Topology in Molecular Biology

9 Euler, Dehn–Sommerville Characteristics, and Their Applications 167

9.4 Homology Groups and Characteristic Classes

Given asimplicial complexK, we fix an order of its verticesα 0 ,α 1 ,...,αm.
Then anr-dim simplex ofKcan be written as [αi 0 ,αi 1 ,...,αir],i 0 <···<ir.
This fixes a canonicalorientationon it.
Suppose that we are also given anabelian groupGwith operation “+”. A
k-dimensionalchainofKwith coefficients inGis a finite linear combination
of distinctk-simplices ofKof the form

ck=

∑

i

giσi,gi∈G.

Chains ofKof dimensionkform an abelian groupCk(K) with the sum of
two chainsckandc′k=

∑

ig

′

iσigiven by

ck+c′k=

∑

i

(gi+gi′)σi.

Theboundaryof ann-simplexσn=[α 0 ,...,αn] is the (n−1)-chain

∂σn=∂[α 0 ,...,αn]:=

∑n

i=0

(−1)iσn(i−)^1 ,

whereσ(ni−)^1 := [α 0 ,...,α̂i,...,αn]. For example,

∂[α 0 ]=0,

∂[α 0 ,α 1 ]=[α 1 ]−[α 0 ],

∂[α 0 ,α 1 ,α 2 ]=[α 1 α 2 ]−[α 0 α 2 ]+[α 0 α 1 ].

For anyn-dim simplex we have

∂∂[α 0 ...αn]=

∑n

i=0

(−1)i∂σn(i−)^1 =0.

• 
  


• 
  


• 
  

  a
  a

  
  

a

0

1

2

Fig. 9.3.Boundary