A detail in Haken's Lemma
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This has been killing me - Suppose $S$ is a planar surface, not a disk, that is incompressible in a 3-dimensional 1-handlebody $H$. Let $alpha$ be an arc connecting some boundary components of $S$. How do I show that $alpha$ is $partial-$parallel to $H$? I can't figure out how to use incompressibility of $S$ here. I know that $S$ is $pi_1$ injective so, for example, if $H$ is a genus 2 handlebody and $S$ is a pair of pants, then $S$ would have to be the "thickened 8" surface cutting $H$ in half.
But if we just have a "bad arc" $alpha$ in $S$ that isn't $partial-$parallel (such an $S$ and such an arc exist, just make a knot in $H$, snip the ends, take the boundary of the regular neighborhood of this snipped knot, then hook the boundary circles up to $partial H$ to get a proper embedding) how do we conclude that $S$ must also be bad (i.e. compressible). Another way to ask the question - how do I get the requisite compressing disk from the bad $alpha$?
manifolds smooth-manifolds
asked Nov 24 at 23:39
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This has been killing me - Suppose $S$ is a planar surface, not a disk, that is incompressible in a 3-dimensional 1-handlebody $H$. Let $alpha$ be an arc connecting some boundary components of $S$. How do I show that $alpha$ is $partial-$parallel to $H$? I can't figure out how to use incompressibility of $S$ here. I know that $S$ is $pi_1$ injective so, for example, if $H$ is a genus 2 handlebody and $S$ is a pair of pants, then $S$ would have to be the "thickened 8" surface cutting $H$ in half.
But if we just have a "bad arc" $alpha$ in $S$ that isn't $partial-$parallel (such an $S$ and such an arc exist, just make a knot in $H$, snip the ends, take the boundary of the regular neighborhood of this snipped knot, then hook the boundary circles up to $partial H$ to get a proper embedding) how do we conclude that $S$ must also be bad (i.e. compressible). Another way to ask the question - how do I get the requisite compressing disk from the bad $alpha$?
manifolds smooth-manifolds
asked Nov 24 at 23:39
Prototank
1,015820
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This has been killing me - Suppose $S$ is a planar surface, not a disk, that is incompressible in a 3-dimensional 1-handlebody $H$. Let $alpha$ be an arc connecting some boundary components of $S$. How do I show that $alpha$ is $partial-$parallel to $H$? I can't figure out how to use incompressibility of $S$ here. I know that $S$ is $pi_1$ injective so, for example, if $H$ is a genus 2 handlebody and $S$ is a pair of pants, then $S$ would have to be the "thickened 8" surface cutting $H$ in half.
But if we just have a "bad arc" $alpha$ in $S$ that isn't $partial-$parallel (such an $S$ and such an arc exist, just make a knot in $H$, snip the ends, take the boundary of the regular neighborhood of this snipped knot, then hook the boundary circles up to $partial H$ to get a proper embedding) how do we conclude that $S$ must also be bad (i.e. compressible). Another way to ask the question - how do I get the requisite compressing disk from the bad $alpha$?
manifolds smooth-manifolds
asked Nov 24 at 23:39
Prototank
1,015820
This has been killing me - Suppose $S$ is a planar surface, not a disk, that is incompressible in a 3-dimensional 1-handlebody $H$. Let $alpha$ be an arc connecting some boundary components of $S$. How do I show that $alpha$ is $partial-$parallel to $H$? I can't figure out how to use incompressibility of $S$ here. I know that $S$ is $pi_1$ injective so, for example, if $H$ is a genus 2 handlebody and $S$ is a pair of pants, then $S$ would have to be the "thickened 8" surface cutting $H$ in half.
But if we just have a "bad arc" $alpha$ in $S$ that isn't $partial-$parallel (such an $S$ and such an arc exist, just make a knot in $H$, snip the ends, take the boundary of the regular neighborhood of this snipped knot, then hook the boundary circles up to $partial H$ to get a proper embedding) how do we conclude that $S$ must also be bad (i.e. compressible). Another way to ask the question - how do I get the requisite compressing disk from the bad $alpha$?
manifolds smooth-manifolds
manifolds smooth-manifolds
asked Nov 24 at 23:39
Prototank
1,015820
asked Nov 24 at 23:39
Prototank
1,015820
edited Nov 24 at 23:47
asked Nov 24 at 23:39
Prototank
1,015820
asked Nov 24 at 23:39
Prototank
1,015820
asked Nov 24 at 23:39
Prototank
1,015820
1,015820
add a comment |
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