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.



Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
Each element in the image of $d\_n$ corresponds to an element of order $p^n$ in the homology with integer coefficients. So it suffices to compute the image
of $d\_1=\beta$ on $H\_\*(X;\mathbb{Z}/p)$ to determine the number of summands isomorphic
to $Z/p$ in $H\_\*(X;\mathbb{Z})$.


