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Regular values of immersion. 
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#1
Mar1309, 12:45 AM

P: 574

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 (orientationreversing) minus the values where JF has positive determinant. I think I am close, but not there. Any Ideas? Thanks. 


#2
Mar1309, 12:49 AM

P: 574

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...) 


#3
Mar2009, 05:14 PM

P: 707

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. 


#4
Mar2009, 05:30 PM

P: 574

Regular values of immersion.
"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 


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