TDA Notes - Unit 01

TDA in pictures

Detecting topological features with a filtration:

“Barcodes” of persistence homology:

Topological spaces: basic notions

01 Theory - Spaces, subspaces, connectedness

Topological space

A topological space is a set $X$ (the ‘space’) together with a topology structure.

A topology structure is a determination of open subsets of $X$.

Formally: open sets are collected together in a set of sets $\tau$, the topology.

Open sets (in $\tau$) must satisfy axioms:

• $\varnothing$ and $X$ are open
• All unions of opens are open
• Finite intersections of opens are open

A set $A\subset X$ is called closed when $X\backslash A$ is open.

Although these axioms for open sets seem rather obscure, a long history (leading up to and in consequence of the axioms) has shown that the concept of open set they determine accurately formalizes our intuitions about spatial connectedness, and the aspect of proximity that underwrites connectedness.

Many geometric sets have natural topology structures:

Some notes:

• The trivial topology has only the minimum of open sets, $\varnothing$ and $X$.
• The discrete topology has the maximum of open sets, $\tau =2^X$. (All subsets are open.)
• Sometimes a set is simultaneously closed and open, thus ‘clopen’. E.g. $\varnothing$ and $X$ are always clopen.

Subspace topology

For any subset $A\subset X$, there is a subspace topology structure on $A$.

The open sets of $A$ are the intersections $A\cap U$ where $U$ is open in $X$.

Beware that open sets $A\cap U$ in a subspace topology might not be open in $X$ itself.

Connectedness

A topological space $X$ is connected when it is impossible to decompose it as a disjoint union of opens:

$$X\ne U\bigsqcup V(\text{for any opens }U,V)$$

02 Illustration

Simple topology - connectedness

Three points: $X=\{A,B,C\}$, with two (nontrivial) open sets. This is not a topology because the intersection $\{B\}$ is not open:

This is a topology, and it is connected:

Continuum - real number interval

The standard topology structure for real numbers is generated by open intervals $(a,b)$.

(“Generated”: it is the smallest topology structure containing all such intervals.)

In $n$D, the open sets are generated by open boxes $(a,b)\times (c,d)$ or (equivalently) open balls ${𝐁}_{\epsilon }(\overrightarrow{c})=\{\overrightarrow{x}\in {ℝ}^n||\overrightarrow{x}-\overrightarrow{c}|<\epsilon \}$.

Open in $S$ but not in ${ℝ}^2$

Consider the subspace $S={[\mathrm{0,1}]}^2\subset {ℝ}^2$ equipped with the subspace topology inherited from the topology structure on ${ℝ}^2$:

An open set in $S$ may be given as the intersection with a ball in ${ℝ}^2$, but this set is not open in ${ℝ}^2$ because the top and right edges would be included. No open ball can be found around a point in this edge and contained in $S$.

03 Theory - Metric, open balls, continuity

Metric space

A metric space is a set $X$ (the ‘space’) together with a metric structure.

A metric structure is a determination of distance between points in the set.

Formally: distance is measured by a function $d:X\times X\to ℝ$ satisfying: Triangle inequality:

• $d(x,z)\le d(x,y)+d(y,z)$

  

and more basic axioms:

• $d(x,y)\ge 0$ for all $x,y\in X$
• $d(x,y)=d(y,x)$
• $d(x,y)=0⟺x=y$

In ${ℝ}^n$ the distance function is, of course, $d(\overrightarrow{x},\overrightarrow{y})=\sqrt{{(x_1-y_1)}^2+\cdots +{(x_n-y_n)}^2}$.

---

Using any distance function, we can define open balls:

$${𝐁}_r(x)=\{y\in X|d(y,x)<r\}$$

Then we can generate a topology on $X$ from these open balls. We are, in effect, using the topology of $ℝ$ in the codomain of $d:X\times X\to ℝ$ to induce a topology back on the space $X$.

Metric space topology

Any metric space $(X,d)$ has a natural metric space topology structure.

The open sets are generated by open balls ${𝐁}_r(x)$. That is, open sets are all unions of open balls:

$$\bigcup _{x,r}{𝐁}_r(x)$$

Every structure of metric space automatically has a natural structure of topological space due to the above definition.

A topology structure formalizes only the notion of connectedness, or one might say “touching vs. separated”, while a metric structure adds a quantitative measure of distance. As such it implies notions like “double the distance” or “$100\times$ the distance.” But a metric structure does imply at least a topology structure, because any points that are separated by some distance are in fact separated.

Topological structures that do not come from metric structures can also be useful and interesting. (E.g. in algebraic geometry.) But the notion of “touching” that they express is rather abstracted from the geometrical one we are accustomed to.

“All unions of open balls”

The collection “all unions of open balls” satisfies the axioms of a topology. The first two axioms clearly hold. To check the third axiom, one would have to formalize the concept of ‘generates’, which is applicable to the concept of ‘subbase’. The essential verification is that for any point inside a finite intersection of open balls, there is another open ball around that point which is contained in the intersection. The triangle axiom is needed to confirm this.

---

LEMMA: open subsets via open balls

In any metric space $(X,d)$, a subset $A\subset X$ is open if and only if:

$$\forall x\in A,\exists r,x\text{with:}x\in {𝐁}_r(x)\subset A$$

In words: $A$ is open IFF any $x$ in $A$ has some open ball around it which is contained in $A$.

This lemma provides a functional characterization of open sets in a metric space.

Proof of LEMMA

Assume the Lemma condition holds; try to prove $A$ is open.

• For each $x$, write $B_x$ for some open ball over $x$ contained in $A$. Then $A={\bigcup }_{x\in A}{B}_x$, and $A$ is a union of open sets, thus open.

Now assume $A$ is open; try to prove the condition holds.

1. $A$ is necessarily the union of some open balls. (Definition of topology structure given a metric.)
2. Select any $x\in A$. By 1., this $x$ is in some open ball ${𝐁}_{r^{\ast }}(x^{\ast })$: $x\in {𝐁}_{r^{\ast }}(x^{\ast })\subset A$
3. Find a ball centered at $x$ which is inside ${𝐁}_{r^{\ast }}(x^{\ast })$, thus inside $A$.
   • Set $m:=d(x,x^{\ast })$
   • Definition of ball and $x\in {𝐁}_{r^{\ast }}(x^{\ast })$ implies $m<r^{\ast }$
   • Set $r:=\frac{1}{2}(r^{\ast }-m)$
   • Triangle Inequality: Every $y\in {𝐁}_r(x)$ satisfies $d(y,x^{\ast })\le d(y,x)+d(x,x^{\ast })$
   • Therefore $d(y,x^{\ast })<r+m<r^{\ast }$

---

Continuous functions

A function between topological spaces $f:X\to Y$ is continuous when:

$$V\subset Y\text{is open}⟹f^{-1}(V)\subset X\text{is open}$$

This definition implies that a connected subset cannot be mapped to a disconnected subset by a continuous $f$. Given any disjoint union of opens $U\bigsqcup V\supset \mathrm{Im}Y$ that ‘disconnects’ the image of $f$, the definition implies that $f^{-1}(U)$ and $f^{-1}(V)$ are disjoint opens covering $X$ and disconnecting it. The fact that the definition of continuity appears to go “the wrong way” (implication from codomain $Y$ to domain $X$) is related to the logical negation inside “no possible disjoint union of opens” in the definition of connectedness.

For example, the graph of a function with a jump discontinuity:

The disconnected image of $f$ can be separated by open sets in the codomain (vertical axis). The domain is not separated by the preimages – the function is not continuous.

---

An equivalent definition of continuity can be cast in terms of the metric structure, if the topology comes from a metric:

Continuity and distance

A map $f:(X,d)\to (Y,d^{\prime })$ of metric spaces is continuous at $x_0$ when:

For all $\epsilon >0$, there exists $\delta >0$ such that:

$$d(x,x_0)<\delta ⟹d(f(x),f(x_0))<\epsilon$$

Then $f$ itself is continuous if it is continuous at every point $x_0$.

04 Illustration

In Calc I we learn that $f:ℝ\to ℝ$ is continuous at $x_0$ when:

$$\forall \epsilon ,\exists \delta \text{ such that:}|x-x_0|<\delta ⟹|f(x)-f(x_0)|<\epsilon$$

Or, in terms of set containment:

$$\forall \epsilon ,\exists \delta \text{ such that:}f((x_0-\delta ,x_0+\delta ))\subset (f(x_0)-\epsilon ,f(x_0)+\epsilon )$$

Let us compare this to the definition in terms of open sets:

Open-set continuity agrees with $\epsilon$-$\delta$ continuity

Assume that $f$ is continuous according to the topological definition.

1. Select any $x_0\in ℝ$. Let $\epsilon >0$ be given.
2. The interval $V=(f(x_0)-\epsilon ,f(x_0)+\epsilon )$ is open.
3. By the topological definition of continuity, the preimage $f^{-1}(V)$ is open. It contains $x_0$.
4. By the Lemma for metric space continuity, find an open ball around $x_0$ inside $f^{-1}(V)$.
5. This ball is an interval $(x_0-\delta ,x_0+\delta )$.
6. All elements of this interval are mapped into $V$, confirming the $\epsilon$-$\delta$ condition.

Assume that $f$ is continuous according to the $\epsilon$-$\delta$ definition.

1. Select any $V\subset ℝ$ open set. We must show $f^{-1}(V)$ is open.
2. Select any $x_0\in f^{-1}(V)$. Define $y=f(x_0)$.
3. Then $y_0\in V$, and so $y_0\in {𝐁}_{\epsilon }(y_0)\subset V$ for some $\epsilon >0$ by metric space continuity.
4. This ${𝐁}_{\epsilon }(y_0)$ is an interval: $y\in {𝐁}_{\epsilon }(y_0)⟺|y-y_0|<\epsilon$
5. By the $\epsilon$-$\delta$ definition of continuity, produce some $\delta >0$ so that: $|x-x_0|<\delta ⟹|f(x)-y_0|<\epsilon$
6. Therefore, whenever $x\in {𝐁}_{\delta }(x_0)$, we know $f(x)\in V$.
7. So ${𝐁}_{\delta }(x_0)$ is open (by definition), is contained in $f^{-1}(V)$, and contains $x_0$, confirming the topological continuity condition.

05 Theory - Compactness, homeomorphism, continuity, homotopy

Compactness

A topological space is compact when every open cover has a finite subcover.

The entire real line $ℝ$ is not compact because the open cover $\{(n-1,n+1)|n\in \mathbb{Z}\}$ has no finite subset which also covers $ℝ$.

The closed interval $[\mathrm{0,1}]$ is compact.

1. Let $𝒰={\{U_{\alpha }\}}_{\alpha }$ be any open cover of $[\mathrm{0,1}]$.
2. Define $A\subset [\mathrm{0,1}]$:

$$A=\{x\in [\mathrm{0,1}]|[0,x]\text{ can be covered finitely out of }𝒰\}$$

3. Let $\ell$ be the supremum of $A$.
4. We show $\ell =1$.
   • Suppose $\ell <1$ and seek a contradiction.
   • Take some open set $U^{\ast }$ in $𝒰$ containing $\ell$, and an open ball ${𝐁}_{\epsilon }(\ell )\subset U^{\ast }$ around $\ell$ and contained in $U^{\ast }$.
   • Move leftwards to $\ell -\frac{1}{2}\epsilon$. There is a finite cover of $[0,\ell -\frac{1}{2}\epsilon ]$ from $𝒰$.
   • Add $U^{\ast }$ to this cover, and get a finite cover of $[0,\ell +\frac{1}{2}\epsilon ]$.
   • This contradicts the supremum property of $\ell$. So we know $\ell =1$.
5. Take some open set $U^{†}$ in $𝒰$ containing $1$, and an open ball ${𝐁}_{{\epsilon }^{†}}(1)\subset U^{†}$.
6. There is a finite cover of $[0,1-\frac{1}{2}{\epsilon }^{†}]$ because $\ell =1$.
7. Add $U^{†}$ to this cover to get a finite cover of $[\mathrm{0,1}]$. QED

---

Homeomorphism

A homeomorphism is a map between topological spaces, $f:X\to Y$, satisfying:

• $f$ is bijective
• $f$ is continuous
• $f^{-1}$ is continuous

PROP: ‘Homeomorphism’ simplifies for compact metric spaces

If $f:X\to Y$ is a continuous bijection between metric spaces, and $X$ and $Y$ are both compact, then $f$ is necessarily a homeomorphism.

$$\text{metric space}+\text{compact}⟹\text{don’t check }{f}^{-1}$$

Homotopy

A homotopy is a morphism between continuous maps.

Given $f_0,f_1:X\to Y$ continuous maps, a homotopy is a continuous function:

$$F:X\times [\mathrm{0,1}]\to Y,F(x,0)=f_0\text{and}F(x,1)=f_1$$

We say $f_0$ and $f_1$ are homotopic if any homotopy exists between them.

A homotopy is a continuous deformation of $f_0$ into $f_1$. Define $f_t(x)=F(x,t)$ to view it this way.

For example, suppose we have two maps $f_0,f_1:{𝕊}^1\to {𝕊}^1$:

The “winding number” is preserved by $F$, meaning that each $f_t$ has the same winding number.

Topological spaces: further notions

06 Theory - Quotients, gluing

Quotient topology

For any partition of $𝒫$ of $X$ into cells $Y_{\alpha }$, $\alpha \in 𝒜$, there is a quotient topology structure on the set of cells $Y={\{Y_{\alpha }\}}_{𝒜}$.

The open sets of $Y$ are the collections of cells whose unions are open in $X$.

A quotient space is therefore given by ‘collapsing’ some sets of points in $X$ into single objects $Y_{\alpha }$, and the open sets come from open sets in $X$ that are compatible with the partition, meaning they do not cross partition boundaries, or that they contain only whole cells.

There are many equivalent formulations of this notion, and these two are most common:

• Using an equivalence relation $\sim$ on $X$, whereby the equivalence classes $[x]$ determine cells of the partition. Then the quotient space is written $X/\sim$.
  • Recall that partitions and equivalence relations are interchangeable notions.
• Using a surjective function $f:X\twoheadrightarrow Y$, whereby the preimage fibers $f^{-1}(y)$ determine cells of the partition. Then $Y$ has the quotient topology when open sets are given by $V$ where $f^{-1}(V)$ is open in $X$.

Gluing

Two spaces $X_1$ and $X_2$ can be glued along specified subsets $A_1\subset X_1$ and $A_2\subset X_2$ using a bijection $\phi :A_1\to A_2$ by defining the quotient topology $X_1\bigsqcup X_2/\sim$ where $\sim$ identifies $a_1\in A_1$ with $\phi (a_1)\in A_2$.

07 Illustration

Gluing shapes

• Glue a pair of triangles along an edge.
• Glue a line segment to a triangle, endpoint to vertex.

Real projective space ${ℝ\mathbb{P}}^n$

Consider the unit sphere ${𝕊}^2\subset {ℝ}^3$. Antipodal points of ${𝕊}^2$ are those determining a straight line passing through the origin.

Define ${ℝ\mathbb{P}}^2$ as the quotient ${𝕊}^2/\sim$ where equivalence classes of $\sim$ are the pairs of antipodal points.

Define ${ℝ\mathbb{P}}^n$ similarly by gluing together antipodal points in ${𝕊}^n$.

We can also describe ${ℝ\mathbb{P}}^2$ using a disk with antipodally glued boundary:

This surface is also derivable by gluing a disk to a Mobius band:

08 Theory - Embedding, limit points, closure, boundary

Embedding

An embedding $f:A\hookrightarrow Y$ is a continuous injective map that induces a homeomorphism $f:A\to f(A)$.

It is important to emphasize that $f(A)$ should have the subspace topology inherited from $Y$.

For example, a knot in ${ℝ}^3$ is an embedding ${𝕊}^1\hookrightarrow {ℝ}^3$.

Exercise - Touching upon

An open interval is mapped into ${ℝ}^2$ by $f$ such that one end has a limit point on the image of the middle.

The map $f$ is continuous and bijective. Show that $f^{-1}$ is not continuous.

---

Limit points

Given a subspace $S\subset X$, a point $p\in S$ is a limit point of $S$ when every open set around $p$ touches $S$ somewhere besides $p$.

$$p\in U\text{open}⟹(U\cap S)\backslash \{p\}\ne \varnothing$$

Notes:

• Limit points are sometimes called accumulation points.
• The limit point $p$ might be outside $S$!
• In a metric space, this can be reformulated: $p$ is a limit point of $S$ when there is some sequence $x_n\in S$ such that $d(x_n,p)\to 0$ as $n\to \infty$. This is written $x_n\to p$.
  • In other words, $p$ is “approached” from inside $S$.

---

Interior and exterior

The interior of a set $S\subset X$ is the set of points $p\in S$ with an open neighborhood contained in $S$.

The exterior of a set $S\subset X$ is the interior of the complement of $S$.

Closure of a set

The closure $\overline{S}$ of a set $S\subset X$ is the union of $S$ with all limit points of $S$.

• Equivalently: the closure $\overline{S}$ is the complement of the union of all open sets in $X$ that don’t touch $S$.
• Equivalently: the closure $\overline{S}$ is the intersection of all closed sets containing $S$.

Boundary of a set

The boundary $\mathrm{Bd}(S)$ of a set $S\subset X$ is the set of points in both $\overline{S}$ and $\overline{(X\backslash S)}$.

• Equivalently: the boundary $\mathrm{Bd}(S)$ is the set of points for which every open neighborhood touches both the interior and the exterior of $S$.
• Boundary points are not always in $S$!

Simplicial complexes

A simplicial complex is an assembly of simplices, which generalize triangles and tetrahedra. Simplices provide “building block” objects in each dimension that may be glued together to form a complex.

Simplices provide a combinatorial method of representing a large class of topological spaces, since the ‘assembly data’ encoding the gluing of simplices into a complex is combinatorial in nature.

Simplicial homology is a homology theory (topological invariant) that is naturally defined in terms of a given simplicial complex. It is easy to find this homology for a space when a representation as simplicial complex is known.

Simplicial homology is isomorphic to other homology theories.

• Singular homology: abstract ‘naturality’; proving invariance is easier
• Simplicial homology: combinatorial; build complexes by gluing faces ‘directly’; stricter gluing rules
• Delta homology: combinatorial, like simplicial; more relaxed gluing rules
• CW homology: combinatorial, like simplicial but gluing cells (balls) not simplices; build CW complexes inductively; very relaxed gluing rules

For smooth manifolds, we also have:

• De Rham cohomology: uses differential forms “closed but not exact”

09 Theory - Simplices and complexes

$n=0$:

• A $0$-simplex is a point (a vertex).

$n=1$:

• A $1$-simplex is a line segment.

$n=2$:

• A $2$-simplex is a triangle (interior included).

$n=3$:

• A $3$-simplex is a tetrahedron.

Geometric simplex

Vertices: $v_0,v_1,\dots ,v_n\in {ℝ}^{n+1}$, required to be affinely independent.

Geometric simplex:

$$\begin{array}{c}\sigma =[v_0,v_1,\dots ,v_n]\\ \\ =\{x\in {ℝ}^{n+1},x=\sum t_i{v}_i|\forall i,t_i\ge 0\text{and}\sum t_i=1\}\end{array}$$

• Thus $\sigma$ is the “convex hull” of $v_0,\dots ,v_n$. In other words: all weighted averages of the vertices.
• Affinely independent means: $v_0,\dots ,v_n$ do not lie in any hyperplane (dimension $n$) of ${ℝ}^{n+1}$; equivalently that $v_1-v_0,\dots ,v_n-v_0$ are linearly independent, or that they generate a vector space of dimension $n$.
• A simplex failing this criterion is called degenerate.

• Standard $n$-simplex ${\sigma }_n$: vertices are the unit vectors in ${ℝ}^{n+1}$.

For example, with $n=1$ the standard simplex is:

$${\sigma }_1=\{\overrightarrow{v}\in {ℝ}^2|\overrightarrow{v}=t_0{𝐞}_x+(1-t_0){𝐞}_y,0\le t_0\le 1\}$$

This defines the line segment from $(\mathrm{1,0})$ to $(\mathrm{0,1})$.

Similarly, with $n=2$ we find that ${\sigma }_2$ is parametrized by $\overrightarrow{v}=t_0{𝐞}_x+t_1{𝐞}_y+(1-t_0-t_1){𝐞}_z$.

---

A simplicial complex can be characterized in several ways:

• Combinatorially
• Geometrically
• Topologically

Simplicial complex - Combinatorial

A combinatorial simplical complex $K$ with a set of ‘vertices’ $𝒱=\{v_0,v_1,\dots ,v_N\}$ is a collection of subsets of $𝒱$, called ‘simplices’, satisfying:

• All faces included: If $\sigma \in K$, then all faces of $\sigma$ are in $K$. (And thus faces of faces, all the way down to vertices.)
• Touching only at a face: If $\sigma ,\tau \in K$, and $\sigma \cap \tau =\varnothing$, then $\sigma \cap \tau$ is a face of $\sigma$ and a face of $\tau$.

For $\sigma =\{w_1,\dots ,w_n\}\subset 𝒱$, the faces of $\sigma$ are the subsets of $\sigma$ lacking a single vertex of $\sigma$.

A simplicial complex has a topology. This is given by gluing the disjoint union of constituent simplices along the faces that intersect. (The quotient topology.)

If the simplicial complex is given a geometric representation in ${ℝ}^n$, then the glued/quotient topology should match the one inherited by the complex as a subspace of ${ℝ}^n$.

---

10 Illustration

For example, given a simplex $[v_0,v_1,v_2]$, the faces are $[v_0,v_1]$ and $[v_1,v_2]$ and $[v_0,v_2]$:

---

A simplicial complex $𝒦$ with $3$D subcomplex ${\sigma }^3$ and $2$D subcomplex ${\tau }^2$ and $1$D subcomplex ${\gamma }^1$.

Simplices of dimension $0$ are called vertices, simplices of dimension $1$ are called edges, and simplices of dimension $n-1$ in an $n$D complex are called faces.

The simplices of top dimension are sometimes called facets (i.e. the biggest ones are called ‘little faces’).

---

The intersection of two sub-simplices is allowed to be a single face, not two faces (vertices, in this case):

Torus as simplicial complex

It is easy to create a torus by gluing the edges of a square:

But it is not this easy to represent a torus as a simplicial complex. Initial attempts fail:

One can do it with a little more work and complexity:

---

Every finite simple undirected graph is a simplex.

• No “parallel edges”
• No “one-loops”

---

In a geometric representation of a simplicial complex, edges should not cross, etc.

Kuratawki: Graph planarity

The graph planarity theorem of Kuratowski says that a graph is planar (can be drawn in ${ℝ}^2$ without self intersection) if and only if it does not contain a copy of $K_5$ or $K_{\mathrm{3,3}}$:

11 Theory - Subdivision and triangulation

A single topological space (up to homeomorphism) can be represented in many different ways as a simplicial complex:

One can perform “barycentric subdivision” upon a given simplicial complex. This produces a simplicial complex that is homeomorphic to the original one, but the edges are shorter and areas of faces smaller.

---

A triangulation is a simplicial complex that is homeomorphic to a given smooth manifold. A triangulated manifold is a simplicial complex together with a homeomorphism to a given smooth manifold.

For example, the tetrahedron obtained by gluing four copies of the standard $2$-simplex ${\sigma }_2$ is homeomorphic to ${𝕊}^2$ and is therefore a triangulation of ${𝕊}^2$.

Triangulations of spheres

For ${𝕊}^2$, all possible triangulation types have been found by Tutte.

For ${𝕊}^3$, there is not a known enumeration of triangulation types.

Simplicial homology

Simplicial homology can be defined with “coefficients in $\mathbb{Z}$” or with “coefficients in $ℝ$” (or another ring). Using $\mathbb{Z}$ gives abelian groups, while using $ℝ$ gives vector spaces. Vector spaces are nicer to work with, but some information is lost. (Generators of finite order are lost, for example.)

12 Theory - Boundaries, chain complexes

Fix a simplicial complex $𝒦$ which is $n$-dimensional. (Highest dimension of a simplex in $𝒦$ is $n$.)

Chain groups

For each $k\in \mathbb{N}$, $k\le n$, define the $k^{\text{th}}$ chain group $C_k(𝒦)$:

$$C_k(𝒦)=\begin{array}{c}\text{free abelian group}\\ \text{generated by}\\ k\text{-simplices in }𝒦\end{array}$$

In other words:

$$C_k=\{\sum _{i\in I\text{finite}}{n}_i{\sigma }_i|n_i\in \mathbb{Z},{\sigma }_i\in 𝒦,\dim ({\sigma }_i)=k\}$$

An element of the $k$-chain group, written generically as $n_1{\sigma }_1+\cdots +n_{\ell }{\sigma }_{\ell }$, is called a $k$-chain.

• Sometimes $𝒦$ is dropped and we write just $C_k$ when the complex is clear from context.

By allowing combinations of simplices with coefficients, we can define a boundary operator with alternating signs. This alternation leads to cancellation when boundaries of boundaries are taken. The cancellation in turn enables the definition of homology, which is generated by cycles of simplices that are not boundaries.

Boundary operator

The boundary operator ${\partial }_k$ maps simplices to their boundaries. It is defined on simplices and extended linearly to chains:

$$\begin{array}{cc}{\partial }_k:C_k& ⟶C_{k-1}\\ {\partial }_k:[v_0,\dots ,v_k]& ⟼\sum _{i=0}^n{(-1)}^i[v_0,{\dots ,\widehat{v_i},\dots ,}^{\begin{array}{c}{i}^{\text{th}}\text{ omitted}\\ \phantom{x}\end{array}}{v}_k]\end{array}$$

Examples:

• A $1$-simplex:

$$\partial ([v_0,v_1])\text{⨠}\text{⨠}[\widehat{v_0},v_1]-[v_0,\widehat{v_1}]\text{⨠}\text{⨠}{v}_1-v_0$$

• A $2$-simplex:

$$\partial ([v_0,v_1,v_2])\text{⨠}\text{⨠}[v_1,v_2]-[v_0,v_2]+[v_0,v_1]$$

Orientation and vertex ordering

A simplex was initially defined as the convex closure of a set of points. The points were not ordered! The simplex was not ordered!

The boundary map was defined for an ordered list of points (“$i^{\text{th}}$ omitted”). These definitions are not compatible!

Let us upgrade both definitions to employ oriented sets of points.

Hereafter, the notation $[v_0,\dots ,v_n]$ will indicate an ordered list of vertex points up to the equivalence relation that identifies lists which are related by an even permutation.

For example: $[v_0,v_1,v_2]=[v_1,v_2,v_0]$, but: $[v_1,v_0,v_2]\ne [v_0,v_1,v_2]$.

• With this convention, the boundary map is well-defined.
• Notice that the orientation of the vertex list determines the sign of its boundary:

$$\begin{array}{cc}\partial ([v_0,v_1])& \text{⨠}\text{⨠}{v}_0-v_1\\ & \\ \text{but:}\partial ([v_1,v_0])& \text{⨠}\text{⨠}{v}_1-v_0\end{array}$$

• Therefore: one may interpret the sign of a coefficient in $C_k$ as indicating the orientation of the simplex it is attached to.
• Thus the boundary of a 2-simplex (filled triangle) is an oriented path around the triangle:

$$\begin{array}{cc}\partial ([v_0,v_1,v_2])& \text{⨠}\text{⨠}[v_1,v_2]-[v_0,v_2]+[v_0,v_1]\\ & \\ & \text{⨠}\text{⨠}[v_0,v_1]+[v_1,v_2]+[v_2,v_0]\end{array}$$

---

LEMMA: Boundary of boundary is empty

For all $k\in \mathbb{N}$, we have:

$${\partial }_{k-1}\circ {\partial }_k=0$$

Proof of LEMMA

Apply the double boundary to a generator of $C_k$:

$$\begin{array}{c}{C}_k\ni [v_0,\dots ,v_k]\stackrel{{\partial }_k}{⟼}\sum _{i=0}^k{(-1)}^i[v_0,{\dots ,\widehat{v_i},\dots ,}^{\begin{array}{c}{i}^{\text{th}}\text{ omitted}\\ \phantom{x}\end{array}}{v}_k]\\ \\ \stackrel{{\partial }_{k-1}}{⟼}\sum _{i=0}^k\sum _{j<i}{(-1)}^i{(-1)}^j[v_0,\dots ,\widehat{v_j},\dots ,\widehat{v_i},\dots ,v_k]+\\ \\ +\sum _{i=0}^k\sum _{j>i}{(-1)}^i{(-1)}^{j-1}[v_0,\dots ,\widehat{v_i},\dots ,\widehat{v_j},\dots ,v_k]\\ \\ =0\end{array}$$

The second summation duplicates the terms of the first summation, but has one fewer power of $-1$, so the whole sum vanishes. Both summations cover all $(k-2)$-simplices omitting some two vertices.

Using this fact we construct the chain complex of group homomorphisms:

Chain complex

Given a simplicial complex $𝒦$ of maximal dimension $n$, we define the chain complex of $𝒦$ as the sequence of morphisms:

$$0\to C_n\stackrel{{\partial }_n}{⟶}{C}_{n-1}\stackrel{{\partial }_{n-1}}{⟶}{C}_{n-2}⟶\cdots ⟶C_1\stackrel{{\partial }_1}{⟶}{C}_0⟶0$$

This sequence is a complex because the Lemma ensures that

$$\mathrm{Im}({\partial }_k)\subset \mathrm{Ker}({\partial }_{k-1})\text{for every}k.$$

The image and kernel groups have names:

• $B_k:=\mathrm{Im}({\partial }_{k+1})$ — called the boundary chains
• $Z_k:=\mathrm{Ker}({\partial }_k)$ — called the cycles

Homology

The $k^{\text{th}}$ homology of $𝒦$ is defined as the quotient group of cycles modulo boundary chains:

$$H_k(𝒦)=\frac{\mathrm{Ker}({\partial }_k)}{\mathrm{Im}({\partial }_{k+1})}$$

Notice that $k$ in $H_k$ appears in the numerator. So $H_k$ is generated by chains of $k$-simplices.

13 Illustration

Homology of a triangle

Consider the 1D simplicial complex consisting of a triangle:

The chain complex is:

$$\begin{array}{cc}0& ⟶C_1⟶C_0⟶0\\ & \\ 0& ⟶{\mathbb{Z}}^3\stackrel{{\partial }_1}{⟶}{\mathbb{Z}}^3⟶0\end{array}$$

The group $C_1$ is generated by the 1-simplices:

$$C_1=⟨[v_0,v_1],[v_1,v_2],[v_0,v_2]⟩$$

The group $C_0$ is generated by the 0-simplices:

$$C_0=⟨v_0,v_1,v_2⟩$$

The boundary operator:

$$\begin{array}{cc}[v_0,v_1]& \stackrel{{\partial }_1}{⟶}{v}_1-v_0\\ & \\ [v_1,v_2]& \stackrel{{\partial }_1}{⟶}{v}_2-v_1\\ & \\ [v_0,v_2]& \stackrel{{\partial }_1}{⟶}{v}_2-v_0\end{array}$$

The kernel of ${\partial }_1$ is freely generated by $[v_0,v_1]+[v_1,v_2]-[v_0,v_2]$. The image of ${\partial }_1$ is isomorphic to $\mathbb{Z}$ via the map determined by $v_0\mapsto 1$.

So we have $H_1\cong \mathbb{Z}$ and $H_0\cong \mathbb{Z}$. Higher homology groups vanish.

Homology counts holes by dimension

Consider the space $X$, the wedge sum of three copies of ${𝕊}^1$ and one copy of ${𝕊}^2$ obtained by gluing all spaces at a single point:

The homology of this space is:

$$H^i(X)=\{\begin{array}{cc}0& i>3\\ {\mathbb{Z}}^2& i=2\\ {\mathbb{Z}}^3& i=1\\ \mathbb{Z}& i=0\end{array}$$

Homology independent loops in a graph

The “nullity” of a connected graph is the number of independent cycles in the graph. If $H^1(\mathrm{\Gamma })={\mathbb{Z}}^c$ for $\mathrm{\Gamma }$ a graph considered as a simplicial complex, then $c$ is the nullity.

So $H^1(\mathrm{\Gamma })=6$ for $\mathrm{\Gamma }$ the following graph: