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Such an $f$ only exists in two cases: when $k=1$ (when $\beta Sq^1 = 0$) and when $k=0$ (when $\beta Sq^0 = \beta$). Here is a proof, which will take a little work with the Steenrod algebra. Let's su …

The cup-i products are not associative for $i > 0$. For example, Steenrod's cup-1 product has the following description (taken mod 2 for expedience). For cocycles $f$ of degree $p$ and $g$ of degree $ …

As near as I've been able the find, the primary reference for a proof is probably Kristensen and Madsen's "On the structure of the operation algebra for certain cohomology theories." This result (in f …

This is not necessarily true. The first warning one should give is that there are multiple definitions of "$\cup_{i}$" on cocycles, and different ones satisfy different identities. However, this is un …

What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $\Sigma_2$ and operations, or the rel …

The behavior of the Steenrod squaring operations in the Serre spectral sequence was determined by Araki and independently by Vázquez (whose article I cannot locate online). However, it's a little work …