Why Is the Inverse Image of a Regular Value in an Immersion a Finite Set?

[WWGD]

Hi:
This problem should be relatively simple, but I have been going in circles, without
figuring out a solution:

If f:X->R^2k is an immersion

and a is a regular value for the differential map F_*: T(X) -> R^2k, where

F(x,v) = df_x(v). Then show F^-1 (a) is a finite set.

I have tried using the differential topology def. of degree of a map , where we calculate

the degree by substracting the number of points where the Jacobian has negative

determinant (orientation-reversing) minus the values where JF has positive determinant.

I think I am close, but not there.

Any Ideas?

Thanks.

 

[WWGD]

Is it enough to say that for regular values, since f is a local diffeomorphism, it is

a covering map?. ( but we could always have infinitely many sheets in the cover...)

 

[wofsy]

Hi:
This problem should be relatively simple, but I have been going in circles, without
figuring out a solution:

If f:X->R^2k is an immersion

and a is a regular value for the differential map F_*: T(X) -> R^2k, where

F(x,v) = df_x(v). Then show F^-1 (a) is a finite set.

I have tried using the differential topology def. of degree of a map , where we calculate

the degree by substracting the number of points where the Jacobian has negative

determinant (orientation-reversing) minus the values where JF has positive determinant.

I think I am close, but not there.

Any Ideas?

Thanks.

Your question confuses me. An immersion I think by definition has all regular values. It Jacobian is a maximal rank everywhere. Maybe you are using a different definition of regular value.

You are right that since your map is a local diffeomorphism the inverse image of any point must be discrete. If X is compact then any discrete subset must be finite. If X is not compact this is not true. For instance the covering of the circle by the real line x -> exp(ix) is an immersion but the inverse image of each point is infinite.

 

[WWGD]

"Your question confuses me. An immersion I think by definition has all regular values. It Jacobian is a maximal rank everywhere. Maybe you are using a different definition of regular value."

I was just considering the immersion to be in "standard position" for the inverse image
of a regular value to be a manifold, but I admit I did not explain that clearly.

Still, please put up with some innacuracies for a while, since I am still an analyst in Algebraic topology exile. Hope not for too long