I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.

How about for multiple summands in the integral homology, say $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. How does the Bockstein Spectral Sequence detect them? Merely knowledge that the differential is nonzero doesn't seem to be enough to detect the multiple summand.

Thanks for any help.

Cohomology Operations and Applications in Homotopy Theory. $\endgroup$ – John Palmieri Nov 3 '17 at 18:01