From: John Baez <baez@math.mit.edu>
Newsgroups: sci.math.research
Subject: Re: framed manifolds
Date: Wed, 11 Jan 95 18:17:08 GMT

In article <1995Jan10.223029.8916@galois.mit.edu>,
John Baez  <baez@math.mit.edu> wrote:
>Is it true that any (smooth, compact) n-manifold whose normal bundle
>is stably trivial can be embedded in R^{n+k} in such a manner that its 
>normal bundle is trivial, when k >= n+2?  

I did a terrible job here of asking the question I was really
interested in, but Steve Ferry still managed to help me out.  He told
me that if a k-dimensional bundle over an n-complex is stably trivial,
then it is trivial when k>n.  But there's something more fancy 
that seems to require k >= n+2, and let me try to explain it with 
an example where our n-manifold is a circle.

There are lots of non-isotopic framed embeddings of the circle in R^3,
where by framed embedding I mean an embedding together with a
trivialization of the normal bundle, and isotopy means the hopefully
obvious sort of smooth 1-parameter family of such framed embeddings.
(Note that my usage of "framing" here is not the common one in knot
theory, since my sort of framed embedding of a circle in R^3
determines a "framed oriented knot" in the knot theory jargon where a
"framing" simply refers to a homotopy class of nonzero *sections* of
the normal bundle, rather than an actual trivialization.  In fact,
isotopy classes of framed embeddings of the circle in R^3 should be
the same as isotopy classes of "framed oriented knots".)

Now such a framed embedding in R^3 automatically gives a framed
embedding in R^4 --- just stick on another coordinate.  We thus get a
map from {isotopy classes of framed embeddings of a circle in R^3} to
{isotopy classes of framed embeddings of a circle in R^4}.  This map
is very far from being one-to-one, since I think the latter set has
only two elements.  You can untie all knots in R^4, and I think there
are just two isotopy classes of framed embeddings of a circle in R^4,
represented by any embedding of the circle equipped with two different
trivializations of its normal bundle: such a trivialization gives an
element of pi_1(GL(3)) in a pretty natural way, and this is Z_2.

Now similarly we get maps 

F: {isotopy classes of framed embeddings of a circle in R^m} -> 
{isotopy classes of framed embeddings of a circle in R^{m+1}}

for all m, but I think these are 1-1 and onto for m >= 4.  Things settle
down, in other words, when there is enough room to maneuver.  

So more generally I'm wondering if

F: {isotopy classes of framed embeddings of X in R^{n+k}} -> 
{isotopy classes of framed embeddings of X in R^{n+k+1}}

is 1-1 and onto when k >= n+2, where X is a compact n-manifold.

Hmm, now that I've taken the trouble to figure out how to ask the
question correctly, I can probably figure out the answer.  Funny how
that works.  :-)  But anyway, that was supposed to be the warmup for
trickier questions where X represented a framed cobordism, or a sort
of "framed cobordism between framed cobordisms", and the embeddings
were into R^{n+k-1} x [0,1], or R^{n+k-2} x [0,1] x [0,1], in a manner
respecting that fact.