In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.
If
X
Xn
… \to{Cn
where
X-1
The group
{Cn
is free abelian, with generators that can be identified with the
n
X
\alpha | |
e | |
n |
n
X
\alpha | |
\chi | |
n |
:\partial
\alpha | |
e | |
n |
\congSn\toXn-1
\alpha\beta | |
\chi | |
n |
: Sn\stackrel{\cong}{\longrightarrow} \partial
\alpha | |
e | |
n |
\alpha | |
\stackrel{\chi | |
n |
where the first map identifies
Sn
\partial
\alpha | |
e | |
n |
\alpha | |
\Phi | |
n |
\alpha | |
e | |
n |
\beta | |
e | |
n-1 |
(n-1)
q
Xn\setminus
\beta | |
e | |
n-1 |
\beta | |
e | |
n-1 |
Sn
Xn/\left(Xn\setminus
\beta | |
e | |
n-1 |
\right)
Sn
\beta | |
\Phi | |
n-1 |
\beta | |
e | |
n-1 |
The boundary map
\partialn:{Cn
is then given by the formula
{\partialn
where
\deg\left(
\alpha\beta | |
\chi | |
n |
\right)
\alpha\beta | |
\chi | |
n |
(n-1)
X
{Cn
The following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.
The n-dimensional sphere Sn admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from
Sn-1
{Ck
{Ck
k=0,n,
Hence for
n>1
...b\overset{\partialn+2
but then as all the boundary maps are either to or from trivial groups, they must all be zero, meaning that the cellular homology groups are equal to
n) | |
H | |
k(S |
=\begin{cases}Z&k=0,n\ \{0\}&otherwise.\end{cases}
n=1
\partial1
n
\Sigmag
\Sigmag
4n
\Sigmag
2n
4n
S0
… \to0\xrightarrow{\partial3}Z\xrightarrow{\partial2}Z2g\xrightarrow{\partial1}Z\to0,
Hk(\Sigmag)=\begin{cases}Z&k=0,2\ Z2g&k=1\ \{0\}&otherwise.\end{cases}
The n-torus
(S1)n
n=0,1,2,3
Thus,
1) | |
H | |
k((S |
n)\simeq\Z\binom{n{k}}
If
X
CW | |
H | |
n |
(X)
n
PnC
nC)= | |
H | |
k(P |
\Z
k=0,2,...,2n
RPn
k
ek
k\in\{0,1,...,n\}
k
\varphik\colonSk\toRPk
k
R
n | |
P | |
k |
\congRPk
k\in\{0,1,...,n\}
Ck(R
n | |
P | |
k, |
R
n | |
P | |
k-1 |
)\congZ
k\in\{0,1,...,n\}
To compute the boundary map
\partialk\colonCk(R
n | |
P | |
k, |
R
n | |
P | |
k-1 |
)\toCk(R
n | |
P | |
k-1 |
,R
n | |
P | |
k-2 |
),
k-1 | |
\chi | |
k \colon S |
\overset{\varphik}{\longrightarrow} RPk\overset{qk}{\longrightarrow} RPk/RPk\cong Sk.
-1 | |
\varphi | |
k |
(RPk)=Sk\subseteqSk
x\inRPk\setminusRPk
\varphi-1(\{x\})
Sk\setminusSk
\chik
\chik
Bk
\tildeBk
Sk\setminusSk
\chik|
Bk |
\chik|
\tildeBk |
\chik|
\tildeBk |
=\chik|
Bk |
\circA
A
Sk
(-1)k
\chik
Bk
1
\chik
\tildeBk
(-1)k
\deg(\chik) = 1+(-1)k = \begin{cases} 2&ifkiseven, \\ 0&ifkisodd. \end{cases}
\partialk
\deg(\chik)
We thus have that the CW-structure on
RPn
0 \longrightarrow Z \overset{\partialn}{\longrightarrow} … \overset{2}{\longrightarrow} Z \overset{0}{\longrightarrow} Z \overset{2}{\longrightarrow} Z \overset{0}{\longrightarrow} Z \longrightarrow 0,
\partialn=2
n
\partialn=0
n
RPn
Hk(RPn) = \begin{cases} Z&ifk=0andk=nodd, \\ Z/2Z&if0<k<nodd, \\ 0&otherwise. \end{cases}
One sees from the cellular chain complex that the
n
{Hk
for
k<n
CPn
0\leqk\leqn
{H2
and
{H2
The Atiyah–Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.
For a cellular complex
X
Xj
j
cj
j
{Cj
X
\chi(X)=
n | |
\sum | |
j=0 |
(-1)jcj.
The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of
X
\chi(X)=
n | |
\sum | |
j=0 |
(-1)j\operatorname{Rank}({Hj
This can be justified as follows. Consider the long exact sequence of relative homology for the triple
(Xn,Xn,\varnothing)
… \to{Hi
Chasing exactness through the sequence gives
n | |
\sum | |
i=0 |
(-1)i\operatorname{Rank}({Hi
The same calculation applies to the triples
(Xn,Xn,\varnothing)
(Xn,Xn,\varnothing)
n | |
\sum | |
i=0 |
(-1)i \operatorname{Rank}({Hi