description/proof of that for \(2\) continuous maps with same domain and codomain and equivalence relations on domain and codomain, if each class is mapped into class and maps are homotopic relative to subset that contains multi-points classes, induced maps between quotient spaces is homotopic relative to quotient of subset
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of maps homotopic relative to subset of domain.
- The reader knows a definition of equivalence relation on set.
- The reader knows a definition of quotient topology on set with respect to map.
- The reader admits the proposition that for any continuous map from the product of any topological space and any locally compact Hausdorff topological space and any equivalence relations on the 1st space and the codomain, if each 1st space equivalence class is mapped into any codomain equivalence class, the induced map between the product of the quotient space and the 2nd space and the quotient space is continuous.
Target Context
- The reader will have a description and a proof of the proposition that for any \(2\) continuous maps with any same domain and codomain and any equivalence relations on the domain and the codomain, if each equivalence class is mapped into an equivalence class and the maps are homotopic relative to any subset that contains all the multi-points equivalence classes, the induced maps between the quotient spaces is homotopic relative to the quotient of the subset.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(\in T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(f'\): \(\in T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(\sim_1\): \(\in \{\text{ the equivalence relations on } T_1\}\)
\(\sim_2\): \(\in \{\text{ the equivalence relations on } T_2\}\)
\(T_1 / \sim_1\): \(= \text{ the quotient topological space }\)
\(T_2 / \sim_2\): \(= \text{ the quotient topological space }\)
\(S_1\): \(\subseteq T_1\), such that \(\forall t_1 \in T_1 \setminus S_1 ([t_1]_1 = \{t_1\})\)
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Statements:
(
\(\forall t_1, t'_1 \in T_1 \text{ such that } t_1 \sim_1 t'_1 (f (t_1) \sim_2 f (t'_1) \land f' (t_1) \sim_2 f' (t'_1))\)
\(\land\)
\(f \simeq f' rel S_1\)
)
\(\implies\)
\(\widetilde{f} \simeq \widetilde{f'} rel \{[s_1]_1 \vert s_1 \in S_1\} \subseteq T_1 / \sim_1\) where \(\widetilde{f}: T_1 / \sim_1 \to T_2 / \sim_2, [t_1]_1 \mapsto [f (t_1)]_2\) and \(\widetilde{f'}: T_1 / \sim_1 \to T_2 / \sim_2, [t_1]_1 \mapsto [f' (t_1)]_2\)
//
2: Note
\(\{[s_1]_1 \vert s_1 \in S_1\}\) is called "the quotient of the subset" in Title and Target Context, but it is not really so, because "the quotient of the subset" is \(S_1 / \sim_1\) but \(\{[s_1]_1 \vert s_1 \in S_1\}\) is a subset of \(T_1 / \sim_1\), but anyway, \(\{[s_1]_1 \vert s_1 \in S_1\}\) and \(S_1 / \sim_1\) are canonically 'sets - maps' isomorphic.
A typical case that this proposition applies is an adjunction space, \(T_1 = T_{1, 1} + T_{1, 2}\) and \(T_1 / \sim_1 = T_{1, 2} \cup_g T_{1, 1}\) with \(g: S \to T_{1, 2}\) and \(S_1 = S \cup T_{1, 2}\).
The reason why "\(rel S_1\)" is required is that otherwise, \(F\) would not necessarily induce \(\widetilde{F}\) in Proof.
3: Proof
Whole Strategy: Step 1: see that \(\widetilde{f}\) and \(\widetilde{f'}\) are well-defined; Step 2: take any homotopy between \(f\) and \(f'\) relative to \(S_1\), \(F\); Step 3: see that \(\widetilde{F}: (T_1 / \sim_1) \times I \to T_2 / \sim_2\) is induced from \(F\); Step 4: see that \(\widetilde{F}\) is a homotopy between \(\widetilde{f}\) and \(\widetilde{f'}\) relative to \(\{[s_1]_1 \vert s_1 \in S_1\}\).
Step 1:
Let us see that \(\widetilde{f}\) and \(\widetilde{f'}\) are well-defined.
Let \(t_1, t'_1 \in T_1\) be any such that \([t_1]_1 = [t'_1]_1\).
That means that \(t_1 \sim_1 t'_1\).
By the supposition, \(f (t_1) \sim_2 f (t'_1)\).
That means that \([f (t_1)]_2 = [f (t'_1)]_2\).
So, for each \([t_1]_1 \in T_1 / \sim_1\), \([f (t_1)]_2\) is uniquely determined independent of the choice of \(t_1\).
So, \(\widetilde{f}\) is well-defined.
\(\widetilde{f'}\) is well-defined, likewise.
Step 2:
As \(f \simeq f' rel S_1\), there is a homotopy, \(F: T_1 \times I \to T_2\), such that for each \(t_1 \in T_1\), \(F (t_1, 0) = f (t_1)\) and \(F (t_1, 1) = f' (t_1)\) and for each \(s_1 \in S_1\), for each \(r \in I\), \(F (s_1, r) = f (s_1) = f' (s_1)\).
Step 3:
Let us see that \(\widetilde{F}: (T_1 / \sim_1) \times I \to T_2 / \sim_2, ([t_1]_1, r) \mapsto [F (t_1, r)]_2\) is well-defined.
Let \(t_1, t'_1 \in T_1\) be any such that \([t_1]_1 = [t'_1]_1\).
When \(t_1 = t'_1\), \(F (t_1, r) = F (t'_1, r)\), so, \([F (t_1, r)]_2 = [F (t'_1, r)]_2\).
Let us suppose that \(t_1 \neq t'_1\).
\(t_1, t'_1 \in S_1\), because if \(t_1 \in T_1 \setminus S_1\), \([t_1]_1 = \{t_1\}\), by the supposition, so, \(t'_1 \in [t_1]\) would imply that \(t'_1 = t_1\), a contradiction, and likewise for \(t'_1 \in T_1 \setminus S_1\).
So, \(F (t_1, r) = f (t_1) = f' (t_1)\) and \(F (t'_1, r) = f (t'_1) = f' (t'_1)\).
But \([f (t_1)]_2 = [f (t'_1)]_2\), by the supposition.
So, \([F (t_1, r)]_2 = [f (t_1)]_2 = [f (t'_1)]_2 = [F (t'_1, r)]_2\).
So, for each \(([t_1]_1, r) \in (T_1 / \sim_1) \times I\), \([F (t_1, r)]_2\) is uniquely determined independent of the choice of \(t_1\).
So, \(\widetilde{F}\) is well-defined.
Step 4:
\(\widetilde{F}\) is continuous, by the proposition that for any continuous map from the product of any topological space and any locally compact Hausdorff topological space and any equivalence relations on the 1st space and the codomain, if each 1st space equivalence class is mapped into any codomain equivalence class, the induced map between the product of the quotient space and the 2nd space and the quotient space is continuous.
\(\widetilde{F} ([t_1]_1, 0) = [F (t_1, 0)]_2 = [f (t_1)]_2 = \widetilde{f} ([t_1]_1)\) and \(\widetilde{F} ([t_1]_1, 1) = [F (t_1, 1)]_2 = [f' (t_1)]_2 = \widetilde{f'} ([t_1]_1)\).
For each \([s_1]_1 \in \{[s_1]_1 \vert s_1 \in S_1\} \subseteq T_1 / \sim_1\), for each \(r \in I\), \(\widetilde{F} ([s_1]_1, r) = [F (s_1, r)]_2 = [f (s_1)]_2 = [f' (s_1)]_2\), but \([f (s_1)]_2 = \widetilde{f} ([s_1]_1)\) and \([f' (s_1)]_2 = \widetilde{f'} ([s_1]_1)\).
That means that \(\widetilde{F}\) is a homotopy between \(\widetilde{f}\) and \(\widetilde{f'}\) relative to \(\{[s_1]_1 \vert s_1 \in S_1\}\).
So, \(\widetilde{f} \simeq \widetilde{f'} rel \{[s_1]_1 \vert s_1 \in S_1\}\).
