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4

$begingroup$

Let $M^n$ ($n=3$ if needed) be a smooth manifold and $K subset M$ be a compact submamifold diffeomorphic to the $n$-disk, satisfying $K subset U$, for some chart $(varphi,U).$

Consider a compact submanifold $N^nsubset M^n $ with boundary such that

1. $text Int N^n cap text IntK neq emptyset,$


2. 
   $N^n cap K$ is connected.

I would like to know if there exists a compact smooth manifold $Lsubset U $ with boundary such that the following properties hold

1. 
   $Ksubset L$,


2. 
   $L cup N $ is a smooth manifold with boundary,


3. 
   $Lcap N $ is a smooth manifold with boundary.

This result seems true, however, I was not able to prove it. Can anyone help me?

I think it is always possible to do something like the picture below

manifolds differential-topology smooth-manifolds

share|cite|improve this question

edited Mar 30 at 1:08

Matheus Manzatto

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626

$endgroup$

add a comment | 

4

$begingroup$

Let $M^n$ ($n=3$ if needed) be a smooth manifold and $K subset M$ be a compact submamifold diffeomorphic to the $n$-disk, satisfying $K subset U$, for some chart $(varphi,U).$

Consider a compact submanifold $N^nsubset M^n $ with boundary such that

1. $text Int N^n cap text IntK neq emptyset,$


2. 
   $N^n cap K$ is connected.

I would like to know if there exists a compact smooth manifold $Lsubset U $ with boundary such that the following properties hold

1. 
   $Ksubset L$,


2. 
   $L cup N $ is a smooth manifold with boundary,


3. 
   $Lcap N $ is a smooth manifold with boundary.

This result seems true, however, I was not able to prove it. Can anyone help me?

I think it is always possible to do something like the picture below

manifolds differential-topology smooth-manifolds

share|cite|improve this question

edited Mar 30 at 1:08

Matheus Manzatto

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626

$endgroup$

add a comment | 

$begingroup$

Let $M^n$ ($n=3$ if needed) be a smooth manifold and $K subset M$ be a compact submamifold diffeomorphic to the $n$-disk, satisfying $K subset U$, for some chart $(varphi,U).$

Consider a compact submanifold $N^nsubset M^n $ with boundary such that

1. $text Int N^n cap text IntK neq emptyset,$


2. 
   $N^n cap K$ is connected.

I would like to know if there exists a compact smooth manifold $Lsubset U $ with boundary such that the following properties hold

1. 
   $Ksubset L$,


2. 
   $L cup N $ is a smooth manifold with boundary,


3. 
   $Lcap N $ is a smooth manifold with boundary.

This result seems true, however, I was not able to prove it. Can anyone help me?

I think it is always possible to do something like the picture below

manifolds differential-topology smooth-manifolds

share|cite|improve this question

edited Mar 30 at 1:08

Matheus Manzatto

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626

$endgroup$

share|cite|improve this question

edited Mar 30 at 1:08

Matheus Manzatto

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626

share|cite|improve this question

edited Mar 30 at 1:08

Matheus Manzatto

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626

asked Mar 29 at 16:30

Matheus ManzattoMatheus Manzatto

1,2991626