Final step of homotopy lemma

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In proving Homotopy lemma in Milnor Topology from the differential viewpoint we consider $V_1 cap V_2$ where $V_1$ is a neighbourhood of $y$ in which $card f^-1( y)$ is constant. Similarly $V_2$ s a neighbourhood of $y$ in which $card g^-1( y)$ is constant. If $F$ is smooth homotopy between $f$ and $g$ then he chooses $zin V_1cap V_2$ such that $z$ is regular value of $F$. How do we know such $z$ exists ?

Let me know if more information is needed.

general-topology differential-topology

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asked Jul 16 at 13:53

mathemather

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In proving Homotopy lemma in Milnor Topology from the differential viewpoint we consider $V_1 cap V_2$ where $V_1$ is a neighbourhood of $y$ in which $card f^-1( y)$ is constant. Similarly $V_2$ s a neighbourhood of $y$ in which $card g^-1( y)$ is constant. If $F$ is smooth homotopy between $f$ and $g$ then he chooses $zin V_1cap V_2$ such that $z$ is regular value of $F$. How do we know such $z$ exists ?

Let me know if more information is needed.

general-topology differential-topology

share|cite|improve this question

asked Jul 16 at 13:53

mathemather

805420

add a comment | 

up vote
0
down vote

favorite

1

up vote
0
down vote

favorite

1

1

In proving Homotopy lemma in Milnor Topology from the differential viewpoint we consider $V_1 cap V_2$ where $V_1$ is a neighbourhood of $y$ in which $card f^-1( y)$ is constant. Similarly $V_2$ s a neighbourhood of $y$ in which $card g^-1( y)$ is constant. If $F$ is smooth homotopy between $f$ and $g$ then he chooses $zin V_1cap V_2$ such that $z$ is regular value of $F$. How do we know such $z$ exists ?

Let me know if more information is needed.

general-topology differential-topology

share|cite|improve this question

asked Jul 16 at 13:53

mathemather

805420

In proving Homotopy lemma in Milnor Topology from the differential viewpoint we consider $V_1 cap V_2$ where $V_1$ is a neighbourhood of $y$ in which $card f^-1( y)$ is constant. Similarly $V_2$ s a neighbourhood of $y$ in which $card g^-1( y)$ is constant. If $F$ is smooth homotopy between $f$ and $g$ then he chooses $zin V_1cap V_2$ such that $z$ is regular value of $F$. How do we know such $z$ exists ?

Let me know if more information is needed.

general-topology differential-topology

share|cite|improve this question

asked Jul 16 at 13:53

mathemather

805420

share|cite|improve this question

share|cite|improve this question

asked Jul 16 at 13:53

mathemather

805420

asked Jul 16 at 13:53

mathemather

805420

asked Jul 16 at 13:53

mathemather

805420

805420

add a comment | 

add a comment | 

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accepted

$V_1 cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)

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edited Jul 16 at 14:14

answered Jul 16 at 14:09

Randall

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

$V_1 cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)

share|cite|improve this answer

edited Jul 16 at 14:14

answered Jul 16 at 14:09

Randall

7,2471825

add a comment | 

up vote
1
down vote



accepted

$V_1 cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)

share|cite|improve this answer

edited Jul 16 at 14:14

answered Jul 16 at 14:09

Randall

7,2471825

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

$V_1 cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)

share|cite|improve this answer

edited Jul 16 at 14:14

answered Jul 16 at 14:09

Randall

7,2471825

$V_1 cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)

share|cite|improve this answer

edited Jul 16 at 14:14

answered Jul 16 at 14:09

Randall

7,2471825

share|cite|improve this answer

share|cite|improve this answer

edited Jul 16 at 14:14

edited Jul 16 at 14:14

edited Jul 16 at 14:14

answered Jul 16 at 14:09

Randall

7,2471825

answered Jul 16 at 14:09

Randall

7,2471825

answered Jul 16 at 14:09

Randall

7,2471825

7,2471825

add a comment | 

add a comment | 

 

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