# Higher order differentials of Bockstein spectral sequence

The Bockstein SS is obtained from the exact sequence $$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$ with $$E_1^p=H^p(X,\mathbb{Z}/2)$$ and the differential $$d_1=Sq^1$$.

How to identify the differentials $$d_2$$ for the $$E_2$$-page without knowing $$H^*(X,\mathbb{Z})$$ in advance?

• $d_2$ is described by a “secondary” cohomology operation. You can read quite a bit about it in May’s A general algebraic approach to Steenrod operations (whose title I may be imperfectly remembering), where the relevant operation is called $\beta_2$. 2 days ago

The $$E_1$$ page does not tell you what the higher differentials will be, and you will have to know at least something about the integral cohomology. Consider the case when $$X$$ is a Moore space $$M(1,\mathbb{Z}/2^n)$$ which you may prefer to think of as a circle with a disc attached via a $$\times 2^n$$ map. In each case the $$E_1$$-page is the same, independent of $$n$$. If $$n=1$$, the differential $$d_1$$ is non-zero. If $$n>1$$ then the differential $$d_1=Sq^1$$ is zero. The only differential that is non-zero in this case is $$d^n$$. The cases $$n=2,3,4,\ldots$$ cannot be distinguished by knowing the $$E_1$$-page and $$d_1$$.
• Is the $d_2$ some kind of cohomological operation for which some references could be found? 2 days ago
• In general, higher differentials are higher cohomology operations, defined only on the kernels of the differentials below them. If $a$ is a mod-2 cohomology class with $Sq^1(a)=0$, this says that the coboundary $b$ of an integral lift of $a$ is divisible by~2. The secondary operation in this case looks at $b/2$ as a mod-2 cocycle; this is defined only because $b$ was 0 mod-2.