Notes of Topology

Lesson 1: Topological Spaces

Topological Spaces

Definition of topological spaces

A topological space is a pair $(X,\mathcal{T})$ where $X$ is a set and $\mathcal{T}$ is a collection of subsets of $X$ such that:

• $\mathrm{\varnothing }\in \mathcal{T}$ and $X\in \mathcal{T}$,
• for every infinite collection $\{O_{\alpha }{\}}_{\alpha \in A}\subset \mathcal{T}$, we have $\bigcup _{\alpha \in A}{O}_{\alpha }\in \mathcal{T}$,
• for every finite collection $\{O_i{\}}_{1\le i\le n}\subset \mathcal{T}$, we have $\bigcap _{1\le i\le n}{O}_i\in \mathcal{T}$.

The set $\mathcal{T}$ is called a topology on $X$. The elements of $\mathcal{T}$ are called the open sets.

Definition via closed sets

Let $(X,\mathcal{T})$ be a topological space. For every open set $O\in \mathcal{T}$, its complement ${}^{c}O=\{x\in X,x\notin O\}$ is called a closed set.

In other words, a set $A\subset X$ is closed iff ${}^{c}A$ is open.

Topology of $R^n$

Open balls of $R^n$

Let $x\in R^n$ and $r>0$. The open ball of center $x$ and radius $r$, denoted $\mathcal{B}(x,r)$, is defined as:

$$\mathcal{B}(x,r)=\{y\in R^n,\parallel x-y\parallel <r\}$$

Euclidean topology

Let $A\subset R^n$ be a subset. Let $x\in A$.

We say that $A$ is open around $x$ if there exists $ϵ>0$ such that $\mathcal{B}(x,ϵ)\subset A$.

We say that $A$ is open if for every $x\in A$, $A$ is open around $x$.

We denote the set of such open set by ${\mathcal{T}}_{R^n}$, the Euclidean topology on $R^n$.

Topology of Subsets of $R^n$

Subspace topology

Let $(X,\mathcal{T})$ be a topological space, and $Y\subset X$ a subset. We define the subspace topology on $Y$ as the following set:

$${\mathcal{T}}_{|Y}=\{O\cap Y,O\in \mathcal{T}\}$$

Continuous Maps

Continuous maps

$(X,\mathcal{T})$ $(Y,\mathcal{U})$

Let $f:X\to Y$ be a map. We say that $f$ is continuous if for every $O\in \mathcal{U}$, the preimage $f^{-1}(O)=\{x\in X,f(x)\in O\}$ is in $\mathcal{T}$.

Lesson 2: Homeomorphisms

Homeomorphic Topological Spaces

Definition of homeomorphism

Let $(X,\mathcal{T})$ and $(Y,\mathcal{U})$ be two topological spaces, and $f:X\to Y$ a map.

We say that $f$ is a homeomorphism if

• $f:X\to Y$ is continuous,
• $f$ is a bijection,
• $f^{-1}:Y\to X$ is continuous.

If there exist such a homeomorphism, we say that the two topological spaces are homeomorphic.

Homeomorphism equivalence relation

Let us write $X\simeq Y$ if the two topological spaces $X$ and $Y$ are homeomorphic, i.e., if there exists a homeomorphism $f:X\to Y$.

For any $X$, we have $X\simeq X$.

Moreover, we have $X\simeq Y⟺Y\simeq X$.

We also have a third property $X\simeq Y$ and $Y\simeq X⟹X\simeq Z$.

Connected Components

Connectedness

Let $(X,\mathcal{T})$ be a topological space. We say that $X$ is connected if for every open sets $O$, $O^{\prime }\in \mathcal{T}$ such that $O\cap O^{\prime }=\mathrm{\varnothing }$, we have $X=O\cup O^{\prime }⟹O=\mathrm{\varnothing }$ or $O^{\prime }=\mathrm{\varnothing }$.

In other words, a connected topological space cannot be divided into two non-empty disjoint open sets.

Connectedness as an invariant

Invariant property

Two homeomorphic topological spaces admit the same number of connected components.

Dimension

Invariance of domain

If $m\ne n$, the Euclidean spaces $R^m$ and $R^n$ are not homeomorphic.

Dimension

Let $(X,\mathcal{T})$ be a topological space, and $n\ge 0$. We say that it has dimension $n$ if the following is true: for every $x\in X$, there exists an open set $O$ such that $x\in O$, and a homeomorphism $O\to R^n$.

Dimension Invariant

Let $X$, $Y$ be two homeomorphic topological spaces. If $X$ has dimension $n$, then $Y$ also has dimension $n$.

Lesson 3: Homotopies

Homotopy Equivalence between Maps

Definition

Let $(X,\mathcal{T})$ and $(Y,\mathcal{U})$ be two topological spaces, and $f,g:X\to Y$ two continuous maps. A homotopy between $f$ and $g$ is a map $F:X\times [0,1]\to Y$ such that:

• $F(\cdot ,0)$ is equal to $f$,
• $F(\cdot ,1)$ is equal to $g$,
• $F:X\times [0,1]\to Y$ is continuous.

If such a homotopy exists, we say that the maps $f$ and $g$ are homotopic.

For any $t\in [0,1]$, the notation $F(\cdot ,t)$ refers to the map

$$\begin{array}{rl}F(\cdot ,t):X& ⟶Y\\ x& ⟼F(x,t)\end{array}$$

Trivial maps

From a homotopic point a view, a trivial map is a map that is homotopic to a constant map.

Homotopy Equivalence between Topological Spaces

Definition of homotopy equivalence

Let $(X,\mathcal{T})$ and $(Y,\mathcal{U})$ be two topological spaces. A homotopy equivalence between $X$ and $Y$ is a pair of continuous maps $f:X\to Y$ and $g:Y\to X$ such that:

• $g\circ f:X\to X$ is homotopic to the identity map id: $X\to X$,
• $f\circ g:Y\to Y$ is homotopic to the identity map id: $Y\to Y$.

If such a homotopy equivalence exists, we say that $X$ and $Y$ are homotopy equivalent.

Deformation retractions

Let $(X,\mathcal{T})$ be a topological space and $Y\subset X$ a subset, endowed with the subspace topology ${\mathcal{T}}_{|Y}$.

A retraction is a continuous map $r:X\to X$ such that $\mathrm{\forall }x\in X$, $r(x)\in Y$ and $\mathrm{\forall }y\in Y$, $r(y)=y$.

A deformation retraction is a homotopy $F:X\times [0,1]\to Y$ between the identity map id: $X\to X$ and a retraction $r:X\to X$.

Homotopy Equivalence Relation

Link with Homeomorphic Spaces

Homeomorphic implies homotopic

Let $X$, $Y$ be two topological spaces. If they are homeomorphic, then they are homotopic equivalent.

Invariants

Number of connected components

Two homotopy equivalent topological spaces admit the same number of connected components.

Lesson 4: Simplicial Complexes

Combinatorial Simplicial Complexes

Standard simplices

单纯形 - 维基百科，自由的百科全书

The standard simplex of dimension $n$ is the following subset of $R^{n+1}$

$${\mathrm{\Delta }}_n=\{x=(x_1,\dots ,x_{n+1})\in R^{n+1},x_1,\dots ,x_{n+1}\ge 0and{x}_1+\cdots +x_{n+1}=1\}$$

凸包 - 维基百科，自由的百科全书

For any collection of points $a_1,\dots ,a_k\in R^n$, their convex hull is defined as:

$$conv(\{a_1\dots a_k\})=\{\sum _{1\le i\le k}{t}_i{a}_i,t_1+\cdots +t_k=1,t_1,\dots ,t_k\ge 0\}$$

We can say that ${\mathrm{\Delta }}_n$ is the convex hull of the vectors $e_1,\dots ,e_{n+1}$ of ${\text{R}}^{n+1}$, where $e_i=(0,\dots ,1,0,\dots ,0)$

Simplicial complexes

单纯复形 - 维基百科，自由的百科全书

Let $V$ be a set (called the set of vertices). A simplicial complex over $V$ is a set $K$ of subsets of $V$ (called the simplices) such that, for every $\sigma \in K$ and every non-empty $\tau \subset \sigma$, we have $\tau \in K$.

Topology

Topological realization

Let $K$ be a simplicial complex, with vertex $V=\{1,\dots ,n\}$.

In $R^n$, consider, for every $i\in [1,n]$, the vector $e_i=(0,\dots ,1,0,\dots ,0)$.

Let $|K|$ be the subset of $R^n$ defined as:

$$|K|=\bigcup _{\sigma \in K}conv(\{e_j,j\in \sigma \})$$

where conv represent the convex hull of points.

Endowed with the subspace topology, $(|K|,{\mathcal{T}}_{|K|})$ is a topological space, that we call the topological realization of $K$.

Triangulations

Let $(X,\mathcal{T})$ be a topological space. A triangulation of $X$ is a simplicial complex $K$ such that its topological realization $|K|$ is homeomorphic to $X$.

Euler Characteristic

Euler characteristic

Let $K$ be a simplicial complex of dimension $n$. Its Euler characteristic is the integer:

$$\chi (K)=\sum _{0\le i\le n}(-1{)}^i\cdot (numberofsimplicesofdimensioni)$$

Let $X$ be a topological space. Its Euler characteristic is defined as the Euler characteristic of any triangulation of it.

Euler characteristic is an invariant

If $X$ and $Y$ are two homotopy equivalent topological spaces, then $\chi (X)=\chi (Y)$.

Lesson 5: Homological Algebra

Reminder of Algebra

Groups

We recall that a group $(G,+)$ is a set $G$ endowed with an operation

$$\begin{array}{rl}G\times G& ⟶G\\ (g,h)& ⟼g+h\end{array}$$

such that:

• (associativity) $\mathrm{\forall }a,b,c\in G$, $(a+b)+c=a+(b+c)$,
• (identity) $\mathrm{\exists }0\in G$, $\mathrm{\forall }a\in G$, $a+0=0+a=a$,
• (inverse) $\mathrm{\forall }a\in G$, $\mathrm{\exists }b\in G$, $a+b=b+a=0$.

Quotient group

商群 - 维基百科，自由的百科全书

A subgroup of $(G,+)$ is a subset $H\subset G$ such that

$$\mathrm{\forall }a,b\in H,a+b\in H$$

If $H$ is a subgroup of $G$, the operation $+:G\times G\to G$ restricts to an operation $+:H\times H\to H$, making $H$ a group on its own.

Suppose that $\ast \ast G$ is commutative**, and that $\ast \ast H$ is a subgroup of $G$**. We define the following equivalence relation on $G$: $\mathrm{\forall }a,b\in G$,

$$a\sim b⟺a-b\in H$$

Denoted by $G/H$ the quotient set of $G$ under this relation. For any $a\in G$, one shows that the equivalence class of $a$ is equal to $a+H=\{a+h,h\in H\}$

Let $a_0,a_1,\dots ,a_n$ be a choice of representants of equivalence classes of the relation $\sim$.

The quotient set can be written as $G/H=\{0+H,a_1+H,\dots ,a_n+H\}$.

One defines a group structure $\oplus$ on $G/H$ as follows: for any $i,j\in [0,n]$,

$$(a_i+H)\oplus (a_j+H)=(a_i+a_j)+H$$

The group $(G/H,\oplus )$ is called the quotient group.

The group $Z/2Z$

The subgroup $2Z\subset Z$ consists of all even numbers.

The relation $a\sim b⟺a-b\in 2Z$ admits two equivalence classes: $2Z=\{2n,n\in Z\}$ and $1+2Z=\{1+2n,n\in Z\}$

The quotient group can be seen as the group $Z/2Z=\{0,1\}$ with the operation

$$\begin{gathered} 0+0=0 \\ 0+1=1 \\ 1+0=1 \\ 1+1=0 \end{gathered}$$

For any $n\ge 1$, the product group $((Z/2Z{)}^n,+)$ is the group whose underlying set is

$$(Z/2Z{)}^n=\{({ϵ}_1,\dots ,{ϵ}_n),{ϵ}_1,\dots ,{ϵ}_n\in Z/2Z\}$$

and whose operation is defined as

$$({ϵ}_1,\dots ,{ϵ}_n)+({ϵ}_1^{\prime },\dots ,{ϵ}_n^{\prime })=({ϵ}_1+{ϵ}_1^{\prime },\dots ,{ϵ}_n+{ϵ}_n^{\prime })$$

Note that the set $(Z/2Z{)}^n$ has $2^n$ elements.

Vector spaces

Let $(\mathbb{F},+,\times )$ be a field. We recall that a vector space over $\mathbb{F}$ is a group $(V,+)$ endowed with an operation:

$$\begin{array}{rl}\mathbb{F}\times V& ⟶V\\ (\lambda ,v)& ⟼\lambda \cdot v\end{array}$$

such that

• (compatibility of multiplication) $\mathrm{\forall }\lambda ,\mu \in \mathbb{F}$, $\mathrm{\forall }v\in V$, $\lambda \cdot (\mu \cdot v)=(\lambda \times \mu )\cdot v$,
• (identity) $\mathrm{\forall }v\in V$, $1\cdot v=v$ where $1$ denotes the unit of $\mathbb{F}$,
• (scalar distributivity) $\mathrm{\forall }\lambda ,\upsilon \in \mathbb{F}$, $\mathrm{\forall }v\in V$, $(\lambda +\upsilon )\cdot v=\lambda \cdot v+\upsilon \cdot v$,
• (vector distributivity) $\mathrm{\forall }\mu \in \mathbb{F}$, $\mathrm{\forall }v,w\in V$, $\lambda \cdot (u+v)=\lambda \cdot v+\upsilon \cdot v$.

Let $\{v_1,\dots ,v_n\}$ be a collection of elements of $V$. We say that it is free if

$$\mathrm{\forall }{\lambda }_1,\dots ,{\lambda }_n\in \mathbb{F},\sum _{1\le i\le n}{\lambda }_i{v}_i=0⟹{\lambda }_1=\cdots ={\lambda }_n=0$$

We say that it is spans $V$ if

$$\mathrm{\forall }v\in V,\mathrm{\exists }{\lambda }_1,\dots ,{\lambda }_n\in \mathbb{F},\sum _{1\le i\le n}{\lambda }_i{v}_i=v$$

If the collection $\{v_1,\dots ,v_n\}$ is free and spans $V$, we say that it is a basis.

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A linear subspace of $(V,+,\cdot )$ is a subset $W\subset V$ such that

$$\mathrm{\forall }u,v\in W,u+v\in Wand\mathrm{\forall }v\in W,\mathrm{\forall }\lambda \in \mathbb{F},\lambda v\in W$$

Just as for groups, we can define an equivalence relation $\sim$ on $V$, and a quotient vector space $V/W$.

Isomorphism & Isomorphic

We have $\dim V/W=\dim V-\dim W$

Let $(V,+,\cdot )$ and $(W,+,\cdot )$ be two vector spaces. A linear map is a map $f:V\to W$ such that

$$\mathrm{\forall }u,v\in V,f(u+v)=f(u)+f(v)and\mathrm{\forall }v\in V,\mathrm{\forall }\lambda \in \mathbb{F},f(\lambda v)=\lambda \cdot f(v)$$

If $f$ is a bijection, it is called an isomorphism, and we say that $V$ and $W$ are isomorphic.

$Z/2Z$-vector spaces

Let $(V,+)$ be a commutative group.

It can be given a $Z/2Z$-vector space structure iff $\mathrm{\forall }v\in V$, $v+v=0$.

Chains, cycles, and boundaries

Skeleton

Let $K$ be a simplicial complex. For any $n\ge 0$, define the $n$-skeleton of $K$:

$$K_n=\{\sigma \in K,\dim (\sigma )\le n\}$$

Also, define

$$K_{(n)}=\{\sigma \in K,\dim (\sigma )=n\}$$

Chains

Let $n\ge 0$. The $n$-chains of $K$ is the set $C_n(K)$ whose elements are the formal sums

$$\sum _{\sigma \in K_{(n)}}{ϵ}_{\sigma }\cdot \sigma where\mathrm{\forall }\sigma \in K_{(n)},{ϵ}_{\sigma }\in Z/2Z$$

Boundary operator

Let $n\ge 1$, and $\sigma =[x_0,\dots ,x_n]\in K_{(n)}$ a simplex of dimension $n$. We define its boundary as the following element of $C_{n-1}(K)$:

$${\partial }_n\sigma =\sum _{\tau \subset \sigma ,|\tau |=|\sigma |-1}\tau$$

We can extend the operator ${\partial }_n$ as a linear map ${\partial }_n:C_n(K)\to C_{n-1}(K)$ as follows: for any element of $C_n(K)$,

$${\partial }_n\sum _{\sigma \in K(n)}{ϵ}_{\sigma }\cdot \sigma =\sum _{\sigma \in K(n)}{ϵ}_{\sigma }\cdot {\partial }_n\sigma$$

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For any $n\ge 1$, for any $c\in C_n(K)$, we have ${\partial }_{n-1}\circ {\partial }_n(c)=0$.

Cycles and boundaries

Let $n\ge 0$. We have a triplet of vector spaces

$$C_{n+1}(K)\to C_n(K)\to C_{n-1}(K)$$

We can consider their kernel and image.

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We define:

Kernel (set theory) - Wikipedia

• The $n$-cycles: $Z_n(K)=Ker({\partial }_n)$,
• The $n$-boundaries: $B_n(K)=Im({\partial }_{n+1})$.

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We say that two chains $c$, $c^{\prime }\in C_n(K)$ are homologous if there exists $b\in B_n(K)$ such that $c=c^{\prime }+b$. Two chains are homologous if they are equal up to a boundary.

Homology Groups

Homology groups

$n^{th}$ homology group of $K$:

$$H_n(K)=Z_n(K)/B_n(K)$$

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$\dim H_n(K)=\dim B_n(K)-\dim Z_n(K)$.

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Let $K$ be a simplicial complex and $n\ge 0$. Its $n^{th}$ Betti number is the integer ${\beta }_n(K)=\dim H_n(K)$.

Homology Groups of Topological Spaces

Invariant property

The homology groups of a topological space are the homology groups of any triangulation of it. We define its Betti numbers similarly.

If $X$ and $Y$ are two homotopy equivalent topological spaces, then for any $n\ge 0$ we have isomorphic homology groups $H_n(X)\simeq H_n(Y)$. As a consequence, ${\beta }_n(X)={\beta }_n(Y)$.

Incremental Algorithm

Incremental Algorithm

Ordering the simplicial complex

Positivity of simplices

Let $i\in [1,n]$, and $d=\dim ({\sigma }^i)$. Recall that $K^i=K^{i+1}\cup \{{\sigma }_i\}$.

The simplex ${\sigma }^i$ is positive if there exists a cycle $c\in Z_d(K^i)$ that contains ${\sigma }^i$.

In other words, there exists $c=\sum _{\sigma \in K_{(n)}^i}{ϵ}_{\sigma }\cdot \sigma \in C_n(K^i)$ such that ${ϵ}_{{\sigma }^i}=1$ and ${\partial }_n(c)=0$. Otherwise, ${\sigma }_i$ is negative.

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Input: an increasing sequence of simplicial complexes $K^1\subset \cdots \subset K^n=K$

Output: the Betti numbers ${\beta }_0(K),\dots ,{\beta }_d(K)$