Compact manifolds and CW structure
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Let $M$ be a compact smooth manifold, with a regular submanifold $N$. Assuming I already have a decomposition of $N$ into a CW complex, can I always extend this decomposition to $M$? To be precise, does $M$ admit a decomposition as a CW complex such that the decomposition of $N$ is a subcomplex?
differential-topology
asked 2 days ago
user09127
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Let $M$ be a compact smooth manifold, with a regular submanifold $N$. Assuming I already have a decomposition of $N$ into a CW complex, can I always extend this decomposition to $M$? To be precise, does $M$ admit a decomposition as a CW complex such that the decomposition of $N$ is a subcomplex?
differential-topology
asked 2 days ago
user09127
424
1
Yes. 1) Show that this is true for $M$ a unit disc bundle over $N$. 2) Show that this is true for $N = partial M$. 3) Glue these together to get the general case.
– Mike Miller
2 days ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $M$ be a compact smooth manifold, with a regular submanifold $N$. Assuming I already have a decomposition of $N$ into a CW complex, can I always extend this decomposition to $M$? To be precise, does $M$ admit a decomposition as a CW complex such that the decomposition of $N$ is a subcomplex?
differential-topology
asked 2 days ago
user09127
424
Let $M$ be a compact smooth manifold, with a regular submanifold $N$. Assuming I already have a decomposition of $N$ into a CW complex, can I always extend this decomposition to $M$? To be precise, does $M$ admit a decomposition as a CW complex such that the decomposition of $N$ is a subcomplex?
differential-topology
differential-topology
asked 2 days ago
user09127
424
asked 2 days ago
user09127
424
asked 2 days ago
user09127
424
asked 2 days ago
user09127
424
asked 2 days ago
user09127
424
424
1
Yes. 1) Show that this is true for $M$ a unit disc bundle over $N$. 2) Show that this is true for $N = partial M$. 3) Glue these together to get the general case.
– Mike Miller
2 days ago
add a comment |
1
Yes. 1) Show that this is true for $M$ a unit disc bundle over $N$. 2) Show that this is true for $N = partial M$. 3) Glue these together to get the general case.
– Mike Miller
2 days ago
1
1
Yes. 1) Show that this is true for $M$ a unit disc bundle over $N$. 2) Show that this is true for $N = partial M$. 3) Glue these together to get the general case.
– Mike Miller
2 days ago
Yes. 1) Show that this is true for $M$ a unit disc bundle over $N$. 2) Show that this is true for $N = partial M$. 3) Glue these together to get the general case.
– Mike Miller
2 days ago
add a comment |
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Yes. 1) Show that this is true for $M$ a unit disc bundle over $N$. 2) Show that this is true for $N = partial M$. 3) Glue these together to get the general case.
– Mike Miller
2 days ago