- Source: Cellular homology
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.
Definition
If
X
{\displaystyle X}
is a CW-complex with n-skeleton
X
n
{\displaystyle X_{n}}
, the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex
⋯
→
C
n
+
1
(
X
n
+
1
,
X
n
)
→
C
n
(
X
n
,
X
n
−
1
)
→
C
n
−
1
(
X
n
−
1
,
X
n
−
2
)
→
⋯
,
{\displaystyle \cdots \to {C_{n+1}}(X_{n+1},X_{n})\to {C_{n}}(X_{n},X_{n-1})\to {C_{n-1}}(X_{n-1},X_{n-2})\to \cdots ,}
where
X
−
1
{\displaystyle X_{-1}}
is taken to be the empty set.
The group
C
n
(
X
n
,
X
n
−
1
)
{\displaystyle {C_{n}}(X_{n},X_{n-1})}
is free abelian, with generators that can be identified with the
n
{\displaystyle n}
-cells of
X
{\displaystyle X}
. Let
e
n
α
{\displaystyle e_{n}^{\alpha }}
be an
n
{\displaystyle n}
-cell of
X
{\displaystyle X}
, and let
χ
n
α
:
∂
e
n
α
≅
S
n
−
1
→
X
n
−
1
{\displaystyle \chi _{n}^{\alpha }:\partial e_{n}^{\alpha }\cong \mathbb {S} ^{n-1}\to X_{n-1}}
be the attaching map. Then consider the composition
χ
n
α
β
:
S
n
−
1
⟶
≅
∂
e
n
α
⟶
χ
n
α
X
n
−
1
⟶
q
X
n
−
1
/
(
X
n
−
1
∖
e
n
−
1
β
)
⟶
≅
S
n
−
1
,
{\displaystyle \chi _{n}^{\alpha \beta }:\mathbb {S} ^{n-1}\,{\stackrel {\cong }{\longrightarrow }}\,\partial e_{n}^{\alpha }\,{\stackrel {\chi _{n}^{\alpha }}{\longrightarrow }}\,X_{n-1}\,{\stackrel {q}{\longrightarrow }}\,X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{\beta }\right)\,{\stackrel {\cong }{\longrightarrow }}\,\mathbb {S} ^{n-1},}
where the first map identifies
S
n
−
1
{\displaystyle \mathbb {S} ^{n-1}}
with
∂
e
n
α
{\displaystyle \partial e_{n}^{\alpha }}
via the characteristic map
Φ
n
α
{\displaystyle \Phi _{n}^{\alpha }}
of
e
n
α
{\displaystyle e_{n}^{\alpha }}
, the object
e
n
−
1
β
{\displaystyle e_{n-1}^{\beta }}
is an
(
n
−
1
)
{\displaystyle (n-1)}
-cell of X, the third map
q
{\displaystyle q}
is the quotient map that collapses
X
n
−
1
∖
e
n
−
1
β
{\displaystyle X_{n-1}\setminus e_{n-1}^{\beta }}
to a point (thus wrapping
e
n
−
1
β
{\displaystyle e_{n-1}^{\beta }}
into a sphere
S
n
−
1
{\displaystyle \mathbb {S} ^{n-1}}
), and the last map identifies
X
n
−
1
/
(
X
n
−
1
∖
e
n
−
1
β
)
{\displaystyle X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{\beta }\right)}
with
S
n
−
1
{\displaystyle \mathbb {S} ^{n-1}}
via the characteristic map
Φ
n
−
1
β
{\displaystyle \Phi _{n-1}^{\beta }}
of
e
n
−
1
β
{\displaystyle e_{n-1}^{\beta }}
.
The boundary map
∂
n
:
C
n
(
X
n
,
X
n
−
1
)
→
C
n
−
1
(
X
n
−
1
,
X
n
−
2
)
{\displaystyle \partial _{n}:{C_{n}}(X_{n},X_{n-1})\to {C_{n-1}}(X_{n-1},X_{n-2})}
is then given by the formula
∂
n
(
e
n
α
)
=
∑
β
deg
(
χ
n
α
β
)
e
n
−
1
β
,
{\displaystyle {\partial _{n}}(e_{n}^{\alpha })=\sum _{\beta }\deg \left(\chi _{n}^{\alpha \beta }\right)e_{n-1}^{\beta },}
where
deg
(
χ
n
α
β
)
{\displaystyle \deg \left(\chi _{n}^{\alpha \beta }\right)}
is the degree of
χ
n
α
β
{\displaystyle \chi _{n}^{\alpha \beta }}
and the sum is taken over all
(
n
−
1
)
{\displaystyle (n-1)}
-cells of
X
{\displaystyle X}
, considered as generators of
C
n
−
1
(
X
n
−
1
,
X
n
−
2
)
{\displaystyle {C_{n-1}}(X_{n-1},X_{n-2})}
.
Examples
The following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.
= The n-sphere
=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
S
n
−
1
{\displaystyle S^{n-1}}
to 0-cell. Since the generators of the cellular chain groups
C
k
(
S
k
n
,
S
k
−
1
n
)
{\displaystyle {C_{k}}(S_{k}^{n},S_{k-1}^{n})}
can be identified with the k-cells of Sn, we have that
C
k
(
S
k
n
,
S
k
−
1
n
)
=
Z
{\displaystyle {C_{k}}(S_{k}^{n},S_{k-1}^{n})=\mathbb {Z} }
for
k
=
0
,
n
,
{\displaystyle k=0,n,}
and is otherwise trivial.
Hence for
n
>
1
{\displaystyle n>1}
, the resulting chain complex is
⋯
⟶
∂
n
+
2
0
⟶
∂
n
+
1
Z
⟶
∂
n
0
⟶
∂
n
−
1
⋯
⟶
∂
2
0
⟶
∂
1
Z
⟶
0
,
{\displaystyle \dotsb {\overset {\partial _{n+2}}{\longrightarrow \,}}0{\overset {\partial _{n+1}}{\longrightarrow \,}}\mathbb {Z} {\overset {\partial _{n}}{\longrightarrow \,}}0{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}0{\overset {\partial _{1}}{\longrightarrow \,}}\mathbb {Z} {\longrightarrow \,}0,}
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
H
k
(
S
n
)
=
{
Z
k
=
0
,
n
{
0
}
otherwise.
{\displaystyle H_{k}(S^{n})={\begin{cases}\mathbb {Z} &k=0,n\\\{0\}&{\text{otherwise.}}\end{cases}}}
When
n
=
1
{\displaystyle n=1}
, it is possible to verify that the boundary map
∂
1
{\displaystyle \partial _{1}}
is zero, meaning the above formula holds for all positive
n
{\displaystyle n}
.
= Genus g surface
=Cellular homology can also be used to calculate the homology of the genus g surface
Σ
g
{\displaystyle \Sigma _{g}}
. The fundamental polygon of
Σ
g
{\displaystyle \Sigma _{g}}
is a
4
n
{\displaystyle 4n}
-gon which gives
Σ
g
{\displaystyle \Sigma _{g}}
a CW-structure with one 2-cell,
2
n
{\displaystyle 2n}
1-cells, and one 0-cell. The 2-cell is attached along the boundary of the
4
n
{\displaystyle 4n}
-gon, which contains every 1-cell twice, once forwards and once backwards. This means the attaching map is zero, since the forwards and backwards directions of each 1-cell cancel out. Similarly, the attaching map for each 1-cell is also zero, since it is the constant mapping from
S
0
{\displaystyle S^{0}}
to the 0-cell. Therefore, the resulting chain complex is
⋯
→
0
→
∂
3
Z
→
∂
2
Z
2
g
→
∂
1
Z
→
0
,
{\displaystyle \cdots \to 0\xrightarrow {\partial _{3}} \mathbb {Z} \xrightarrow {\partial _{2}} \mathbb {Z} ^{2g}\xrightarrow {\partial _{1}} \mathbb {Z} \to 0,}
where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by
H
k
(
Σ
g
)
=
{
Z
k
=
0
,
2
Z
2
g
k
=
1
{
0
}
otherwise.
{\displaystyle H_{k}(\Sigma _{g})={\begin{cases}\mathbb {Z} &k=0,2\\\mathbb {Z} ^{2g}&k=1\\\{0\}&{\text{otherwise.}}\end{cases}}}
Similarly, one can construct the genus g surface with a crosscap attached as a CW complex with 1 0-cell, g 1-cells, and 1 2-cell. Its homology groups are
H
k
(
Σ
g
)
=
{
Z
k
=
0
Z
g
−
1
⊕
Z
2
k
=
1
{
0
}
otherwise.
{\displaystyle H_{k}(\Sigma _{g})={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} ^{g-1}\oplus \mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise.}}\end{cases}}}
= Torus
=The n-torus
(
S
1
)
n
{\displaystyle (S^{1})^{n}}
can be constructed as the CW complex with 1 0-cell, n 1-cells, ..., and 1 n-cell. The chain complex is
0
→
Z
(
n
n
)
→
Z
(
n
n
−
1
)
→
⋯
→
Z
(
n
1
)
→
Z
(
n
0
)
→
0
{\displaystyle 0\to \mathbb {Z} ^{\binom {n}{n}}\to \mathbb {Z} ^{\binom {n}{n-1}}\to \cdots \to \mathbb {Z} ^{\binom {n}{1}}\to \mathbb {Z} ^{\binom {n}{0}}\to 0}
and all the boundary maps are zero. This can be understood by explicitly constructing the cases for
n
=
0
,
1
,
2
,
3
{\displaystyle n=0,1,2,3}
, then see the pattern.
Thus,
H
k
(
(
S
1
)
n
)
≃
Z
(
n
k
)
{\displaystyle H_{k}((S^{1})^{n})\simeq \mathbb {Z} ^{\binom {n}{k}}}
.
= Complex projective space
=If
X
{\displaystyle X}
has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then
H
n
C
W
(
X
)
{\displaystyle H_{n}^{CW}(X)}
is the free abelian group generated by its n-cells, for each
n
{\displaystyle n}
.
The complex projective space
P
n
C
{\displaystyle P^{n}\mathbb {C} }
is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus
H
k
(
P
n
C
)
=
Z
{\displaystyle H_{k}(P^{n}\mathbb {C} )=\mathbb {Z} }
for
k
=
0
,
2
,
.
.
.
,
2
n
{\displaystyle k=0,2,...,2n}
, and zero otherwise.
= Real projective space
=The real projective space
R
P
n
{\displaystyle \mathbb {R} P^{n}}
admits a CW-structure with one
k
{\displaystyle k}
-cell
e
k
{\displaystyle e_{k}}
for all
k
∈
{
0
,
1
,
…
,
n
}
{\displaystyle k\in \{0,1,\dots ,n\}}
.
The attaching map for these
k
{\displaystyle k}
-cells is given by the 2-fold covering map
φ
k
:
S
k
−
1
→
R
P
k
−
1
{\displaystyle \varphi _{k}\colon S^{k-1}\to \mathbb {R} P^{k-1}}
.
(Observe that the
k
{\displaystyle k}
-skeleton
R
P
k
n
≅
R
P
k
{\displaystyle \mathbb {R} P_{k}^{n}\cong \mathbb {R} P^{k}}
for all
k
∈
{
0
,
1
,
…
,
n
}
{\displaystyle k\in \{0,1,\dots ,n\}}
.)
Note that in this case,
C
k
(
R
P
k
n
,
R
P
k
−
1
n
)
≅
Z
{\displaystyle C_{k}(\mathbb {R} P_{k}^{n},\mathbb {R} P_{k-1}^{n})\cong \mathbb {Z} }
for all
k
∈
{
0
,
1
,
…
,
n
}
{\displaystyle k\in \{0,1,\dots ,n\}}
.
To compute the boundary map
∂
k
:
C
k
(
R
P
k
n
,
R
P
k
−
1
n
)
→
C
k
−
1
(
R
P
k
−
1
n
,
R
P
k
−
2
n
)
,
{\displaystyle \partial _{k}\colon C_{k}(\mathbb {R} P_{k}^{n},\mathbb {R} P_{k-1}^{n})\to C_{k-1}(\mathbb {R} P_{k-1}^{n},\mathbb {R} P_{k-2}^{n}),}
we must find the degree of the map
χ
k
:
S
k
−
1
⟶
φ
k
R
P
k
−
1
⟶
q
k
R
P
k
−
1
/
R
P
k
−
2
≅
S
k
−
1
.
{\displaystyle \chi _{k}\colon S^{k-1}{\overset {\varphi _{k}}{\longrightarrow }}\mathbb {R} P^{k-1}{\overset {q_{k}}{\longrightarrow }}\mathbb {R} P^{k-1}/\mathbb {R} P^{k-2}\cong S^{k-1}.}
Now, note that
φ
k
−
1
(
R
P
k
−
2
)
=
S
k
−
2
⊆
S
k
−
1
{\displaystyle \varphi _{k}^{-1}(\mathbb {R} P^{k-2})=S^{k-2}\subseteq S^{k-1}}
, and for each point
x
∈
R
P
k
−
1
∖
R
P
k
−
2
{\displaystyle x\in \mathbb {R} P^{k-1}\setminus \mathbb {R} P^{k-2}}
, we have that
φ
−
1
(
{
x
}
)
{\displaystyle \varphi ^{-1}(\{x\})}
consists of two points, one in each connected component (open hemisphere) of
S
k
−
1
∖
S
k
−
2
{\displaystyle S^{k-1}\setminus S^{k-2}}
.
Thus, in order to find the degree of the map
χ
k
{\displaystyle \chi _{k}}
, it is sufficient to find the local degrees of
χ
k
{\displaystyle \chi _{k}}
on each of these open hemispheres.
For ease of notation, we let
B
k
{\displaystyle B_{k}}
and
B
~
k
{\displaystyle {\tilde {B}}_{k}}
denote the connected components of
S
k
−
1
∖
S
k
−
2
{\displaystyle S^{k-1}\setminus S^{k-2}}
.
Then
χ
k
|
B
k
{\displaystyle \chi _{k}|_{B_{k}}}
and
χ
k
|
B
~
k
{\displaystyle \chi _{k}|_{{\tilde {B}}_{k}}}
are homeomorphisms, and
χ
k
|
B
~
k
=
χ
k
|
B
k
∘
A
{\displaystyle \chi _{k}|_{{\tilde {B}}_{k}}=\chi _{k}|_{B_{k}}\circ A}
, where
A
{\displaystyle A}
is the antipodal map.
Now, the degree of the antipodal map on
S
k
−
1
{\displaystyle S^{k-1}}
is
(
−
1
)
k
{\displaystyle (-1)^{k}}
.
Hence, without loss of generality, we have that the local degree of
χ
k
{\displaystyle \chi _{k}}
on
B
k
{\displaystyle B_{k}}
is
1
{\displaystyle 1}
and the local degree of
χ
k
{\displaystyle \chi _{k}}
on
B
~
k
{\displaystyle {\tilde {B}}_{k}}
is
(
−
1
)
k
{\displaystyle (-1)^{k}}
.
Adding the local degrees, we have that
deg
(
χ
k
)
=
1
+
(
−
1
)
k
=
{
2
if
k
is even,
0
if
k
is odd.
{\displaystyle \deg(\chi _{k})=1+(-1)^{k}={\begin{cases}2&{\text{if }}k{\text{ is even,}}\\0&{\text{if }}k{\text{ is odd.}}\end{cases}}}
The boundary map
∂
k
{\displaystyle \partial _{k}}
is then given by
deg
(
χ
k
)
{\displaystyle \deg(\chi _{k})}
.
We thus have that the CW-structure on
R
P
n
{\displaystyle \mathbb {R} P^{n}}
gives rise to the following chain complex:
0
⟶
Z
⟶
∂
n
⋯
⟶
2
Z
⟶
0
Z
⟶
2
Z
⟶
0
Z
⟶
0
,
{\displaystyle 0\longrightarrow \mathbb {Z} {\overset {\partial _{n}}{\longrightarrow }}\cdots {\overset {2}{\longrightarrow }}\mathbb {Z} {\overset {0}{\longrightarrow }}\mathbb {Z} {\overset {2}{\longrightarrow }}\mathbb {Z} {\overset {0}{\longrightarrow }}\mathbb {Z} \longrightarrow 0,}
where
∂
n
=
2
{\displaystyle \partial _{n}=2}
if
n
{\displaystyle n}
is even and
∂
n
=
0
{\displaystyle \partial _{n}=0}
if
n
{\displaystyle n}
is odd.
Hence, the cellular homology groups for
R
P
n
{\displaystyle \mathbb {R} P^{n}}
are the following:
H
k
(
R
P
n
)
=
{
Z
if
k
=
0
and
k
=
n
odd
,
Z
/
2
Z
if
0
<
k
<
n
odd,
0
otherwise.
{\displaystyle H_{k}(\mathbb {R} P^{n})={\begin{cases}\mathbb {Z} &{\text{if }}k=0{\text{ and }}k=n{\text{ odd}},\\\mathbb {Z} /2\mathbb {Z} &{\text{if }}0
Other properties
One sees from the cellular chain complex that the
n
{\displaystyle n}
-skeleton determines all lower-dimensional homology modules:
H
k
(
X
)
≅
H
k
(
X
n
)
{\displaystyle {H_{k}}(X)\cong {H_{k}}(X_{n})}
for
k
<
n
{\displaystyle k
.
An important consequence of this cellular perspective is that if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complex projective space
C
P
n
{\displaystyle \mathbb {CP} ^{n}}
has a cell structure with one cell in each even dimension; it follows that for
0
≤
k
≤
n
{\displaystyle 0\leq k\leq n}
,
H
2
k
(
C
P
n
;
Z
)
≅
Z
{\displaystyle {H_{2k}}(\mathbb {CP} ^{n};\mathbb {Z} )\cong \mathbb {Z} }
and
H
2
k
+
1
(
C
P
n
;
Z
)
=
0.
{\displaystyle {H_{2k+1}}(\mathbb {CP} ^{n};\mathbb {Z} )=0.}
Generalization
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.
Euler characteristic
For a cellular complex
X
{\displaystyle X}
, let
X
j
{\displaystyle X_{j}}
be its
j
{\displaystyle j}
-th skeleton, and
c
j
{\displaystyle c_{j}}
be the number of
j
{\displaystyle j}
-cells, i.e., the rank of the free module
C
j
(
X
j
,
X
j
−
1
)
{\displaystyle {C_{j}}(X_{j},X_{j-1})}
. The Euler characteristic of
X
{\displaystyle X}
is then defined by
χ
(
X
)
=
∑
j
=
0
n
(
−
1
)
j
c
j
.
{\displaystyle \chi (X)=\sum _{j=0}^{n}(-1)^{j}c_{j}.}
The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of
X
{\displaystyle X}
,
χ
(
X
)
=
∑
j
=
0
n
(
−
1
)
j
Rank
(
H
j
(
X
)
)
.
{\displaystyle \chi (X)=\sum _{j=0}^{n}(-1)^{j}\operatorname {Rank} ({H_{j}}(X)).}
This can be justified as follows. Consider the long exact sequence of relative homology for the triple
(
X
n
,
X
n
−
1
,
∅
)
{\displaystyle (X_{n},X_{n-1},\varnothing )}
:
⋯
→
H
i
(
X
n
−
1
,
∅
)
→
H
i
(
X
n
,
∅
)
→
H
i
(
X
n
,
X
n
−
1
)
→
⋯
.
{\displaystyle \cdots \to {H_{i}}(X_{n-1},\varnothing )\to {H_{i}}(X_{n},\varnothing )\to {H_{i}}(X_{n},X_{n-1})\to \cdots .}
Chasing exactness through the sequence gives
∑
i
=
0
n
(
−
1
)
i
Rank
(
H
i
(
X
n
,
∅
)
)
=
∑
i
=
0
n
(
−
1
)
i
Rank
(
H
i
(
X
n
,
X
n
−
1
)
)
+
∑
i
=
0
n
(
−
1
)
i
Rank
(
H
i
(
X
n
−
1
,
∅
)
)
.
{\displaystyle \sum _{i=0}^{n}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{n},\varnothing ))=\sum _{i=0}^{n}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{n},X_{n-1}))+\sum _{i=0}^{n}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{n-1},\varnothing )).}
The same calculation applies to the triples
(
X
n
−
1
,
X
n
−
2
,
∅
)
{\displaystyle (X_{n-1},X_{n-2},\varnothing )}
,
(
X
n
−
2
,
X
n
−
3
,
∅
)
{\displaystyle (X_{n-2},X_{n-3},\varnothing )}
, etc. By induction,
∑
i
=
0
n
(
−
1
)
i
Rank
(
H
i
(
X
n
,
∅
)
)
=
∑
j
=
0
n
∑
i
=
0
j
(
−
1
)
i
Rank
(
H
i
(
X
j
,
X
j
−
1
)
)
=
∑
j
=
0
n
(
−
1
)
j
c
j
.
{\displaystyle \sum _{i=0}^{n}(-1)^{i}\;\operatorname {Rank} ({H_{i}}(X_{n},\varnothing ))=\sum _{j=0}^{n}\sum _{i=0}^{j}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{j},X_{j-1}))=\sum _{j=0}^{n}(-1)^{j}c_{j}.}
References
Albrecht Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1.
Allen Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the author's homepage.
Kata Kunci Pencarian:
- Protein Wiskott–Aldrich syndrome
- Bakteri
- Protein Kinase B
- Kromosom
- Arkea
- Jalur persinyalan Notch
- PCK1
- Protein
- Reseptor terhubung protein G
- ZAP70
- Cellular homology
- Singular homology
- Homology (mathematics)
- CW complex
- Simplicial homology
- Künneth theorem
- List of algebraic topology topics
- Triangulation (topology)
- Morse theory
- Morse homology
