Lagrangian intersection floer theory: anomaly and obstruction
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Providence, R. I.
American Mathematical Society
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Internformat
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| LEADER | 00000nam a2200000 ca4500 | ||
|---|---|---|---|
| 001 | BV035844528 | ||
| 003 | DE-604 | ||
| 005 | 20110826 | ||
| 007 | t | ||
| 008 | 091125nuuuuuuuu |||| 00||| eng d | ||
| 020 | |a 9780821848319 |c set |9 978-0-8218-4831-9 | ||
| 035 | |a (OCoLC)426147150 | ||
| 035 | |a (DE-599)BVBBV035844528 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 084 | |a SK 380 |0 (DE-625)143235: |2 rvk | ||
| 245 | 1 | 0 | |a Lagrangian intersection floer theory |b anomaly and obstruction |c Kenji Fukaya ... [et al.] |
| 264 | 1 | |a Providence, R. I. |b American Mathematical Society | |
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a AMS/IP Studies in Advanced Mathematics |v ... | |
| 650 | 4 | |a Floer homology | |
| 650 | 4 | |a Lagrangian points | |
| 650 | 4 | |a Symplectic geometry | |
| 650 | 0 | 7 | |a Symplektische Geometrie |0 (DE-588)4194232-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Symplektische Geometrie |0 (DE-588)4194232-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Fukaya, Kenji |d 1959- |e Sonstige |0 (DE-588)142815144 |4 oth | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018702786&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-018702786 | ||
Datensatz im Suchindex
| _version_ | 1804140814312407040 |
|---|---|
| adam_text | Contents
Volume
I
Preface
Chapter
1.
Introduction
1
1.1
What is
Floer (co)homology
1
1.2
General theory of Lagrangian
Floer
cohomology
5
1.3
Applications to symplectic geometry
13
1.4
Relation to mirror symmetry
16
1.5
Chapter-wise outline of the main results
25
1.6
Acknowledgments
35
1.7
Conventions
36
Chapter
2.
Review:
Floer
cohomology
39
2.1
Bordered stable maps and the Maslov index
39
2.1.1
The Maslov index: the relative first Chern number
39
2.1.2
The moduli space of bordered stable maps
43
2.2
The Novikov covering and the action functional
49
2.2.1
The r-equivalence
50
2.2.2
The action functional and the Maslov-Morse index
51
2.3
Review of
Floer
cohomology I: without anomaly
53
2.3.1
The L2-gradient equation of
Λ
53
2.3.2
Floer s definition: Za-coefficients.
57
2.3.3
Bott-Morse
Floer
cohomology
59
2.4
Review of
Floer
cohomology II: anomaly appearance
60
2.4.1
The
Floer cochain
module
61
2.4.2
The
Floer
moduli space
62
2.4.3
The Novikov ring AR(L)
66
2.4.4
Monotone Lagrangian submanifolds
69
2.4.5
Appearance of the primary obstruction
71
Chapter
3.
The
Аж
algebra associated to a Lagrangian submanifold
77
3.1
Outline of Chapter
3 77
3.2
Algebraic framework on filtered
Αχ
algebras
86
3.2.1
Аж
algebras and homomorphisms
86
3.2.2
Filtered Ax algebras and homomorphisms
89
3.3
Algebraic framework on the homotopy unit
94
3.3.1
Definition of the homotopy unit
94
3.3.2
Unital
(resp.
homotopy unital) Aoo homomorphisms
97
3.4
Αχ
deformation of the cup product
97
vi
CONTENTS
3.5
The filtered
A^
algebra associated to a Lagrangian submanifold
102
3.6
Bounding cochains and the
Ам
Maurer-Cartan
equation.
107
3.6.1
Bounding cochains and deformations
108
3.6.2
Obstruction for the existence of bounding cochain 111
3.6.3
Weak unobstructedness and existence of
Floer cohomology
114
3.6.4
The
superpotential
and M{G)
117
3.7
Aoo bimodules and
Floer
cohomology
120
3.7.1
Algebraic framework
120
3.7.2
Aoo bimodule homomorphisms
123
3.7.3
Weak unobstructedness and deformations
125
3.7.4
The filtered
Аж
bimodule C(L
W
,
L1®
;
A0,nov)
126
3.7.5
The
Bott-Morse
case
137
3.7.6
Examples
151
3.7.7
The multiplicative structure on
Floer
cohomology
155
3.8
Inserting marked points in the interior
156
3.8.1
The operator
p
156
3.8.2
Applications to vanishing of the obstruction classes Ok(L)
159
3.8.3
Outline of the construction of the operator
p
161
3.8.4
The operator
q
165
3.8.5
Bulk deformation of filtered
A^
structures
168
3.8.6
Outline of the construction of the operator
q
175
3.8.7
The operator
r
and the
Аж
bimodule
178
3.8.8
Construction of the operator
t
181
3.8.9
Generalization of the operator
p
182
3.8.10
Proof of parts of Theorems
В, С
and
G
188
Chapter
4.
Homotopy equivalence of A^ algebras
191
4.1
Outline of Chapters
4
and
5 191
4.2
Homotopy equivalence of
А^
algebras: the algebraic framework
197
4.2.1
Models of
[0,1]
x C
197
4.2.2
Homotopies between A^ homomorphisms
205
4.2.3
The unital or homotopy-unital cases
208
4.3
Gauge equivalence of bounding cochains
211
4.3.1
Basic properties and the category
ЅјШоо
211
4.3.2
A^weakCC) arjd its homotopy
invariance
215
4.3.3
Aiv/eak,deí(Ľ)
and its homotopy
invariance
216
4.4
Uniqueness of the model of
[0,1]
x C
217
4.4.1
Induction on the number filtration I
218
4.4.2
Ak structures and homomorphisms
219
4.4.3
Induction on the number filtration II
220
4.4.4
Unital case I: the unfiltered version
223
4.4.5
Coderivation and
Hochschild
cohomology
226
4.4.6
Induction on the energy filtration
230
4.4.7
Unital case II: the filtered version
232
4.5
Whitehead theorem in A«, algebras
233
4.5.1
Extending AK homomorphisms to
Ακ+ι
homomorphisms
234
4.5.2
Proof of Theorem
4.2.45
I: the number filtration
236
4.5.3
Unital case: the unfiltered version
237
CONTENTS
vii
4.5.4
Extending filtered 4M homomorphism modulo TXi to modulo
Τλ·+Χ
239
4.5.5
Proof of Theorem
4.2.45
II: the energy filtration
241
4.6
Homotopy equivalence of A^ algebras: the geometric realization
242
4.6.1
Construction of Aoa homomorphisms
242
4.6.2
Homotopies between A^ homomorphisms
249
4.6.3
Compositions
257
4.6.4
Homotopy equivalence and the operator
q
I: changing the cycle
in the interior
259
4.6.5
Homotopy equivalence and the operator
q
II:
invariance
of sym-
plectic diffeomorphisms
1 261
4.6.6
Homotopy equivalence and the operator
q
III:
invariance
of
symplectic diffeomorphisms
2 264
Chapter
5.
Homotopy equivalence of
Αχ,
bimodules
267
5.1
Novikov rings
267
5.1.1
Reduction to universal Novikov ring
267
5.1.2
Hamiltonian independence of the Novikov ring
270
5.1.3
Floer cohomologies
over A(L(o),L(1);.£o) and Anov
272
5.2
Homotopy equivalences of j4oo bimodules: the algebraic framework
275
5.2.1
Weakly filtered
Аж
bimodule homomorphisms
275
5.2.2
Deformations of A«, bimodule homomorphisms
276
5.2.3
Homotopies between
Α^
bimodule homomorphisms
282
5.2.4
Gauge
invariance
and the category Sji&^Ci, Co)
288
5.2.5
Obstructions to defining
Аж
bimodule homomorphisms I
291
5.2.6
Whitehead theorem for
Α,χ,
bimodule homomorphisms
292
5.2.7
Obstructions to defining
A,*,
bimodule homomorphisms II
294
5.3
Homotopy equivalences of Aoo bimodules: the geometric realiza¬
tion
296
5.3.1
Construction of filtered Aqo bimodule homomorphisms
296
5.3.2
Moving Lagrangian submanifolds by Hamiltonian isotopies
306
5.3.3
Homotopies between bimodule homomorphisms
313
5.3.4
Compositions of Hamiltonian isotopies and of bimodule homo¬
morphisms
319
5.3.5
An energy estimate.
321
5.3.6
The operators q,t and homotopy equivalence
326
5.3.7
Wrap-up of the proof of
invariance
of
Floer
cohomologies
327
5.4
Canonical models, formal super schemes and Kuranishi maps
330
5.4.1
Canonical models, Kuranishi maps and bounding cochains
330
5.4.2
The canonical models of filtered
Аж
bimodules
336
5.4.3
Filtered A^ bimodules and complex of coherent sheaves
337
5.4.4
Construction of the canonical model
339
5.4.5
Including the operator
ą
347
5.4.6
Wrap-up of the proofs of Theorems
F, G, M, N
and Corollaries
Ο,Ρ
349
Chapter
6.
Spectral sequences
355
6.1
Statement of the results in Chapter
6 355
CONTENTS
6.1.1
The spectral sequence
355
6.1.2
Non-
vanishing theorem and a Maslov class conjecture
357
6.1.3
Applications to Lagrangian intersections
360
6.2
A toy model: rational Lagrangian submanifolds
362
6.3
The algebraic construction of the spectral sequence
366
6.3.1
c.f.z.
367
6.3.2
d.g.c.f.z. (differential graded c.f.z.)
369
6.3.3
Construction and convergence
371
6.4
The spectral sequence associated to a Lagrangian submanifold
375
6.4.1
Construction
375
6.4.2
A condition for degeneration: proof of (D.3)
375
6.4.3
Non-vanishing theorem: proof of Theorem
6.1.9. 377
6.4.4
Application to the Maslov class conjecture: proofs of Theorems
6.1.15
and
6.1.17 381
6.4.5
Compatibility with the product structure
382
6.5
Applications to Lagrangian intersections
385
6.5.1
Proof of Theorem
H
385
6.5.2
Proof of Theorem I
385
6.5.3
Torsion of the
Floer cohomology
and Hofer distance: Proof of
Theorem
J
388
6.5.4
Floer cohomologies
of Lagrangian submanifolds that do not in¬
tersect cleanly
393
6.5.5
Unobstructedness modulo TE
395
Volume II
Chapter
7.
Transversality
397
7.1
Construction of the Kuranishi structure
398
7.1.1
Statement of the results in Section
7.1 398
7.1.2
Kuranishi charts on
M^ies(ß):
Fredholm
theory
401
7.1.3
Kuranishi charts in the complement of
M^ ieg(ß):
gluing
404
7.1.4
Wrap-up of the proof of Propositions
7.1.1
and
7.1.2 418
7.1.5
The Kuranishi structure of
Mfţf(M ,L ,{Jp}p
: ß;top(p)):
Аса
map analog of Stasheff cells
425
7.2
Multisections and choice of a countable set of chains
435
7.2.1
Transversality at the diagonal
436
7.2.2
Inductive construction of compatible system of multisections in
the
Bott-Morse
case
437
7.2.3
Perturbed moduli space running out of the Kuranishi neighbor¬
hood I
444
7.2.4
Statement of results
445
7.2.5
Proof of Proposition
7.2.35 449
7.2.6
Filtered Anik structures
458
7.2.7
Construction of filtered
Ап^к
structures
461
7.2.8
Perturbed moduli space running out of the Kuranishi neigbor-
hood II
466
7.2.9
Construction of filtered
Αη,κ
homomorphisms
468
CONTENTS
ix
7.2.10
Constructions of filtered
А п,к
homotopies
483
7.2.11
Constructions of filtered
Аж
homotopies I: a short cut
502
7.2.12
Constructions of filtered A» homotopies II: the algebraic frame¬
work on homotopy of homotopies
505
7.2.13
Constructions of filtered
Аж
homotopies III: the geometric real¬
ization of homotopy of homotopies
534
7.2.14
Bifurcation vs cobordism method: an alternative proof
569
7.3
Construction of homotopy unit
574
7.3.1
Statement of the result and the idea of its proof
574
7.3.2
Proof of Theorem
7.3.1 576
7.3.3
Proof of
(3.8.36) 587
7.4
Details of the construction of the operators
p, q
and
r
589
7.4.1
Details of the construction of
p
589
7.4.2
Construction of
q
I: the
Ап>к
version
595
7.4.3
Construction of
q
II:
q
is an
£<»
homomorphism
596
7.4.4
Construction of
q
III: the homotopy
invariance
of Der(jB(C[l]),
ВІСЩ))
601
7.4.5
Construction of
q
IV: wrap-up of the proof of Theorem
3.8.32 621
7.4.6
Proof of Theorem
Y
625
7.4.7
Algebraic formulation of
r
I:
Der
B{C , Co; D) and its homotopy
invariance
631
7.4.8
Algebraic formulation of
r
II: via bifurcation argument
637
7.4.9
Algebraic formulation of
τ
III: via cobordism argument
640
7.4.10
Algebraic formulation of
ρ
I: the cyclic bar complex is an Lx
module
644
7.4.11
Algebraic formulation of
ρ Π: ρ
induces an Loo module homo¬
morphism
647
7.5
Compatibility with rational homotopy theory
650
7.5.1
Statement of results
650
7.5.2
Virtual fundamental chain in
de Rham
theory
652
7.5.3
The Kuranishi structure of
Mftfißo) 654
7.5.4
Construction of the
Αχ
homomorphism I
655
7.5.5
Construction of the
Ακ
homomorphism II
663
7.5.6
The .Aoo map to a topological monoid and N k+1
669
Chapter
8.
Orientation
675
8.1
Orientation of the moduli space of unmarked discs
675
8.1.1
The case of holomorphic discs
675
8.1.2
The example of non-orientable family index
684
8.1.3
The case of connecting orbits in
Floer
theory
686
8.1.4
Change of relatively spin structure and orientation
690
8.2
Convention and preliminaries
691
8.3
Orientation of the moduli space of marked discs and of the singular
strata of the moduli space
698
8.4
Orientation of Me+i
(β;
Pi
, ■ ■ · ,
Pi)
■ 703
8.4.1
Definition of the orientation of Me+i
(β;
Pi
,....
Pt)
703
8.4.2
Cyclic symmetry and orientation
705
8.5
The filtered Ax algebra case
708
8.6
Orientation of the moduli space of constant maps
713
χ
CONTENTS
8.7
Orientation
of the moduli space of connecting orbits
716
8.8
The
Bott-Morse
case
719
8.9
Orientations of the top-moduli spaces and the twp-moduli spaces
731
8.9.1
Orientation of
Mf^f{M ,
Ľ,
{Jp}p
:
β;
top(p))
731
8.9.2
Orientation of Mf%f({ Jp}p
:
β;
twp(p); Vu
...,
Vk)
735
8.10
Homotopy units, the operators p,q, continuous families of pertur¬
bations, etc.
738
8.10.1
Homotopy unit
738
8.10.2
Operators p,
q
738
8.10.3
Continuous families of perturbations
749
Appendices
753
Al
Kuranishi structures
753
Al.l Review of the definition of the Kuranishi structure and multi-
sections
754
A1.2 Fiber products
764
Al.3 Finite group actions and the quotient space
766
Al.
4
A remark on smoothness of coordinate transforms
768
Al.
5
Some counter examples
778
Al.
6
Some errors in the earlier versions and corrections thereof
779
A2 Singular chains with local coefficients
780
A3
Filtered Lqo algebras and symmetrization of filtered Aoo algebras
782
A4
The differential graded Lie algebra homomorphism in Theorem
7.4.132 787
Bibliography
791
Index
801
|
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| id | DE-604.BV035844528 |
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| indexdate | 2024-07-09T22:05:57Z |
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| isbn | 9780821848319 |
| language | English |
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| spelling | Lagrangian intersection floer theory anomaly and obstruction Kenji Fukaya ... [et al.] Providence, R. I. American Mathematical Society txt rdacontent n rdamedia nc rdacarrier AMS/IP Studies in Advanced Mathematics ... Floer homology Lagrangian points Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s DE-604 Fukaya, Kenji 1959- Sonstige (DE-588)142815144 oth Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018702786&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lagrangian intersection floer theory anomaly and obstruction Floer homology Lagrangian points Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd |
| subject_GND | (DE-588)4194232-2 |
| title | Lagrangian intersection floer theory anomaly and obstruction |
| title_auth | Lagrangian intersection floer theory anomaly and obstruction |
| title_exact_search | Lagrangian intersection floer theory anomaly and obstruction |
| title_full | Lagrangian intersection floer theory anomaly and obstruction Kenji Fukaya ... [et al.] |
| title_fullStr | Lagrangian intersection floer theory anomaly and obstruction Kenji Fukaya ... [et al.] |
| title_full_unstemmed | Lagrangian intersection floer theory anomaly and obstruction Kenji Fukaya ... [et al.] |
| title_short | Lagrangian intersection floer theory |
| title_sort | lagrangian intersection floer theory anomaly and obstruction |
| title_sub | anomaly and obstruction |
| topic | Floer homology Lagrangian points Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd |
| topic_facet | Floer homology Lagrangian points Symplectic geometry Symplektische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018702786&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT fukayakenji lagrangianintersectionfloertheoryanomalyandobstruction |