- #1

Kreizhn

- 743

- 1

## Homework Statement

Is there a non-orientable compact m-dimensional boundry-less submanifold of [itex]\mathbb{R}^{m+1}[/itex]?

## The Attempt at a Solution

It should be noted that in the context of the situation, we've assumed that the manifolds we're dealing with are Hausdorff.

But I'm wondering if this isn't a trick question since all compact subspaces of a Hausdorff space are necessarily closed, and as such have a boundary.

And if it isn't a trick question, does anybody have any clues as how I can go about showing that there is/isn't one. I suspect that there isn't since m-dimensional submanifolds are often hypersurfaces that can be expressed as the preimage of a regular point of a homogeneous polynomial, and are therefore automatically orientable.

I've also thought about defining an orientation preserving map between the submanifold, say M, and [itex]\mathbb{R}^{m+1} [/itex] via [itex]\{v_1, ... , v_m\} \rightarrow \{v_1, ..., v_m, N(p) \}[/itex] where N(p) is a normal vector field at a point p mapping M to the tangent bundle. This would show that non-orientability in M would imply non-orientability of [itex]\mathbb{R}^{m+1}[/itex], a contradiction. The only problem here is that we would require N(p) to be everywhere non-vanishing.

Any ideas?