description/proof of that intersection of simplicial complexes is simplicial complex, and underlying space of intersection is contained in but not necessarily equal to intersection of underlying spaces of constituents
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
- 4: Note
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that the intersection of any 2 simplicial complexes is a simplicial complex, and the underlying space of the intersection is contained in but not necessarily equal to the intersection of the underlying spaces of the constituent simplicial complexes.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(V_1\): \(\in \{\text{ the real vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the real vectors spaces }\}\)
\(C_1\): \(\in \{\text{ the simplicial complexes on } V_1\}\)
\(C_2\): \(\in \{\text{ the simplicial complexes on } V_2\}\)
//
Statements:
\(V_1\) and \(V_2\) share the operations on \(V_1 \cap V_2\)
\(\implies\)
(
\(V_1 \cap V_2 \in \{\text{ the real vectors spaces }\}\) with the restriction of the shared operations
\(\land\)
\(C_1 \cap C_2 \in \{\text{ the simplicial complexes on } V_1 \cap V_2\}\)
\(\land\)
\(\vert C_1 \cap C_2 \vert \subseteq \vert C_1 \vert \cap \vert C_2 \vert\)
\(\land\)
Not necessarily \(\vert C_1 \cap C_2 \vert = \vert C_1 \vert \cap \vert C_2 \vert\)
)
//
2: Natural Language Description
For any real vectors spaces, \(V_1, V_2\), such that \(V_1\) and \(V_2\) share the operations on \(V_1 \cap V_2\), any simplicial complexes, \(C_1\), on \(V_1\) and \(C_2\), on \(V_2\), \(V_1 \cap V_2\) is a vectors space with the restriction of the shared operations, \(C_1 \cap C_2\) is a simplicial complex on \(V_1 \cap V_2\), and \(\vert C_1 \cap C_2 \vert \subseteq \vert C_1 \vert \cap \vert C_2 \vert\), but not necessarily \(\vert C_1 \cap C_2 \vert = \vert C_1 \vert \cap \vert C_2 \vert\).
3: Proof
Whole Strategy: Step 1: see that \(V_1 \cap V_2\) is a real vectors space with the restriction of the shared operations; Step 2: see that each \(S \in C_1 \cap C_2\) is an affine simplex on \(V_1 \cap V_2\); Step 3: see that each face of \(S\) is contained in \(C_1 \cap C_2\); Step 4: see that for each \(S_1, S_2 \in C_1 \cap C_2\), \(S_1 \cap S_2\) is a face of \(S_1\) and is a face of \(S_2\); Step 5: see that \(\vert C_1 \cap C_2 \vert \subseteq \vert C_1 \vert \cap \vert C_2 \vert\); Step 6: see an example that \(\vert C_1 \cap C_2 \vert \neq \vert C_1 \vert \cap \vert C_2 \vert\).
Step 1:
\(V_1 \cap V_2\) is a real vectors space with the restriction of the shared operations, by the proposition that for any 2 vectors spaces that share the operations on the intersection, the intersection is a vectors space with the restriction of the shared operations.
Step 2:
For each \(S \in C_1 \cap C_2\), \(S \in C_1\) and \(S \in C_2\). \(S \subseteq V_1\) and \(S \subseteq V_2\), which means that \(S \subseteq V_1 \cap V_2\). \(S\) is an affine simplex in \(V_1 \cap V_2\), because \(V_1 \cap V_2\) is a vectors subspace of \(V_1\), and the set of the vertexes of \(S\) is affine-independent on \(V_1 \cap V_2\).
Step 3:
Each face of \(S\) is contained in \(C_1\), because \(S\) is an element of \(C_1\), and likewise is contained in \(C_2\). So, each face of \(S\) is contained in \(C_1 \cap C_2\).
Step 4:
For each \(S_1, S_2 \in C_1 \cap C_2\), \(S_1 \in C_1\) and \(S_2 \in C_1\). So, \(S_1 \cap S_2\) is a face of \(S_1\). Likewise, \(S_1 \cap S_2\) is a face of \(S_2\).
So, \(C_1 \cap C_2\) is a simplicial complex on \(V_1 \cap V_2\).
Step 5:
Let us prove that \(\vert C_1 \cap C_2 \vert \subseteq \vert C_1 \vert \cap \vert C_2 \vert\).
For each point, \(p \in \vert C_1 \cap C_2 \vert\), there is an \(S \in C_1 \cap C_2\) such that \(p \in S\); as \(S \in C_1\), \(p \in \vert C_1 \vert\), and likewise, \(p \in \vert C_2 \vert\); so, \(p \in \vert C_1 \vert \cap \vert C_2 \vert\).
Step 6:
As an example of not \(\vert C_1 \cap C_2 \vert = \vert C_1 \vert \cap \vert C_2 \vert\), let \(V_1 = V_2 = \mathbb{R}^2\) and \(C_1\) consist of a 2-simplex and \(C_2\) consist of a 2-simplex that shares a vertex and only a part of an edge (which contains the vertex) of the former simplex. Then, \(C_1 \cap C_2\) consists of the 0-simplex as the shared vertex, and \(\vert C_1 \cap C_2 \vert\) consists of only the vertex, while \(\vert C_1 \vert \cap \vert C_2\vert \) consists of the shared part of the edge. The issue is that as the latter 2-simplex is not any element of \(C_1\), the intersection of the 2 2-simplexes does not need to be a face of the former 2-simplex, so, does not need to be contained in \(C_1\).
4: Note
The union, \(C_1 \cup C_2\), is not necessarily a simplicial complex, as is shown in another article.
