A difficulty in understanding a part of a paragraph in Guillemin & Pollack p.60
I do not understand the highlighted part of the paragraph given below:
Could anyone explain it for me please?
general-topology differential-topology manifolds-with-boundary transversality
asked Nov 28 '18 at 2:43
hopefully
129112
add a comment |
I do not understand the highlighted part of the paragraph given below:
Could anyone explain it for me please?
general-topology differential-topology manifolds-with-boundary transversality
asked Nov 28 '18 at 2:43
hopefully
129112
1
There must be a better way to highlight.
– Randall
Nov 28 '18 at 2:54
I am sorry @Randall for being unskillful in highligting
– hopefully
Nov 28 '18 at 3:41
add a comment |
I do not understand the highlighted part of the paragraph given below:
Could anyone explain it for me please?
general-topology differential-topology manifolds-with-boundary transversality
asked Nov 28 '18 at 2:43
hopefully
129112
I do not understand the highlighted part of the paragraph given below:
Could anyone explain it for me please?
general-topology differential-topology manifolds-with-boundary transversality
general-topology differential-topology manifolds-with-boundary transversality
asked Nov 28 '18 at 2:43
hopefully
129112
asked Nov 28 '18 at 2:43
hopefully
129112
asked Nov 28 '18 at 2:43
hopefully
129112
asked Nov 28 '18 at 2:43
hopefully
129112
asked Nov 28 '18 at 2:43
hopefully
129112
129112
1
There must be a better way to highlight.
– Randall
Nov 28 '18 at 2:54
I am sorry @Randall for being unskillful in highligting
– hopefully
Nov 28 '18 at 3:41
add a comment |
1
There must be a better way to highlight.
– Randall
Nov 28 '18 at 2:54
I am sorry @Randall for being unskillful in highligting
– hopefully
Nov 28 '18 at 3:41
1
1
There must be a better way to highlight.
– Randall
Nov 28 '18 at 2:54
There must be a better way to highlight.
– Randall
Nov 28 '18 at 2:54
I am sorry @Randall for being unskillful in highligting
– hopefully
Nov 28 '18 at 3:41
I am sorry @Randall for being unskillful in highligting
– hopefully
Nov 28 '18 at 3:41
add a comment |
1 Answer
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oldest
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For their example map $f$, we have $f^{-1}(Z) = partial H^2$, which is the $x$-axis in the plane. This is boundaryless, so $partial f^{-1}(Z)$ is empty. On the other hand, $f^{-1}(Z) cap partial H^2 = partial H^2$ which is the $x$-axis and not empty. Hence
$partial f^{-1}(Z) = f^{-1}(Z) cap partial H^2$ is false in this example. Since you would rather have the boundary of the preimage be the part of the preimage that lives in the boundary, you will have to make additional assumptions above and beyond transversality.
answered Nov 28 '18 at 2:54
Randall
9,12611129
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
votes
For their example map $f$, we have $f^{-1}(Z) = partial H^2$, which is the $x$-axis in the plane. This is boundaryless, so $partial f^{-1}(Z)$ is empty. On the other hand, $f^{-1}(Z) cap partial H^2 = partial H^2$ which is the $x$-axis and not empty. Hence
$partial f^{-1}(Z) = f^{-1}(Z) cap partial H^2$ is false in this example. Since you would rather have the boundary of the preimage be the part of the preimage that lives in the boundary, you will have to make additional assumptions above and beyond transversality.
answered Nov 28 '18 at 2:54
Randall
9,12611129
add a comment |
For their example map $f$, we have $f^{-1}(Z) = partial H^2$, which is the $x$-axis in the plane. This is boundaryless, so $partial f^{-1}(Z)$ is empty. On the other hand, $f^{-1}(Z) cap partial H^2 = partial H^2$ which is the $x$-axis and not empty. Hence
$partial f^{-1}(Z) = f^{-1}(Z) cap partial H^2$ is false in this example. Since you would rather have the boundary of the preimage be the part of the preimage that lives in the boundary, you will have to make additional assumptions above and beyond transversality.
answered Nov 28 '18 at 2:54
Randall
9,12611129
add a comment |
For their example map $f$, we have $f^{-1}(Z) = partial H^2$, which is the $x$-axis in the plane. This is boundaryless, so $partial f^{-1}(Z)$ is empty. On the other hand, $f^{-1}(Z) cap partial H^2 = partial H^2$ which is the $x$-axis and not empty. Hence
$partial f^{-1}(Z) = f^{-1}(Z) cap partial H^2$ is false in this example. Since you would rather have the boundary of the preimage be the part of the preimage that lives in the boundary, you will have to make additional assumptions above and beyond transversality.
answered Nov 28 '18 at 2:54
Randall
9,12611129
For their example map $f$, we have $f^{-1}(Z) = partial H^2$, which is the $x$-axis in the plane. This is boundaryless, so $partial f^{-1}(Z)$ is empty. On the other hand, $f^{-1}(Z) cap partial H^2 = partial H^2$ which is the $x$-axis and not empty. Hence
$partial f^{-1}(Z) = f^{-1}(Z) cap partial H^2$ is false in this example. Since you would rather have the boundary of the preimage be the part of the preimage that lives in the boundary, you will have to make additional assumptions above and beyond transversality.
answered Nov 28 '18 at 2:54
Randall
9,12611129
answered Nov 28 '18 at 2:54
Randall
9,12611129
answered Nov 28 '18 at 2:54
Randall
9,12611129
answered Nov 28 '18 at 2:54
Randall
9,12611129
9,12611129
add a comment |
add a comment |
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1
There must be a better way to highlight.
– Randall
Nov 28 '18 at 2:54
I am sorry @Randall for being unskillful in highligting
– hopefully
Nov 28 '18 at 3:41