Subject: \Omega\eta
Date: Fri, 20 Feb 2004 10:12:18 -0500
From: Tom Goodwillie <tomg@math.brown.edu>

I have a retraction to make, too. It's a retraction from \Omega S^2
to  \Omega S^3.

We know a number of ways to make a map ("Hopf map")

        \Omega S^2 --H--> \Omega S^3

inducing an isomorphism on \pi_2.

As I argued in my previous post, any such H must be such that the composed map

        \Omega S^3 --{\Omega\eta}--> \Omega S^2 --H--> \Omega S^3

is an equivalence (not just 2-locally) because of the structure of
the integral cohomology ring of \Omega S^3.

But if something composed with {\Omega\eta} is an isomorphism in the
homotopy category then something else composed with {\Omega\eta} is
the identity.

Explicitly,

        {H {\Omega\eta}}^{-1} H

is a left inverse for {\Omega\eta} even before looping.

        - Tom Goodwillie