Orientability and Diffeomorphisms of Manifolds

[MathematicalPhysicist]

I have a few question, I hope you can help me on some of them.

1.Show that if A (subset of R^n) is a submanifold with dimension, n, with boundary then dA (the boundary of A) is orientable.
2. Show that a torus in R^3 is orientable.
3.Show that a mobius band isn't orientable.
4. Let M,N be two connected oriented manifolds. Let f:M->N be a diffeomorphism.
Show that $$df_x:T_x M\rightarrow T_f(x) N$$ either preserves or reverses orientation for all x in M simultaneously.

Here is what I thought of:
1)I think that the standard orientation on R^n is induced to the boundary of A.

4) I need to prove that the determinant of df_x is always positive or negative, now from the definition of orientation on M and N, we have two diffeomorphism $$\psi , \phi$$
such that for every x in M there's a neighbourhood U, such that: psi is a local diffeomorphism of U onto an open set V of R^N, and for every z in U $$d\psi_z : T_z M\rightarrow R^m$$ keeps the orientation, the same for N.
Now if f can be broken into two diffeomorphisms one from $$T_x M \rightarrow R^m$$
the other from $$T_f(x) N \rightarrow R^m$$, then the determinant of df_x would be equal the product of two determinants which both of them have a plus sign cause they keep the orientation.

2. a torus is $$S^1 x S^1$$ where S^1 is a circle, intuitively I understand why it's orintebale but how to prove it rigourosly?
I mean I think I need to show that if I induce the standrad orientation of R^3 onto the torus, it keeps orientation, not sure.

3. the same for 2, just inducing the standard orientation and to show the determinat changes signs from some point.

 

[MathematicalPhysicist]

for 4, I think it should be phrased differently, like this:
Suppose that f:M->N is a diffeomorphism of connected oriented manifolds with boundary, show that if $$df_x$$ for some point x in M, preserves orientation, then f preserves orientation globally.
(the first statement of problem was from my lecturer).

well I think this follows quite immediately if we look at first order approximation of f, like this:
$$f(x+x0)=df_x_0 x +o(||x||)$$
then if we take df_x at some point x in M (other than x0), then its determinant will have the same sign as of $$df_x_0$$ cause the other orders of the taylor expansion are o(1) and thus don't affect its sign.

I feel that it's mambo jambo, can anyone help me on this?

thanks in advance.

 

[olliemath]

1. Using the standard orientation to prove that all closed n-1 dimensional manifolds embedded in $$R^n$$ are orientable? Of course you must then prove that said boundaries are of this form.

2. First prove that $$S^1$$ is orientable.
Can you show that if $$M,N$$ are smooth manifolds then $$M\times N$$ has a canonical differentiable structure? What if the original manifolds were orientable? (This is a standard theorem).

3. I'll have a think :)

4. The restatement of the problem makes sense. Play around with the following..
Take a curve c passing through p, then $$df_p$$ maps the vector $$(\phi\circ c)'(0)$$ to $$(\psi\circ f\circ c)'(0)=(\psi\circ f\circ\phi^{-1}\circ\phi\circ c)'(0)=d(\psi\circ f\circ\phi^{-1})_{\phi(p)}\circ(\phi\circ c)'(0)$$ (where we're using appropriate charts). Because we're dealing with a diffeomorphism, $$d(\psi\circ f\circ\phi^{-1})$$ is invertible, so nonzero, and varies continuously through members of GL(n,R) as we go from point to point.

 

[maze]

For part 1, orientability is a topological invariant, right? If this is so, wouldn't it suffice to prove the result for very simple shapes such as a rectangular n-box with rectangular holes cut out of it?

Part 2 looks like a straightforward application of part 1.

Part 3. For a mobius band, one could take a triangulization with only a few triangles, and then use counting arguments on the edges and faces to show that at least 2 faces next to each other must be oriented oppositely. The orientation of a face may be expressed by directing its edges in a loop, and for it to be orientable is equivalent to having each edge that shares a face be directed oppositely, unlike gears. The outer edge will all have to go the same direction, which will wreak havok with any attempts to antialign the inner edges.
http://img166.imageshack.us/img166/7089/mobiusnonorientablezh8.png

I don't have enough background to understand what part 4 is asking.