# Bockstein homomorphism and Square Operations: Their consistency formulas

Here are various ways to define "Bockstein homomorphism:"

1. Let $$\beta_p:H^*(-,\mathbb{Z}_p) \to H^{*+1}(-,\mathbb{Z}_p)$$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}_p\to\mathbb{Z}_{p^2}\to\mathbb{Z}_p,$$ it is an element of the mod $$p$$ Steenrod algebra $$A_p$$ where $$p$$ is a prime. If $$p=2$$, then $$\beta_2=Sq^1$$.

2. Let $$\beta_p'$$:$$H^*(-,\mathbb Z_p)\to H^{*+1}(-,\mathbb Z)$$ be the Bockstein homomorphism associated to the extension $$\mathbb Z\stackrel{\cdot p}{\to}\mathbb Z\to\mathbb Z_p.$$

3. Let $$\beta_{2^n}: H^*(-,\mathbb Z_{2^n})\to H^{*+1}(-,\mathbb Z_2)$$ be the Bockstein homomorphism associated to the extension $$\mathbb Z_2\to\mathbb Z_{2^{n+1}}\to\mathbb Z_{2^n}.$$ By the naturality of connecting homomorphism, $$\beta_{2^{n+k}}\cdot2^k=\beta_{2^n}$$ where $$\cdot2^k: H^*(-,\mathbb Z_{2^n})\to H^*(-,\mathbb Z_{2^{n+k}})$$ is induced from $$\mathbb Z_{2^n}\stackrel{\cdot2^k}{\to}\mathbb Z_{2^{n+k}}$$.

• question (i) Since $$\beta_2=Sq^1$$ coincides with the Steenrod square, are there other additional coincidences of other "Generalized Square" (Pontryagin Square, Postnikov Square, etc) coincide with the above "Bockstein homomorphism" $$\beta_p$$, $$\beta_p'$$, $$\beta_{2^n}$$?

• question (ii) Are there useful consistency formulas for these above "Bockstein homomorphism"?

For example, for Steenrod square, the total Stiefel-Whitney class $$w=1+w_1+w_2+\cdots$$ is related to the total Wu class $$u=1+u_1+u_2+\cdots$$ through the total Steenrod square $$w=Sq(u),\ \ \ Sq=1+Sq^1+Sq^2+ \cdots .$$ Therefore, $$w_n=\sum_{i=0}^n Sq^i (u_{n-i})$$. The Steenrod squares have the following properties: $$Sq^i(x_j) =0, \ i>j, \ \ Sq^j(x_j) =x_jx_j, \ \ Sq^0=1,$$

Do we have something similar for thse "Bockstein homomorphism?" $$\beta_p$$, $$\beta_p'$$, $$\beta_{2^n}$$?

I'm not sure if this is what you are asking, but you get useful relations among the Bockstein operators whenever you have a map between short exact coefficient sequences. For example, using the map of short exact sequences $$\require{AMScd}$$ $$\begin{CD} \mathbb{Z} @>\cdot p>> \mathbb{Z} @>\rho_p>> \mathbb{Z}/p\\ @V \rho_p V V @VV \rho_{p^2} V @VV = V\\ \mathbb{Z}/p @>>\cdot p> \mathbb{Z}/p^2 @>>\rho_p> \mathbb{Z}/p \end{CD}$$ where $$\rho_n$$ denotes reduction mod $$n$$, and the associated map between long exact sequences of cohomology groups, one sees quickly that $$\beta_p = \rho_p\circ \beta_p'$$.
Similarly from the map of short exact sequences $$\require{AMScd}$$ $$\begin{CD} \mathbb{Z} @>\cdot 2^{k+1}>> \mathbb{Z} @>\rho_{2^{k+1}}>> \mathbb{Z}/2^{k+1}\\ @V \cdot 2 V V @VV = V @VV \rho_{2^k} V\\ \mathbb{Z} @>>\cdot 2^k> \mathbb{Z} @>>\rho_{2^k}> \mathbb{Z}/2^k \end{CD}$$ one can deduce that $$2\beta_{2^{k+1}}' = \beta_{2^k}'\circ \rho_{2^k}$$. Thus for the Pontrjagin square $$P_k:H^{2n}(X;\mathbb{Z}/2^k)\to H^{4n}(X;\mathbb{Z}/2^{k+1}),$$ which satisfies $$\rho_{2^k}\circ P_k(x) = x^2$$ for all $$x\in H^{2n}(X;\mathbb{Z}/2^k)$$, one sees that $$2\beta_{2^{k+1}}'\circ P_k(x) = \beta_{2^k}'(x^2)$$, giving a partial answer to your question (i).