Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by Postnikov (1949). Eilenberg (1952) described a generalization taking classes in $H^t$ to $H^{2t+1}$.

I checked Eilenberg's paper, I find that the generalized Postnikov square requires that $t$ is odd.

In Browder and Thomas's paper: Axioms for the generalized Pontryagin cohomology operations (https://www.maths.ed.ac.uk/~v1ranick/papers/browthom.pdf), they also defined a generalized Postnikov square $$\mathfrak{P}:H^{2n}(-,\mathbb{Z}/2^r)\to H^{4n+1}(-,\mathbb{Z}/{2^{r+1}}).$$

My question: Is there a generalized Postnikov square $$\mathfrak{P}:H^t(-,\mathbb{Z}/p^r)\to H^{2t+1}(-,\mathbb{Z}/{p^{r+1}})$$ for even $t$ and odd prime $p$?

Thank you!

  • can you give Refs for Postnikov (1949). Eilenberg (1952)? – annie heart Oct 23 at 4:04
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    I'm not clear on what properties you want. Browder and Thomas give an explicit formula for their Pontrjagin square: it sends $u$ to $\phi(u \beta u)$ where $\beta$ is the Bockstein and $\phi$ is the inclusion of $\Bbb Z/2^r$ into $\Bbb Z/2^{r+1}$. This formula is still valid at odd primes. In fact, for $t < 2p-3$ the only cohomology operations $H^t \to H^{2t+1}$ are of the form $u \mapsto k \beta'(u^2)$ for some scalar $k$, where $\beta'$ is the Bockstein associated to $0 \to \Bbb Z/p^{r+1} \to \Bbb Z/p^{2r+1} \to \Bbb Z/p^r \to 0$. What do you need from the Postnikov square? – Tyler Lawson Oct 26 at 10:37
  • @TylerLawson Thank you very much! I want to know whether Postnikov square can be defined for even $t$ and odd prime $p$. Can you elaborate on the fact you mentioned above? – Zheyan Wan Oct 26 at 11:15
  • @Tyler Lawson, do you have any Refs? – annie heart Oct 26 at 15:10

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