True or False in differential topology
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I am trying to answer the following questions with a True (and give a proof) or a False (and give a counter example). I really have no idea how to approach this problems or how to start thinking about them. I would appreciate any hints, comments or suggestions on how to decide the answer.
a) Let $M$ and $N$ be embedded sub manifolds of $mathbb{R}^3$. Then $M cap N$ is an embedded sub manifold of $mathbb{R}^3$ iff $M$ and $N$ intersect transversely.
b) Let $F: S^{35} rightarrow mathbb{R}^{36}$ be a smooth map. Then the image $F(S^{35})$ has measure zero in $mathbb{R}^{36}$
f) If $F:M rightarrow N$ is a submersion, then for all $p,q in N$, them manifolds $F^{-1}(p)$ and $F^{-1}(q)$ are diffeomorphic.
Thanks for your help!
differential-topology
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
$endgroup$
add a comment |
$begingroup$
I am trying to answer the following questions with a True (and give a proof) or a False (and give a counter example). I really have no idea how to approach this problems or how to start thinking about them. I would appreciate any hints, comments or suggestions on how to decide the answer.
a) Let $M$ and $N$ be embedded sub manifolds of $mathbb{R}^3$. Then $M cap N$ is an embedded sub manifold of $mathbb{R}^3$ iff $M$ and $N$ intersect transversely.
b) Let $F: S^{35} rightarrow mathbb{R}^{36}$ be a smooth map. Then the image $F(S^{35})$ has measure zero in $mathbb{R}^{36}$
f) If $F:M rightarrow N$ is a submersion, then for all $p,q in N$, them manifolds $F^{-1}(p)$ and $F^{-1}(q)$ are diffeomorphic.
Thanks for your help!
differential-topology
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
$endgroup$
add a comment |
$begingroup$
I am trying to answer the following questions with a True (and give a proof) or a False (and give a counter example). I really have no idea how to approach this problems or how to start thinking about them. I would appreciate any hints, comments or suggestions on how to decide the answer.
a) Let $M$ and $N$ be embedded sub manifolds of $mathbb{R}^3$. Then $M cap N$ is an embedded sub manifold of $mathbb{R}^3$ iff $M$ and $N$ intersect transversely.
b) Let $F: S^{35} rightarrow mathbb{R}^{36}$ be a smooth map. Then the image $F(S^{35})$ has measure zero in $mathbb{R}^{36}$
f) If $F:M rightarrow N$ is a submersion, then for all $p,q in N$, them manifolds $F^{-1}(p)$ and $F^{-1}(q)$ are diffeomorphic.
Thanks for your help!
differential-topology
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
$endgroup$
I am trying to answer the following questions with a True (and give a proof) or a False (and give a counter example). I really have no idea how to approach this problems or how to start thinking about them. I would appreciate any hints, comments or suggestions on how to decide the answer.
a) Let $M$ and $N$ be embedded sub manifolds of $mathbb{R}^3$. Then $M cap N$ is an embedded sub manifold of $mathbb{R}^3$ iff $M$ and $N$ intersect transversely.
b) Let $F: S^{35} rightarrow mathbb{R}^{36}$ be a smooth map. Then the image $F(S^{35})$ has measure zero in $mathbb{R}^{36}$
f) If $F:M rightarrow N$ is a submersion, then for all $p,q in N$, them manifolds $F^{-1}(p)$ and $F^{-1}(q)$ are diffeomorphic.
Thanks for your help!
differential-topology
differential-topology
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
asked Dec 14 '18 at 5:12
BOlivianoperuano84BOlivianoperuano84
1778
1778
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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Hints:
(a) Duplicate for the hard implication: see Why is a transversal intersection of submanifolds a manifold? The other implication is obviously...
(b) Think in the easier cases $F: S^1rightarrowmathbb{R}^2$ or $F: S^2rightarrowmathbb{R}^3$.
(c) What happens if $F^{-1}(p)neemptyset$ and $F^{-1}(q) = emptyset$?
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Hints:
(a) Duplicate for the hard implication: see Why is a transversal intersection of submanifolds a manifold? The other implication is obviously...
(b) Think in the easier cases $F: S^1rightarrowmathbb{R}^2$ or $F: S^2rightarrowmathbb{R}^3$.
(c) What happens if $F^{-1}(p)neemptyset$ and $F^{-1}(q) = emptyset$?
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
$endgroup$
add a comment |
$begingroup$
Hints:
(a) Duplicate for the hard implication: see Why is a transversal intersection of submanifolds a manifold? The other implication is obviously...
(b) Think in the easier cases $F: S^1rightarrowmathbb{R}^2$ or $F: S^2rightarrowmathbb{R}^3$.
(c) What happens if $F^{-1}(p)neemptyset$ and $F^{-1}(q) = emptyset$?
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
$endgroup$
add a comment |
$begingroup$
Hints:
(a) Duplicate for the hard implication: see Why is a transversal intersection of submanifolds a manifold? The other implication is obviously...
(b) Think in the easier cases $F: S^1rightarrowmathbb{R}^2$ or $F: S^2rightarrowmathbb{R}^3$.
(c) What happens if $F^{-1}(p)neemptyset$ and $F^{-1}(q) = emptyset$?
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
$endgroup$
Hints:
(a) Duplicate for the hard implication: see Why is a transversal intersection of submanifolds a manifold? The other implication is obviously...
(b) Think in the easier cases $F: S^1rightarrowmathbb{R}^2$ or $F: S^2rightarrowmathbb{R}^3$.
(c) What happens if $F^{-1}(p)neemptyset$ and $F^{-1}(q) = emptyset$?
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
answered Dec 14 '18 at 7:44
Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla
35.4k42972
35.4k42972
add a comment |
add a comment |
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