From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Forms as a function of dimensions
Date: 16 Jun 1999 05:05:40 GMT
Newsgroups: sci.math
Keywords: embedding dimensions of compact smooth manifolds into R^n

In article <7k6179$t9f@mcmail.cis.McMaster.CA>,
Zdislav V. Kovarik <kovarik@mcmail.cis.McMaster.CA> wrote:
> There is a theorem (Whitney) that (IIRC) every compact n-dimensional
>smooth manifold can be embedded in the (2*n+1) Euclidean space. 

Whitney later proved the rather harder result that every such  M^n  can
be embedded in  R^(2n). This explains this observation:
> This estimate is not sharp individually: Klein bottle can be embedded in
>the 4-space, although the general estimate calls for dimension 5.

Now, I guess I had always thought this bound to be optimal in general, while
I knew a sort of cottage industry had developed, computing the minimum
embedding dimensions of, say, M = RP^n. But in response to this question:

>Is this estimate sharp over the class of all such manifolds? And if it
>is, is there an "algorithmic" sequence of examples exhibiting this
>sharpness?

I did a quick search of MathSciNet and found only a good but dated
reference:

   47 #4275 57D40
   Gitler, S.
   Immersion and embedding of manifolds. Algebraic topology (Proc.
   Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970),
   pp. 87--96.
   Amer. Math. Soc., Providence, R.I., 1971.
   
   This is a comprehensive and very readable account of known results on
   the problem whether, for given $M\sp m$ and $N\sp n$, there exists an
   immersion (or an embedding) of $M$ in $N$. 
[...]
   In addition, the article is liberally sprinkled with conjectures, the
   most interesting of which are that every $n$-manifold immerses in
   ${R}\sp {2n-\alpha(n)}$ (where $\alpha(n)=$ the number of ones in the
   dyadic expansion of $n$) and embeds in ${R}\sp {2n-\alpha(n)+1}$.

                          Reviewed by E. C. Turner
   © Copyright American Mathematical Society 1999

If this last conjecture is true, it states the bound  "2n" is never
optimal except when  n  is a power of  2! 

I don't have those tables handy any more but IIRC the embedding dimensions
of projective spaces and related constructs are known and look something
like the conjectured minimum, so these would qualify as
'..."algorithmic" sequence of examples exhibiting this sharpness'.

dave