2025.01.13 - Lecture 3

← Lecture 2 · Lecture 4 →

In this lecture (13 January)

• What homology is for
• Chain groups $C_p(K)$
• The boundary operator ${\partial }_p$ and the boundary of a boundary is zero (Proposition 13)
• Cycles, boundaries and holes: why $H_p(K)$ counts holes
• Betti numbers via matrix ranks (Proposition 14)
• Worked example: the pyramid with open base

Recall from Lecture 2: a chain complex $\mathcal{C}=(V_p,{\partial }_p)$ is a sequence of vector spaces with linear maps ${\partial }_p:V_p\to V_{p-1}$ such that ${\partial }_p\circ {\partial }_{p+1}=0$. Its $p$-th homology group is

$$H_p(\mathcal{C})=\frac{\ker {\partial }_p}{\mathrm{Im}{\partial }_{p+1}}.$$

Today we build a chain complex out of a simplicial complex $K$; its homology is the simplicial homology $H_p(K)$. We always work with coefficients in ${\mathbb{Z}}_2$.

What homology is for: counting holes

The purpose of $H_p(K)$ is to count the $p$-dimensional holes of $K$.

A hole is a cavity that is “surrounded” by something, and its dimension is the dimension of what surrounds it:

• 0-dimensional holes: gaps between connected components (a component is “surrounded” by points);
• 1-dimensional holes: tunnels/loops, i.e. a closed curve that does not bound a filled region (e.g. a hollow triangle);
• 2-dimensional holes: cavities enclosed by a closed surface (e.g. the inside of a hollow tetrahedron or of a sphere).

So “dim(hole) = dim(the cycle that surrounds it)“.

Chain groups of a simplicial complex

Definition – Chain group $C_p(K)$

$C_p(K)$ is the set of formal linear combinations with coefficients in ${\mathbb{Z}}_2$ of the $p$-simplices of $K$:

$$C_p(K)=\{\sum _i{a}_i{\sigma }_i:a_i\in {\mathbb{Z}}_2,{\sigma }_{i}p\text{-simplex of }K\}.$$

Its elements are the $p$-chains. It is a vector space over ${\mathbb{Z}}_2$ whose natural basis is the set of $p$-simplices, so

$$\dim C_p(K)=n_p:=\#\{p\text{-simplices of }K\},|C_p(K)|=2^{n_p}.$$

Over ${\mathbb{Z}}_2$ a $p$-chain simply selects a set of $p$-simplices, and the sum of two chains is the symmetric difference of the two sets.

Running example ($K$ = triangle $\alpha =⟨A,B,C⟩$ with edges $a=BC$, $b=AC$, $c=AB$, plus the edge $d=BD$):

• $C_0$: all combinations of $A,B,C,D$, e.g. $1A+0B+0C+0D=A$. These are 16 elements:

$$C_0=\{0,A,B,C,D,A+B,A+C,A+D,B+C,B+D,C+D,A+B+C,A+B+D,A+C+D,B+C+D,A+B+C+D\},$$

with $|C_0|=2^{n_0}=2^4$ and $\dim C_0=4$.

• $C_1=\{0,a,b,c,d,a+b,\dots ,a+b+c+d\}$: $2^4=16$ elements, $\dim C_1=4$.

• $C_2=\{0,\alpha \}$: only one triangle, $\dim C_2=1$.

• $C_p=\{0\}$, the null space, for $p\ge 3$: there are no simplices of those dimensions.

• $C_{-1}$: the only $(-1)$-simplex is the empty set. It depends on the convention:

  • reduced homology (the one we use for now): $C_{-1}=\{0,\varnothing \}$, generated by the empty simplex, so $\dim C_{-1}=1$ (two elements);
  • classical homology: $C_{-1}=\{0\}$.

  The two conventions are compared in Lecture 4.

Notation: $0$ vs $\varnothing$

The handwritten notes often write $\varnothing$ for the zero chain (e.g. "${\partial }^2=\varnothing$", "$2u_0+2u_1+2u_2=\varnothing$"). Here we keep them distinct:

• $0$ is the null vector of $C_p(K)$ (the empty selection of simplices);
• $\varnothing$ is the empty simplex, a basis vector of $C_{-1}$ in reduced homology.

They are different objects: $\varnothing \ne 0$ in $C_{-1}$, but $\varnothing +\varnothing =0$.

The boundary operator

The boundary of a triangle should be the sum of its three edges:

$${\partial }_2⟨u_0,u_1,u_2⟩=⟨{\hat{u}}_0,u_1,u_2⟩+⟨u_0,{\hat{u}}_1,u_2⟩+⟨u_0,u_1,{\hat{u}}_2⟩=⟨u_1,u_2⟩+⟨u_0,u_2⟩+⟨u_0,u_1⟩,$$

where the hat ${\hat{u}}_i$ means “remove $u_i$”.

Definition – Boundary operator

${\partial }_p:C_p(K)\to C_{p-1}(K)$ is the linear map defined on the natural basis of $C_p(K)$ by

$${\partial }_p⟨u_0,\dots ,u_p⟩:=\sum _{i=0}^p⟨u_0,\dots ,{\hat{u}}_i,\dots ,u_p⟩$$

(the sum of the $(p-1)$-faces of the simplex), and extended by linearity: ${\partial }_p\left({\sum }_i{a}_i{\sigma }_i\right)={\sum }_i{a}_i{\partial }_p{\sigma }_i$.

In low dimensions:

• ${\partial }_1⟨u_0,u_1⟩=u_0+u_1$: the boundary of an edge is the pair of its endpoints;
• ${\partial }_0⟨u_0⟩=⟨⟩=\varnothing$ in reduced homology ($=0$ in classical homology);
• ${\partial }_p=0$ when $C_p=0$.

In the running example ${\partial }_2\alpha =a+b+c$, ${\partial }_{1}a=B+C$, and ${\partial }_1(a+d)=(B+C)+(B+D)=C+D$: $B$ cancels because $1+1=0$.

The boundary of a boundary is zero

Example on a triangle:

$$\begin{array}{rl}{\partial }_1{\partial }_2⟨u_0,u_1,u_2⟩& ={\partial }_1⟨u_1,u_2⟩+{\partial }_1⟨u_0,u_2⟩+{\partial }_1⟨u_0,u_1⟩\\ & =(u_1+u_2)+(u_0+u_2)+(u_0+u_1)=2u_0+2u_1+2u_2=0,\end{array}$$

because in ${\mathbb{Z}}_2$ we have $2=1+1=0$ (2 mod 2 = 0). Each vertex appears twice, once for each edge that contains it.

Proposition 13 – ${\partial }^2=0$

For every $p$, ${\partial }_{p-1}\circ {\partial }_p=0$. Hence $(C_p(K),{\partial }_p{)}_{p\in \mathbb{Z}}$ is a chain complex.

Proof

By linearity it suffices to check a simplex $\sigma =⟨u_0,\dots ,u_p⟩$. For $i<j$ let

$${\sigma }_{ij}=⟨u_0,\dots ,{\hat{u}}_i,\dots ,{\hat{u}}_j,\dots ,u_p⟩\in C_{p-2}(K)$$

(remove both $u_i$ and $u_j$), and set ${\sigma }_{ij}=0$ if $i\ge j$. Then

$$\begin{array}{rl}{\partial }_{p-1}({\partial }_p\sigma )& =\sum _{i=0}^p{\partial }_{p-1}⟨u_0,\dots ,{\hat{u}}_i,\dots ,u_p⟩\\ & =\sum _{i=0}^p(\underset{\text{remove a vertex before }{u}_i}{\underbrace{\sum _{j=0}^{i-1}⟨\dots ,{\hat{u}}_j,\dots ,{\hat{u}}_i,\dots ⟩}}+\underset{\text{remove a vertex after }{u}_i}{\underbrace{\sum _{j=i+1}^p⟨\dots ,{\hat{u}}_i,\dots ,{\hat{u}}_j,\dots ⟩}})\\ & =\sum _{i=0}^p(\sum _{j=0}^{i-1}{\sigma }_{ji}+\sum _{j=i+1}^p{\sigma }_{ij})=\sum _{i=0}^p\sum _{j=0}^p{\sigma }_{ji}+\sum _{i=0}^p\sum _{j=0}^p{\sigma }_{ij}\\ & =2\sum _{i=0}^p\sum _{j=0}^p{\sigma }_{ij}=0.\end{array}$$

In the fourth equality we extended the sums to all $j$, which adds only null terms since ${\sigma }_{ji}=0$ for $j\ge i$ and ${\sigma }_{ij}=0$ for $j\le i$. In the fifth we renamed the indices of the first double sum. In words: each $(p-2)$-face (remove $u_i$ and $u_j$) is obtained exactly twice, removing $u_i$ first and then $u_j$ or the other way round, and $2=0$ in ${\mathbb{Z}}_2$.

Intuition: “the boundary of a boundary is empty”. The boundary of a triangle is a closed loop, and a closed loop has no endpoints.

So for a simplicial complex $K$ we can define its homology

$$H_p(K)=H_p(C(K))=\frac{\ker {\partial }_p}{\mathrm{Im}{\partial }_{p+1}}=\frac{Z_p(K)}{B_p(K)}.$$

Cycles, boundaries and holes

Definition – Cycles and boundaries

• $Z_p(K)=\ker {\partial }_p$: the $p$-cycles, chains with zero boundary (closed chains, e.g. closed loops of edges);
• $B_p(K)=\mathrm{Im}{\partial }_{p+1}$: the $p$-boundaries, chains that bound a $(p+1)$-chain;
• $B_p\subseteq Z_p$ by ${\partial }^2=0$: every boundary is a cycle.

Why do homology groups count $p$-holes? Imagine a point cloud turned into a complex $K$: the holes of $K$ are what we want to detect.

A cycle that is also a boundary. $\partial (a+b+c)=0$, so $a+b+c$ is a cycle; it is the boundary of the filled triangle $\alpha$, so it surrounds nothing empty.

A chain that is not a cycle. A path is not closed: its boundary is the pair of its endpoints, $\partial (a+b)=(A+B)+(B+C)=A+2B+C=A+C\ne 0$.

A cycle that is not a boundary. $\partial (d+e+f)=0$, so $d+e+f$ is still a cycle. The three edges $d,e,f$ bound the right triangle, which is not filled (not shaded: it is not a 2-simplex of the complex), so $d+e+f$ is not the boundary of any 2-chain. It detects a hole.

Key idea

A hole is a cycle that is not a boundary.

Different cycles, same hole. Two loops going around the same hole should count as one hole, not two:

Think of the tent as a triangular tube with filled walls. The loop $z_1$ around the front opening and the loop $z_2$ around the back opening are two different cycles, but they surround the same tunnel: their sum $z_1+z_2$ is the boundary of the walls between them (the cycles “have a common boundary”), so they are equivalent. This is what the quotient does:

$$[z_1]=[z_2]\text{ in }{H}_p(K)⟺z_1+z_2\in B_p(K)$$

(over ${\mathbb{Z}}_2$, $z_1-z_2=z_1+z_2$). Such cycles are called homologous. In $H_p(K)=Z_p/B_p$:

• cycles that are boundaries become $0$ (no hole);
• homologous cycles become the same element (same hole).

So $\dim H_p(K)$ is the number of independent $p$-holes.

Computing homology with linear algebra

We will use linear algebra to compute homology groups. Define

symbol	meaning	name
$n_p$	number of $p$-simplices $=\dim C_p$	$p$-chains
$z_p$	$\dim Z_p=\dim \ker {\partial }_p$	$p$-cycles
$b_p$	$\dim B_p=\dim \mathrm{Im}{\partial }_{p+1}$	$p$-boundaries
${\beta }_p$	$\dim H_p(K)=\dim Z_p/B_p$	$p$-th Betti number ($p$-holes)

Since $H_p(K)$ is a vector space over ${\mathbb{Z}}_2$, it is determined up to isomorphism by its dimension: $H_p(K)\cong {\mathbb{Z}}_2^{{\beta }_p}$. Computing ${\beta }_p$ is equivalent to computing the homology group.

Proposition 14

$${\beta }_p=n_p-b_p-b_{p-1}.$$

Proof

We use the rank–nullity theorem twice ($\dim (\text{domain})=\dim \ker +\dim \mathrm{Im}$).

• (A) The projection $\pi :Z_p\to Z_p/B_p$, $\pi (z)=[z]$, is surjective with $\ker \pi =B_p$:

$$\dim Z_p=\dim \ker \pi +\dim \mathrm{Im}\pi ⟹z_p=b_p+{\beta }_p.$$

• (B) For ${\partial }_p:C_p\to C_{p-1}$:

$$\dim C_p=\dim \ker {\partial }_p+\dim \mathrm{Im}{\partial }_p⟹n_p=z_p+b_{p-1}$$

(because $\mathrm{Im}{\partial }_p=B_{p-1}$).

Substituting (A) in (B): $n_p=b_p+{\beta }_p+b_{p-1}$, i.e. ${\beta }_p=n_p-b_p-b_{p-1}$.

$n_p$ is obtained by counting simplices. For $b_p=\dim \mathrm{Im}{\partial }_{p+1}$: let $A_{p+1}$ be the matrix of ${\partial }_{p+1}$ with respect to the natural bases of $C_{p+1}$ and $C_p$. It has one row per $p$-simplex and one column per $(p+1)$-simplex, and the entry is $1$ iff the row simplex is a face of the column simplex. Then

$$b_p=\mathrm{rank}(A_{p+1}),$$

computed with Gaussian elimination mod 2.

Compute ranks over ${\mathbb{Z}}_2$, not over $\mathbb{R}$

Example: the hollow triangle (vertices $u_0,u_1,u_2$, edges $u_0{u}_1$, $u_1{u}_2$, $u_0{u}_2$) has

$$A_1=\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 0& 1& 1\end{array}\right).$$

Over $\mathbb{R}$ its determinant is $2$, so the rank is 3. Over ${\mathbb{Z}}_2$ the three rows sum to $0$ and the rank is 2. The correct value is $b_0=2$, which gives ${\beta }_1=3-0-2=1$: one hole. With the real rank we would wrongly find no hole. Working mod 2 is what allows us to forget orientations.

First example (the running $K$). $n_0=4$, $n_1=4$, $n_2=1$, $n_{-1}=1$ (reduced).

• $b_1=\mathrm{rank}(A_2)=1$: $A_2$ is the single non-zero column $\partial \alpha =a+b+c$;
• $b_0=\mathrm{rank}(A_1)=3$: every column has two 1s, so the 4 rows sum to $0$ and the rank is $\le 3$; the edges $b,c,d$ form a spanning tree, so they are independent;
• $b_{-1}=1$ (every vertex has boundary $\varnothing$) and $b_2=0$ (no 3-simplices).

Then ${\beta }_2=1-0-1=0$, ${\beta }_1=4-1-3=0$, ${\beta }_0=4-3-1=0$: no holes of any dimension. Indeed $|K|$ is a filled triangle with a tail, which can be shrunk to a point.

Worked example: pyramid with open base

$K$ is the surface of a square-based pyramid without the base, plus the diagonal $AC$ of the base: 4 triangles + 1 extra segment. What is $H_p(K)$?

Labels (deduced from the boundary matrices of the handwritten notes; dashed edges are hidden behind the surface):

• vertices: $A,B,C,D$ (the base square) and the apex $E$;
• edges: $a=AB$, $b=BC$, $c=CD$, $d=DA$ (base square), $e=AE$, $f=BE$, $g=CE$, $h=DE$ (lateral edges), $i=AC$ (diagonal of the open base);
• triangles: $\alpha =ADE$, $\beta =ABE$, $\gamma =BCE$, $\delta =CDE$.

Counting: $n_{-1}=1$, $n_0=5$, $n_1=9$, $n_2=4$, $n_p=0$ for $p\ge 3$ (19 simplices in total).

$b_1=\mathrm{rank}(A_2)$, where $A_2$ is the matrix of ${\partial }_2:C_2\to C_1$ (rows: edges, columns: triangles):

$$A_2=\begin{array}{ccccc}& \alpha & \beta & \gamma & \delta \\ a& 0& 1& 0& 0\\ b& 0& 0& 1& 0\\ c& 0& 0& 0& 1\\ d& 1& 0& 0& 0\\ e& 1& 1& 0& 0\\ f& 0& 1& 1& 0\\ g& 0& 0& 1& 1\\ h& 1& 0& 0& 1\\ i& 0& 0& 0& 0\end{array}$$

Each column has a “private” row: $d$ only in $\alpha$, $a$ only in $\beta$, $b$ only in $\gamma$, $c$ only in $\delta$. So the columns are independent and $\mathrm{rank}(A_2)=4$: $b_1=4$. In the notes the same result is obtained by moving one column and reading off 4 pivots.

$b_0=\mathrm{rank}(A_1)$, the matrix of ${\partial }_1:C_1\to C_0$ (rows: vertices, columns: edges):

$$A_1=\begin{array}{cccccccccc}& a& b& c& d& e& f& g& h& i\\ A& 1& 0& 0& 1& 1& 0& 0& 0& 1\\ B& 1& 1& 0& 0& 0& 1& 0& 0& 0\\ C& 0& 1& 1& 0& 0& 0& 1& 0& 1\\ D& 0& 0& 1& 1& 0& 0& 0& 1& 0\\ E& 0& 0& 0& 0& 1& 1& 1& 1& 0\end{array}$$

Every column contains exactly two 1s (the endpoints), so the five rows sum to $0$ and the rank is $\le 4$. The columns $e,f,g,h$ are independent (each one has a private row among $A,B,C,D$). So $b_0=4$. In general, for a graph, $\mathrm{rank}(A_1)=\#\text{vertices}-\#\text{connected components}$.

The other ranks. $b_{-1}=\mathrm{rank}(A_0)=1$: in reduced homology $A_0=(11111)$, since ${\partial }_{0}P=\varnothing$ for every vertex. $b_2=\mathrm{rank}(A_3)=0$: there are no 3-simplices, so $\mathrm{Im}{\partial }_3=0$. Also $b_{-2}=b_3=0$.

Betti numbers with ${\beta }_p=n_p-b_p-b_{p-1}$:

$$\begin{array}{rlrl}{\beta }_2& =n_2-b_2-b_1=4-0-4=0& & \text{no cavity}\\ {\beta }_1& =n_1-b_1-b_0=9-4-4=1& & H_1(K)\cong {\mathbb{Z}}_2:\text{one 1-dimensional hole}\\ {\beta }_0& =n_0-b_0-b_{-1}=5-4-1=0& & \text{connected (reduced homology)}\\ {\beta }_{-1}& =n_{-1}-b_{-1}-b_{-2}=1-1-0=0& & \\ {\beta }_3& =n_3-b_3-b_2=0-0-0=0& & \end{array}$$

Interpretation. The hole is the open base. The base contains two triangular loops, $a+b+i$ (around $ABC$) and $c+d+i$ (around $ACD$). Both are cycles and neither is a boundary, but

$$(a+b+i)+(c+d+i)=a+b+c+d=\partial (\alpha +\beta +\gamma +\delta ),$$

since $\partial (\alpha +\beta +\gamma +\delta )=(d+e+h)+(a+e+f)+(b+f+g)+(c+g+h)=a+b+c+d$. The two loops are homologous (they differ by the boundary of the lateral surface), so they represent the same hole: one hole, not two. The result is drawn in Lecture 4.

Recap

• $C_p(K)$: ${\mathbb{Z}}_2$-combinations of $p$-simplices, $\dim C_p=n_p$.
• ${\partial }_p⟨u_0,\dots ,u_p⟩={\sum }_i⟨u_0,\dots ,{\hat{u}}_i,\dots ,u_p⟩$ and ${\partial }^2=0$ (Prop. 13).
• $Z_p=\ker {\partial }_p$ (cycles), $B_p=\mathrm{Im}{\partial }_{p+1}$ (boundaries), $H_p(K)=Z_p/B_p$. A hole is a cycle that is not a boundary, and homologous cycles are the same hole.
• ${\beta }_p=\dim H_p=n_p-b_p-b_{p-1}$ (Prop. 14), with $b_p={\mathrm{rank}}_{{\mathbb{Z}}_2}(A_{p+1})$.

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Handwritten source: tdafull.pdf, page 3