On 11/25/06, Daniel Asimov <dasimov@earthlink.net> wrote:
<< Finally, turning a previous comment on its head, this result seems to give a very cheap proof of a weaker version of the manifold embedding theorem: just triangulate a d-manifold arbitrarily densely, then continuity ensures it can be embedded in (2d+1)-space! [Have I overlooked anything here?] Now then, how might the extra dimension be unloaded, I wonder ...
Only that with a little more work, it can be seen that d-manifolds embed in 2d-dimensional space.
That's what I meant --- but for instance does any sufficiently dense triangulation of a given manifold also embed in 2d-space? Or does the extra dimension somehow shrivel away faster than the others, say as the square of the edge-length?
But I suspct there's a simplicial complex whose highest-dimensional simplex is d, but which doesn't embed in 2d-space.
Try the set of d-faces of a (2d+2)-simplex --- e.g. for d = 1, the 10 edges of a pentatope constitute a non-planar graph (Kuratowski). But don't ask me to prove it ... WFL