Chapter 03 - Complexes, Homotopy and Betti Numbers

Delta, Nerve, Čech complexes

Theory - Delta complexes

Delta complexes

The $\Delta$-complex structure is a generalization of simplicial complex structure. Instead of gluing simplices from the same ${\mathbb{R}}^{n+1}$ by causing them to intersect at faces, one glues simplices from disjoint copies of ${\mathbb{R}}^{n+1}$ using a formal quotient topology construction.

This technique allows many common spaces to be represented using fewer simplices. For example, a circle may be given as a $\Delta$-complex by identifying the two vertices of the standard $1$-simplex. The torus may be given using just two $2$-simplices by identifying each of the pairs of sides.

Why not simplicial?

These spaces cannot be formed as simplicial complexes by gluing the given simplices.

They are not literally simplicial complexes because the circle is round and the torus is not a subset of ${\mathbb{R}}^2$.

They are not homeomorphic to simplicial complexes (with the indicated decompositions) because the circle has a 1-loop while the torus has simplices which intersect in more than one face. (Both are impossible features for the homeomorphism type of simplicial complexes with the given decompositions.)

$\Delta$-Complex - Combinatorial

A combinatorial $\Delta$-complex is a sequence of simplex sets $\{S_n{\}}_{n\in \mathbb{N}}$ together with attaching maps $d_i^n:S_{n+1}\to S_n$, for $i=0,1,\dots ,n+1$, one for each facet of a simplex in $S_{n+1}$.

Attaching maps join the faces of each simplex to simplices of lower dimension. This works geometrically when the attaching maps preserve adjacency of faces of faces. Preserving adjacency is equivalent to this combinatorial condition:

$$d_i^n\circ d_j^{n+1}=d_{j-1}^n\circ d_i^{n+1}\text{when}i<j$$

The adjacency condition may be illustrated with a 2-simplex:

Faces are numbered by removing a vertex in the order of vertices. So $[\widehat{v_0},v_1,v_2]$ is edge 0 of the 2-simplex $[v_0,v_1,v_2]$, and $[v_0,\widehat{v_1},v_2]$ is edge 1, and so on. Thus $v_1$ is face 0 of the 1-simplex $[v_0,v_1]$, and $v_0$ is face 1, and so on for the other edges.

Now interpret the adjacency condition:
Take as input the 2-simplex.

• With $i=0,j=1$
  • The condition says “vertex 0 of edge 1 equals vertex $1-1$ of edge 0.”
  • These vertices are the point $v_2$.
• With $i=0,j=2$
  • The condition says “vertex 0 of edge 2 equals vertex $2-1$ of edge 0.”
  • These vertices are the point $v_1$.
• With $i=1,j=2$
  • The condition says “vertex 1 of edge 2 equals vertex $2-1$ of edge 1.”
  • These vertices are the point $v_0$.

$\Delta$-Complex - Topological

A $\Delta$-complex $\mathcal{K}$ is the topological space formed from a combinatorial $\Delta$-complex $\mathcal{S}=\{S_n{\}}_{n\in \mathbb{N}}$ as the disjoint union of simplices enumerated by elements of $\mathcal{S}$, glued together according to the combinatorial attaching maps with canonically defined linear functions interpolating between glued vertices.

$$\mathcal{K}=\left(\underset{n\in \mathbb{N}}{⨆}{S}_n\times {\sigma }_n\right)/\sim$$

where $(\sigma ,{\iota }^{i}x)\sim (d_i\sigma ,x)$, with ${\iota }^i:{\sigma }_{n-1}\hookrightarrow {\sigma }_n$ is the inclusion of the $i^{\text{th}}$ face of the standard simplex ${\sigma }_n$, and $x\in {\sigma }_{n-1}$ is any point in that face.

Simplicial vs. $\Delta$ complexes

How best to think about simplicial vs. $\Delta$- complexes?

• Simplicial complex:
  • Collection of geometric simplices in the same background space ${\mathbb{R}}^{n+1}$.
  • A sub-simplex brought to touch another sub-simplex only in one congruent face, where they are glued.
• $\Delta$-complex:
  • Collection of abstract geometric simplices in distinct background spaces (disjoint union).
  • Simplices in $S_{n+1}$ are attached to the lower-dimensional structure by gluing in the faces of new $(n+1)$-simplices, preserving only the adjacency relation.

Underlying space

Given a complex $\mathcal{K}$ (simplicial complex or $\Delta$-complex), the underlying topological space of $\mathcal{K}$ is denoted $|\mathcal{K}|$.

$\Delta$-homology

Homology may be defined for $\Delta$-complexes using the same technique as for simplicial complexes.

When a simplicial complex is also a $\Delta$-complex, their homology groups are canonically isomorphic.

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Theory - Nerve complex

Nerve complex

Nerve complex

Suppose $\mathcal{U}=\{U_i{\}}_{i\in I}$ is a collection of sets indexed by $I$. The nerve complex $\mathcal{N}(\mathcal{U})$ is the collection of finite subsets $J\subset I$ such that $\bigcap _{j\in J}{U}_j$ is nonempty.

That is, the nerve is the complex of indices of finite collections of mutually overlapping subsets from $\mathcal{U}$.

Notes:

• The definition does not assume that the $U_i$ are subsets of a topological space. If they are, we do not assume that they are open.
• The nerve complex is a combinatorial complex: the elements are sets of indices, not the subsets themselves.
• Suppose $J\in \mathcal{N}(\mathcal{U})$ is a subset in the nerve.
  • Then all subsets of $J$ are also in the nerve.
  • I.e. the nerve is closed under taking subsets.
• Therefore: the nerve is a combinatorial simplicial complex.

As a simplicial complex:

• vertices of the nerve correspond to individual sets $U_i\in \mathcal{U}$
• $k$-simplices in the nerve correspond to nonempty subsets having $k$ elements

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Theorem: Geometric realization

Every combinatorial simplicial complex with $N$ vertices can be realized as a geometric simplicial complex in ${\mathbb{R}}^N$.

In fact, every combinatorial simplicial complex with maximum simplex dimension equal to $d$ can be realized as a geometric simplicial complex in ${\mathbb{R}}^{2d+1}$.

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Theory - Čech complex

Čech complex

Čech complex

Suppose $(X,d)$ is a metric space. Suppose $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ is a finite set of points in $X$.

Fix $r>0$. Define $U_i$ as the open balls:

$$U_i={\mathbf{B}}_r(p_i),r=1,2,\dots ,n$$

The nerve of $\mathcal{U}=\{U_1,U_2,\dots ,U_n\}$ is called the Čech complex of $\mathcal{P}$.

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Illustration

Illustration - Čech complexes of four points

Let $\mathcal{P}$ be four points in ${\mathbb{R}}^2$ as in the figure.

The Čech complex created using $r=2$ is 1-dimensional – three of the points are linked.

The Čech complex created using $r=5$ is 2-dimensional – the three linked points now span a 2-simplex. The fourth point is also connected.

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Homotopy type

Theory - Homotopy type and homology

Homotopy type and homology

Homology can be used to distinguish topological spaces or to measure the topological complexity of a space. From the perspective of computational complexity theory, homology is very easy to compute.

Theorem: Homology depends functionally on $|\mathcal{K}|$

The isomorphism types of the homology groups $H_i(\mathcal{K};\mathbb{Z})$ of a simplicial complex $\mathcal{K}$ depend only on $|\mathcal{K}|$, the underlying topological space.

More precisely, a homeomorphism of topological spaces induces an isomorphism of homology groups.

This theorem implies that any two homeomorphic spaces have isomorphic homology groups.

• %& For a topological space $X$, we write $H_i(X;\mathbb{Z})$ for the homology groups of $X$ computed using any simplicial complex structure provided on $X$.
  • If $X$ does not have a simplicial complex structure, or $\Delta$-complex structure, one can still define singular homology for $X$, and this homology still depends functionally only on the homeomorphism type.
  • Singular homology is isomorphic to simplicial and $\Delta$-homology when the latter can be defined.

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There is an equivalence relation between spaces that is much ‘coarser’ than homeomorphism, namely homotopy equivalence.

Homotopy equivalence

A homotopy equivalence between topological spaces $X$ and $Y$ is a pair $(f,g)$ of continuous maps:

such that $f\circ g$ is homotopic to ${\mathrm{Id}}_X$ and $g\circ f$ is homotopic to ${\mathrm{Id}}_Y$.

We write $X\simeq Y$ when $X$ and $Y$ are homotopy equivalent.

A homotopy from $f\circ g$ to ${\mathrm{Id}}_X$ would be a continuous map $F:X\times [0,1]\to X$ such that $F(x,0)=(f\circ g)(x)$ and $F(x,1)={\mathrm{Id}}_X(x)=x$. Analogously for a homotopy $g\circ f$ to ${\mathrm{Id}}_Y$.

A homeomorphism is actually the special case of a homotopy equivalence where the homotopies $f\circ g\stackrel{\sim }{⟶}{\mathrm{Id}}_X$ and $g\circ f\stackrel{\sim }{⟶}{\mathrm{Id}}_Y$ are both the identity maps.

Theorem: Homology depends functionally on homotopy type

The isomorphism types of the homology groups $H_i(\mathcal{K};\mathbb{Z})$ of a topological space $|\mathcal{K}|$ depend only on the homotopy type of the space.

More precisely, a homotopy equivalence of topological spaces induces an isomorphism of homology groups.

We will see a partial justification of this fact later, after discussing simplicial maps and chain maps.

Contractible spaces

A topological space is contractible if it is homotopy equivalent to a point.

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Illustration

Contractibility of Rn

Contractibility of ${\mathbb{R}}^n$

We show that ${\mathbb{R}}^n$ is contractible.

Let $p=0\in {\mathbb{R}}^n$ be the origin. Consider the maps $f$ and $g$:

Then $f\circ g:{\mathbb{R}}^n\to p$ is the constant map sending everything in ${\mathbb{R}}^n$ to the origin.

Define $F(x,t)=tx$. Then $F(x,0)\equiv 0$, which agrees with $f\circ g$, while $F(x,1)=x={\mathrm{Id}}_{{\mathbb{R}}^n}$. Therefore $f\circ g$ is homotopic to ${\mathrm{Id}}_{{\mathbb{R}}^n}$.

Now consider that $g\circ f={\mathrm{Id}}_p$ already. No additional homotopy is needed.

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Illustration - convex sets and simplices are contractible

The same homotopy formula, namely $F(x,t)=tx$, can be used to show that any ball in ${\mathbb{R}}^n$ is contractible, and indeed any convex set in ${\mathbb{R}}^n$ is contractible.

• ! Therefore, any simplex is contractible.

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Cylinder is homotopic to circle

Cylinder is homotopic to circle

A cylinder is given by $X={\mathbb{S}}^1\times [-1,1]$. Consider the ‘collapse’ map $f:X\to {\mathbb{S}}^1$ given by projecting to the circle along straight lines, and the map $g:{\mathbb{S}}^1\hookrightarrow X$ given by inclusion:

The map $f\circ g$ is equal to the collapse map, and the map $g\circ f$ is the identity on ${\mathbb{S}}^1$.

Now construct a homotopy from the collapse map to ${\mathrm{Id}}_X$:

$$\begin{array}{rl}F:X\times [0,1]& ⟶X\\ & \\ F:({\mathbb{S}}^1\times [-1,1])\times [0,1]& ⟶{\mathbb{S}}^1\times [-1,1]\\ & \\ ((x,h),t)& ⟼(x,th)\end{array}$$

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Rank of homology

Theory - Betti numbers, Euler Characteristic

Betti numbers, Euler Characteristic

Rank of an abelian group

Every finitely generated abelian group $G$ is isomorphic to a group with this form:

$${\mathbb{Z}}^r⨁({\oplus }_i\mathbb{Z}/k_i\mathbb{Z})$$

• The ${\mathbb{Z}}^r$ summands give a free abelian part and contain all infinite-order generators.
• The $\mathbb{Z}/k_i\mathbb{Z}$ summands give torsion and contain all finite-order generators.

The rank of $G$ is the number $r$.

Betti numbers

The Betti numbers ${\beta }_i$ of a topological space $X$ are the ranks of the homology groups:

$${\beta }_i(X)=\mathrm{rank}(H_i(X;\mathbb{Z}))$$

These Betti numbers may be combined into a single number, the Euler characteristic:

Euler characteristic

The Euler characteristic $\chi (X)$ of a topological space $X$ is the alternating sum of Betti numbers:

$$\chi (X)=\sum _i(-1{)}^i{\beta }_i(X)$$

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Illustration

Illustration - Euler characteristic of polyhedra and simplicial complexes

For convex polyhedra, Euler’s formula for the Euler characteristic is simply vertices minus edges plus faces:

$$\chi =V-E+F$$

For nonconvex polyhedra, this formula can give values other than the actual Euler characteristic defined using Betti numbers.

Name	Figure	Vertices	Edges	Faces	$\chi =V-E+F$
Tetrahedron		4	6	4	2
Cube		8	12	6	2
Octahedron		6	12	8	2
Dodecahedron		20	30	12	2
Icosahedron		12	30	20	2
Tetrahemihexahedron		6	12	7	1
Cubohemioctahedron		12	24	10	-2
Small stellated dodecahedron		12	30	12	-6

For a simplicial complex, the Euler characteristic can always be computed using a formula analogous to the Euler formula: count the simplices of each dimension:

$$\chi =k_0-k_1+k_2-k_3+\cdots \pm k_n$$

Here $k_i$ is the number of $i$-simplices in the simplicial complex.

Theorem: Euler $=$ Euler

For a simplicial complex, the same Euler characteristic may be calculated using Betti numbers or using the counts of simplices.

We do not prove this here, although the proof is a straightforward homology calculation.

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