Unstable homotopy theory 2 — Determine the v_1 -exponents for the spheres.

Definitions and mathematical statement

Hovey asks to determine $v_1$-exponents of spheres, while explicitly saying that he is not sure of the precise statement or an expected conjectural formula. The motivating comparison is the reported Cohen--Moore--Neisendorfer result: for an odd prime $p$ and $n\geq1$, the $p$-primary torsion in $\pi_*(S^{2n+1})$ is annihilated by $p^n$, with $p^{n-1}$ insufficient. This is the standard exponent reading of the source's plain-text "not p\^{}n-1," not an assertion about the nontorsion fundamental sphere class.

The source suggests maps from a Moore space so an Adams self-map acts. For example, after making the necessary suspension and prime choices, let $V$ be a based Moore space and $v:\Sigma^dV\to V$ an Adams $v_1$-self-map. Precomposition gives maps \[v^*:[\Sigma^jV,X]\to[\Sigma^{j+d}V,X].\] The Moore exponent, unstable realization of the self-map and group/degree range must be fixed; a stable self-map alone does not automatically define every proposed unstable operation.

There are at least two materially different exponent questions. One can consider $v$-torsion classes (killed by some iterate of $v^*$) and ask for a uniform power $(v^*)^N$ killing all of them. Alternatively one can invert $v$ in the directed system and ask for the $p$-power exponent of the resulting periodic groups. The first is a $v$-torsion exponent and the second a $p$-exponent after periodic localization; they are not interchangeable. The summary adopts the latter language, but Hovey's uncertain text does not decide between them, nor specify which Moore spaces/coefficients should be used. Accordingly no one of these is silently chosen as the problem, and no numeric formula is inserted.

The broad sphere range, possible prime-two version and expected bound remain unspecified; only the motivating ordinary-torsion comparison explicitly concerns odd primes. The final source request is for clarification and an expected conjecture, not a declared value.

Short English statement

Clarify and determine the intended $v_1$-exponent of spheres using Moore-space Adams self-maps. The source itself leaves open whether this means a $v_1$-torsion bound or a $p$-exponent of periodic groups.

Sources

• Local UnsolvedMath registry, permanent ID $11100061$, exact source code catalogue entry.
• Mark Hovey, Hovey's Algebraic Topology Problem List, Unstable homotopy theory, item 2, surviving mirror: https://www-users.cse.umn.edu/~tlawson/hovey/. Registry access 2026-07-29; html-ordered-list extraction.
• Cohen--Moore--Neisendorfer are credited for the ordinary odd-primary exponent comparison. The summary mentions Bousfield, Davis--Mahowald, Toda and a truncated Davis lead without precise citations; its self-corrected names and placeholder exponent expression are not usable theorem statements.