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. It's a generalization of the assertion that if m <= n, then there is a subset of m-tuples of vectors in R^n that are linearly independent, i.e. the rank of a random mXn matrix is min (m,n)---which easily follows from induction, if you have (m-1) linearly independent vectors, then the mth just has to avoid an m-dimensional plane spanned by the others. From this it follows that a random (k)-dimensional subspace and a random (m-k)-dimensional subspace intersect only at the origin --- just use the first k of your m-tuple to give one subspace, and the last (m-k) to give the other. If we switch to affine subspaces, algebraically given by linear combinations of vectors where the sum of the coefficients is 1, then the same argument says that a random set of (n+1) vectors is affinely independent, and two random affine subspaces of R^n whose dimensions sum to less than n are disjoint. More generally, if you look at pairs of affine subspaces A and B at random subject to the condition that there intersection contains a third subspace C, then if dim(A) + dim(B) - dim(C) < n there is a set of full measure that intersects only in C. So, among countable sequences of points in R^n, there is a subset of full measure such that all affine subspaces spanned by k-tuples are disjoint, provided k < n/ 2. This gives an embedding of any k-dimensional simplicial complex with vertex set these points. The set where this condition holds is also "generic" or Baire second category: an intersection of open dense subsets. Bill