On the same N-type conjecture for the suspension of the infinite complex projective space

Let [Φ_ik, [Φ_ik-1,..., [Φ_i1, Φ_i2],...]] be an iterated commutator of self-maps Φ_ij. on the suspension of the infinite complex projective space. In this paper, we produce useful self-maps of the form I +[Φ_ik, [Φ_ik-1,..., [Φ_i1, Φ_i2],...]], where + means the addition of maps on the suspension structure of ΣℂP^∞. We then give the answer to the conjecture saying that the set of all the same homotopy n-types of the suspension of the infinite complex projective space is the one element set consisting of a single homotopy type.