Subject: Re: two postings From: Mark Hovey Date: 08 Jul 2005 07:22:14 -0400 This is a reply to John Klein's question. Recall John Klein asked for which spectra E we have F(E,E)=maps(E,E) being a sphere. I have a partial answer. I will assume we are in the p-local stable hmootopy category for some p. 1. Let E be the Brown-Comenetz dual of the sphere. Then F(E,E) is the p-completion of the sphere. This is an indication that there might be exotic answers to John's question, though I have not found them. 2. If F(E,E) is any sphere at all, it must be the 0-sphere S, and the unit map S --> F(E,E) of the ring spectrum F(E,E) must be an equivalence. This is because the only sphere that can be a ring spectrum is S and it can only be a ring spectrum in one way. 3. If E is an R-module spectrum for some ring spectrum R, then the unit map pi_* S --> pi_* R is a split monomorphism of rings. The retraction is given by pi_* R --> pi_* F(E,E) = pi_* S. The map pi_* R --> pi_* F(E,E) takes v to the self-map of E induced by smashing with v. This of course rules out massive numbers of potential E, like all BP-module spectra, for example. 4. In particular, if E is a ring spectrum, then E=S. (because then the retraction pi_* E --> pi_* F(E,E) is a monomorphism). 5. I have also been able to prove that the rational homology of E is just Q concentrated in a single degree, but I am not sure this helps. Mark Hovey