Algebraic Topology and Topology Invariants - Mathematical Tools for Shape Analysis and Description - page 57

We denote by C q .K/ the group of q -chains of K with respect to the addition; for q larger than n or
for q smaller than 0 , we set C q .K/ equal to the trivial group. On the group C q .K/ , we can define
the boundary operator @ q WC q .K/!C q1 .K/ . is is defined as the trivial homomorphism if
q0 , while for q > 0 it acts on each q -simplex via:
@ q A 0 ;A 1 ;:::;A q D q X
iD0

O

.1/ i A 0 ;A 1 ;:::;A i1 ; A i ;A iC1 ;:::;A q

O

where A 0 ;A 1 ;:::;A i1 ; A i ;A iC1 ;:::;A q is the .q1/ -simplex obtained by eliminating the

vertex A i ; the boundary map @ q extends by linearity to arbitrary q -chains. In Figure 7.5 , the
boundary operator is evaluated on some elementary simplices. e arrows represent the orien-
tation of the simplices. Notice that changing the orientation of the simplices implies a different
result for the boundary operator.
A chain z2C q .K/ is called a cycle if the boundary operator sends z to zero, i.e., @ q zD0 ;
a chain b2C q .K/ is called a boundary if it is the image, through the boundary operator, of a
chain of dimension greater by one, i.e., there exists c2C qC1 .K/ such that bD@ qC1 c . e sets
of cycles and boundaries form two subgroups of C q .K/ :
Z q .K/Dfz2C q .K/j@ q zD0gD ker @ q
B q .K/Dfb2C q .K/jbD@ qC1 c; for some c2C qC1 .K/gD Im @ qC1 ;
where ker and Im denote the kernel and the image of the map, respectively. It holds that B q .K/
Z q .K/ , since @ q @ qC1 D0 . e q - th simplicial homology group of K is then the quotient group:

H q .K/DZ q .K/=B q .K/:

Figure 7.5: e boundary operator on elementary q -simplices [ 24 ].