Here we follow the article "Persistent homology - a Survey," by Herbert Edelsbrunner and John Harer, published in 2008 in "Surveys on discrete and computational geometry," Volume 453.
Consider the sphere, which has known homology groups. Consider a slightly bent embedding of the sphere in $\R^3$, call it $M$, as in the diagram below (imagine it as a hollow blob, whose outline is drawn below). Let $f:M\to \R$ be the height function, measuring the distance from a point in $M$ to a plane just below $M$, coming out of the page. Then we have some critical values $t_0,t_1,t_2,t_3$, as indicated below. Note we have embedded the shape so that no two critical points of $f$ have the same value.
This is remniscent of Morse theory. Set $M_i = f^{-1}[0,t_i]$ and $b_i = \dim(H_i)$ the $i$th Betti number. Then we may easily calculate the Betti numbers of the $M_j$, as in the table below.
\[
\renewcommand\arraystretch{1.3}
\begin{array}{r|c|c|c|c|c}
& M_0 & M_1 & M_2 & M_3 & M \\\hline
b_0 & 1 & 2 & 1 & 1 & 1 \\\hline
b_1 & 0 & 0 & 0 & 0 & 0 \\\hline
b_2 & 0 & 0 & 0 & 1 & 1
\end{array}
\renewcommand\arraystretch{1}
\]
Definition: In the context above, suppose that there is some $p$ and $j>i$ such that:
- $b_p(M_i)=b_p(M_{i-1})+1$,
- $b_p(M_j)=b_p(M_{j-1})-1$, and
- the generator of $H_p$ introduced at $t_i$ is the same generator of $H_p$ that disappears at $t_j$.
Then $(i,j)$ (or ($t_i,t_j$)) is called a
persistence pair and the
persistence of $(i,j)$ is $j-i$ (or $f(j)-f(i)$).
For $i$ not in a persistence pair, we say that $i$ represents an
essential cycle, or that the persistence of $i$ is infinite. In the example considered, the only persistence pair is $(1,2)$. This may be presented in a
persistence diagram, with the indices of critical points on both axes, and the persistence measured as a vertical distance.
If we put a simplicial complex structure on $M$, we may also calculate the homology (and persistence pairs, although they may be different than the ones found above). To make calculations easier, we instead describe a CW structure on our embedded sphere $M$ (with $X_i$ the $i$-skeleton, and the ordering of the $i$-cells as indicated). The results will be the same as for a simplicial complex structure.
This gives one 0-cell, two 1-cells, and three 2-cells (with the obvious gluings), allowing us to construct the chain groups $C_p$ as well as maps between them. The map $d_p:C_p\to C_{p-1}$ as a matrix has size $\dim(C_{p-1})\times \dim(C_p)$, and has entry $(i,j)$ equal to the number of times, counting multiplicity, that the $i$th $(p-1)$-cell is a face of the $j$th $p$-cell. Calculations are done in $\Z/2\Z$.
\[
d_2\ :\ C_2\to C_1
\hspace{.5cm}\text{is}\hspace{.5cm}
\begin{bmatrix}
1 & 0 & 1 \\ 0 & 1 & 1
\end{bmatrix}
\hspace{2cm}
d_1\ :\ C_1\to C_0
\hspace{.5cm}\text{is}\hspace{.5cm}
\begin{bmatrix}
0 & 0
\end{bmatrix}
\]
The Betti numbers are then $b_p = \dim(C_p) - \text{rk}(d_p)-\text{rk}(d_{p+1})$. From above, it is immediate that $\text{rk}(d_1)=0$, $\text{rk}(d_2) = 2$, and $\text{rk}(d_p)=0$ for all other $p$. This tells us that
\begin{align*}
b_0 & = \dim(C_0) - \text{rk}(d_0) - \text{rk}(d_1) = 1 - 0 - 0 = 1, \\
b_1 & = \dim(C_1) - \text{rk}(d_1) - \text{rk}(d_2) = 2 - 0 - 2 = 0, \\
b_2 & = \dim(C_2) - \text{rk}(d_2) - \text{rk}(d_3) = 3 - 2 - 0 = 1, \\
\end{align*}
as expected. To find the persistence pairs, we introduce a filtration on the simplices (equivalently, on the cells) by always having the faces of a cell precede the cell, as well as lower-dimensional cells preceding higher-dimensional cells. Using the same ordering as described above, consider the following filtration:
\begin{align*}
K_0 & = \{\}, \\
K_1 & = \{e^0_1\}, \\
K_2 & = \{e^0_1,e^1_1,e^1_2\} ,\\
K_3 & = \{e^0_1,e^1_1,e^1_2,e^2_1,e^2_2,e^2_3\},
\end{align*}
so $\emptyset = K_0\subset K_1\subset K_2\subset K_3 = M$. This gives an ordering on all the cells of $M$, namely
\[
\sigma_1 = e^0_1,\
\sigma_2 = e^1_1,\
\sigma_3 = e^1_2,\
\sigma_4 = e^2_1,\
\sigma_5 = e^2_2,\
\sigma_6 = e^2_3.
\]
Construct the
boundary matrix $D$, with the $(i,j)$ entry of $D$ equal to the number of times, counting multiplicity, modulo 2, that $\sigma_i$ is a codimension 1 face of $\sigma_j$. In the case of our example sphere, we get the matrix
\[
D = \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\ \ \sim\ \
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
in its reduced form (call it $\tilde D$). With respect to the matrix $\tilde D$, define the following numbers:
\begin{align*}
low(j) & = \text{the row number of the lowest non-zero entry in column $j$} ,\\
zero(p) & = \text{the number of zero columns that correspond to $p$-simplices} ,\\
one(p) & = \text{the number of 1s in rows that correspond to $p$-simplices}.
\end{align*}
We calculate all the relevant values of these expressions to be as below.
\begin{align*}
low(1) & = 0 & zero(0) & = 1 & one(0) & = 0 \\
low(2) & = 0 & zero(1) & = 2 & one(1) & = 2 \\
low(3) & = 0 & zero(2) & = 1 & one(2) & = 0 \\
low(4) & = 2 \\
low(5) & = 3 \\
low(6) & = 0
\end{align*}
For persistence, we have
- if $low(j)=i\neq 0$, then $(i,j)$ is a persistence pair,
- if $low(j)=0$ and there is no $k$ such that $low(k)=j$, then $j$ is an essential cycle.
For our sphere example, we get two persistence pairs $(2,4)$ and $(3,5)$, and two essential cycles 1 and 6. Note that this is different from the persistence pairs found by the height function $f:M\to \R$ earlier (but there are still two essential cycles), because there we were comparing the homologies $H_p(M_j)$, but here we are comparing $H_p(K_\ell)$. The persistence diagram is as below.
As an added feature, from the numbers above we may calculate the homology and relative homology groups. Construct the relative chain groups $C_p(M,K_\ell) = C_p(M)/C_p(K_\ell)$ and set $zero(p,\ell)$ to be $zero(p)$ for the lower right submatrix of $\tilde D$ corresponding to the cells in $M-K_\ell$ (and similarly for $one(p,\ell)$). We find these numbers for the bent sphere to be as below.
\begin{align*}
zero(0,0) & = 1 & zero(0,1) & = 0 & zero(0,2) & = 0 & zero(0,3) & = 0 \\
zero(1,0) & = 2 & zero(1,1) & = 2 & zero(1,2) & = 0 & zero(1,3) & = 0 \\
zero(2,0) & = 1 & zero(2,1) & = 1 & zero(2,2) & = 1 & zero(2,3) & = 0 \\[10pt]
one(0,0) & = 0 & one(0,1) & = 0 & one(0,2) & = 0 & one(0,3) & = 0 \\
one(1,0) & = 2 & one(1,1) & = 2 & one(1,2) & = 0 & one(1,3) & = 0 \\
one(2,0) & = 0 & one(2,1) & = 0 & one(2,2) & = 0 & one(2,3) & = 0
\end{align*}
Note that $zero(p,0)=zero(p)$ and $one(p,0)=one(p)$, as well as $zero(p,3)=one(p,3)=0$. The above numbers are useful in calculating
\begin{align*}
\dim(H_p(M)) & = zero(p)-one(p), \\
\dim(H_p(M,K_\ell)) & = zero(p,\ell) - one(p,\ell).
\end{align*}
References: Edelsbrunner and Harer (Persistent homology - a Survey)
