Diffeomorphisms and regular values

[TheHup]

Homework Statement

1) If f:M -> N, where M is a compact and boundaryless manifold, N is a connected manifold has regular values y and z. And h:N -> N is a diffeomorphism which is smoothly isoptopic to the identity and carries y to z, then why is z a regular value of the composition h o f?

Homework Equations

The Attempt at a Solution

1) Have been trying to think it through but it's just not clicking.

 

[lavinia]

Homework Statement

1) If f:M -> N, where M is a compact and boundaryless manifold, N is a connected manifold has regular values y and z. And h:N -> N is a diffeomorphism which is smoothly isoptopic to the identity and carries y to z, then why is z a regular value of the composition h o f?

2) Also if f,g:M -> N (M,N as above) are smoothly homotopic to each either, then why does Sard's Theorem imply that there exists an element y in N which is a regular value for both f and g?

Homework Equations

The Attempt at a Solution

1) Have been trying to think it through but it's just not clicking.

2) I know Sard's theorem tells me that the set of regular values of f is dense in N. Similarly for g, however the way I've had dense sets defined to me isn't convincing me that there has to be a point that is a regular value for both f and g.

1) dh is an isomorphism of tangent spaces. So dim ker f is preserved by dh.

2) the regular values of f and g are not only dense but they are open. I think this is enough to prove it.

 

[TheHup]

1) dh is an isomorphism of tangent spaces. So dim ker f is preserved by dh.

2) the regular values of f and g are not only dense but they are open. I think this is enough to prove it.

Thanks for the help.

For 1) would it be correct to say that d(h o f) = dh o df.

df is non-singular as y is a regular value and as dh is an isomorphism of tangent spaces when it is applied to df it remains non-singular. (This maybe makes no sense at all linear algebra has always been a failing for me).

2) Is the regular values being open a consequence of Sard's theorem or something else?

Thanks again.

 

[lavinia]

Thanks for the help.


2) Is the regular values being open a consequence of Sard's theorem or something else?

Thanks again.

it is the continuity of the derivative, df. If f is smooth then so is its differential so if it is non-singular at a point it must be non singular in a neighborhood of that point. Since the manifold is compact there are only finitely many preimages so there is an open neighborhood of a regular value consisting totally of regular values.