Subject: Re: 2 responses and question Date: Fri, 13 Feb 2004 14:50:40 -0600 From: Brayton Gray To: Don Davis CC: Brayton Gray > >> From: "Neil Strickland" >> >> Let \eta : S^3 --> S^2 be the Hopf map, and let >> H : \Omega S^2 --> \Omega S^3 be the Hopf invariant. >> The map >> >> \Omega^3\eta >> \Omega^3 S^3 ---------------> \Omega^3 S^2 >> >> is an equivalence, and the map >> >> \Omega^2 H >> \Omega^3 S^2 ---------------> \Omega^3 S^3 >> >> is a 2-local equivalence. Can anyone tell me if the >> composite is 2-locally homotopic to the identity? >> >> Neil >> This certainly depends on what you mean by the Hopf invariant. Even the James Hopf invariant is dependent on a choice of lexicographic ordering, of which there are at least 2. As a variation on the second Hopf invariant(H_2), for example, we could add the composition of \Omega\eta^2 with H_4, the fourth James Hopf invariant. This would add a term to the composition, which in homotopy would add \eta^3 composed with H_4. I don't have an example off hand where this perturbation is not zero, but suspect that such examples exist.