Cellular homology - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, cellular homology in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...

is a homology theory

Homology theory

In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...

for CW-complexes. It agrees with singular homology

Singular homology

In algebraic topology, a branch of mathematics, 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....

, and can provide an effective means of computing homology modules.

Definition

If X is a CW-complex with n-skeleton

N-skeleton

In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n...

X_n, the cellular homology modules are defined as the homology groups of the cellular chain complex

Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

\cdots \to  H_{n+1}( X_{n+1}, X_n ) \to H_n( X_n, X_{n-1} ) \to H_{n-1}( X_{n-1}, X_{n-2} ) \to \cdots .

The module

H_n( X_n, X_{n-1} ) \,

is free

Free module

In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...

, with generators which can be identified with the n-cells of X. Let

e_n^{\alpha}

be an n-cell of X, let

\chi_n^{\alpha} : \partial e_n^{\alpha}\cong S^{n-1} \to X_{n-1}

be the attaching map, and consider the composite maps

\chi_n^{\alpha\beta}:S^{n-1} \to X_{n-1} \to X_{n-1}/(X_{n-1}-e_{n-1}^{\beta})\cong S^{n-1}

where

e_{n-1}^{\beta}

is an

(n-1)

-cell of X and the second map is the quotient map identifying

(X_{n-1}-e_{n-1}^{\beta})

to a point. The boundary map

d_n:H_n(X_n,X_{n-1}) \to H_{n-1}(X_{n-1},X_{n-2}) \,

is then given by the formula

d_n(e_n^{\alpha})=\sum_{\beta}deg(\chi_n^{\alpha\beta})e_{n-1}^{\beta}\,

where

deg(\chi_n^{\alpha\beta})

is the degree

Degree of a continuous mapping

In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping...

\chi_n^{\alpha\beta}

and the sum is taken over all

(n-1)

-cells of X, considered as generators of

H_{n-1}(X_{n-1},X_{n-2})\,

Other properties

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

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

CPⁿ 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 .

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

Homology theory

Euler characteristic

For a cellular complex X, let X_j be its j-th skeleton, and c_j be the number of j-cells, i.e. the rank of the free module H_j(X_j, X_j-1). The Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

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

Betti number

In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

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

Relative homology

In algebraic topology, a branch of mathematics, the 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...

for the triple (X_n, X_{n - 1}, ∅):

\cdots \to  H_i( X_{n-1}, \empty) \to H_i( X_n, \empty) \to H_i( X_{n}, X_{n-1} ) \to \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)

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