Floer Homology Language Note 3 Grothendieck Group

Floer Homology Language

TANAKA Akio

   

Note 3

Grothendieck Group


1
Topological space    X
All the sets of continuous families' isomorphic classes ofZ2 graded Hermitian vector bundle over X  KH(X)  
Set thatKH (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.
ΦX : Vect(X) →K(X)
3
(Definition : Grothendieck Group)
Commutative semigroup    S
Commutative group     AS
Homomorphism as semigroup  ΦA : S →AS    
Arbitrary module     R
Arbitrary homomorphism     ΦR : S →R 
Homomorphism      Ψ : AS→R    (1) 
ΦR = ΨΦs    (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), ΦX ) 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), ΦX ).
Distance of word is identified with Hermitian vector bundle.

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

supp F= 

All the KH(X)'s subsets that support are compact      KC(X)
8
(Theorem)
When X is locally compact Hausdorff,  arbitrary element F of KC(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