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)
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, 
)
Riemann surface of genus g with k marked points (Σ, 
)
(Σ, 
) 
M g,k
Automorphism 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
Sekinan Research Field of Language
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