How do we count how many holes a shape has? We can answer this question using homology groups, whose order's count the number of $n$-dimensional holes.

Outline

• Simplexes

• Simplicial Complexes

• Orientation

• Chain groups

• Boundary Homomorphism

• Homology Groups

Simplexes

Definitions

Let $v^0,\dots,v^n$ be $n+1$ vectors in $\euc{N},n\geq 1$. They are called affinely linearly independent if $v^1 - v^0,\dots,v^n - v^0$ are linearly independent. By convention if $n= 0$ then $v^0$ is affinely linearly independent even if its 0.

Let $v^0,\dots,v^n$ be vectors in $\euc{N}$. A vector $v$ is said to be affinely dependent on them if there exist real numbers $\lambda_0,\dots,\lambda_n$ such that

Let $v^0,\dots,v^n$ be a-independent. The (closed) simplex with vertices $v^0,\dots,v^n$ is the set of points a-dependent on $v^0,\dots,v^n$ and with every barycentric coordinate greater than or equal to 0. The simplex is said to be spanned by $v^0,\dots,v^n$. The points with barycentric coordinate greater than 0 are said to be the interior of the simplex, and the set of interior points is sometimes called the open simplex with vertices $v^0,\dots,v^n$. The boundary of the simplex consists of those points which are not interior. We denote a (closed) simplex with a-indepenent vertices $v^0,\dots,v^n$ by $(v^0\dots v^n)$ and often represent it by a symbol $s_n$ or just $s$. The integer $n$ is called the dimension of $(v^0 \dots v^n)$. A simplex of dimension $n$ is called an $n$-simplex.

Let $s_n = (v^1 \dots v^n)$ be a simplex. A face of $s_n$ is a simplex whose vertices form a (non-empty) subset of $\{v^0,\dots,v^n\}$. If the subset is proper we say the face is a proper face. If $s_p$ is a face of $s_n$ we write $s_p < s_n$ or $s_n > s_p$. Note $s_n < s_n$ for any simplex $s_n$.

Ordered and Oriented Simplexes

Let $s_n = (v^0 \dots v^n)$ be a simplex. An orientation for $s_n$ is a collection of orderings for the vertices consisting of a particular ordering and all even permuations of it. An oriented n-simplex $\sigma_n$ is an $n$-simplex $s_n$, together with an orientation for $s_n$.

We write $\sigma_n = (v^0 \dots v^n)$ to mean $\sigma_n$ is the oriented $n$-simplex with vertices $(v^0,\dots,v^n)$ and orientation given by the ordering displayed and all even permutations of it. It will always be clear from the context wether $(v^0 \dots v^n)$ means simplex or orientated simplex. When $n > 0$ we write $-\sigma_n$ for the oriented simplex consisting of the same $n$-simplex and the collection of other possible orderings of the vertices. Thus by definition $-(-\sigma_n) = \sigma_n$.

Simplical Complexes

A simplical complex is a finite set $K$ of simplexes in $\euc{N}$ with the following two properties:

- if ``s \in K`` and ``t < s`` (``t`` is a face of ``s``) then ``t \in s``

- intersection condition: if ``s \in K`` and ``t \in K`` then ``s \cap t`` is
  either empty or else a face of both ``s`` and ``t``

The dimension of $K$ is the largest dimension of any simplex in $K$. The underlying space of $K$, denoted $\abs{K}$, is the set of points in $\euc{N}$ which belong to at least one simplex of $K$.

A subcomplex of a simplical complex $K$ is a subset $L$ of $K$ which is itself a simplical complex. (This is if and only if $L$ has the property that)

The subcomplex is called proper if $L \neq K$.

An oriented simplical complex is a simplical complex in which every simplex is provided an orientation.

Abstract Simplicial Complexes

An abstract simplical complex is a pair $X = (V,S)$ where $V$ is a finite set whose elements are called the vertices of $X$ and $S$ is a set of non-empty subsets of $V$. Each element $s \in S$ is called a simplex of $X$ and if $s \in S$ has precisely $n+1$ elements, $s$ is called an $n$-simplex. $S$ is required to statisfy the following two conditions

1. $v \in V \Rightarrow \{v\} \in S$

2. $s \in S, t\subset s \Rightarrow t \in S$

The dimension of $X$ is so long as $V \neq \emptyset$ the largest $n$ for which $S$ contains an $n$-simplex.

Two abstract simplical complexes $X = (V,S), X' = (V',S')$ are said to be isomorphic if there is a bijection $f: V \rightarrow V'$ with the property that

A realization of an abstract simplical complex $X$ is a simplical complex whose corresponding abstract simplical complex is isomorphic to $X$.

Theorem: Every abstract simplical complex of dimension $n$ has a realization in $\euc{2n + 1}$.

Chain Groups and Boundary Homomorphisms

Let $K$ be an oriented simplical complex of dimension $n$, and let $\alpha_p$ be the number of $p$-simplexes of $K$. For $0 \leq p \leq n$ let $\sigma_p^1,\dots,\sigma_p^{\alpha_p}$ be the oriented $p$-simplexes of $K$; for such $p$ the $p$th chain group of $K$, denoted $C_p(K)$ is the free abelian group on the set $\{\sigma_p^1,\dots,\sigma_p^{\alpha_p}\}$. Thus an element of $C_p(K)$ is a linear combination

with $\lambda$'s integers. This is called a $p$-chain on $K$ and two $p$-chains are added by adding the corresponding coefficients. When writing down $p$-chains it is customary to omit $p$-simplexes whose coefficient is zero, unless they are all zero, in which case we write 0.

Let $\sigma = (v^0 \dots v^p)$ be an oriented $p$-simplex of $K$ for some $p > 0$. The boundary of $\sigma$ is the $(p-1)$-chain

For $p = 0$ the boundary is defined to be 0. The boundary homomorphism $\partial = \partial_p : C_p(K) \rightarrow C_{p-1}(K)$ is defined by

for $0 \leq p \leq n$ and is defined to be the trivial homomorphism otherwise.

The augmentation

is the homomorphism defined by

the sequences

and

are called the chain complex of $K$ and the augmented chain complex of $K$, respectively.

Proposition: For any $p$, the homomorphism

is trivial. Also $\epsilon \circ \partial_1:C_1(K) \rightarrow \mathbb{Z}$ is trivial.

Corollary: For any $p$

and also

Homology Groups

Consider the sequence

$\Ker \partial_p$ is denoted by $Z_p$ or $Z_p(K)$ and the elements of $Z_p$ are called $p$-cycles. $Z_p$ is a free, finitely generated group. $\Ima \partial_{p+1}$ is denoted by $B_p$ or $B_p(K)$ and the elements of $B_p$ are called $p$-boundaries. $B_p$ is a free finitely generated group.

Since $B_p \subset Z_p$ there is a quotient group $H_p(K) = Z_p / B_p(K)$ called the $p$th homology group of $K$.

The quotient group $\Ker \epsilon / B_0(K)$ is denoted $\tilde{H}_0(K)$ and is called the reduced zeroth homology group of $K$.

An element of $H_p(K)$ is a coset $\bar{z} = z + B_p(K)$ where $z \in Z_p(K)$. We use the notation $\{z\}$ for this coset. It is called the homology class of the cycle $z$. Any cycle $z' \in \{z\}$ is called a representative cycle for $\{z\}$. We say that $z$ and $z'$ are homologous if $z - z' \in B_p(K)$ and write $z \sim z'$

Proposition: $v \sim v'$ if and only $v$ and $v'$ lie in the same connected component of $K$

Proposition: Let $K$ be a non-empty oriented simplical complex with $k$ components $K_1,\dots,K_k$ and let $v^i$ be a vertex in $K_i$. Then $H_0(K)$ is freely generated by the homology class $\{v^i\}$ for $i = 1,\dots,k$. Thus $H_0(K) \simeq \mathbb{Z}^k$.

$H_0(K) \cong \mathbb{Z}^k$ where $k$ is the number of connected components of $K$.