Subject: cones and suspensions
Date: Thu, 8 Feb 2001 04:58:41 +0000
From: Tom Goodwillie <tomg@math.brown.edu>
To: dmd1@Lehigh.EDU (DON DAVIS)

(OK, I promise this is my last word on the subject! Couldn't resist sharing
this new improved version; in the theorem below I'm saying homeomorphic
where I previously said weakly homotopy equivalent.)

Here is a contractible space whose reduced suspension is not simply connected.

(Of course, to speak of the reduced suspension I need the space to be
based; but  by "contractible" I mean in the unbased sense; if the space
were based contractible then the same would be true of its suspension.)

The example will be the unreduced cone CX on a certain space X.

Theorem: For any based space X, the reduced suspension of the cone on X is
naturally homeomorphic to the mapping cone of the canonical map from the
unreduced suspension of X to the reduced suspension of X.

(So to get the desired example you need a space X such that that map from
one suspension to the other is bad. I choose X={0,1,1/2,1/3,...}.)

Recall that CX is defined as the pushout of

* <- 0xX -> IxX.

If X is nonempty then CX is a quotient of X. CX is always contractible; in
fact it is based contractible if the basepoint is taken to be the "cone
point". But my X has a basepoint x, and I am using (1,x) as basepoint in
CX.

The unreduced suspension SY of a space Y is the pushout of

* <- 0xY -> IxY <- 1xY -> *.

If Y is nonempty then SY is a quotient of IxY. If Y has basepoint y then
the reduced suspension can be defined as a quotient of SY by identifying
Ix{y} to a point.

In particular the reduced suspension of CX can be identified with a
quotient of IxCX, therefore with a quotient of IxIxX.

On the other hand, that mapping cone mentioned in the theorem is a quotient
of IxSX, therefore a quotient of IxIxX. Specifically, it is what you get if
you identify Ix0xX to X and Ix1xX to another copy of X and 1xIx{x} to a
point.

Now, I claim that each of these two quotients of IxIxX may be identified
with the following: Let D be a closed 2-disk whose boundary consists of two
arcs A and B meeting only at two points p and q. Make the pushout of

A <- AxX -> DxX <- Bx{x} -> *

Our first quotient of IxIxX is homeomorphic with this quotient of DxX by
means of a map from IxI to D that takes Ix0 to A, Ix1 to B, 0xI to p, and
1xI to q, and the interior to the interior.

Our second quotient of IxIxX is homeomorphic with this quotient of DxX by
means of a map from IxI to D that takes 1xI to B, Ix0 to half of A, Ix1 to
the other half of A, 0xI to a point in the middle of A, and the interior to
the interior.

Tom Goodwillie

Tom Goodwillie