description/proof of that composition of \(C^\infty\) embedding after diffeomorphism or diffeomorphism after \(C^\infty\) embedding is \(C^\infty\) embedding
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary.
- The reader knows a definition of \(C^\infty\) embedding.
- The reader admits the proposition that any finite composition of injections is an injection.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
- 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 any restriction of any continuous map on the domain and the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that the composition of any \(C^\infty\) embedding after any diffeomorphism or any diffeomorphism after any \(C^\infty\) embedding is a \(C^\infty\) embedding.
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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_3\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(f_1\): \(: M_1 \to M_2\), \(\in \{\text{ the diffeomorphisms }\}\)
\(f_2\): \(: M_2 \to M_3\), \(\in \{\text{ the } C^\infty \text{ embeddings }\}\)
\(f_2 \circ f_1\): \(: M_1 \to M_3\)
\(f'_1\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ embeddings }\}\)
\(f'_2\): \(: M_2 \to M_3\), \(\in \{\text{ the diffeomorphisms }\}\)
\(f'_2 \circ f'_1\): \(: M_1 \to M_3\)
//
Statements:
\(f_2 \circ f_1 \in \{\text{ the } C^\infty \text{ embeddings }\}\)
\(\land\)
\(f'_2 \circ f'_1 \in \{\text{ the } C^\infty \text{ embeddings }\}\)
//
2: Proof
Whole Strategy: Step 1: see that \(f_2 \circ f_1\) is injective; Step 2: see that \(f_2 \circ f_1\) is \(C^\infty\); Step 3: see that \(f_2 \circ f_1\) is a \(C^\infty\) immersion; Step 4: see that the codomain restriction of \(f_2 \circ f_1\) is homeomorphic; Step 5: see that \(f'_2 \circ f'_1\) is injective; Step 6: see that \(f'_2 \circ f'_1\) is \(C^\infty\); Step 7: see that \(f'_2 \circ f'_1\) is a \(C^\infty\) immersion; Step 8: see that the codomain restriction of \(f'_2 \circ f'_1\) is homeomorphic.
Step 1:
\(f_2 \circ f_1\) is injective, by the proposition that any finite composition of injections is an injection: any diffeomorphism or any \(C^\infty\) embedding is injective.
Step 2:
\(f_2 \circ f_1\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Step 3:
Let us see that \(f_2 \circ f_1\) is a \(C^\infty\) immersion.
For each \(p \in M_1\), \(d_p (f_2 \circ f_1) = d_{f_1 (p)} f_2 \circ d_p f_1\).
\(d_p f_1\) is injective (in fact, bijective) because \(f_1\) is a diffeomorphism and \(d_{f_1 (p)} f_2\) is injective because \(f_2\) is a \(C^\infty\) embedding.
So, \(d_p (f_2 \circ f_1)\) is injective, by the proposition that any finite composition of injections is an injection.
So, \(f_2 \circ f_1\) is a \(C^\infty\) immersion.
Step 4:
Let us see that the codomain restriction of \(f_2 \circ f_1\), \(\overline{f_2 \circ f_1}: M_1 \to f_2 \circ f_1 (M_1) \subseteq M_3\), is a homeomorphism.
\(f_2 \circ f_1\) 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.
\(\overline{f_2 \circ f_1}\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
As \(\overline{f_2 \circ f_1}\) is a bijection (\(f_2 \circ f_1\) is injective and the codomain is restricted to be surjective), there is the inverse, \(\overline{f_2 \circ f_1}^{-1}: f_2 \circ f_1 (M_1) \subseteq M_3 \to M_1\).
\(\overline{f_2 \circ f_1} = \overline{f_2} \circ f_1\), where \(\overline{f_2}: M_2 \to f_2 (M_2) \subseteq M_3\) is the codomain restriction of \(f_2\).
So, \(\overline{f_2 \circ f_1}^{-1} = f_1^{-1} \circ \overline{f_2}^{-1}\), but \(\overline{f_2}^{-1}\) is continuous because \(f_2\) is a \(C^\infty\) embedding and \(f_1^{-1}\) is continuous because \(f_1\) is a diffeomorphism, and so, \(\overline{f_2 \circ f_1}^{-1}\) is continuous.
Step 5:
\(f'_2 \circ f'_1\) is injective, by the proposition that any finite composition of injections is an injection: any \(C^\infty\) embedding or any diffeomorphism is injective.
Step 6:
\(f'_2 \circ f'_1\) is \(C^\infty\), by the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Step 7:
Let us see that \(f'_2 \circ f'_1\) is a \(C^\infty\) immersion.
For each \(p \in M_1\), \(d_p (f'_2 \circ f'_1) = d_{f'_1 (p)} f'_2 \circ d_p f'_1\).
\(d_p f'_1\) is injective because \(f'_1\) is a \(C^\infty\) embedding and \(d_{f'_1 (p)} f'_2\) is injective (in fact, bijective) because \(f'_2\) is a diffeomorphism.
So, \(d_p (f'_2 \circ f'_1)\) is injective, by the proposition that any finite composition of injections is an injection.
So, \(f'_2 \circ f'_1\) is a \(C^\infty\) immersion.
Step 8:
Let us see that the codomain restriction of \(f'_2 \circ f'_1\), \(\overline{f'_2 \circ f'_1}: M_1 \to f'_2 \circ f'_1 (M_1) \subseteq M_3\), is a homeomorphism.
\(f'_2 \circ f'_1\) 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.
\(\overline{f'_2 \circ f'_1}\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
As \(\overline{f'_2 \circ f'_1}\) is a bijection (\(f'_2 \circ f'_1\) is injective and the codomain is restricted to be surjective), there is the inverse, \(\overline{f'_2 \circ f'_1}^{-1}: f'_2 \circ f'_1 (M_1) \subseteq M_3 \to M_1\).
\(\overline{f'_2 \circ f'_1} = \overline{f'_2} \circ \overline{f'_1}\), where \(\overline{f'_1}: M_1 \to f'_1 (M_1) \subseteq M_2\) is the codomain restriction of \(f'_1\) and \(\overline{f'_2}: f'_1 (M_1) \subseteq M_2 \to f'_2 \circ f'_1 (M_1) \subseteq M_3\) is the domain and the codomain restriction of \(f'_2\).
So, \(\overline{f'_2 \circ f'_1}^{-1} = \overline{f'_1}^{-1} \circ \overline{f'_2}^{-1}\), but \(\overline{f'_2}^{-1}\) is continuous because \(f'_2\) is a diffeomorphism, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous, and \(\overline{f'_1}^{-1}\) is continuous because \(f'_1\) is a \(C^\infty\) embedding. So, \(\overline{f_2 \circ f_1}^{-1}\) is continuous.
