Effective torus actions on exotic spheres that do not bound spin manifolds: symmetry rank and degree of symmetry

Lawson and Yau (1974) proved that a compact manifold admitting a smooth effective action of a compact, connected, non-abelian Lie group has a metric of positive scalar curvature, and deduced that on an exotic n-sphere that does not bound a spin manifold — equivalently one with nonzero Atiyah–Bott–Shapiro alpha-invariant, possible only in dimensions $n \equiv 1, 2 \pmod 8$ with $n \geq 9$ — every compact connected transformation group is a torus of dimension at most $\lfloor (n+1)/2\rfloor$ (improved to less than $n/2$ for even $n$ by Wiemeler). The problem is to determine which tori actually act: the sharp symmetry rank — which for these spheres equals the degree of symmetry — and the classification of smooth effective torus actions on non-spin-bounding exotic spheres, including which individual spheres admit even a circle action.

Background

A homotopy n-sphere is a smooth closed n-manifold homotopy equivalent to the standard sphere $S^n$; by the generalized Poincaré conjecture it is homeomorphic to $S^n$, and it is called exotic if it is not diffeomorphic to $S^n$. Kervaire and Milnor (1963) showed that for $n \geq 5$ the diffeomorphism classes of homotopy n-spheres form a finite abelian group $\Theta_n$ under connected sum. The degree of symmetry of a smooth closed manifold is the largest dimension of a compact Lie group acting smoothly and effectively (that is, with no non-identity element acting trivially) on it, a notion introduced for homotopy spheres by Hsiang (1967) and Hsiang–Hsiang (1969); the symmetry rank is the analogous maximum taken only over tori $T^r = U(1)^r$. The standard sphere carries the full rotation group $SO(n+1)$, whose maximal torus — a maximal connected abelian subgroup, acting by simultaneous rotations in $\lfloor (n+1)/2\rfloor$ coordinate planes — has dimension $\lfloor (n+1)/2\rfloor$, so exotic spheres are the natural setting in which the degree of symmetry can drop below that of $S^n$. Every homotopy sphere of dimension $n \geq 3$ is spin (its tangent bundle admits a spin structure), hence represents a class in the spin bordism group $\Omega^{spin}_n$ (closed spin n-manifolds modulo those that bound spin manifolds), and the Atiyah–Bott–Shapiro alpha-invariant — the $KO$-theoretic index of the Dirac operator of the unique spin structure — gives a homomorphism $\Theta_n \to KO_n$ whose kernel is exactly the subgroup $bSpin^{n+1}$ of spheres bounding spin manifolds (see e.g. Wiemeler 2016, who records the equivalence). By the Lichnerowicz formula, a spin manifold with a metric of positive scalar curvature (the mildest curvature condition: the pointwise trace of the Ricci tensor) has vanishing alpha-invariant, a result due to Hitchin (1974), and Gromov–Lawson (1980) and Stolz (1992) proved the converse for simply connected spin manifolds of dimension at least 5. Consequently, among homotopy spheres exactly those with nonzero alpha-invariant admit no positive-scalar-curvature metric; such spheres exist only in dimensions $n = 8k+1$ and $n = 8k+2$ with $k \geq 1$, where they constitute half of $\Theta_n$ (the quotient $\Theta_n/bSpin^{n+1}$ is $\mathbb{Z}/2$; see Joseph 1981, Section 1, and Wiemeler 2016). Lawson and Yau (1974) proved that if a compact manifold admits a smooth effective action of a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. Since the alpha-invariant is unchanged under diffeomorphism, they concluded: if $\Sigma^n$ is an exotic n-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of $\Sigma^n$ are tori of dimension at most $\lfloor (n+1)/2\rfloor$. (A compact Lie group has the same dimension as its identity component, which here must be a torus; hence for these particular spheres the degree of symmetry equals the symmetry rank.) Wiemeler (2016, Corollary 2.7) sharpened the bound: a compact connected group acting effectively on a closed manifold without positive-scalar-curvature metrics and with nonzero Euler characteristic is a torus of dimension strictly less than half the dimension of the manifold; since $\chi(\Sigma^{8k+2}) = 2 \neq 0$, this applies to the even-dimensional non-spin-bounding exotic spheres. Rank-one actions are known on broad families of these spheres (Bredon 1967; Schultz 1975; Joseph 1981), while no effective torus action of rank at least 2 on any such sphere is known, no universal upper bound sharper than these limits is known, and which individual such spheres admit an effective circle action remains open (Schultz 1985; Quigley 2024; Botvinnik–Quigley 2023; Bauer–Quigley 2026). What these structural results leave completely undetermined is which torus ranks actually occur on such spheres — the question considered here.

Problem Statement

Let $\Sigma^n$ be an exotic n-sphere with nonzero Atiyah–Bott–Shapiro alpha-invariant, i.e. one that does not bound a spin manifold; such spheres exist exactly in dimensions $n \equiv 1, 2 \pmod 8$ with $n \geq 9$, where they form half of the Kervaire–Milnor group $\Theta_n$. By Lawson–Yau (1974), every compact connected Lie group acting smoothly and effectively on $\Sigma^n$ is a torus of dimension at most $\lfloor (n+1)/2\rfloor$; by Wiemeler (2016), at most $n/2 - 1$ when $n$ is even. Determine which tori actually act. Concretely: (1) Realizability of ranks — for which $r$ with $2 \leq r$ below the applicable bound does there exist an exotic sphere $\Sigma^n$ with $\alpha(\Sigma^n) \neq 0$ admitting a smooth effective $T^r$-action, and what is the maximal such $r$ (the symmetry rank, which for these spheres equals their degree of symmetry)? (2) Sphere-by-sphere classification — for a fixed exotic n-sphere with $\alpha \neq 0$, which tori act on it; in particular, does every such sphere admit a nontrivial smooth circle action (the $\alpha \neq 0$ case of Schultz's 1985 question whether every exotic n-sphere, $n \geq 5$, supports a nontrivial smooth circle action)? Equivalently: realize the ranks that occur on spheres with $\alpha \neq 0$, and exclude the ranks that do not occur on any such sphere.

References

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