Floer Homology Language
Note6
Homology Structure of Word
§ 1
1
Compact manifold in small diameter M
Inner product space h (M)
Map h (M)
k→ h (M)
2
A model and B model by Witten, E.
A model M as symplectic structure
B model M as complex structure
3
(A model)
(Definition)
H kA (M) = H k(M;C)
H *A (M) =
H kA (M)
Inner product <. , .>A <u, v>A =
( 
; cup product)
mA, 02(u, v) =
4
(Theorem)
( H *A (M), mA, 02, <. , .>A ) is Frobenius algebra.
5
Oriented 2-dimensional manifold with genus g Σg
6
J (Σg) = {JΣg | Smooth complex structure over Σg }
Integer over 0 k
Different k-number points over Σg z1, ..., zk (Gathered points are expressed by
. )
Diff (Σg,) = {ψ : Σg → Σg | ψ(
) = , ψ is differential homeomorphism. }
Quotient space M g,k = J (Σg) / Diff (Σg,)
Rieman surface of genus g with k marked points (Σ,)
(Σ,) 
M g,k
Autmorphism group Aut (Σ,) = { ψ : Σ → Σ | ψ is biregular.ψ() = }
Compactification of M g,k CM g,k
§2
１
Symplectic manifold M
Differential 2-form over M wM
Well-formed almost complex structure with wM JM
β
H2(M; Z)
(
is pseudoholomorphic.)
(Σ,, φ) 
2
[(Σ,, φ)]
Evaluation map ev[(Σ,, φ)]=
3
Forgetting map fg :
Enlarged Forgetting map fg :
3
(Definition)
(Gromov-Witten invariant)
(ev, fg)*
Gromov-Witten invariant is expressed by GWg,k(M, wM; β)
4
(Theorem)
Sumset
is compact.
5
(Associative law)
(Theorem)
[Image]
Meaning minimum of word is identified with
.
Word is identified with
Commutativity of meaning minimums in word guaranteed by theorem of associative law.
[References]
§ 1
1
Compact manifold in small diameter M
Inner product space h (M)
Map h (M)
k→ h (M) 2
A model and B model by Witten, E.
A model M as symplectic structure
B model M as complex structure
3
(A model)
(Definition)
H kA (M) = H k(M;C)
H *A (M) =
H kA (M) Inner product <. , .>A <u, v>A =
( ; cup product)mA, 02(u, v) =
4
(Theorem)
( H *A (M), mA, 02, <. , .>A ) is Frobenius algebra.
5
Oriented 2-dimensional manifold with genus g Σg
6
J (Σg) = {JΣg | Smooth complex structure over Σg }
Integer over 0 k
Different k-number points over Σg z1, ..., zk (Gathered points are expressed by
. )Diff (Σg,
) = {ψ : Σg → Σg | ψ() = , ψ is differential homeomorphism. }Quotient space M g,k = J (Σg) / Diff (Σg,
) Rieman surface of genus g with k marked points (Σ,
) (Σ,
) M g,k Autmorphism group Aut (Σ,
) = { ψ : Σ → Σ | ψ is biregular.ψ() = }Compactification of M g,k CM g,k
§2
１
Symplectic manifold M
Differential 2-form over M wM
Well-formed almost complex structure with wM JM
β
H2(M; Z) (is pseudoholomorphic.)(Σ,
, φ) 2
[(Σ,
, φ)] Evaluation map ev[(Σ,
, φ)]=3
Forgetting map fg :
Enlarged Forgetting map fg :
3
(Definition)
(Gromov-Witten invariant)
(ev, fg)*
Gromov-Witten invariant is expressed by GWg,k(M, wM; β)
4
(Theorem)
Sumset
is compact.5
(Associative law)
(Theorem)
[Image]
Meaning minimum of word is identified with
.Word is identified with
Commutativity of meaning minimums in word guaranteed by theorem of associative law.
[References]
Homology on Language / Symmetry Flow Language / Tokyo May 15, 2007
From Cell to Manifold / Cell Theory / Tokyo June 2, 2007
Deep Fissure between Word and Sentence / Algebraic Linguistics / Linguistic Result / Tokyo September 10, 2007
Reversion Analysis Theory / Tokyo June 8, 2008
Reversion Analysis Theory 2 / Tokyo June 12, 2008
Holomorphic Meaning Theory 10th for KARCEVSKIJ Sergej / Tokyo June 15, 2008
Holomorphic Meaning Theory 11th for KARCEVSKIJ Sergej / Tokyo June 19, 2008
Word and Meaning minimum / Energy Distance Theory / Conjecture 1 / Tokyo September 22, 2008
Geometry of Word / Energy Distance Theory / Conjecture2 / Tokyo November 23, 2008
Amplitude of meaning minimum / Complex Manifold Deformation Theory / Conjecture A4 / Tokyo December 17, 2008
Tokyo June 16, 2009
From Cell to Manifold / Cell Theory / Tokyo June 2, 2007
Deep Fissure between Word and Sentence / Algebraic Linguistics / Linguistic Result / Tokyo September 10, 2007
Reversion Analysis Theory / Tokyo June 8, 2008
Reversion Analysis Theory 2 / Tokyo June 12, 2008
Holomorphic Meaning Theory 10th for KARCEVSKIJ Sergej / Tokyo June 15, 2008
Holomorphic Meaning Theory 11th for KARCEVSKIJ Sergej / Tokyo June 19, 2008
Word and Meaning minimum / Energy Distance Theory / Conjecture 1 / Tokyo September 22, 2008
Geometry of Word / Energy Distance Theory / Conjecture2 / Tokyo November 23, 2008
Amplitude of meaning minimum / Complex Manifold Deformation Theory / Conjecture A4 / Tokyo December 17, 2008
Tokyo June 16, 2009
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