From: Tom Goodwillie <tomg@math.brown.edu>
Subject: Re: suspensions
Date: Tue, 6 Feb 2001 04:40:37 +0000


>
>Is the reduced suspension of a path connected space always simply
>connected?
>
>Brayton Gray

No.

Let X be a based space.

Let CX be the unreduced cone on X.
It is a based space containing X as a based subspace.

Let SCX be the reduced suspension of CX.

CX is always contractible, but SCX is not always simply connected.

In fact, SCX is the union of two open sets U and V as follows:

U is the complement of one point in SCX, and it contains SX as a
deformation retract.

V is the complement of SX in SCX, and it is contractible; it is
homeomorphic to the product of the open cone on X with the line.

the intersection of U and V is (weakly?) homotopy equivalent to the unreduced
suspension of CX.

So for example if X is the set {0,1,1/2,1/3,...,1/n,...} (making SX the
good old Hawaiian earring), then H_1(U intersect V) is countable while
H_1(U) is
uncountable, so H_1(SCX) is nontrivial.)

Tom Goodwillie

Tom Goodwillie