Note 5
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(Σ)
Σ.
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 : RT k+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
.
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]
Note 6 / Tokyo April 26, 2009
Tokyo June 11, 2009
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