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. ◻
Objective. Study “nice” spaces that appear in geometry up to homeomorphism.
Create a “list” of these nice spaces
The spaces will be built by gluing simplices
We'll give a “finite”, “combinatorial” description of these spaces
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:
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
\(∀v∈V, \{v\}∈Σ\)
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:
\(∀σ∈Σ\), if \(|σ|=n+1\), take a copy of \(Δ^n\).
Denote this by \(Δ_σ\) and label its vertices with elements of \(σ\).
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. \(V =\{1, 2, 3\}, Σ =\{\{1\}, \{2\}, \{3\}, \{1, 2\}, \{2, 3\}\}\) Error! Click to view log.
Example. \(V =\{1, 2, 3, 4\}, Σ =\{\{1\}, \{2\}, \{3\}, \{4\}, \{1, 2, 3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{3, 4\}\}\)Error! Click to view log.
Example. Error! Click to view log. is Not a simplicial complex. \(\{1, 2\}\) gives one simplex
Example. Error! Click to view log. 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. Error! Click to view log.is \(|K|\)
\(∃ϕ:|K|→S^2\) homeomorphismError! Click to view log.
Example. Triangularization of the torus \(𝕋^2\)Error! Click to view log.
At first sight, this triangulation of the torus may seem be needlessly complicated. Might we have been able to use fewer simplices?Error! Click to view log.andError! Click to view log.are 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 \(\{v_1, v_2\}, \{v_2, v_3\},…\{v_n, v_1\}\)