Subject: I was wrong
Date: Wed, 9 Oct 2002 22:40:09 -0400
From: Tom Goodwillie
OK, my answer to Belegradek last week was all wrong.
(I thought I knew the Whitney trick pretty well,
but it seems that I've been warped by too many years
of mostly working with codimension >2 phenomena.)
Let me set the record straight.
The question had to do with this:
It's known (Wall) that if W is a compact smooth manifold
of dimension n and M is a codimension zero submanifold of
its boundary such that the pair (W,M) is r-connected
then W may be obtained from MxI by attaching handles of
index > r, provided r5.
Question: Are these dimension restrictions necessary?
I thought the answer was no, r=n-3 will work.
Recall the usual strategy for proving the h-cobordism theorem.
(although Belegradek wasn't asking about h-cobordisms;
he was asking about cobordisms that are highly connected
relative to one end.)
First eliminate the 0-handles, then the 1-handles, and so on,
stopping at some index k. Working from the other end, cancel n-handles
then n-1 and so on. This gets you to the situation of
just two indices k and k+1. Except for the cases of index 0 and 1
(and n and n-1) the key is the Whitney trick, which allows you to cancel an
r-handle and an (r+1)-handle by arranging an r-sphere in an
(n-1)-manifold in such a way that it meets a certain codimension r sphere once
transversely and others not at all (if the intersection numbers
are favorable).
I thought that since this argument even works when r=2 it must also
work when n-1-r=2; in both cases you are working with 2-spheres and
codimension 2 spheres. That's where I was wrong. In the r=2 case
you have some extra info, namely that cutting out those codimension 2
spheres does not alter the fundamental group of the (n-1)-manifold.
(This has to do with the fact that the 2-handles are attached trivially
to MxI.) In the r=n-3 case you don't have that.
How to make counterexamples to the assertion I made:
If you know a finite 3-dimensional CW complex such that its cohomology
vanishes above dimension 2 but it is not homotopy equivalent to a 2-dimensional
complex, then by taking n big enough you get a compact n-manifold
of that homotopy type which can be built out of handles of index 0, 1, 2, 3
but not 0, 1, 2.
(I don't know an example of such a complex, but I have a feeling they are
"well-known". Are they?)
Note that for any k>2 it is a theorem (of Wall again, maybe?)
that every CW complex whose cohomology vanishes above k (for all coeffs) is
equivalent to a k-dimensional complex.
There is also a relative theorem of the same kind: a pair (X,A) with
cohomological dimension k>2 is equivalent rel A to a pair (Y,A) with
relative dimension k.
For this last statement but with k=2 I gather from Belegradek that
there are old counterexamples due to Casson if k=2.
In fact there are finite complexes A with perfect fundamental group
such that the plus construction on A cannot be achieved with just
1- and 2-cells, because the group cannot be killed by adding a number of new
generators and an equal number of new relations.
Getting back to the Whiney trick stuff, I want to repeat that
sometimes, even in a simply connected manifold M, an embedded 2-sphere S
cannot be isotoped disjoint from a codimension 2 manifold T even if
there is no homology obstruction. And I want to point out that
the reason is that there can be an obstruction to homotoping the map
S -> M into M-T; pi_2(M,M-T) can be nonabelian and bigger than H_2(M,M-T),
even if M is 1-connected.
Tom Goodwillie