Hatcher, Corollary 3E.4

Question

http://pi.math.cornell.edu/~hatcher/AT/AT.pdf is a link to the book, for convenience.

How does the underlined statement hold? Here $\rho$ is the ring homomorphism induced by the homomorphism $\Bbb Z\to \Bbb Z_p$.

Answer 1

You know that one has the LES $$\to H^*(X;\Bbb Z) \to H^*(X;\Bbb Z) \to H^*(X;\Bbb Z/p) \to H^{*+1}(X;\Bbb Z) \to$$ The first map is multiplication by $p$, the next map is reduction mod p $\rho$, the last map is the bockstein $\beta$.

If there is an integral class $x$ which is order $p^2$, then $\rho(px) = 0$ by exactness, where $px$ is a p-torsion class. So the existence of $p^2$-torsion implies that $\rho$ is not injective on p-torsion.

Conversely if $\rho$ fails to be injective on p-torsion (say $\rho(x) = 0$) then exactness in the middle says that there is a class $y$ with $x = py$. Thus $y$ is a class of order $p^2$.

This proves the equivalence.