Floer Homology Language Note 6 Homology Structure of Word
Floer Homology Language
TANAKA Akio
Note 6
Homology Structure of Word
§ 1
1
Compact manifold in small diameter M
Inner product space h (M)
Map h (M) 
k→ 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)
(A model)
(Definition)H kA (M) = H k(M;C)H *A (M) = 
H kA (M)
H kA (M) Inner product <. , .>A <u, v>A = 
( 
; cup product)mA, 02(u, v) = 
( ; 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. }
) = {ψ : Σg → Σg | ψ() = , ψ is differential homeomorphism. }Quotient space M g,k = J (Σg) / Diff (Σg, 
)
) Riemann surface of genus g with k marked points (Σ, 
) (Σ, 
) 
M g,k
) (Σ, ) M g,k Automorphism group Aut (Σ, 
) = { ψ : Σ → Σ | ψ is biregular.ψ(
) = 
}
) = { 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 JMβ
H2(M; Z)
(
is pseudoholomorphic.)(Σ,
, φ) 
2[(Σ,
, φ)] 
Evaluation map ev[(Σ,
, φ)]=
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)
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.
is compact.5(Associative law)(Theorem)
[Image]
Meaning minimum of word is identified with
.Word is identified with 
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 / Conjecture 2 / Tokyo November 23, 2008
Amplitude of meaning minimum / Complex Manifold Deformation Theory / Conjecture A4 / Tokyo December 17, 2008
Tokyo June 16, 2009
Sekinan Research Field of Language
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