Tuesday, October 13, 2020

Solutions To Algebraic Homology Problems

 1)  Given   = { 1 1 ,  2 ,   1 }   

a) Find:  dim o ,   dim 1  ,   dim  Z1 , dim B 1  

Thence or otherwise, find:  H1 = Z/ B 1 

Solutions:

dim C o = the number of nodes = n  =  3

dim C 1  = the number of branches = b =  3

dim Z 1 = the number of cycles = [b + 1 – n] =   3 +1 – 3 = 1

dim B 1  = 1  (The number of boundaries for the given cycle, i.e. every boundary is a cycle.).    Then the quotient space is:

H1 = Z/ B 1   =   dim Z 1  -  dim B 1  =  1 - 1 = 0

b)Write expressions for each branch (or chain, or 1 - simplex) for the figure.

(B – A)  =  11

(C – B)  =  12

(A – C)  =  13

c) This (triangle space )group we can denote by B 1 (D ).

 Which also implies the group of n-cycles or  Z n (D )  

Write an expression for the 1-cycle   1  and thence show we may write:

1 (Z 1)  =    a  +   b    +   g

We may write for the one cycle:

Z 1 =  AB + BC + CA    Or:

1 (Z 1)  =    (B – A)  +   (C – B) +   (A – C)  

Or:  1 (Z 1)  =    a  +   b   +   g

Letting:     =   (B – A) ;   = (C – B);   g =  (A – C)  

Thereby confirming Z 1 is a 1-cycle.

d) Let each node be expressed:  A =   0 1         B =   0 2         C =   0 3          

 Then rewrite each as vectors of the chain space  C o 

Let us take the set of nodes  S o    first for which elements applied to the previous triangle ABC we have::

A =   0 1         B =   0 2              C =   0 3          

Then each of these can alternatively be denoted by vectors of the chain space  C o 

A =   0 1       =   [1, 0,  0] T

B =   0 2       =   [0, 1,  0] T

C =   0 3       =   [0, 0,  1] T


e)  Repeat for the 1-simplexes (chains)  as vectors related to the chain space C :

By the same token each of the simplexes can be specified by vectors related to the chain space C 1 :

   : 11   =  a  =  (B - A) =   [0, 1,  0] T   -  [1, 0,  0] T      etc.

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