Subject: handles and connectivity
Date: Sun, 6 Oct 2002 19:42:07 -0400
From: Tom Goodwillie <tomg@math.brown.edu>
To: Don Davis <dmd1@lehigh.edu>

>
>Let M be a boundary component of a cobordism W such that the
>inclusion M-->W is r-connected. It is known that if r<dim(W)-3,
>then W can be obtained from M x I by attaching handles of
>indices >r.
>
>(See e.g. C. T. C. Wall, Geometrical connectivity I, J. London.
>Math. Society., 3, (1971), 597-604).
>
>Is the bound r<dim(W)-3 optimal? I am especially interested in
>the case r=1, dim(W)=4, but I wish to know about higher
>dimensions as well.
>

In higher dimensions (dim(W)>5) I'm quite sure that
the case r=dim(W)-3 is also OK, the argument being as in
the proof of the s-cobordism theorem: if there are no handles
of index <r then to set up index r+1 handles to cancel the
r-handles you use the Whitney trick in a codimension 1 submanifold
that lies between the index r and index r+1 handles.

Tom Goodwillie