849: Composition of C∞ Embedding After Diffeomorphism or Diffeomorphism After C∞ Embedding Is C∞ Embedding

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that composition of $C^{\mathrm{\infty }}$ embedding after diffeomorphism or diffeomorphism after $C^{\mathrm{\infty }}$ embedding is $C^{\mathrm{\infty }}$ embedding

---

Topics

About: $C^{\mathrm{\infty }}$ manifold

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof

---

Starting Context

• The reader knows a definition of diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary.
• The reader knows a definition of $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, 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 $C^{\mathrm{\infty }}$ embedding after any diffeomorphism or any diffeomorphism after any $C^{\mathrm{\infty }}$ embedding is a $C^{\mathrm{\infty }}$ 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:
$M_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_3$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$f_1$: $:M_1\to M_2$, $\in \{\text{ the diffeomorphisms }\}$
$f_2$: $:M_2\to M_3$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
$f_2\circ f_1$: $:M_1\to M_3$
$f_1^{\prime }$: $:M_1\to M_2$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
$f_2^{\prime }$: $:M_2\to M_3$, $\in \{\text{ the diffeomorphisms }\}$
$f_2^{\prime }\circ f_1^{\prime }$: $:M_1\to M_3$
//

Statements:
$f_2\circ f_1\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
$\wedge$
$f_2^{\prime }\circ f_1^{\prime }\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
//

---

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 $C^{\mathrm{\infty }}$; Step 3: see that $f_2\circ f_1$ is a $C^{\mathrm{\infty }}$ immersion; Step 4: see that the codomain restriction of $f_2\circ f_1$ is homeomorphic; Step 5: see that $f_2^{\prime }\circ f_1^{\prime }$ is injective; Step 6: see that $f_2^{\prime }\circ f_1^{\prime }$ is $C^{\mathrm{\infty }}$; Step 7: see that $f_2^{\prime }\circ f_1^{\prime }$ is a $C^{\mathrm{\infty }}$ immersion; Step 8: see that the codomain restriction of $f_2^{\prime }\circ f_1^{\prime }$ 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 $C^{\mathrm{\infty }}$ embedding is injective.

Step 2:

$f_2\circ f_1$ is $C^{\mathrm{\infty }}$, by the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

Step 3:

Let us see that $f_2\circ f_1$ is a $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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:

$f_2^{\prime }\circ f_1^{\prime }$ is injective, by the proposition that any finite composition of injections is an injection: any $C^{\mathrm{\infty }}$ embedding or any diffeomorphism is injective.

Step 6:

$f_2^{\prime }\circ f_1^{\prime }$ is $C^{\mathrm{\infty }}$, by the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

Step 7:

Let us see that $f_2^{\prime }\circ f_1^{\prime }$ is a $C^{\mathrm{\infty }}$ immersion.

For each $p\in M_1$, $d_p(f_2^{\prime }\circ f_1^{\prime })=d_{f_1^{\prime }(p)}{f}_2^{\prime }\circ d_p{f}_1^{\prime }$.

$d_p{f}_1^{\prime }$ is injective because $f_1^{\prime }$ is a $C^{\mathrm{\infty }}$ embedding and $d_{f_1^{\prime }(p)}{f}_2^{\prime }$ is injective (in fact, bijective) because $f_2^{\prime }$ is a diffeomorphism.

So, $d_p(f_2^{\prime }\circ f_1^{\prime })$ is injective, by the proposition that any finite composition of injections is an injection.

So, $f_2^{\prime }\circ f_1^{\prime }$ is a $C^{\mathrm{\infty }}$ immersion.

Step 8:

Let us see that the codomain restriction of $f_2^{\prime }\circ f_1^{\prime }$, $\overline{f_2^{\prime }\circ f_1^{\prime }}:M_1\to f_2^{\prime }\circ f_1^{\prime }(M_1)\subseteq M_3$, is a homeomorphism.

$f_2^{\prime }\circ f_1^{\prime }$ 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.

$\overline{f_2^{\prime }\circ f_1^{\prime }}$ is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

As $\overline{f_2^{\prime }\circ f_1^{\prime }}$ is a bijection ($f_2^{\prime }\circ f_1^{\prime }$ is injective and the codomain is restricted to be surjective), there is the inverse, ${\overline{f_2^{\prime }\circ f_1^{\prime }}}^{-1}:f_2^{\prime }\circ f_1^{\prime }(M_1)\subseteq M_3\to M_1$.

$\overline{f_2^{\prime }\circ f_1^{\prime }}=\overline{f_2^{\prime }}\circ \overline{f_1^{\prime }}$, where $\overline{f_1^{\prime }}:M_1\to f_1^{\prime }(M_1)\subseteq M_2$ is the codomain restriction of $f_1^{\prime }$ and $\overline{f_2^{\prime }}:f_1^{\prime }(M_1)\subseteq M_2\to f_2^{\prime }\circ f_1^{\prime }(M_1)\subseteq M_3$ is the domain and the codomain restriction of $f_2^{\prime }$.

So, ${\overline{f_2^{\prime }\circ f_1^{\prime }}}^{-1}={\overline{f_1^{\prime }}}^{-1}\circ {\overline{f_2^{\prime }}}^{-1}$, but ${\overline{f_2^{\prime }}}^{-1}$ is continuous because $f_2^{\prime }$ is a diffeomorphism, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous, and ${\overline{f_1^{\prime }}}^{-1}$ is continuous because $f_1^{\prime }$ is a $C^{\mathrm{\infty }}$ embedding. So, ${\overline{f_2\circ f_1}}^{-1}$ is continuous.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>