Homology (mathematics)

In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, there is the homology of a chain complex, a sequence of abelian groups, called homology groups, which are regarded as fundamental invariants of the chain complex. Secondly, when one can associate a chain complex to a different mathematical object, one can also associate its homology to that object. Distinct procedures of associating chain complexes to a given object are grouped into homology theories. Finally, homology is important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology groups, one can define a single homology of a topological space. This last notion of homology is closely related to topological ideas frequently discussed in popular mathematics such as the holes of a surface or the cycles of a graph. There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space.

Homology of chain complexes

We start with a chain complex, which is a sequence

(

C

∙

,

d

∙

)

{\displaystyle (C_{\bullet },d_{\bullet })}

of abelian groups

C

n

{\displaystyle C_{n}}

(whose elements are called chains) and group homomorphisms

d

n

{\displaystyle d_{n}}

(called boundary maps):

⋯

⟶

C

n

+

1

⟶

d

n

+

1

C

n

⟶

d

n

C

n

−

1

⟶

d

n

−

1

⋯

{\displaystyle \cdots \longrightarrow C_{n+1}{\stackrel {d_{n+1}}{\longrightarrow }}C_{n}{\stackrel {d_{n}}{\longrightarrow }}C_{n-1}{\stackrel {d_{n-1}}{\longrightarrow }}\cdots }

,

such that the composition of any two consecutive maps is zero:

d

n

∘

d

n

+

1

=

0.

{\displaystyle d_{n}\circ d_{n+1}=0.}

The

n

{\displaystyle n}

th group of cycles,

Z

n

{\displaystyle Z_{n}}

, is given by the kernel subgroup

Z

n

=

ker

⁡

d

n

=

{

c

∈

C

n

|

d

n

(

c

)

=

0

}

{\displaystyle Z_{n}=\ker d_{n}=\{c\in C_{n}\,|\;d_{n}(c)=0\}}

,

and the

n

{\displaystyle n}

th group of boundaries,

B

n

{\displaystyle B_{n}}

, is given by the image subgroup

B

n

:=

i

m

d

n

+

1

=

{

d

n

+

1

(

c

)

|

c

∈

C

n

+

1

}

{\displaystyle B_{n}:=\mathrm {im} \,d_{n+1}=\{d_{n+1}(c)\,|\;c\in C_{n+1}\}}

.