On 11/24/06, Bill Thurston <wpt4@cornell.edu> wrote:
Any finite *simplicial* complex with n vertices is a subcomplex of the (n-1)-simplex, since each simplex in the complex is determied by its set of vertices. Therefore it embeds affinely in R^(n-1).
Good point, which gets obscured by talking about identifying vertices ...
It seems that by a general position argument you can embed a k- dimensional complex affinely in (2k+1)-space. You just need to make sure that a bunch of linear subspaces don't have
Something gone missing here! But yes, e.g. a complex of vertices and edges is surely embeddable in 3-space. Since this number is one greater than the dimension in the theorem Dan quoted about embedding a k-manifold, is it possible to employ a similar argument to prove it?
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