Mixing
Homotopy fibers and preimages of regular values.
1
$begingroup$
Let $f colon M to N$ be a smooth map between two manifolds. Let $q_0 in N$ be a regular value of $f$. Is there a relation between the homotopy fiber of $f$ and the submanifold $f^{-1}(q_0)$?
algebraic-topology differential-topology homotopy-theory
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
$endgroup$
add a comment |
1
$begingroup$
Let $f colon M to N$ be a smooth map between two manifolds. Let $q_0 in N$ be a regular value of $f$. Is there a relation between the homotopy fiber of $f$ and the submanifold $f^{-1}(q_0)$?
algebraic-topology differential-topology homotopy-theory
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
$endgroup$
$begingroup$
Not if $f$ is just an arbitrary map, I don't think. The different point-preimages of $f$ can be very different, but the homotopy fibre of a map is determined up to homotopy equivalence. However if $f$ is surjective, a submersion (ALL points are regular) and proper, then Ehresmann's fibration theorem says it's a locally trivial fibration, and so the homotopy and point-fibres are homotopy equivalent.
$endgroup$
– William
Jan 27 at 21:03
$begingroup$
If you know some Morse Theory it's a good example of when point pre-images do and don't change. Given a Morse function $fcolon M to mathbb{R}$, if $I = [a, b]$ is a closed interval of regular values in $mathbb{R}$ then $f^{-1}I$ is diffeomorphic to a cylinder $f^{-1}(a) times I$. However if $[a, b]$ contains a unique non-degenerate critical point $c$, then the diffeomorphism types of $f^{-1}(a)$ and $f^{-1}(b)$ differ by a handle whose index is the Morse index of c.
$endgroup$
– William
Jan 27 at 21:17
add a comment |
1
1
1
$begingroup$
Let $f colon M to N$ be a smooth map between two manifolds. Let $q_0 in N$ be a regular value of $f$. Is there a relation between the homotopy fiber of $f$ and the submanifold $f^{-1}(q_0)$?
algebraic-topology differential-topology homotopy-theory
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
$endgroup$
Let $f colon M to N$ be a smooth map between two manifolds. Let $q_0 in N$ be a regular value of $f$. Is there a relation between the homotopy fiber of $f$ and the submanifold $f^{-1}(q_0)$?
algebraic-topology differential-topology homotopy-theory
algebraic-topology differential-topology homotopy-theory
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
asked Jan 27 at 19:15
Wilhelm L.Wilhelm L.
41117
41117
$begingroup$
Not if $f$ is just an arbitrary map, I don't think. The different point-preimages of $f$ can be very different, but the homotopy fibre of a map is determined up to homotopy equivalence. However if $f$ is surjective, a submersion (ALL points are regular) and proper, then Ehresmann's fibration theorem says it's a locally trivial fibration, and so the homotopy and point-fibres are homotopy equivalent.
$endgroup$
– William
Jan 27 at 21:03
$begingroup$
If you know some Morse Theory it's a good example of when point pre-images do and don't change. Given a Morse function $fcolon M to mathbb{R}$, if $I = [a, b]$ is a closed interval of regular values in $mathbb{R}$ then $f^{-1}I$ is diffeomorphic to a cylinder $f^{-1}(a) times I$. However if $[a, b]$ contains a unique non-degenerate critical point $c$, then the diffeomorphism types of $f^{-1}(a)$ and $f^{-1}(b)$ differ by a handle whose index is the Morse index of c.
$endgroup$
– William
Jan 27 at 21:17
add a comment |
$begingroup$
Not if $f$ is just an arbitrary map, I don't think. The different point-preimages of $f$ can be very different, but the homotopy fibre of a map is determined up to homotopy equivalence. However if $f$ is surjective, a submersion (ALL points are regular) and proper, then Ehresmann's fibration theorem says it's a locally trivial fibration, and so the homotopy and point-fibres are homotopy equivalent.
$endgroup$
– William
Jan 27 at 21:03
$begingroup$
If you know some Morse Theory it's a good example of when point pre-images do and don't change. Given a Morse function $fcolon M to mathbb{R}$, if $I = [a, b]$ is a closed interval of regular values in $mathbb{R}$ then $f^{-1}I$ is diffeomorphic to a cylinder $f^{-1}(a) times I$. However if $[a, b]$ contains a unique non-degenerate critical point $c$, then the diffeomorphism types of $f^{-1}(a)$ and $f^{-1}(b)$ differ by a handle whose index is the Morse index of c.
$endgroup$
– William
Jan 27 at 21:17
$begingroup$
Not if $f$ is just an arbitrary map, I don't think. The different point-preimages of $f$ can be very different, but the homotopy fibre of a map is determined up to homotopy equivalence. However if $f$ is surjective, a submersion (ALL points are regular) and proper, then Ehresmann's fibration theorem says it's a locally trivial fibration, and so the homotopy and point-fibres are homotopy equivalent.
$endgroup$
– William
Jan 27 at 21:03
$begingroup$
Not if $f$ is just an arbitrary map, I don't think. The different point-preimages of $f$ can be very different, but the homotopy fibre of a map is determined up to homotopy equivalence. However if $f$ is surjective, a submersion (ALL points are regular) and proper, then Ehresmann's fibration theorem says it's a locally trivial fibration, and so the homotopy and point-fibres are homotopy equivalent.
$endgroup$
– William
Jan 27 at 21:03
$begingroup$
If you know some Morse Theory it's a good example of when point pre-images do and don't change. Given a Morse function $fcolon M to mathbb{R}$, if $I = [a, b]$ is a closed interval of regular values in $mathbb{R}$ then $f^{-1}I$ is diffeomorphic to a cylinder $f^{-1}(a) times I$. However if $[a, b]$ contains a unique non-degenerate critical point $c$, then the diffeomorphism types of $f^{-1}(a)$ and $f^{-1}(b)$ differ by a handle whose index is the Morse index of c.
$endgroup$
– William
Jan 27 at 21:17
$begingroup$
If you know some Morse Theory it's a good example of when point pre-images do and don't change. Given a Morse function $fcolon M to mathbb{R}$, if $I = [a, b]$ is a closed interval of regular values in $mathbb{R}$ then $f^{-1}I$ is diffeomorphic to a cylinder $f^{-1}(a) times I$. However if $[a, b]$ contains a unique non-degenerate critical point $c$, then the diffeomorphism types of $f^{-1}(a)$ and $f^{-1}(b)$ differ by a handle whose index is the Morse index of c.
$endgroup$
– William
Jan 27 at 21:17
add a comment |
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$begingroup$
Not if $f$ is just an arbitrary map, I don't think. The different point-preimages of $f$ can be very different, but the homotopy fibre of a map is determined up to homotopy equivalence. However if $f$ is surjective, a submersion (ALL points are regular) and proper, then Ehresmann's fibration theorem says it's a locally trivial fibration, and so the homotopy and point-fibres are homotopy equivalent.
$endgroup$
– William
Jan 27 at 21:03
$begingroup$
If you know some Morse Theory it's a good example of when point pre-images do and don't change. Given a Morse function $fcolon M to mathbb{R}$, if $I = [a, b]$ is a closed interval of regular values in $mathbb{R}$ then $f^{-1}I$ is diffeomorphic to a cylinder $f^{-1}(a) times I$. However if $[a, b]$ contains a unique non-degenerate critical point $c$, then the diffeomorphism types of $f^{-1}(a)$ and $f^{-1}(b)$ differ by a handle whose index is the Morse index of c.
$endgroup$
– William
Jan 27 at 21:17