description/proof of that for continuous map from 1st space into 2nd space and continuous map from 2nd space into 3rd space, if 1st map and composition of 2nd map after 1st map are homotopy equivalences, 2nd map is homotopy equivalence
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of homotopy equivalence.
- The reader admits the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
- The reader admits the proposition that for any homotopic maps from any 1st topological space into any 2nd topological space and any homotopic maps from the 2nd topological space into any 3rd topological space, the compositions of the homotopic maps are homotopic with a homotopy as this.
- The reader admits the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation.
Target Context
- The reader will have a description and a proof of the proposition that for any continuous map from any 1st topological space into any 2nd topological space and any continuous map from the 2nd space into any 3rd space, if the 1st map and the composition of the 2nd map after the 1st map are some homotopy equivalences, the 2nd map is a homotopy equivalence.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(T_3\): \(\in \{\text{ the topological spaces }\}\)
\(f_1\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\(f_2\): \(: T_2 \to T_3\), \(\in \{\text{ the continuous maps }\}\)
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Statements:
\(f_1 \in \{\text{ the homotopy equivalences }\} \land f_2 \circ f_1 \in \{\text{ the homotopy equivalences }\}\)
\(\implies\)
\(f_2 \in \{\text{ the homotopy equivalences }\}\)
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2: Note
\(f_2\) needs to be presupposed to be continuous, for this proposition, because Proof does not prove that \(f_2\) is continuous, which is a necessity for \(f_2\) to be a homotopy equivalence.
3: Proof
Whole Strategy: Step 1: take \(\widetilde{f_2}: T_2 \to T_1\) such that \(\widetilde{f_2} \circ f_1 \simeq id_{T_1}\) and \(f_1 \circ \widetilde{f_2} \simeq id_{T_2}\) and \(\widetilde{\widetilde{f_3}}: T_3 \to T_1\) such that \(\widetilde{\widetilde{f_3}} \circ f_2 \circ f_1 \simeq id_{T_1}\) and \(f_2 \circ f_1 \circ \widetilde{\widetilde{f_3}} \simeq id_{T_3}\); Step 2: see that \(f_2 \circ f_1 \circ \widetilde{\widetilde{f_3}} \simeq id_{T_3}\) and \(f_1 \circ \widetilde{\widetilde{f_3}} \circ f_2 \simeq id_{T_2}\).
Step 1:
There is a continuous \(\widetilde{f_2}: T_2 \to T_1\) such that \(\widetilde{f_2} \circ f_1 \simeq id_{T_1}\) and \(f_1 \circ \widetilde{f_2} \simeq id_{T_2}\), by Note for the definition of homotopy equivalence.
There is a continuous \(\widetilde{\widetilde{f_3}}: T_3 \to T_1\) such that \(\widetilde{\widetilde{f_3}} \circ f_2 \circ f_1 \simeq id_{T_1}\) and \(f_2 \circ f_1 \circ \widetilde{\widetilde{f_3}} \simeq id_{T_3}\), likewise.
Step 2:
Let us take \(f_1 \circ \widetilde{\widetilde{f_3}}: T_3 \to T_2\).
\(f_1 \circ \widetilde{\widetilde{f_3}}\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
\(f_2 \circ f_1 \circ \widetilde{\widetilde{f_3}} \simeq id_{T_3}\), which has been seen above.
\(f_1 \circ \widetilde{\widetilde{f_3}} \circ f_2 = f_1 \circ \widetilde{\widetilde{f_3}} \circ f_2 \circ id_{T_2} \simeq f_1 \circ \widetilde{\widetilde{f_3}} \circ f_2 \circ f_1 \circ \widetilde{f_2}\), because \(id_{T_2} \simeq f_1 \circ \widetilde{f_2}\), by the proposition that for any homotopic maps from any 1st topological space into any 2nd topological space and any homotopic maps from the 2nd topological space into any 3rd topological space, the compositions of the homotopic maps are homotopic with a homotopy as this, \(= f_1 \circ (\widetilde{\widetilde{f_3}} \circ f_2 \circ f_1) \circ \widetilde{f_2} \simeq f_1 \circ id_{T_1} \circ \widetilde{f_2}\), because \(\widetilde{\widetilde{f_3}} \circ f_2 \circ f_1 \simeq id_{T_1}\), as before, \(= f_1 \circ \widetilde{f_2} \simeq id_{T_2}\).
By the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation, \(f_1 \circ \widetilde{\widetilde{f_3}} \circ f_2 \simeq id_{T_2}\).
So, \(f_2\) is a homotopy equivalence.
