Two responses to embedding question........DMD ___________________________________ Date: Thu, 18 Jan 2001 09:15:30 -0500 From: John Klein Subject: Re: embedding question I don't know the most general result, but I do know the most general conjecture for this class of manifolds: the Hirsch conjecture says that M^n ought to embed in (roughly) R^{3/2 n}. If I remember correctly, the following reference establishes the Hirsch conjecture when M is [n/4]-connected: De Sapio, Rodolfo Embedding $\pi $-manifolds. Ann. of Math. (2) 82 1965 213--224. I don't know if the status has improved since the 1960s. John ________________________________ From: Mark Mahowald Subject: Re: embedding question Date: Thu, 18 Jan 2001 11:49:32 -0600 (CST) It is clear from Whitney's embedding result that such an M^n would embed in R^{2n-1} since it is orientable. RP^3 shows that such a result is best possible. It is possible to play with other examples. For example, if one desuspends the spherical tangent bundle to S^15 as far as possible, on gets a 21 dimensional manifold whose best embedding dimension (with trivial normal bundle, at least) is R^{21+9}. Examples of this sort are in some papers of Hsiang and Szczarba (Ann of Math vol 80). There are some results (unpublished) concerning embedding exotic spheres in dimensions just below the 3/2 dimension. The general question, though, is the statement about RP^3 Mark Mahowald > > Subject: Embeddings > From: Bill Dwyer > Date: 17 Jan 2001 15:52:20 -0500 > > One of my colleagues had a question for me. > It's this. Suppose that M is a stably > parallelizable manifold of dimension n. What's the most you can say > about the embedding dimension of M? Off the top of my head I couldn't > think of any way in which the stable parallelizability would let you > improve on 2n. > > Regards, > > Bill