Response to Frank Connolly's comments on embeddings from last week.....DMD ______________________________ From: Mark Mahowald Subject: Re: embeddings Date: Mon, 22 Jan 2001 09:54:10 -0600 (CST) Frank, I found the reference I wanted. Massey and Peterson, in a paper submitted to the Mexican Bol. in the early 60's and titled,"On the dual Stiefel-Whitney classes" make the following claims: Among the more striking results obtained by combining our theorems with those of Haefliger and Hirsch are the following: (1) A compact, orientable, differentiable n-manifold can be imbedded differentially in Euclidean (2n-1)-space (with the possible exception of the case n = 4). (2) If n is not a power of 2, then any compact, non-orientable n-manifold can be differentially imbedded in Euclidean (2n-1)-space (with the possible exception of n = 3). (3) A compact, simply connected n-manifold can be differentially imbedded in Euclidean (2n-2) space provided n is not of the form 2^k or 2^k + 1 (with possible exception of the case n = 6). The results of Haefliger are in his major Comm. Hev. paper of 1961. The final results are in Haefliger and Hirsch, Vol. 2 of Topology. In this last paper they prove that the obstruction to embedding in 2n-1 is the dual w_{n-1} and Massey and Peterson settle that. So I did remember things correctly (except for the n /ne 4). Mark