- Cellular homology
In
mathematics , cellular homology inalgebraic topology is ahomology theory forCW-complex es.It agrees with
singular homology , and can provide an effective means of computing homology modules. If "X" is a CW-complex withn-skeleton "Xn", the cellular homology modules are defined as thehomology group s of the cellularchain complex ::cdots o H_{n+1}( X_{n+1}, X_n ) o H_n( X_n, X_{n-1} ) o H_{n-1}( X_{n-1}, X_{n-2} ) o cdots .
The module
:H_n( X_n, X_{n-1} ) ,
is free, with generators which can be identified with the "n"-cells of "X". The
boundary map s:H_n(X_n,X_{n-1}) o H_{n-1}(X_{n-1},X_{n-2}) ,
can be determined by computation of the degrees of the
attaching map s of the cells.One sees from the cellular chain complex that the "n"-skeleton determines all lower-dimensional homology:
:H_k(X) cong H_k(X_n)
for "k" < "n".
An important consequence of the cellular perspective is that if a CW-complex has no cells in consecutive dimensions, all its homology modules are free. For example,
complex projective space CP"n" has a cell structure with one cell in each even dimension; it follows that for 0 ≤ "k" ≤ "n",:H_{2k}(mathbb{CP}^n; mathbb{Z}) cong mathbb{Z}
and
:H_{2k+1}(mathbb{CP}^n) = 0 .
Euler characteristic
For a cellular complex "X", let "Xj" be its "j"-th skeleton, and "cj" be the number of "j"-cells, i.e. the rank of the free module "Hj"("Xj", "X""j"-1). The
Euler characteristic of "X" is defined by:chi (X) = sum _0 ^n (-1)^j c_j.
The Euler characteristic is a homotopy invariant. In fact, in terms of the
Betti number s of "X",:chi (X) = sum _0 ^n (-1)^j ; mbox{rank} ; H_j (X).
This can be justified as follows. Consider the long exact sequence of
relative homology for the triple ("Xn", "X""n" - 1 , ∅)::cdots o H_i( X_{n-1}, empty) o H_i( X_n, empty) o H_i( X_{n}, X_{n-1} ) o cdots .
Chasing exactness through the sequence gives
:sum_{i = 0} ^n (-1)^i ; mbox{rank} ; H_i (X_n, empty)
= sum_{i = 0} ^n (-1)^i ; mbox{rank} ; H_i (X_n, X_{n-1}) ; + ; sum_{i = 0} ^n (-1)^i ; mbox{rank} ; H_i (X_{n-1}, empty).
The same calculation applies to the triple ("X""n" - 1, "X""n" - 2, ∅), etc. By induction,
:
sum_{i = 0} ^n (-1)^i ; mbox{rank} ; H_i (X_n, empty)
= sum_{j = 0} ^n ; sum_{i = 0} ^j (-1)^i ; mbox{rank} ; H_i (X_j, X_{j-1})
= sum_{j = 0} ^n (-1)^j c_j.
Wikimedia Foundation. 2010.