# Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?

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.

• If the differential is nonzero on two linearly independent classes, then each of those classes corresponds to a different summand. So I suppose you want to know the rank of the differential as a linear map; the rank should be the number of summands. Nov 2, 2017 at 14:29
• Thanks. Is there any reference that mentions this? I checked McCleary's book, it doesn't seem to have it. Nov 2, 2017 at 14:32
• @JohnPalmieri Just to check regarding the dimension. Is the following true: if there is one summand $\mathbb{Z}/p$ in $d^r E^r_n$, then there is one summand $\mathbb{Z}/p^r$ in $H_{n-1}(X)$. Is that correct? Thanks! Nov 3, 2017 at 2:11
• I'm not sure about the grading. Regarding references, look at the description of the spectral sequence in Mosher and Tangora, Cohomology Operations and Applications in Homotopy Theory. Nov 3, 2017 at 18:01

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 $H_*(X;\mathbb{Z})$.