Homology Structure of Word

sekinanlogoshome

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)     
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
βH2(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
Sekinan Research Field of Language



[Related Note / June 18, 2009]
Potential of Language / Floer Homology Language / June 18, 2009