The $\mathbb{Z}_2$ and $\mathbb{Z}_p$ Periodicity Conjectures: extending Adams's Hopf invariant one theorem from singly generated to periodic cohomology

Adams's 1960 theorem on singly generated cohomology rings — the key step in his resolution of the Hopf invariant one problem — states that if $H^*(X;\mathbb{Z}_2)\cong\mathbb{Z}_2[x]$ then $|x|\in\{1,2,4\}$ (with degree 8 possible only in the truncated case $x^3=0$), and Liulevicius, Shimada, and Yamanoshita proved the odd-prime analogue. Kennard (2013), motivated by Wilking's connectedness theorem in positively curved Riemannian geometry, generalized the first step (Adem's theorem) to spaces whose mod-$p$ cohomology is periodic: at $p=2$ the minimal period is a power of two, and at odd primes it has the form $k=2\lambda p^a$ with $\lambda\mid p-1$, and posed the conjecture that the Adams-level restrictions survive. Harper and Kennard (2024) restated the conjectures in the non-truncated form — minimal period $k\in\{1,2,4\}$ at $p=2$ and $k=2\lambda$, $\lambda\mid p-1$, at odd primes — and proved them under the extra hypothesis that $\mathrm{Sq}^{k/2}(x)=0$ (respectively $H^{2(p-1)p^{a-1}}(X;\mathbb{Z}_p)=0$), recovering Adams's theorem as the case of no intermediate cohomology. The general case — whether $\mathrm{Sq}^{k/2}(x)$ must vanish for a minimal-period class, or whether genuine counterexamples with minimal period at least 8 (a power of two) or divisible by $p$ exist — remains open, with the smallest structurally analyzed candidate being a $k=16$ ring proposed in Question 7.3 of the primary source.

Background

A central chapter of algebraic topology begins with the Hopf invariant one problem: for which $n$ does there exist a map $f\colon S^{2n-1}\to S^n$ of Hopf invariant one, i.e., a map whose mapping cone $C_f$ has mod-2 cohomology ring $H^*(C_f;\mathbb{Z}_2)\cong\mathbb{Z}_2[x]/(x^2)$ with $|x|=n$? J.F. Adams resolved this in 1960 using secondary cohomology operations: such maps exist only for $n\in\{1,2,4,8\}$, realized by the Hopf fibrations built from the real, complex, quaternionic, and octonionic division algebras. Here the Steenrod squares $\mathrm{Sq}^i$ are the natural stable cohomology operations at the prime 2 — they act on mod-2 cohomology, satisfy $\mathrm{Sq}^i(x)=x^2$ for $i=|x|$, and together form the Steenrod algebra; secondary cohomology operations are operations defined only on classes annihilated by suitable primary (Steenrod) operations, with values well-defined only modulo an indeterminacy. Adams's argument in ring-theoretic form (Theorem 1.1 of the primary source) states: if $H^*(X;\mathbb{Z}_2)\cong\mathbb{Z}_2[x]$, then $|x|\in\{1,2,4\}$; the theorem also holds for truncated polynomial rings $\mathbb{Z}_2[x]/(x^{q+1})$ with $q\geq 2$, where $|x|=8$ is additionally allowed but only when $x^3=0$. Adem had earlier shown, from the relations among Steenrod squares bearing his name, that $|x|$ must be a power of two. At odd primes, the analogous result — $|x|=2\lambda$ with $\lambda$ dividing $p-1$ — is due to Liulevicius and to Shimada–Yamanoshita. All the allowed degrees are realized by the classifying spaces $B\mathsf{O}(1)=\mathbb{RP}^\infty$, $B\mathsf{U}(1)=\mathbb{CP}^\infty$, and $B\mathsf{Sp}(1)=\mathbb{HP}^\infty$ (the classifying space of a topological group $G$ is the base of its universal bundle and carries polynomial cohomology on the characteristic classes), their truncated skeleta, and, in the truncated degree-8 case, the Cayley plane $\mathbb{OP}^2$ (the 16-dimensional octonionic projective plane, with $H^*(\mathbb{OP}^2;\mathbb{Z}_2)\cong\mathbb{Z}_2[x]/(x^3)$, $|x|=8$).

Singly generated cohomology is a special case of periodic cohomology. Following the primary source: a nonzero class $x\in H^k(X;\mathbb{Z}_p)$ with $k\geq 1$ induces periodicity if cup product with $x$ induces isomorphisms $H^i(X;\mathbb{Z}_p)\to H^{i+k}(X;\mathbb{Z}_p)$ for all $i\geq 0$, and $H^*(X;\mathbb{Z}_p)$ is $k$-periodic when such a class exists; the intermediate groups $H^i$ for $0<i<k$ are arbitrary. Besides the singly generated examples, $\mathbb{S}^1\times\mathbb{CP}^\infty$ and $N^2\times\mathbb{HP}^\infty$, $N^3\times\mathbb{HP}^\infty$ (for manifolds $N$ of dimension 2 or 3) have 4-periodic mod-2 cohomology without singly generated rings. Kennard, generalizing Adem's theorem to this setting, proved that at $p=2$ the minimal period must be a power of two, and at odd primes must have the form $k=2\lambda p^a$ with $\lambda\mid p-1$ and $a\geq 0$, and posed the conjecture that the secondary-operation restrictions survive: minimal period in $\{1,2,4\}$ at $p=2$ and $a=0$ at odd primes. The question is not merely formal: any counterexample must be an infinite-dimensional space (a finite complex cannot satisfy $H^i\cong H^{i+k}$ for all $i\geq 0$ with $H^0\neq 0$), the obvious octonionic construction of a period-8 analogue of $\mathbb{HP}^\infty$ fails because the octonions are non-associative so no classifying-space-style construction is available, and 8-periodicity does occur in the truncated sense ($\mathbb{OP}^2\times N^m$ for $m\leq 7$), which the separate truncated-case analogue of the conjecture explicitly permits.

The problem also has a Riemannian-geometric source: Wilking's connectedness theorem implies that positively curved manifolds with large isometric torus actions have (truncated) periodic integral cohomology, and Kennard's four-periodicity theorem and its refinements — by Nienhaus via characteristic classes, and by Kennard–Wiemeler–Wilking in their verification of Hopf's 1930s conjecture that even-dimensional positively curved manifolds have positive Euler characteristic under symmetry-rank assumptions — are direct applications of the periodic-cohomology machinery. The Harper–Kennard paper proves the conjecture at $p=2$ under the hypothesis $\mathrm{Sq}^{k/2}(x)=0$ (implied by $H^{k/2}(X;\mathbb{Z}_2)=0$; Adams's theorem is recovered when $H^i=0$ for $0<i<k$), and at odd primes shows that a counterexample with $k=2\lambda p^a$, $a\geq 1$, must have $H^{2(p-1)p^{a-1}}(X;\mathbb{Z}_p)\neq 0$. The authors state that proving $\mathrm{Sq}^{k/2}(x)=0$ in general would complete the $p=2$ conjecture, and they analyze the structure of potential counterexamples with minimal period $k=16$, prescribing in Question 7.3 an explicit candidate cohomology ring $\mathbb{Z}_2[y_4,y_8,y_{12},x_{16}]$ with nilpotent generators and prescribed Steenrod action.

Problem Statement

Let $p$ be a prime and $X$ a connected topological space. A nonzero class $x\in H^k(X;\mathbb{Z}_p)$ with $k\geq 1$ induces periodicity if cup product with $x$ induces isomorphisms $H^i(X;\mathbb{Z}_p)\to H^{i+k}(X;\mathbb{Z}_p)$ for every $i\geq 0$; the smallest degree of such a class is the minimal period of $H^*(X;\mathbb{Z}_p)$. Determine, in full generality, which minimal periods can occur:

(1) ($\mathbb{Z}_2$ Periodicity Conjecture, Conjecture 1.5 of Harper–Kennard) If $H^*(X;\mathbb{Z}_2)$ is $k$-periodic with minimal period $k$, then $k\in\{1,2,4\}$.

(2) ($\mathbb{Z}_p$ Periodicity Conjecture, Conjecture 1.6 of Harper–Kennard) If $p$ is odd and $H^*(X;\mathbb{Z}_p)$ is $k$-periodic with minimal period $k$, then $k=2\lambda$ for some divisor $\lambda$ of $p-1$.

The two conjectures carry one scientific objective: extend Adams's Hopf invariant one theorem from singly generated to periodic mod-$p$ cohomology by classifying the possible minimal periods, with no finiteness, dimension, truncation, or manifold assumptions on $X$ and periodicity required in all degrees. The question is to settle both (1) and (2); each conjecture is resolved either by a proof of it as stated or by an explicit space realizing a minimal period excluded by that conjecture. By Kennard's periodicity theorem the minimal period is necessarily a power of two at $p=2$ and of the form $k=2\lambda p^a$, $\lambda\mid p-1$, $a\geq 0$, at odd primes, so the concrete remaining content is: at $p=2$, exclude — or realize — minimal periods $k=2^a\geq 8$; at odd primes, exclude — or realize — minimal periods with $a\geq 1$.

References

1. John R. Harper and Lee Kennard, “Extending Adams' theorem from singly generated to periodic cohomology”, arXiv:2412.16340 (2024), DOI 10.48550/arXiv.2412.16340, https://arxiv.org/abs/2412.16340 (primary source)
2. Lee Kennard, “On the Hopf conjecture with symmetry”, Geometry & Topology 17 (2013) 563–593, DOI 10.2140/gt.2013.17.563, https://doi.org/10.2140/gt.2013.17.563 (origin of the conjecture, stated there as Conjecture 6.3 in a bounded-range form)
3. J. F. Adams, “On the non-existence of elements of Hopf invariant one”, Annals of Mathematics (2) 72 (1960) 20–104, DOI 10.2307/1970147, https://doi.org/10.2307/1970147
4. José Adem, “The iteration of the Steenrod squares in algebraic topology”, Proceedings of the National Academy of Sciences U.S.A. 38 (1952) 720–726, DOI 10.1073/pnas.38.8.720, https://doi.org/10.1073/pnas.38.8.720
5. Arunas Liulevicius, “The factorization of cyclic reduced powers by secondary cohomology operations”, Memoirs of the American Mathematical Society 42 (1962), DOI 10.1090/memo/0042, https://doi.org/10.1090/memo/0042
6. Nobuo Shimada and Tsuneyo Yamanoshita, “On triviality of the mod p Hopf invariant”, Japanese Journal of Mathematics 31 (1961) 1–25, DOI 10.4099/jjm1924.31.0_1, https://doi.org/10.4099/jjm1924.31.0_1
7. Lee Kennard, Lawrence Mouillé, and Jan Nienhaus, “On Hopf's conjecture and positive second intermediate Ricci curvature”, arXiv:2507.16936 (2025), DOI 10.48550/arXiv.2507.16936, https://arxiv.org/abs/2507.16936
8. Jan Nienhaus, “An improved four-periodicity theorem and a conjecture of Hopf with symmetry”, arXiv:2211.13151 (2022), DOI 10.48550/arXiv.2211.13151, https://arxiv.org/abs/2211.13151
9. Lee Kennard, Michael Wiemeler, and Burkhard Wilking, “Splitting of torus representations and applications in the Grove symmetry program”, arXiv:2106.14723 (2021), DOI 10.48550/arXiv.2106.14723, https://arxiv.org/abs/2106.14723
10. Lee Kennard, “Fundamental groups of manifolds with positive sectional curvature and torus symmetry”, Journal of Geometric Analysis 27 (2017) 2894–2925, DOI 10.1007/s12220-017-9787-2, https://doi.org/10.1007/s12220-017-9787-2