Homology Groups

p-chain

A p-chain (for $p\le \text{dim}K$), of $K$ is a formal sum of the $p$-simplices ${\sigma }_1,\dots ,{\sigma }_m$ in $K$

$$c=\sum _{i=1}^m{a}_i{\sigma }_i$$

where $a_i$ are coefficients in some underlying group $G$.

Typically the underlying group $G$ is ${\mathbb{Z}}_2$ or $\mathbb{Z}$.

One may think of $p$-chains as vectors with elements in $G$. In fact, we can easily construct a group by defining the operation to be as we expect:

$$c+c^{\prime }=\sum _{i=1}^m(a_i+a_i^{\prime }){\sigma }_i.$$

Under this operation we get $C_p(K)$, the group of $p$-chains of $K$. For a simplicial complex $K$, there always exists $C_p(K)$ for $0\le p\le \text{dim}K$. For $p<0$ or $p>\dim K$, we define $C_p$ to be the trivial group.

When a simplicial complex $K$ has finitely many simplices $m$, the $p$-chains group $C_p$ is finitely generated by the elementary p-chains $e_1,\dots ,e_m$ where

$$e_i={\sigma }_i.$$

Boundary Map

We can connect chain groups $C_p$ with the boundary map ${\partial }_p:C_p\to C_{p-1}$.

Boundary

The boundary of a $p$-simplex is the $(p-1)$-chain formed by taking the sum of all $(p-1)$-faces, i.e., if $\sigma =[v_0{v}_1\dots v_p]$, then

$${\partial }_p\sigma =\sum _{j=0}^p(-1{)}^j[v_0\dots \widehat{v_i}\dots v_p],$$

where $\widehat{v_i}$ denote an omitted vertex.

Note that when working over ${\mathbb{Z}}_2$, $(-1{)}^j=1$ for all $j$. So the boundary becomes

$${\partial }_p\sigma =\sum _{j=0}^p[v_0\dots \widehat{v_i}\dots v_p].$$

Thus, given a $p$-chain $c={\sum }_{i=1}^m{a}_i{\sigma }_i$, the p-th boundary map is defined by

$${\partial }_{p}c=\sum _{i=1}^m{a}_i({\partial }_p{\sigma }_i).$$

One may verify that ${\partial }_p$ is in fact a homomorphism, that is, ${\partial }_p(c+c^{\prime })={\partial }_{p}c+{\partial }_p{c}^{\prime }$.

Since ${\partial }_p$ is defined for $1\le p\le \dim K$, we get a chain (called a chain complex) of boundary homomorphisms:

$$\cdots \stackrel{{\partial }_{p+2}}{\to }{C}_{p+1}\stackrel{{\partial }_{p+1}}{\to }{C}_p\stackrel{{\partial }_p}{\to }{C}_{p-1}\stackrel{{\partial }_{p-1}}{\to }\cdots$$

Note that the chain complex forms an abstract simplicial complex.

Cycles and Boundaries

p-cycle

A p-cycle is a $p$-chain $c$ with empty boundary, i.e., ${\partial }_{p}c=0$.

The set of $p$-cycles, $Z_p$, is a subgroup of $C_p$. More specifically,

$$Z_p=\text{Ker}{\partial }_p.$$

Typically, $Z_p$ is a proper subgroup of $C_p$, however for any 0-simplex $v_j$, ${\partial }_0{v}_j=0$. Thus, when $p=0$ we get that $Z_0=C_0$.

p-boundary

A p-boundary $b$ is a $p$-chain that is the boundary of some $(p+1)$-chain, i.e., $b={\partial }_{p+1}c$ for some $c\in C_{p+1}$.

The set of $p$-boundaries, $B_p$ is a subgroup of $C_p$. More specifically,

$$B_p=\text{Im}{\partial }_{p+1}.$$

In fact, $B_p$ is actually a subgroup of $Z_p$ due to the fundamental law of homology:

Fundamental Law of Homology

${\partial }_p{\partial }_{p+1}c=0$ for all $p\in \mathbb{Z}$

Therefore, our chain complex

$$\cdots \stackrel{{\partial }_{p+2}}{\to }{C}_{p+1}\stackrel{{\partial }_{p+1}}{\to }{C}_p\stackrel{{\partial }_p}{\to }{C}_{p-1}\stackrel{{\partial }_{p-1}}{\to }\cdots$$

is exact, i.e., $\text{Im}{\partial }_{p+1}=\text{Ker}{\partial }_p$.

Homology Groups

Homology Group

The p-th homology group is the quotient group

$$H_p=Z_p/B_p.$$

and is composed of classes of cycles that are not boundaries.

Each element of $H_p$ is given by adding $p$-boundaries to a given $p$-cycle: $z+B_p$. Another $z^{\prime }$ is in the same class as $z$ if $z^{\prime }+B_p=z+B_p$. In such a case, $z$ and $z^{\prime }$ are called homologous which is denoted $z\sim z^{\prime }$.

The operation in $H_p$ is the typical quotient group operation:

$$(z+B_p)+(z^{\prime }+B_p)=(z+z^{\prime })+B_p.$$

Therefore, $H_p$ is abelian as our underlying group is either $\mathbb{Z}$ or ${\mathbb{Z}}_2$.