If \(q:X→Y\) is a quotient map and \(ℛ\) is given by the partition \(\{q^{-1}(y):y∈Y\}\) then \(X /ℛ≃y\)

Definition. \(f:X→Y\) is a quotient map if \(f\) is surjective and \(U⊂Y\) is open iff \(f^{-1}(U)\) is open.

Remark. If \(f:X→Y\) is continuous surjective open, then \(f\) is a quotient map.

Example. \(ℝ/ℤ\) is homeomorphic to \(S^1\)

Proof. \(f:ℝ→S^1, f (x) =e^{2πix}\)

\(x∼y⇔f (x) =f (y)⇔e^{2πi (x-y)} =1⇔x-y∈ℤ\)

So the equivalence relation of \(ℝ/ℤ\) is given by \(\{f^{-1}(x):x∈S^1\}\)

\(f\) is surjective, continuous.

Also \(f\) is open, since \(f ((x-ε, x+ε))\) is an open arc on \(S^1\).

So by Proposition, \(ℝ/ℤ≃S^1\). ◻

Example. Define ∼ on \(ℝ^n\) by \(x∼y⇔x-y∈ℤ^n\). Then \(ℝ^n /ℤ^n≃𝕋^n =\underbrace{S^1×⋯×S^1}_n\)

Proof. \(F (x_1,…x_n) =(e^{2πix_1},…e^{2πix_n})\)

\(F\) continuous, surjective, reduces ∼ and open, so \(F\) is a quotient map. ◻

Geometric Topology

Objective. Study “nice” spaces that appear in geometry up to homeomorphism.

Simplicial complexes

Definition. The standard $n$-simplex is the set

\[Δ^n =\left\{(x_1,…x_{n+1})∈ℝ^{n+1}:x_i≥0∀i \text{ and } \sum_{i =1}^{n+1} x_i =1 \right\}\]

The non-negative integer \(n\) is the dimension of the simplex.

Its vertices denoted by \(V (Δ^n)\) are points \((x_1,…x_{n+1})\) where some \(x_i =1\), all the others are 0.

\(∀A \subseteq \{1,…n+1\}\) there is a corresponding face of \(Δ^n\) which is

\[\{(x_1,…x_{n+1})∈Δ^n:x_i =0 \text{ if } i \not\in A\}\]

Eg. \(Δ^1\) has 3 faces.

The inside of a simplex is

\[\operatorname{Inside} (Δ^n) =\left\{(x_1,…x_{n+1})∈ℝ^{n+1}:x_i > 0∀i \text{ and } \sum_{i =1}^{n+1} x_i =1 \right\}\]

Remark. The vertices \(\{v_1,…v_{n+1}\}\) of the standard simplex \(Δ^n⊂ℝ^{n+1}\) are the basis elements of the standard basis of \(ℝ^{n+1}\). If \(\{e_1,…e_{k+1}\}\) is standard basis of \(ℝ^{k+1} (k<n)\) and we define \(T (e_i) =V_{n_i}\) where \(V_{n_1},…V_{n_{k+1}}\) are all distinct, then \(T\) extends uniquely to a 1-1 linear map \(T:ℝ^{k+1}→ℝ^{n+1}\) and \(T (Δ^k)\) is a face of \(Δ^n\).

Definition. A face inclusion of a standard \(m\)-simplex \(Δ^m\) into a standard \(n\)-simplex \(Δ^n (m<n)\) is a function \(Δ^m→Δ^n\) that is a restriction of an injective linear map that sends the vertices of \(Δ^m\) to vertices of \(Δ^n\).

Note. Any 1-1 map \(V (Δ^m)→V (Δ^n)\) extends to a unique face inclusion.

Eg. For \(Δ^1→Δ^2\), \(V (Δ^1) =\{1, 2\}→V (Δ^2) =\{1, 2, 3\}\) have 6 possible maps:unitsize(1cm);pair A=dir(90),B=dir(-150),C=dir(-30),A1=A+W,B1=B+W;dot(A);dot(B);dot(C);dot(A1);dot(B1);draw(A--B--C--cycle^^A1--B1);draw((A1+B1)/2+.2E--(A1+B1)/2+.8E,arrow=Arrow(TeXHead),red);

Definition. An abstract simplicial complex is a pair \((V, Σ)\) where \(V\) is a set called vertices and \(Σ\) is a set of non-empty subset of \(V\) called set of simplices such that

  1. \(∀v∈V, \{v\}∈Σ\)

  2. If \(σ∈Σ\), then any non-empty \(τ⊂σ\) lies in \(Σ\) as well.

We say \((V, Σ)\) is finite if \(V\) is finite.

Definition. The topological realization \(|K|\) of an abstract simplicial complex \(K =(V, Σ)\) is the space obtained by the following procedure:

  1. \(∀σ∈Σ\), if \(|σ|=n+1\), take a copy of \(Δ^n\).

    Denote this by \(Δ_σ\) and label its vertices with elements of \(σ\).

  2. whenever \(σ⊂τ∈Σ\) identify \(Δ_σ\) with a subset of \(Δ_{τ}\) via the face inclusion that sends the elements of \(σ\) to the corresponding elements of \(τ\).

In other words, \(|K|=\bigsqcup_{σ∈Σ} Δ_σ /ℛ\) where \(ℛ\) is described in 2.

Whenever we refer to a simplicial complex, we will mean either an abstract simplicial complex or its topological realisation.

Example. \(V =\{1, 2, 3\}, Σ =\{\{1\}, \{2\}, \{3\}, \{1, 2\}, \{2, 3\}\}\) 

Example. \(V =\{1, 2, 3, 4\}, Σ =\{\{1\}, \{2\}, \{3\}, \{4\}, \{1, 2, 3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{3, 4\}\}\)

Example. is Not a simplicial complex. \(\{1, 2\}\) gives one simplex

Example. is Not a simplicial complex. \(\{1, 2\}\) gives a unique simplex.

Definition. A triangularization of a space \(X\) is a simplicial complex \(K\) together with a homeomorphism \(f:|K|→X\). (Note: not unique!)

Example. is \(|K|\)

\(∃ϕ:|K|→S^2\) homeomorphism

Example. Triangularization of the torus \(𝕋^2\)

At first sight, this triangulation of the torus may seem be needlessly complicated. Might we have been able to use fewer simplices?

andare not allowed because \(Δ^2 =\{1, 2\}\) must determine a single edge

Definition. A simplicial circle is a simplicial complex \(K\) with vertices \(\{v_1,…v_n\}\) and 1-simplices \(\{v_1, v_2\}, \{v_2, v_3\},…\{v_n, v_1\}\)

Eg.