Homology (mathematics) - Construction of Homology Groups

Construction of Homology Groups

The construction begins with an object such as a topological space X, on which one first defines a chain complex C(X) encoding information about X. A chain complex is a sequence of abelian groups or modules C0, C1, C2, ... connected by homomorphisms which are called boundary operators. That is,

\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n \overset{\partial_n}{\longrightarrow\,}C_{n-1} \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} C_1 \overset{\partial_1}{\longrightarrow\,} C_0\overset{\partial_0}{\longrightarrow\,} 0

where 0 denotes the trivial group and for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,

i.e., the constant map sending every element of Cn + 1 to the group identity in Cn - 1. This means .

Now since each Cn is abelian all its subgroups are normal and because and are both subgroups of Cn, is a normal subgroup of and one can consider the factor group

called the n-th homology group of X.

We also use the notation and, so

Computing these two groups is usually rather difficult since they are very large groups. On the other hand, we do have tools which make the task easier.

The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with C(X)n the free abelian group generated by the n-simplices of X. The singular homology groups Hn(X) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex.

A chain complex is said to be exact if the image of the (n + 1)-th map is always equal to the kernel of the nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.

Cohomology groups are formally similar: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted dn point in the direction of increasing n rather than decreasing n; then the groups and follow from the same description and

as before.

Sometimes, reduced homology groups of a chain complex C(X) are defined as homologies of the augmented complex

\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n \overset{\partial_n}{\longrightarrow\,}C_{n-1} \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} C_1 \overset{\partial_1}{\longrightarrow\,} C_0\overset{\epsilon}{\longrightarrow\,} \Z {\longrightarrow\,} 0

where

for a combination Σ niσi of points σi (fixed generators of C0). The reduced homologies coincide with for i≠0.

Read more about this topic:  Homology (mathematics)

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