homotopy unit
94(3)
Unital (resp. homotopy unital) A∞ homomorphisms
97(1)
A∞ deformation of the cup product
97(5)
The filtered A∞ algebra associated to a Lagrangian submanifold
102(5)
Bounding cochains and the A∞ Maurer-Cartan equation
107(13)
Bounding cochains and deformations
108(3)
Obstruction for the existence of bounding cochain
111(3)
Weak unobstructedness and existence of Floer cohomology
114(3)
The superpotential and M(C)
117(3)
A∞ bimodules and Floer cohomology
120(36)
Algebraic framework
120(3)
A∞ bimodule homomorphisms
123(2)
Weak unobstructedness and deformations
125(1)
The filtered A∞ bimodule C(L(1), L(0);Λ0, nov)
126(11)
The Bott-Morse case
137(14)
Examples
151(4)
The multiplicative structure on Floer cohomology
155(1)
Inserting marked points in the interior
156(35)
The operator
156(3)
Applications to vanishing of the obstruction classes ok (L)
159(2)
Out line of the construction of the operator
161(4)
The operator
165(3)
Bulk deformation of filtered A∞ Structures
168(7)
Out line of the construction of the operator
175(3)
The operator and the A∞ bimodule
178(3)
Construction of the operator
181(1)
Generalization of the operator
182(6)
Proof of parts of Theorems B, C and G
188(3)
Homotopy equivalence of A∞ algebras
191(76)
Outline of
Chapters 4 and 5
191(6)
Homotopy equivalence of A∞ algebras: the algebraic framework
197(14)
Models of [ 0, 1] x C
197(8)
Homotopies between A∞ homomorphisms
205(3)
The unital or homotopy-unital cases
208(3)
Gauge equivalence of bounding cochains
211(6)
Basic properties and the category
211(4)
Mweak (C) and its homotopy invariance
215(1)
Mweak, def(L) and its homotopy invariance
216(1)
Uniqueness of the model of [ 0, 1] x C
217(16)
Induction on the number filtration I
218(1)
AK structures and homomorphisms
219(1)
Induction on the number filtration II
220(3)
Unital case I: the unfiltered version
223(3)
Coderivation and Hochschild cohomology
226(4)
Induction on the energy filtration
230(2)
Unital case II: the filtered version
232(1)
Whitehead theorem in A∞ algebras
233(9)
Extending AK homomorphisms to AK+1 homomorphisms
234(2)
Proof of Theorem 4.2.45 I: the number filtration
236(1)
Unital case: the unfiltered version
237(2)
Extending filtered A∞ homomorphism modulo Ti to modulo Ti+1
239(2)
Proof of Theorem 4.2.45 II: the energy filtration
241(1)
Homotopy equivalence of A∞ algebras: the geometric realization
242(25)
Construction of A∞ homomorphisms
242(7)
Homotopies between A∞ homomorphisms
249(8)
Compositions
257(2)
Homotopy equivalence and the operator I: changing the cycle in the interior
259(2)
Homotopy equivalence and the operator II: invariance of symplectic diffeomorphisms 1
261(3)
Homotopy equivalence and the operator III: invariance of symplectic diffeomorphisms 2
264(3)
Homotopy equivalence of A∞ bimodules
267(88)
Novikov rings
267(8)
Reduction to universal Novikov ring
267(3)
Hamiltonian independence of the Novikov ring
270(2)
Floer cohomologies over Λ(L(0), L(1); 0) and Λnov
272(3)
Homotopy equivalences of A∞ bimodules: the algebraic framework
275(21)
Weakly filtered A∞ bimodule homomorphisms
275(7)
Deformations of A∞ bimodule homomorphisms
282(1)
Homotopies between A∞ bimodule homomorphisms
282(6)
Gauge invariance and the category (C1, C0)
288(3)
Obstructions to defining. A∞ bimodule homomorphisms I
291(1)
Whitehead theorem for A∞ bimodule homomorphisms
292(2)
Obstructions to defining A∞ bimodule homomorphisms II
294(2)
Homotopy equivalences of A∞ bimodules: the geometric realization
296(34)
Construction of filtered A∞ bimodule homomorphisms
296(10)
Moving Lagrangian submanifolds by Hamiltonian isotopies
306(7)
Homotopies between bimodule homomorphisms
313(6)
Compositions of Hamiltonian isotopies and of bimodule homomorphisms
319(2)
An energy estimate
321(5)
The operators and homotopy equivalence
326(1)
Wrap-up of the proof of invariance of Floer cohomologies
327(3)
Canonical models, formal super schemes and Kuranishi maps
330(25)
Canonical models, Kuranishi maps and bounding cochains
330(6)
The canonical models of filtered A∞ bimodules
336(1)
Filtered A∞ bimodules and complex of coherent sheaves
337(2)
Construction of the canonical model
339(8)
Including the operator
347(2)
Wrap-up of the proofs of Theorems F, G, M, N and Corollaries O, P
349(6)
Spectral sequences
355(42)
Statement of the results in
Chapter 6
355(7)
The spectral sequence
355(2)
Non-vanishing theorem and a Maslov class conjecture
357(3)
Applications to Lagrangian intersections
360(2)
A toy model: rational Lagrangian submanifolds
362(4)
The algebraic construction of the spectral sequence
366(9)
c.f.z.
367(2)
d.g.e.f.z. (differential graded c.f.z.)
369(2)
Construction and convergence
371(4)
The spectral sequence associated to a Lagrangian submanifold
375(10)
Construction
375(1)
A condition for degeneration: proof of (D.3)
375(2)
Non-vanishing theorem: proof of Theorem 6.1.9.
377(4)
Application to the Maslov class conjecture: proofs of Theorems 6.1.15 and 6.1.17
381(1)
Compatibility with the product structure
382(3)
Applications to Lagrangian intersections
385(12)
Proof of Theorem H
385(1)
Proof of Theorem I
385(3)
Torsion of the Floer cohomology and Hofer distance: Proof of Theorem J
388(5)
Floer cohomologies of Lagrangian submanifolds that do not intersect cleanly
393(2)
Unobstructedness modulo TE
395(2)
Transversality
397(278)
Construction of the Kuranishi structure
398(37)
Statement of the results in Section 7.1
398(3)
Kuranishi charts on Mmain, reg k+1 (β): Fredholm theory
401(3)
Kuranishi charts in the complement of Mmain, reg k+1 (β): gluing
404(14)
Wrap-up of the proof of Propositions 7.1.1 and 7.1.2
418(7)
The Kuranishi structure of Mmain k+1 (M', L', {Jρ}ρ: βtop(ρ)): A∞ map analog of Stasheff cells
425(10)
Multisections and choice of a countable set of chains
435(139)
Transversality at the diagonal
436(1)
Inductive construction of compatible system of multisections in the Bott-Morse case
437(7)
Perturbed moduli space running out of the Kuranishi neighborhood I
444(1)
Statement of results
445(4)
Proof of Proposition 7.2.35
449(9)
Filtered An, k structures
458(3)
Construction of filtered An, k structures
461(5)
Perturbed moduli space running out of the Kuranishi neigborhood II
466(2)
Construction of filtered An, K homomorphisms
468(15)
Constructions of filtered An, K homotopies
483(19)
Constructions of filtered A∞ homotopies I: a short cut
502(3)
Constructions of filtered A∞ homotopies II: the algebraic framework on homotopy of homotopies
505(29)
Constructions of filtered A∞ homotopies III: the geometric realization of homotopy of homotopies
534(35)
Bifurcation vs cobordism method: an alternative proof
569(5)
Construction of homotopy unit
574(15)
Statement of the result and the idea of its proof
574(2)
Proof of Theorem 7.3.1
576(11)
Proof of (3.8.36)
587(2)
Details of the construction of the operators , and
589(1)
Details of the construction of
589(61)
Construction of I: the An, K version
595(1)
Construction of II: is an L∞ homomorphism
596(5)
Construction of III: the homotopy invariance of Der((C[ 1]), (C[ 1]))
601(20)
Construction of IV: wrap-up of the proof of Theorem 3.8.32
621(4)
Proof of Theorem Y
625(6)
Algebraic formulation of I: Der B(C1, C0; D) and its homotopy invariance
631(6)
Algebraic formulation of II: via bifurcation argument
637(3)
Algebraic formulation of III: Via cobordism argument
640(4)
Algebraic formulation of I: the cyclic bar complex is an L∞ module
644(3)
Algebraic formulation of II: induces an L∞ module homomorphism
647(3)
Compatibility with rational homotopy theory
650(1)
Statement of results
650(25)
Virtual fundamental chain in de Rham theory
652(2)
The Kuranishi structure of Mmain k+1 (β0)
654(1)
Construction of the AK homomorphism I
655(8)
Construction of the AK homomorphism II
663(6)
The A∞ map to a topological monoid and N'k+1
669(6)
Orientation
675(78)
Orientation of the moduli space of unmarked discs
675(16)
The case of holomorphic discs
675(9)
The example of non-orientable family index
684(2)
The case of connecting orbits in Floer theory
686(4)
Change of relatively spin structure and orientation
690(1)
Convention and preliminaries
691(7)
Orientation of the moduli space of marked discs and of the singular strata of the moduli space
698(5)
Orientation of M+1 (β P1,..., P)
703(5)
Definition of the orientation of M+1 (β P1,..., P)
703(2)
Cyclic symmetry and orientation
705(3)
The filtered A∞ algebra case
708(5)
Orientation of the moduli space of constant maps
713(3)
Orientation of the moduli space of connecting orbits
716(3)
The Bott-Morse case
719(12)
Orientations of the top-moduli spaces and the twp-moduli spaces
731(7)
Orientations of Mmain k+1 (M', L', {Jρ}ρ: β;top(ρ))
731(4)
Orientation of Mmain k+1 ({Jρ}ρ: βtwp(ρ);1,..., k)
735(3)
Homotopy units, the operators, continuous families of perturbations, etc.
738(15)
Homotopy unit
738(1)
Operators
738(11)
Continuous families of perturbations
749(4)
Appendices
753(38)
Kuranishi structures
753(27)
Review of the definition of the Kuranishi structure and multisections
754(10)
Fiber products
764(2)
Finite group actions and the quotient space
766(2)
A remark on smoothness of coordinate transforms
768(10)
Some counter examples
778(1)
Some errors in the earlier versions and corrections thereof
779(1)
Singular chains with local coefficients
780(2)
Filtered L∞ algebras and symmetrization of filtered A∞ algebras
782(5)
The differential graded Lie algebra homomorphism in Theorem 7.4.132
787(4)
Bibliography 791(10)
Index 801