Homotopy and orientation preserving

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  • #1
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This is actually a problem from Lee's Introduction to smooth manifolds 14-21:
Let M and N be compact, connected, oriented, smooth manifolds. and suppose F,
G:M->N are diffeomorphisms. If F and G are homotopic, show that they are either
both orientation-preserving or both orientation-reversing.
The hint given in book suggests to use Whitney approximation and Stokes' Theorem
on MxI to prove, however I don't see how should I apply both theorems to solve the prob.
 

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  • #2
mathwonk
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it helps to know your definitions of orientation preserving and so on. I.e. the quick proof apparently using words you don't have, is that homotopic maps induce the same map on homology, and that determines orientation behavior according to whether the preferred top homology generator goes to the preferred one of the other manifold, or not.
 
  • #3
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orientation preserving is defined as if for each p in M, F_* takes the oriented bases of TpM to oriented bases of TF(p)N
And I don't think it requires to use homology related concepts to solve this problem
 
  • #4
mathwonk
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well your definition only make sense for differentiable maps whereas homotopy makes sense for continuous maps. Are your homotopies differentiable? i.e. what is the definition for a homotopy for you?
 
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I think here the homotopy should be smooth as well, since both F and G are diffeomorphisms.
 
  • #6
quasar987
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Here however, you will no doubt want to use the characterization of orientation preserving diffeomorphism according to which a diffeo is orientation preserving iff its pullback takes the orientation form to some positive multiple of the orientation form.
 
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But I just don't see what's the point of using Whitney approximation theorem in this prob.
 
  • #8
quasar987
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You must not assume that the homotopy between F and G is smooth.

But assume it is. Then can you do the problem? Let H be such a smooth homotopy between F and G, and let [itex]\Omega_N[/itex] be an orientation form on N. Compute the integral of [itex]H^*\Omega_N[/itex] on [itex]\partial(I\times M)=\{1\}\times M - \{0\}\times M[/itex].

For the general case, approximate H by a smooth homotopy H'. That is to say, by Whitney's approximation theorem, there exists a homotopy J: I x I x M between H and some smooth homotopy H'(t,x)=J(1,t,x). Since H is already smooth on the closed set [itex]A:=\{1\}\times M \cup \{0\}\times M[/itex], the homotopy J can be taken to be rel A, meaning in particular that H'(0,.)=F and H'(1,.)=G. So now you've reduced to the previous case.
 
  • #9
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Thanks for clarification on smooth homotopy part, then with smooth homotopy established, we just use Stokes's theorem, since any n-form on N representing orientation will be closed, since N is compact Hausdorff space, then d\Omega_N=0, which gives integral of F^*\Omega_N on M is equal to G^*\Omega_N on M, and since M is connected, then F and G must both preserve orientation or reverse it.
 
  • #10
mathwonk
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An orientation form is the smooth version of a generator of top cohomology (via deRham cohomology), and Stokes' theorem is the usual tool to show (smooth) homotopy invariance. So this is the smooth version of the same argument I gave.
 
  • #11
I want to know whether homotopy analysis method (HAM) is a localize method or globalize method?
I also want to know how one expresses the nonlinear equation in terms of set of base functions in homotopy analysis method?
Is it true that in HAM we choose linear part of equation as a linear operator & convergence region of h cut can extend from -2 to 0.
 
  • #12
IN homotopy analysis method how one can define the set of base functions? what is criterion on which we choose base functions?
IS HAM A LOCALIZE OR A GLOBALIZE METHOD?
 
  • #13
lavinia
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This is actually a problem from Lee's Introduction to smooth manifolds 14-21:
Let M and N be compact, connected, oriented, smooth manifolds. and suppose F,
G:M->N are diffeomorphisms. If F and G are homotopic, show that they are either
both orientation-preserving or both orientation-reversing.
The hint given in book suggests to use Whitney approximation and Stokes' Theorem
on MxI to prove, however I don't see how should I apply both theorems to solve the prob.

There are several proofs that rely on the theorem that the homotopy between the two maps can be chosen to be smooth. Once you agree with that then the proof depends upon which definition of orientation that you use.
 

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