On 11/25/06, Bill Thurston <wpt4@cornell.edu> wrote:
On Nov 24, 2006, at 8:54 PM, Fred lunnon wrote:
I should have added that if you have a countable d-dimensional simplicial complex, then a random map of the vertices into R^(2d+1) extends to an affine embedding. Bill
Is this obvious [tho not to me] or are you quoting a theorem? Does d = k here? WFL
This is obvious once you've met an argument of this sort. ...
Sure --- once I'd managed to fill in the line missing from the previous post, I realised what you'd been trying to say. All the same, I'd like to see some more work to provide a geometric or algebraic construction. For example, each time we need to choose a new vertex, we can make it the centroid of one of the convex hulls into which the set of all primes (hyperplanes) partitions the space; except for multiple intersections at the previously chosen vertices, all these hulls have properly of maximum dimension, by induction. Alternatively, we might ask for an explicit d x oo matrix such that every subset of d rows has maximum rank. I can't see how to do this off-hand. 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 ... Fred Lunnon