Problem: Let $p$ be an odd prime, let $n>2$, and let $M(\mathbb{Z}/p,n)$ be the cofibre of the degree $p$ map $f:S^n \rightarrow S^n$. In what range of dimensions (depending on $n$) does $f$ induce multiplication by $p$ on homotopy groups?
My first question is how is $M(\mathbb{Z}/p,n)$ the cofibre? I thought the cofibre of $A \hookrightarrow X$ was $X/A$ always. Is $M(\mathbb{Z}/p,n)=S^n/pS^n$ ? Why or why not?
Wouldn't a map of degree $p$ induce multiplication by $p$ on all homotopy groups? Based on how the question is worded, I'm sure I don't have the right idea here. Does this have something to do with the stable homotopy groups of spheres?