Major problems 10 — Once again, I am not sure whether this problem deserves to be called major, but it is annoying that th…

Definitions and mathematical statement

Major problem 10 (Mark Hovey). Establish the intended permanence of the classes $h_0h_i$ in the Adams spectral sequence at every prime $p>3$, despite the flaw in the proof attributed by the source to R. Cohen--Goerss.

In the customary mod-$p$ sphere interpretation, the classical Adams spectral sequence has \[ E_2^{s,t}=\operatorname{Ext}_{\mathcal A_p}^{s,t}({\mathbb{F}}_p,{\mathbb{F}}_p) \ \Longrightarrow\ \pi_{t-s}(S^0)^{\wedge}_p, \] where $\mathcal A_p$ is the mod-$p$ Steenrod algebra. The symbols $h_i$ are the source's Adams classes, and $h_0h_i$ is their product on the appropriate page. To be a nonzero permanent cycle means surviving all later differentials and incoming boundaries to give a nonzero $E_\infty$ class, not merely being an initial-page cycle.

The uniform target is: for every prime $p>3$ and every intended index $i$ for which the specified nonzero product is defined, $h_0h_i$ survives. The excerpt does not define the $h_i$, give their bidegrees or index range, or explicitly name the spectrum and Adams convention. The sphere spectral sequence above is therefore a customary interpretation requiring confirmation, not an additional supplied hypothesis. The summary's phrase "all $i$" cannot repair these omissions or make zero/undefined products into substantive permanence assertions.

Hovey reports that Minami found a flaw, apparently fatal to the cited R. Cohen--Goerss proof, and calls the problem still open at the time of the list. This is historical source testimony, not an independent present-day review of that proof. The summary mentions $\beta$ and $\alpha\beta$ families and partial progress, but supplies neither an exact comparison to $h_0h_i$ nor citations establishing the required uniform statement.

Short English statement

Prove that the intended $h_0h_i$ classes survive the Adams spectral sequence at every prime greater than three. The source reports a fatal proof gap, while leaving the class convention and index range unstated.

Sources

• Local UnsolvedMath registry, permanent ID 11100010, source code catalogue entry; attributed to Mark Hovey, no proposal year supplied.
• Hovey's Algebraic Topology Problem List, Major problems, item 10; surviving mirror https://www-users.cse.umn.edu/~tlawson/hovey/. The registry records HTML-ordered-list extraction and access on 2026-07-29; the mirror has 65 top-level items on nine category pages.
• The source names R. Cohen--Goerss for the flawed proof and Minami for finding the gap, without precise bibliographic citations. The summary names Ravenel and others for related families, also without a specific reference.