Orientability of hypersurfaces in 𝑅ⁿ

Abstract

HANS SAMELSON1We give an elementary proof for the fact that a hypersurface of codimension 1 in R" is always orientable [a surface of the type we consider is, by definition, a closed, not necessarily compact, subset of Rn that near each of its points is the set of zeros of some realvalued C°°-function with nonzero gradient].Another way to say this: a nonorientable differentiable manifold (without boundary) of dimension ra -1 cannot be C°°-embedded as closed subset of R"; here "closed" is needed as the Möbius strip (without boundary) in R3 shows.This is usually proved (for compact manifolds, without any differentiability hypothesis) by invoking Alexander duality [2].Suppose we had such a nonorientable hypersurface M in R".We take "nonorientable" to mean: there exists a loop (closed curve) in M such that the normal to M, when transported around the loop in a continuous fashion, comes back with the opposite direction.By considering a point on the normal a small distance off M, moving it around the loop and then connecting along the normal from one side of M to the other, we manufacture a closed C°°-curve y in R" that meets M in exactly one point, and is transversal to M at this point [the tangent to y is not tangent to M], y can be contracted to a point in Rn; this amounts to a C°°-map / of the unit disk D2 into Rn that on the boundary Sl of D2 yields y.We now make / transversal to M, i.e., we bring it into "general position" relative to M, so that f(D2) is not tangent to M at any common point; this can be done without changing / on Sl, since / is already transversal there (see [l], [3], [4], [5]).[The general transversality theorem looks formidable, but is really quite simple, particularly in the case of codimension 1 ; one modifies /, locally, by adding suitable affine-linear functions, multiplied by cutoff functions; in the analogous piecewise linear case one shifts the ¡mages of the vertices of some triangulation of D2 into general position.]

Locations

• Proceedings of the American Mathematical Society - View - PDF