Mixing
map to a product is a submersion
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
add a comment |
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 '18 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 '18 at 19:59
add a comment |
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
differential-geometry smooth-manifolds
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
edited Nov 22 '18 at 19:58
Praphulla Koushik
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
asked Nov 22 '18 at 12:06
Praphulla KoushikPraphulla Koushik
25216
25216
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 '18 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 '18 at 19:59
add a comment |
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 '18 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 '18 at 19:59
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 '18 at 19:25
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 '18 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 '18 at 19:59
@Laz I have added the details.
– Praphulla Koushik
Nov 22 '18 at 19:59
add a comment |
1 Answer
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I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
add a comment |
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1 Answer
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1 Answer
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I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
add a comment |
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
add a comment |
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
answered Nov 30 '18 at 3:04
Dante GrevinoDante Grevino
94319
94319
add a comment |
add a comment |
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What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 '18 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 '18 at 19:59