In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, cellular homology in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
is a
homology theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
for the category of
CW-complexes. It agrees with
singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
, and can provide an effective means of computing homology modules.
Definition
If
is a CW-complex with
''n''-skeleton , the cellular-homology modules are defined as the
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s ''H
i'' of the cellular
chain complex
:
where
is taken to be the empty set.
The group
:
is
free abelian, with generators that can be identified with the
-cells of
. Let
be an
-cell of
, and let
be the attaching map. Then consider the composition
:
where the first map identifies
with
via the characteristic map
of
, the object
is an
-cell of ''X'', the third map
is the quotient map that collapses
to a point (thus wrapping
into a sphere
), and the last map identifies
with
via the characteristic map
of
.
The
boundary map
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
:
is then given by the formula
:
where
is the
degree of
and the sum is taken over all
-cells of
, considered as generators of
.
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 ''S
n'' 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
to 0-cell. Since the generators of the cellular chain groups
can be identified with the ''k''-cells of ''S
n'', we have that
for
and is otherwise trivial.
Hence for
, the resulting chain complex is
:
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
:
When
, it is not very difficult to verify that the boundary map
is zero, meaning the above formula holds for all positive
.
Genus ''g'' surface
Cellular homology can also be used to calculate the homology of the
genus ''g'' surface . The
fundamental polygon of
is a
-gon which gives
a CW-structure with one 2-cell,
1-cells, and one 0-cell. The 2-cell is attached along the boundary of the
-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
to the 0-cell. Therefore, the resulting chain complex is
:
where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by
:
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
Torus
The n-torus
can be constructed as the CW complex with 1 0-cell, n 1-cells, ..., and 1 n-cell. The chain complex is
and all the boundary maps are zero. This can be understood by explicitly constructing the cases for
, then see the pattern.
Thus,
.
Complex projective space
If
has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then
is the free abelian group generated by its n-cells, for each
.
The complex projective space
is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus
for
, and zero otherwise.
Real projective space
The
real projective space admits a CW-structure with one
-cell
for all
.
The attaching map for these
-cells is given by the 2-fold covering map
.
(Observe that the
-skeleton
for all
.)
Note that in this case,
for all
.
To compute the boundary map
:
we must find the degree of the map
:
Now, note that
, and for each point
, we have that
consists of two points, one in each connected component (open hemisphere) of
.
Thus, in order to find the degree of the map
, it is sufficient to find the local degrees of
on each of these open hemispheres.
For ease of notation, we let
and
denote the connected components of
.
Then
and
are homeomorphisms, and
, where
is the antipodal map.
Now, the degree of the antipodal map on
is
.
Hence, without loss of generality, we have that the local degree of
on
is
and the local degree of
on
is
.
Adding the local degrees, we have that
:
The boundary map
is then given by
.
We thus have that the CW-structure on
gives rise to the following chain complex:
:
where
if
is even and
if
is odd.
Hence, the cellular homology groups for
are the following:
:
Other properties
One sees from the cellular chain complex that the
-skeleton determines all lower-dimensional homology modules:
:
for
.
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 has a cell structure with one cell in each even dimension; it follows that for
,
:
and
:
Generalization
The
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet ...
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
, let
be its
-th skeleton, and
be the number of
-cells, i.e., the rank of the free module
. The
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of
is then defined by
:
The Euler characteristic is a homotopy invariant. In fact, in terms of the
Betti numbers of
,
:
This can be justified as follows. Consider the long exact sequence of
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
for the triple
:
:
Chasing exactness through the sequence gives
:
The same calculation applies to the triples
,
, etc. By induction,
:
References
*
Albrecht Dold: ''Lectures on Algebraic Topology'', Springer .
*
Allen Hatcher Allen, Allen's or Allens may refer to:
Buildings
* Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee
* Allen Center, a skyscraper complex in downtown Houston, Texas
* Allen Fieldhouse, an indoor sports arena on the Univer ...
: ''Algebraic Topology'', Cambridge University Press {{ISBN, 978-0-521-79540-1. A free electronic version is available on th
author's homepage
Homology theory