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I got stuck when reading Differential topology 46 years later in the last section of the article ("Further details"). It is a summary of what is known about stable homotopy groups of spheres $\Pi_n$. It is stated that "the most difficult part is the $2$-primary component" and also $p$-primary components are mentioned.

What is a "$p$-primary component" and what does it have to do with a homotopy group of spheres? Recently I started reading different papers and articles about differential topology but it is the first time that I encounter this concept but it is not explained in the article.

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From Adams' Infinite Loop Spaces book: Let $X$ be a simply connected CW complex and let $A$ be a subring of the rational numbers. Then there is a simply connected $CW$-complex $Y$ and a universal map $i : X \rightarrow Y$ such that $i$ localizes homotopy, in the sense that $i_{*} : \pi_{r}(X) \rightarrow \pi_{r}(Y)$ induces an isomorphism $\pi_{r}(X) \otimes A \rightarrow \pi_{r}(Y)$. The term localization is used as this should remind you of localizations of modules.

Now let $p$ be a prime and let $A_{p} \subset \mathbb{Q}$ be the subring of all rationals with denominator not a multiple of $p$. Then for any group $G$, the group $G \otimes A_{p}$ consists of the $p$-torsion elements of $G$ only. Thus if we localize a topological space at this ring we obtain a space which we call "$X$ localised at $p$" whose homotopy groups consist of the $p$-torsion parts of the homotopy groups of $X$. By working one prime at a time you can calculate the entire torsion part of the homotopy groups of $X$. Since we know that the stable homotopy groups of spheres are finite for all $r > 0$, this allows them to be calculated if we know the $p$-primary parts.

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  • $\begingroup$ Here $A_p$ is what is called $p$-primary component? Thank you for also including a reference to a book. When you write we localize the topological space at this ring, are you talking of the homology (or homotopy?) groups of the space? I am familiar with the concept of localization of a module so I assume one produces a module from the topological space? $\endgroup$
    – snailspace
    Jul 11, 2013 at 12:53
  • $\begingroup$ I think the $p$-primary component refers to the associated space $Y$ constructed with the homotopy groups $\pi_{r}(X) \otimes A_{p}$. That is, $\pi_{r}(Y) = \pi_{R}(X)$ considered as a $\mathbb{Z}$-module and localized at $p$. Note that in Adams' book this is treated for homotopy and homology. $\endgroup$
    – tharris
    Jul 11, 2013 at 12:59

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