Monday 22 September 2014

Homology Structure of Word / Floer Homology Language / 16 June 2009


Floer Homology Language

   

Note6

Homology Structure of Word


§ 1
1
Compact manifold in small diameter      M
Inner product space     h (M)
Map      h (Mk→  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)
kA (M) = H k(M;C)

*A (M) = kA (M
Inner product <. , .>A     <uv>A =      (  ; cup product)
mA, 02(uv) = 
4
(Theorem)
*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      g,k J (Σg) / Diff (Σg
Rieman surface of genus g with k marked points     (Σ
(Σg,k 
Autmorphism group     Aut (Σ= { ψ : Σ → Σ | ψ is biregular.ψ() = }
Compactification of  g,k      CM g,k 

§2

Symplectic manifold    M
Differential 2-form over M     wM
Well-formed almost complex structure with wM     JM
βH2(MZ)
 (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, wMβ)
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]
Sekinan Research Field of Language


[Related Note / June 18, 2009]

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