Subject: RE: 2 questions
Date: Sat, 5 Oct 2002 15:06:00 +0800
From: "Berrick A J" <matberic@nus.edu.sg>
To: "Don Davis" <dmd1@lehigh.edu>

Dear Don,

        A response to Tom Goodwillie's question (assuming I've correctly read an old paper of mine).  Applying the plus-construction to  K(G,1)  for the torsion-free group  G  below yields  K(Z,2) .  So  G  has the required cohomology ring, more generally than just over the integers.

        The group  G  is a split extension of groups as follows.  Let  Q'  be the linearly ordered set of rational numbers strictly between  0  and  1 , and let  A  be the group of all finitely supported functions from  Q'  to  Z  (infinite cyclic group).  Then the split extension defining  G  has as kernel  A x A  and as quotient the upper-unitriangular group of integral matrices whose entries are indexed by the order  Q' .

        For further details and arguments, see
A J Berrick: Two functors from abelian groups to perfect groups, J Pure Applied Algebra 44 (1987), 35-43.

Best regards,
Jon.
_____________________________________________________________________

Prof A J Berrick
Department of Mathematics                 Tel. : +65 6874 2747
National University of Singapore          Fax  : +65 6779 5452
2 Science Drive 2
Singapore 117543                             Email: berrick@math.nus.edu.sg
SINGAPORE                            http://www.math.nus.edu.sg/~matberic/
_____________________________________________________________________

-----Original Message-----
From: Don Davis [mailto:dmd1@lehigh.edu]
Sent: Thursday, October 03, 2002 8:44 PM
To: (Don Davis)
Subject: 2 questions

Two new questions posted today..........DMD
_________________________________________________

Subject: question for list
Date: Wed, 2 Oct 2002 12:15:53 -0400 (EDT)
From: Tom Goodwillie <tomg@math.brown.edu>

Is there a torsion-free discrete group G such that the
integral cohomology ring of BG is isomorphic to that of
infinite complex projective space?

Tom Goodwillie
___________________________________________________

Subject: question for discussion list
Date: Wed, 2 Oct 2002 17:54:27 -0700 (PDT)
From: Igor Belegradek <ibeleg@its.caltech.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.

Thank you very much,

Igor Belegradek
ibeleg@caltech.edu