Grothendieck Group

sekinanlogoshome

Floer Homology Language

TANAKA Akio

   

Note3

Grothendieck Group


1
Topological space     X
All the sets of continuous families' isomorphic classes ofZ₂ graded Hermitian vector bundle over X  K_H(X)  
Set thatK_H (X)  is divided by equivalence relation as homotopy     K(X)
K(X) is called X's K-group.
2
Compact Haussdorff space    X 
Set of  all the isomorphic classes of complex vector bundle over X     Vect(X)
Vect(X) has structure of commutative semigroup by direct sum.
Φ_X : Vect(X) →K(X)
3
(Definition : Grothendieck Group)
Commutative semigroup     S
Commutative group     A_S
Homomorphism as semigroup  Φ_A : S →A_S    
Arbitrary module     R
Arbitrary homomorphism     Φ_R : S →R 
Homomorphism      Ψ : A_S→R    (1) 
Φ_R = ΨΦ_s    (2)
(1) and (2) are uniquely existent.
4
If there exists Grothendieck group, it is unique.
5
(Theorem)
When X is compact Hausdorff space, (K(X), Φ_X ) is Vect(X)'s Grothendieck group.
6
(Identification)
Language is identified with Vect(X). 
Language is commutative by associative law.
Word is identified with (K(X), Φ_X ).
Distance of word is identified with Hermitian vector bundle.

<Addition>
7
Element of K_H(X)     F
Support of F     supp F

supp F= 
All the K_H(X)'s subsets that support are compact       K_C(X)
8
(Theorem)
When X is locally compact Hausdorff,  arbitrary element F of K_C(X) is globally expressed.
9
Manifold is locally compact Hausdorff.

[References]
Distance Theory / Tokyo May 5, 2004
Actual Language and Imaginary Language / Tokyo September 23, 2004
Distance / Distance Theory Algebraically Supplemented / Tokyo Ocotber 26, 2007
Finsler Manifold and Distance / Energy Distance Theory / Tokyo November 7, 2008
Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008


Tokyo May 30, 2009
Sekinan Research Field of Language

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