Tuesday 23 September 2014

Homology Generation of Language / Floer Homology Language / 11 June 2009



Floer Homology Language

   

Note5

Homology Generation of Language


1
Isomorphism class of manifold     moduli space
Moduli space     M
2
Compactification of Mg, k       CMg, k         
3
Compact Hausdorff space     Σ
Riemann surface that has finite sum that is not intersected each other     
Continuous map     π :  →Σ
There exists finite set S(ΣΣ.
π's restriction  is homeomorphism.
 consists of 2 points over 
Structure over Σ is abbreviatedly  called semistable curve.
4
z1, ..., zk is different regular point.     (Σ) is called k pointed semistable curve.
3
Pointed semistable curve (Σ) is stable.     Automorphism Aut(Σ) group is finite group. 
5
Simply-connected and connected compact 1-dimensional simplicial complex       tree
Stable curve that has genus 0       Σ
Tree       ΓΣ       
6
Tree       Γ
Vertex number that each vertex has only one side      k+1
Tree that has k+1 vertexes        TRk+1
Γ TRk+1
Stable curve Σ that is ΓΣ = Γ       CM (Γ)
7
(Theorem, Knudsen-Keel)
Homology group  is generated from fundamental group of CM (Γ).
8
Γ TRm+1
shh : Rk+1 TRm+1
h0 : {1, 2, 3, 4} →{1, ..., k}
shh0 is expressed by sh.
9
Forgetting map     fg : Mg,k (M,JM;β) →Mg,k

10
There exists next equality at .
         (1)
11
(Theorem, Knudsen-Keel)
 is generated by  .
 is vector space expressed by relation (1).

[Image]
It is seemed to be existential that language is expressed by homology.



[References]
Tokyo June 11, 2009

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