Page 288 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

APPENDIX 263

Addition of k-chains is commutative and associative, and
q
k k k k
C = C + C +· · · + C (5.8)
i 1 2 q
1
k
k
is the set of k-cells that belong to an odd number of the k-chains C .For any C ,
i
k
k
k
k
k
C + C = 0 (the “zero chain”), hence the equation C + X = C is satisfied by
1 2
k
k
k
X = C + C andnoother k-chain. Thus the k-chain on G form a commutative
1 2
group under the operation of addition modulo 2.
k
k
The boundary, ∂C ,of the k-chain C on G (for k = 1, 2) is the set of (k − 1)-
k
cells of G that are contained in an odd number of k-cells of C . (The boundary
of a 0-chain is not defined.)
Theorem 5.9.2. [110, V.2.1]
k
k
k
∂(C + C ) = ∂C + ∂C 2 k (k = 1, 2)
2
1
1
2
2
Since ∂ = 0, it follows from Theorem 5.9.2 that for any C ,
2 −1
2
2
∂(C ) = ∂(C + ) = ∂C 2
A k-cycle,for k = 1, 2, is a k-chain whose boundary is zero; a 0-cycle is a
0-chain with an even number of 0-cells. The sum mod 2 of any set of k-cycle is
a k-cycle (by Theorem 5.9.2, or directly from k = 0).
Theorem 5.9.3. [110, V.2.2]. The boundary of any k-chain is a (k − 1)-cycle
(k = 1, 2).
k
The k-chain C (a finite set whose members are k-cells) is to be distinguished
k
from the union of its k-cells, a set of points denoted by |C | and called the locus
k
k
of C ,or the set of points covered by C .
k
k
k
k
k
k
k
k
Clearly, |C + C |⊂|C |∪ |C | in all cases, and |C + C |=|C |∪ |C | if
1 2 1 2 1 2 1 2
k k 2 −1
and only if the C and C have no common k-cells. Note that whereas |C |
1 2
2 −1
2 −1
2 −1
is an open set, |(C ) | is a closed set, and in fact |(C ) |= |C | .
k
k
A k-chain C is, by definition, connected if its locus |C | is connected. The
k
maximal connected k-chains contained in any k-chain C are called the compo-
k
k
nents of C . They have as loci the components of |C |.
k
k
k
Theorem 5.9.4. [110, V.3.1] If K is a component of C , ∂K is the part of ∂C k
k
in K (k> 0).
1
Theorem 5.9.5. If x and y form the boundary of a 1-chain C , they are connected
1
in |C |.