TITLE: Order of the identity map of a Moore space.
QUESTION [8 upvotes]: Write $M_n = S^n \cup_2 D^{n+1}$.  I know, as a matter of folklore, that the identity map $\mathrm{id}_M$, considered as an element of the group $[M_n, M_n]$, has order $4$ (for $n > 3$, let's say).  
What is a reference for this?  And is there a simple and pretty argument?

REPLY [6 votes]: I don't know who first proved this fact, but a standard reference would be this paper of Toda's. It is stated as Theorem 4.1. Toda's proof is essentially the same as Tom's answer.