More (co)homological constructions

 Preliminary exam prep

Recall a previous post (2016-09-16, "Complexes and their homology") that focused on constructing topological spaces in different ways and recovering the homology. Here we complete that task, introducing cellular homology. Recall a cell complex (or CW complex) $X$ was a sequence of skeleta $X_k$ for $k=0,\dots,\dim(X)$ consisting of $k$-cells $e^k_i$ and their attaching maps to the $(k-1)$-skeleton.

Cellular homology

Definition: The long exact sequence in relative homology for the pair $X_k,X_{k-1}$ shares terms with the long exact sequence for the pair $X_{k+1},X_k$, as well as $X_{k-1},X_{k-2}$. By letting $d_k$ be the composition of maps in different long exact sequences, for $k>1$, that make the diagram

commute, we get a complex of equivalence classes of chains
\[
\cdots \to H_{k+1}(X_{k+1},X_k) \tov{d_{k+1}} H_k(X_k,X_{k-1})\tov{d_k} H_{k-1}(X_{k-1},X_{k-2})\to \cdots \to H_1(X_1,X_0)\tov{d_1} H_0(X_0) \tov{d_0} 0,
\]
whose homology $H_k^{CW}(X) = \ker(d_k)/\text{im}(d_{k-1})$ is called the cellular homology of $X$. The map $d_1$ is the connecting map in the long exact sequence of the pair $X_1,X_0$, and $d_0=0$.

This seems quite a roundabout way of defining homology groups, but it turns out to be very useful. Note that for $k=1$, the map $d_1$ is the same as for a simplicial complex, hence

Theorem: In the context above,

1. for $k\>0$, $H^{CW}_k(X)\cong H_k(X)$;
2. for $k\>1$, $H_k(X_k,X_{k-1})=\Z^\ell$, where $\ell$ is the number of $k$-cells in $X$; and
3. for $k\>2$, $d_k(e^k_i) = \displaystyle\sum_j\deg(\underbrace{\dy e^k_i}_{S^{k-1}}\tov{f_{k,i}} X_{k-1}\tov{\pi} \underbrace{X_{k-1}/X_{k-1}-e^{k-1}_j}_{S^{k-1}})e^{k-1}_j$.

Example: Real projective space $\R\P^n$ has a cell decomposition with one cell in each dimension, and 2-to-1 attaching maps $\dy(e_k) =2X_{k-1}$ for $k>1$. This gives us a construction
\[
X_0 = e_0,
\hspace{1cm}
X_1 = e_1 \bigsqcup_{\dy(e_1)=e_0} X_0,
\hspace{1cm}
X_2 = e_2 \bigsqcup_{\dy(e_2)=2e_1} X_1,
\hspace{1cm}
X_3 = e_3 \bigsqcup_{\dy(e_3)=2e_2} X_2, \dots
\]It is immediate that $d_0=d_1=0$, and for higher degrees, we have
\[
d_k(e^k) = \deg(S^{k-1}\to \R\P^{k-1}\to S^{k-1})e^{k-1}.\]
Since this is a map between spheres, we may apply local degree calculations. The first part is the 2-to-1 cover, where every point in $\R\P^{k-1}$ is covered by two points from $S^{k-1}$, one in each hemisphere. One covers it via the identity, the other via the antipodal map. As long as we choose a point not in $\R\P^{k-2}\subset \R\P^{k-1}$, the second step doesn't affect these degree calculations. The antipodal map $S^{k-1}\to S^{k-1}$ has degree $(-1)^k$, hence for $a$ the antipodal map, the composition has degree
\[
\deg(S^{k-1}\to \R\P^{k-1}\to S^{k-1}) = \deg(\id_{S^{k-1}}) + \deg(a_{S^{k-1}}) = 1+(-1)^k = \begin{cases} 2 & k\text{ even}, \\ 0 & k \text{ odd.}\end{cases}
\]

Products in (co)homology

Recall that an $n$-chain on $X$ is a map $\sigma:\Delta^n\to X$, where $\Delta^n=[v_0,\dots,v_n]$ is an $n$-simplex. These form the group $C_n$ of $n$-chains. An $n$-cochain is an element of $C^n = \Hom(C_n,\Z)$, though the coefficient group does not need to be $\Z$ necessarily.

Definition: The diagonal map $X\to X\times X$ induces a map on cohomology $H^*(X\times X)\to H^*(X)$, and by Kunneth, this gives a map $H^*(X)\otimes H^*(X)\to H^*(X)$, and is called the cup product.

For $a\in H^p(X)$ and $b\in H^q(X)$, representatives of the class $a$ are in $\Hom(C_p,\Z)$ and representatives of the class $b$ are in $\Hom(C_q,\Z)$, though we will conflate the notation for the class with that of a representative. Hence for a $(p+q)$-chain $\sigma$ the cup product of $a$ and $b$ acts as
\[
(a\smile b)\sigma = a\left(\sigma|_{[v_0,\dots,v_p]}\right)\cdot b\left(\sigma|_{[v_p,\dots,v_{p+q}]}\right).
\]
Definition: The cap product combines $p$-cochains with $q$-chains to give $(q-p)$-chains, by
\[
\begin{array}{r c l}
\frown\ :\ H^p(X) \times H_q(X) & \to & H_{q-p}(X), \\\
(a, \sigma) & \mapsto & a\left(\sigma|_{[v_0,\dots,v_p]}\right)\cdot \sigma|_{[v_p,\dots,v_q]}.
\end{array}\]
The cap product with the orientation form of an orientable manifold $X$ gives the isomorphism of Poincare duality.

Remark: Given a map $f:X\to Y$, the cup and cap products satisfy certain identities via the induced map on cohomology groups. Let $a,b\in H^*(Y)$ and $c\in H_*(X)$ be cochain and chain classes, for which
\[f^*(a\smile b) = f^*(a)\smile f^*(b),
\hspace{1cm}
a\frown f_*c = f_*(f^*a\frown c).\]
The first identity asserts that $f^*$ is a ring homomorphism and the second describes the commutativity of an appropriate diagram. The cup and cap products are related by the equation
\[a(b\frown \sigma) = (a\smile b)\sigma,\]for $a\in H^p$, $b\in H^q$ and $\sigma\in C_{p+q}$.

References: Hatcher (Algebraic topology, Chapter 2.2), Prasolov (Elements of homology theory, Chapter 2)

Basic topological constructions

 Preliminary exam prep

Let $X,Y$ be topological spaces based at $x_0,y_0$, respectively, and $I=[0,1]$ the unit interval.
\[
\begin{array}{r r c l}
\text{cone} & CX & = & X\times I / X\times \{0\} \\[5pt]
\text{suspension} & \Sigma X & = & X\times I / X\times \{0\}, X\times \{1\}\\[5pt]
\text{reduced suspension} & \widetilde\Sigma X & = & X\times I/X\times\{0\}, X\times \{0\}, \{x_0\}\times I \\[5pt]
\text{wedge} & X\vee Y & = & X\sqcup Y / \{x_0\} \sim \{y_0\} \\[5pt]
\text{smash} & X\wedge Y & = & X\times Y / X\times \{y_0\}, \{x_0\}\times Y \\[5pt]
\text{join} & X * Y & = & X\times Y \times I \left/\begin{array}{l l}
X\times \{y\}\times \{0\} & \forall\ y\in Y \\
\{x\}\times Y \times \{1\} & \forall\ x\in X
\end{array}\right. \\[5pt]
\text{connected sum} & X \# Y & = & (X\setminus D^n_X)\sqcup (Y\setminus D^n_Y) / \partial D^n_X \sim \partial D^n_Y
\end{array}
\]
In the last description, $X$ and $Y$ are assumed to be $n$-manifolds, with $D^n_X$ a closed $n$-dimensional disk in $X$ (similarly for $Y$). The quotient identification may also be made via some non-trivial map. In fact, only the interior of each $n$-disk is removed from $X$ and $Y$, so that the quotient makes sense.

Remark: Some of the above constructions may be expressed in terms of others, for example
\[
X\wedge Y = X\times Y / X\vee Y,
\hspace{1cm}
X*Y = \Sigma(X\wedge Y).
\]
The first is clear by viewing $X = X\times \{y_0\}$ and $Y = \{x_0\}\times Y$ as sitting inside $X\times Y$. The second is clear by letting $X\times \{y\}\times \{0\}$ be identified to $\{x_0\}\times\{y\}\times \{0\}$ for every $y\in Y$, and analogously with $Y$.

Example: Here are some of the constructions above applied to some common spaces.
\begin{align*}
CX & \simeq \text{pt} & \Sigma S^n & = S^{n+1} & S^n \wedge S^m & = S^{n+m}\\
\Sigma X & = S^1 \wedge X & S^n * S^m & = S^{n+m+1}\end{align*}
Remark: We may also calculate the homology of the new spaces in terms of the old ones.
\[
\begin{array}{r c l l}
\widetilde H_k(CX) & = & 0 & \text{via homotopy} \\
\widetilde H_k(\Sigma X) & = & \widetilde H_{k-1}(X) & \text{via Mayer--Vietoris} \\
\widetilde H_k(X\vee Y) & = & \widetilde H_k(X)\oplus \widetilde H_k(Y) & \text{via Mayer--Vietoris}\\
\widetilde H_k(X\wedge S^\ell) & = & \widetilde H_{k-\ell}(X) & \text{via Kunneth} \\
\widetilde H_k(X\# Y) & = & \widetilde H_k(X) \oplus \widetilde H_k(Y) & \text{via Mayer--Vietoris and relative homology}
\end{array}
\]
The last equality holds for $k<n-1$, for $M$ and $N$ both $n$-manifolds, and for $k=n-1$ when at least one of them is orientable.

References: Hatcher (Algebraic Topology, Chapters 0, 2)

Tools of (co)homology

 Preliminary exam prep

Let $X,Y$ be topological spaces, $G$ a group, and $R$ a unital commutative ring.

Defining homology groups

Theorem: If $(X,A)$ is a good pair (there exists a neighborhood $U\subset X$ of $A$ such that $U$ deformation retracts onto $A$), then for $i:A\hookrightarrow X$ the inclusion and $q:X\twoheadrightarrow X/A$ the quotient maps, there exists a long exact sequence of reduced homology groups
\[
\cdots \to \widetilde H_n(A) \tov{i_*} \widetilde H_n(X) \tov{q_*} \widetilde H_n(X/A) \to \cdots.
\]

Theorem: For any pair $(X,A)$, there exists a long exact sequence of homology groups
\[
\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to \cdots,
\]
where the last is called a relative homology group. Hence $H_n(X,A)\cong \widetilde H_n(X/A)$ for a good pair $(X,A)$.

Theorem (Excision): For any triple of spaces $(Z,A,X)$ with $\text{cl}(Z)\subset \text{int}(A)$, there is an isomorphism $H_n(X-Z,A-Z)\cong H_n(X,A)$.

For any $x\in X$, the local homology of $X$ at $x$ is the relative homology groups $H_n(X,X-\{x\})$. By excision, these are isomorphic to $H_n(U,U-\{x\})$ for $U$ any neighborhood of $x$. If $X$ is nice enough around $x$ (that is, if $U\cong \R^k$), then these groups are isomorphic to $H_n(\R^k,\R^k-\{x\})\cong H_n(D^k,\dy D^k) = H_n(S^k)$.

Theorem (Mayer-Vietoris): For $X=A\cup B$, there is a long exact sequence of homology groups
\[
\cdots \to H_n(A\cap B) \to H_n(A)\oplus H_n(B) \to H_n(X) \to \cdots,
\]

and if $A\cap B$ is non-empty, there is an analogous sequence for reduced homology groups.

Extending with coefficients

Recall the $\Tor$ and $\Ext$ groups, which were, respectively, the left and right derived functors of, respectively, $\otimes$ and $\Hom$ (see post "Exactness and derived functors," 2016-03-20). Here we only need $\Tor_1$ and $\Ext^1$, which are given by, for any groups (that is, $\Z$-modules) $A$, $B$,
\[
\begin{array}{r c c c l}
\Tor(A,B) & = & H_1(\text{projres}(A)\otimes B) & = & H_1(A\otimes \text{projres}(B)), \\
\Ext(A,B) & = & H^1(\Hom(A,\text{injres}(B))) & = & H^1(\Hom(\text{projres}(A),B)).
\end{array}
\]
Note that $\Tor$ is symmetric in its arguments, while $\Ext$ is not. Recall that $\Tor_0(A,B)=A\otimes B$ and $\Ext^0(A,B) = \Hom(A,B)$.

Theorem (Universal coefficient theorem): There exist isomorphisms
\[
\begin{array}{r c c c l}
H_n(X;G) & \cong & \Hom(H^n(X),G)\oplus \Ext(H^{n+1}(X),G) & \cong & H_n(X)\otimes G\ \oplus\ \Tor(H_{n-1}(X),G),  \\
H^n(X;G) & \cong & \Hom(H_n(X),G)\oplus \Ext(H_{n-1}(X),G) & \cong & H^n(X)\otimes G\ \oplus\ \Tor(H^{n+1}(X),G).
\end{array}
\]

Here are some common $\Hom$, $\Tor$, and $\Ext$ groups:
\begin{align*}
\Hom(\Z,G) & = G & \Tor(\Z,G) & = 0 & \Ext(\Z,G) & = 0 \\
\Hom(\Z_m,\Z) & = 0 & \Tor(G,\Z) & = 0 & \Ext(\Z_m,\Z) & = \Z_m \\
\Hom(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} & \Tor(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} & \Ext(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} \\
\Hom(\Q,\Z_n) & = 0 & & & \Ext(\Q,\Z_n) & = 0 \\
\Hom(\Q,\Q) & = \Q & & & \Ext(G,\Q) & = 0
\end{align*}

Theorem (Künneth formula): For $X,Y$ CW-complexes, $F$ a field, and $H^k(Y;G)$ or $H^k(X;G)$ finitely generated for all $k$, there are isomorphisms, for all $k$,

\[
H_k(X\times Y;F) \cong \bigoplus_{i+j=k} H_i(X;F)\otimes_FH_j(Y;F),
\hspace{1cm}
H^k(X\times Y;G) \cong \bigoplus_{i+j=k} H^i(X;G)\otimes_GH^j(Y;G)
\]

Dualities

Theorem (Poincaré duality): For $X$ a closed $n$-manifold (compact, without boundary) that is $R$-orientable (consistent choice of $R$-generator for each local homology group), for $k=0,\dots,n$ there are isomorphisms

\[
H^k(X;R)\cong H_{n-k}(X;R).
\]

Note that a simply orientable manifold means $\Z$-orientable. A manifold that is not $\Z$-orientable is always $\Z_2$-orientable (in fact all manifolds are $\Z_2$-orientable).

Theorem (Alexander duality): For $X\subsetneq S^n$ a non-empty closed locally contractible subset, for $k=0,\dots,n-1$ there are isomorphisms
\[
\widetilde H^k(X) \cong \widetilde H_{n-k-1}(S^n-X).
\]

References: Hatcher (Algebraic topology, Chapters 2, 3), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 7)

Degree and orientation

 Preliminary exam prep

Topology

 Recall that a topological manifold is a Hausdroff space in which every point has a neighborhood homeomorphic to $\R^n$ for some $n$. An orientation on $M$ is a choice of basis of $\R^n$ in each neighborhood such that every path in $M$ keeps the same orientation in each neighborhood. Every manifold $M\owns x$ appears in a long exact sequence (via relative homology) with three terms
\[
H_n(M-\{x\}) \tov{f} H_n(M) \tov{g} H_n(M,M-\{x\}).
\]
The first term is 0, because removing a point from an $n$-dimensional space leaves only its $(n-1)$-skeleton, which is at most $(n-1)$-dimensional. For $U$ a neighborhood of $x$ in $M$, the last term (via excision) is
\[
H_n(M-U^c, M-\{x\}-U^c) = H_n(U, U-\{x\})\cong H_n(\R^n, \R^n-\{x\}) \cong H_n(\R^n, S^{n-1}),
\]
which in turn fits into a long exact sequence whose interesting part is
\[
H_n(\R^n)\to H_n(\R^n,S^{n-1}) \to H_{n-1}(S^{n-1})\to H_{n-1}(\R^n),
\]
and since the first and last terms are zero, $H_n(M,M-\{x\})=\Z$. Since $f$ is zero, $g$ into $\Z$ must be injective, meaning that $H_n(M)=\Z$ or 0.

Theorem: Let $M$ be a connected compact (without boundary) $n$-manifold. Then

1. if $M$ is orientable, $g$ is an isomorphism for all $x\in M$, and
2. if $M$ is not orientable, $g=0$.

Definition: Let $f:M\to N$ be a map of connected, oriented $n$-manifolds. Since $H_n(M)=H_n(N)$ is infinite cyclic, the induced homomorphism $f_*:H_n(M)\to H_n(N)$ must be of the form $x\mapsto dx$. The number $d$ is called the degree of $f$.

In the special case when we are computing the degree for a map $f:S^n\to S^n$, by excision we get
\[
\deg(f) = \sum_{x_i\in f^{-1}(y)} \deg\left(H_n(U_i,U_i-x_i)\tov{f_*} H_n(V,V-y)\right),
\]for any $y\in Y$, some neighborhood $V$ of $y$, and preimages $U_i$ of $V$. This is called the local degree of $f$.

Geometry

Let $M$ be a smooth $n$-manifold. Recall $\Omega_M^r$ is the space of $r$-forms on $M$ and $d^r:\Omega^r_M\to \Omega^{r+1}_M$ is the differential map. Also recall the de Rham cohomology groups $H^r(M) = \text{ker}(d^r)/\text{im}(d^{r-1})$.

Definition: An $n$-manifold $M$ is orientable if it has a nowhere-zero $n$-form $\omega\in \Omega^n_M$. A choice of $\omega$ is called an orientation of $M$.

We also have a map $H^n(M)\to \R$, given by $\alpha\mapsto \int_M\alpha$, where the integral is normalized by the volume of $M$, so that integrating 1 across $M$ gives back 1. It is immediate that this doesn't make sense when $M$ is not compact, but when $M$ is compact and orientable, we get that $H^n(M)\neq0$. Indeed, if $\eta\in \Omega^{n-1}_M$ with $d\eta =\omega$, by Stokes' theorem we have
\[
\int_M\omega = \int_Md\eta = \int_{\dy M}\eta = \int_\emptyset\eta = 0,
\]
as $M$ has no boundary (since it is a manifold). But $\omega$ is nowhere-zero, meaning the first expression on the left cannot be zero. Hence $\omega$ is not exact and is a non-trivial element of $H^n(M)$.

Theorem: Let $M$ be a smooth, compact, orientable manifold of dimension $n$. Then $H^n(M)$ is one-dimensional.

Proof: The above discussion demonstrates that $\dim(H^n(M))\>1$. We can get an upper bound on the dimension by noting that the space of $n$-forms on $M$, given by $\Omega^n_M = \bigwedge^n(TM)^*$, has elements described by $dx_{i_1}\wedge \cdots \wedge dx_{i_n}$, with $\{i_1,\dots,i_n\}\subset \{1,\dots,n\}$. By rearranging the order of the $dx_{i_j}$, every element looks like $\alpha dx_1\wedge \cdots\wedge dx_n$ for some real number $\alpha$. Hence $\dim(\Omega^n_M) \leqslant 1$, so $\dim(H^n(M))$ is either 0 or 1. Therefore $\dim(H^n(M))=1$. $\square$

Definition: Let $f:M\to N$ be a map of smooth, compact, oriented manifolds of dimension $n$. Since $H^n(M)$ and $H^n(N)$ are 1-dimensional, the induced map $f^*:H^n(N)\to H^n(M)$ must be of the form $x\mapsto dx$. The number $d$ is called the degree of $f$. Equivalently, for any $\omega\in \Omega^n_N$,
\[
\int_M f^*\omega = d\int_N\omega
\]

References: Hatcher (Algebraic Topology, Chapters 2, 3.3), Lee (Introduction to Smooth Manifolds, Chapter 17)

Complexes and their homology

 Preliminary exam prep

Here I'll present complexes from the most restrictive to the most general. Recall the standard $n$-simplex is
\[
\Delta^n = \{x\in \R^{n+1}\ :\ \textstyle\sum x_i = 1, x_i\>0\}.
\]

Definition: Let $V$ be a finite set. A simplicial complex $X$ on $V$ is a set of distinct subsets of $V$ such that if $\sigma\in X$, then all the subsets of $\sigma$ are in $X$.

Every $n$-simplex in a simplicial complex is uniquely determined by its vertices, hence no pair of lower dimensional faces of a simplex may be identified with each other.

Definition: Let $A,B$ be two indexing sets. A $\Delta$-complex (or delta complex) $X$ is
\[
X = \left.\bigsqcup_{\alpha\in A} \Delta^{n_\alpha}_\alpha \right/\left\{\mathcal F_{\beta}^{k_\beta}\right\}_{\beta\in B}\ ,
\hspace{1cm}
\mathcal F_\beta^{k_\beta} = \{\Delta_1^{k_\beta},\dots,\Delta_{m_\beta}^{k_\beta}\},
\]
such that if $\sigma$ appears in the disjoint union, all of its lower dimensional faces also appear. The identification of the $k$-simplices in $\mathcal F^k$ is done in the natural (linear) way, and restricting to identified faces gives the identification of the $\mathcal F^{k-1}$ where the faces appear.

To define simplicial homology of a simplicial or $\Delta$-complex $X$, fix an ordering of the set of 0-simplices (which gives an ordering of every $\sigma\in X$), define $C_k$ to be the free abelian group generated by all $\sigma\in X$ of dimension $k$ (defined by $k+1$ 0-simplices), and define a boundary map
\[
\begin{array}{r c l}
\partial_k\ :\ C_k & \to & C_{k-1}, \\\
[v_0,\dots,v_k] & \mapsto & \sum_{i=0}^k(-1)^i[v_0,\dots,\widehat{v_i},\dots,v_k].
\end{array}
\]
Then $H_k(X):= \text{ker}(\partial_k)/\text{im}(\partial_{k+1})$.

Recall the standard $n$-cell is $e^n = \{x\in \R^n\ :\ | x| \leqslant 1\}$, also known as the $n$-disk or $n$-ball.


Definition: Let $X_0$ be a finite set. A cell complex (or CW complex) is a collection $X_0,X_1,\dots$ where
\[
X_k := \left.X_{k-1}\bigsqcup_{\alpha\in A_k} e^k_\alpha \right/\left\{\partial e^k_\alpha\sim f_{k,\alpha}(\dy e^k_\alpha)\right\}_{\alpha\in A_k},
\]
where the $f_{k,\alpha}$ describe how to attach $k$-cells to the $(k-1)$-skeleton $X_{k-1}$, for $k\>1$. $X_k$ may also be described by pushing out $e^k\sqcup_{\dy e^k}X_{k-1}$. Note that $\dy e^k = S^{k-1}$, the $(k-1)$-sphere.

To define cellular homology, we need more tools (relative homology and excision) that require a blog post of their own.

References: Hatcher (Algebraic topology, Chapter 2.1)

Persistent homology (an example)

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)