Hopf conjecture for positively curved manifolds with discrete abelian symmetry: the residual regimes of the Su–Wang theorem

A conjecture attributed to Heinz Hopf in the 1930s states that a closed, even-dimensional Riemannian manifold of positive sectional curvature has positive Euler characteristic. Su and Wang (2008) proved this for closed, even-dimensional positively curved manifolds admitting an isometric action of an elementary abelian $p$-group $\mathbb{Z}_p^k$ when the prime satisfies $p\geq p(n)$ (a constant depending only on $n$, equal to $2c(n)$, twice Gromov's bound $c(n)$ on the total Betti number) and the rank $k$ satisfies any one of three linear conditions: (i) $k\geq(n-4)/8$ with $n\neq12,18,20$; (ii) $k\geq(n-2)/10$ with $n\equiv0\bmod4$ and $n\neq12,20$; (iii) $k\geq(n+4)/12$ with $n\equiv0,4$ or $12\bmod20$ and $n\neq20$. Their theorem leaves gaps: the excluded dimensions $n=18$ and $n=20$ at every rank and $n=12$ at rank $k=1$ (clause (iii) covers $n=12$ for $k\geq2$); and all primes $p<p(n)$, for which no fixed-point theorem is available. This record states the derived problem of completing that program: prove $\chi(M)>0$, or exhibit a counterexample, in each of these residual regimes, with a non-empty fixed-point set assumed in the small-prime regime, as in the subsequent $\mathbb{Z}_p$-torus literature. The unrestricted Hopf conjecture without symmetry remains open and is not part of this problem's window.

Background

The Euler characteristic of a closed manifold, $\chi(M)=\sum_i(-1)^i b_i$ where $b_i$ are the Betti numbers (the ranks of the cohomology groups), is the most basic numerical topological invariant. Sectional curvature measures how geodesic triangles in a Riemannian manifold deviate from Euclidean ones; a manifold has positive sectional curvature when the curvature of every tangent 2-plane is strictly positive, the strongest standard curvature sign condition. A conjecture attributed to Heinz Hopf in the 1930s asserts that a closed, even-dimensional manifold admitting such a metric must satisfy $\chi(M)>0$. It is known in dimensions two and four by Gauss–Bonnet-type results and is open in every higher even dimension.

Because few examples of and few obstructions to positive curvature are known, Grove proposed in the 1990s to study positively curved metrics with large isometry groups. An isometric action of a compact group $G$ on $M$ is a homomorphism into the isometry group; it is effective if only the identity element acts trivially. The symmetry rank of $M$ is the largest rank $r$ of a torus $T^r=(S^1)^r$ acting effectively and isometrically. The natural discrete analogue of a torus is an elementary abelian $p$-group $\mathbb{Z}_p^k=(\mathbb{Z}/p\mathbb{Z})^k$, often called a $\mathbb{Z}_p$-torus. Fixed-point sets $\mathrm{Fix}(G,M)$ of isometric actions are finite unions of totally geodesic (containing every geodesic through each of their points) submanifolds that inherit positive curvature. For even-dimensional $M$, a classical fixed-point theorem of Berger from 1966 (as discussed in the introduction of Kennard–Khalili Samani–Searle cited below) guarantees a non-empty fixed-point set for every isometric torus action; this fails for $\mathbb{Z}_p$-tori, since for example even-dimensional spheres admit free isometric $\mathbb{Z}_2$-actions.

The key analytic input is Wilking's connectedness lemma (2003): if $N$ is a closed, positively curved $n$-manifold and $N'\subset N$ a totally geodesic embedded submanifold of codimension $d$, then the inclusion $N'\hookrightarrow N$ is roughly $(n-2d+1)$-connected, so low-codimension fixed-point sets strongly constrain the cohomology of the ambient manifold. With this tool, Püttmann and Searle (2002) verified the Hopf conjecture for manifolds of low cohomogeneity (i.e. with large generic orbits; formally, the orbit space $M/\mathrm{Isom}(M)$ has small dimension) or high torus symmetry rank, and Rong and Su (2005), together with the companion thresholds of Su and Wang (2008), established it under the linear torus-rank bounds $r\geq n/8$ for $n$ even and $r\geq n/10$ for $n\equiv0\bmod4$ (as cited by Kennard 2013). On the discrete side, Fang and Rong (2004) classified positively curved manifolds whose discrete symmetry rank is maximal.

Su–Wang (2008) prove the discrete-abelian Hopf conclusion at these linear scales. Write $c(n)$ for Gromov's bound (1981) on the total Betti number of any closed non-negatively curved $n$-manifold. Their main theorem states that if $M^n$ is a closed, even-dimensional manifold of positive sectional curvature and $\mathbb{Z}_p^k$ acts isometrically with $p\geq p(n)$ — a constant depending only on $n$, equal to $2c(n)$ — and $k$ satisfies any one of the conditions

• (i) $k\geq(n-4)/8$ and $n\neq12,18,20$;
• (ii) $k\geq(n-2)/10$, $n\equiv0\bmod4$, and $n\neq12,20$;
• (iii) $k\geq(n+4)/12$, $n\equiv0,4$ or $12\bmod20$, and $n\neq20$,

then $\chi(M)>0$. Since clause (i) covers every even $n\notin\{12,18,20\}$, the dimensions at which no clause applies at any rank are exactly $n=18$ and $n=20$, and among the excluded dimensions the single pair $(n,k)=(12,1)$ also lies outside the theorem, clause (iii) covering $n=12$ for $k\geq2$ (as $(12+4)/12=4/3$). At covered dimensions the theorem is likewise silent when $k$ falls below the applicable clause threshold; that low-rank range belongs to the broader bounded-rank question discussed in the Problem Statement and is not part of the residual regimes here. The proof has two stages. First, when $p\geq p(n)$, a counting argument based on Gromov's Betti bound forces $\mathrm{Fix}(\mathbb{Z}_p^k,M)\neq\emptyset$. Second, each fixed component $F$ is analyzed through an ascending chain of fixed-point components $F=N_0\subset N_1\subset\cdots\subset N_k=M$, where $N_i$ is a fixed component of a subgroup $\mathbb{Z}_p^i$ and each inclusion has even codimension $d_i$; the connectedness lemma and an induction over these increments force $\chi(F)>0$, and a fixed-point identity valid under their hypotheses (for the manifolds occurring in the induction, $\chi$ equals the Euler characteristic of the $\mathbb{Z}_p^k$-fixed set) transfers positivity to $\chi(M)$. For primes $p<p(n)$ the first stage fails outright, since no fixed-point theorem is available in that range.

Subsequent work splits into two strands. On the connected side, Kennard (2013) used the Steenrod algebra (the algebra of natural stable operations on mod-$p$ cohomology) to prove the conjecture under a logarithmic symmetry-rank bound when $n\equiv0\bmod4$; Kennard–Wiemeler–Wilking (2021) reduced the torus rank to 5 in all even dimensions via a splitting theorem for torus representations with connected isotropy groups; Nienhaus (2022) reached rank 4. None of this transfers to $\mathbb{Z}_p^k$-actions, which need not extend to torus actions and induce no group action on their fixed-point components. On the discrete side, Kennard–Khalili Samani–Searle (2026), Ghazawneh (2024), Bosgraaf–Escher–Searle (2024), and Abdullah–Searle (2025) built a $\mathbb{Z}_p$-torus theory under an assumed fixed point, obtaining classification results at ranks from about $n/2$ down to about $3n/8$ with the aid of error-correcting-code bounds; these imply $\chi(M)>0$ for even $n$ in those high-rank, fixed-point regimes, but the moderate-rank regimes of Su–Wang and their residual cases remain untouched.

Problem Statement

Let $p$ be prime, $n$ even, and let $M^n$ be a closed, simply connected Riemannian manifold of positive sectional curvature admitting an effective, isometric action of the elementary abelian $p$-group $\mathbb{Z}_p^k$. Prove that $\chi(M)>0$, or produce a counterexample within the stated hypotheses, in each of the following regimes left open by Su–Wang (2008):

(1) Excluded dimensions: $p\geq p(n)$ — the constant of the Su–Wang theorem (equal to $2c(n)$, twice Gromov's bound $c(n)$ on total Betti numbers) — and $(n,k)$ one of the dimension-excluded pairs of that theorem: $n=18$ or $n=20$ with any $k\geq1$ (at these dimensions no clause applies at any rank), or $(n,k)=(12,1)$ (clause (iii) covers $n=12$ for $k\geq2$). Ranks below the applicable clause threshold at covered dimensions are not part of this regime; they belong to the broader bounded-rank question stated below.

(2) Small primes: $p<p(n)$ with $\mathrm{Fix}(\mathbb{Z}_p^k,M)\neq\emptyset$ assumed (necessary, since no fixed-point theorem holds in this range), and $(n,k)$ either one of the excluded-dimensional pairs of regime (1) — $n=18$ or $n=20$ with any $k\geq1$, or $(n,k)=(12,1)$ — or such that $k$ satisfies at least one of the clause conditions (i)–(iii) quoted in Background (each with its attached dimension and congruence restrictions); prove $\chi(M)>0$ or exhibit a counterexample satisfying the fixed-point hypothesis.

At the excluded dimensions $n=18$ and $n=20$ the high ranks already covered by the $\mathbb{Z}_p$-torus classification results cited in Background (with the fixed point forced when $p\geq p(n)$ and assumed when $p<p(n)$) need only be cited, not reproved; the open content of those dimensions is the complementary moderate ranks.

Whether the discrete-abelian Hopf threshold can be pushed down to bounded rank $k$ independent of $n$ is a broader open question and is not part of this statement. The unrestricted Hopf conjecture (no symmetry assumption) is a strictly more general open problem and is not part of this statement.

References

1. Xiaole Su and Yusheng Wang, "The Hopf conjecture for positively curved manifolds with discrete abelian group actions", Differential Geometry and its Applications 26 (2008) 313–322. DOI: 10.1016/j.difgeo.2007.11.023. https://doi.org/10.1016/j.difgeo.2007.11.023 (Zbl 1144.53052)
2. Xiaochun Rong and Xiaole Su, "The Hopf conjecture for manifolds with abelian group actions", Communications in Contemporary Mathematics 7 (2005) 121–136. DOI: 10.1142/S0219199705001660. https://doi.org/10.1142/S0219199705001660
3. Thomas Püttmann and Catherine Searle, "The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank", Proceedings of the American Mathematical Society 130 (2002) 163–166. DOI: 10.1090/S0002-9939-01-06039-7. https://doi.org/10.1090/S0002-9939-01-06039-7
4. Fuquan Fang and Xiaochun Rong, "Positively curved manifolds with maximal discrete symmetry rank", American Journal of Mathematics 126 (2004) 227–245. DOI: 10.1353/ajm.2004.0012. https://doi.org/10.1353/ajm.2004.0012
5. Michael Gromov, "Curvature, diameter and Betti numbers", Commentarii Mathematici Helvetici 56 (1981) 179–195. DOI: 10.1007/BF02566208. https://doi.org/10.1007/BF02566208
6. Burkhard Wilking, "Torus actions on manifolds of positive sectional curvature", Acta Mathematica 191 (2003) 259–297. DOI: 10.1007/BF02392966. https://doi.org/10.1007/BF02392966
7. Lee Kennard, "On the Hopf conjecture with symmetry", Geometry & Topology 17 (2013) 563–593. DOI: 10.2140/gt.2013.17.563. arXiv:1203.3808. https://arxiv.org/abs/1203.3808
8. Lee Kennard, Michael Wiemeler, and Burkhard Wilking, "Splitting of torus representations and applications in the Grove symmetry program", arXiv:2106.14723 (2021). https://arxiv.org/abs/2106.14723
9. Jan Nienhaus, "An improved four-periodicity theorem and a conjecture of Hopf with symmetry", arXiv:2211.13151 (2022). https://arxiv.org/abs/2211.13151
10. Lee Kennard, Elahe Khalili Samani, and Catherine Searle, "Positive curvature and discrete abelian symmetry", Mathematische Annalen 395 (2026). DOI: 10.1007/s00208-026-03509-2. arXiv:2110.13345. https://arxiv.org/abs/2110.13345
11. Farida Ghazawneh, "Z_2-actions on positively curved manifolds", arXiv:2409.15392 (2024). https://arxiv.org/abs/2409.15392
12. Muhammad Abdullah and Catherine Searle, "Z_p-torus actions on positively curved manifolds", arXiv:2510.26853 (2025). https://arxiv.org/abs/2510.26853
13. Austin Bosgraaf, Christine Escher, and Catherine Searle, "On Fixed-Point Sets of Z_2-Tori in Positive Curvature", arXiv:2411.00665 (2024). https://arxiv.org/abs/2411.00665