Positive sectional curvature on exotic spheres in infinitely many dimensions
Abstract
We show that there exist exotic spheres with metrics of strictly positive sectional curvature in infinitely many dimensions: in every dimension n≡3pmod4 with n≥11 and nnot≡255pmod256. Previously, every known closed simply connected manifold of positive sectional curvature in a dimension above $24$ was diffeomorphic to a compact rank-one symmetric space. The spheres we consider are certain homotopy spheres arising as quotients of special S3-S3 bundles introduced by Sperança. These homotopy spheres are determined by an orthogonal representation ρ of S3 and an S3-conjugation-equivariant map β, and are denoted Σρ,β. By recent constructions, such spheres admit Riemannian metrics of strictly positive sectional curvature. In this paper we consider the spheres Σρb,β of dimension 4b+3, b≥2, where ρb is the representation of S3 on ℝ3⊕ℍb by left multiplication on the second summand. For these spheres, we show that the Eells-Kuiper invariant is the product of an integer k(β) depending only on β and a rational number rb given by one coefficient of a power series built from the Â- and L-genera. We then show that, for every b≥2 with bnot≡63pmod64, there is a map β for which the Eells-Kuiper invariant does not vanish, and thus the corresponding sphere is exotic. A computer calculation also gives the dimensions n≡255pmod256 below 216.
Related
- Hitchin's α-invariant and positive sectional curvature on homotopy $9$-spheres
- Positive Sectional Curvature on All Smooth 7-spheres
- Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres
- Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres
- Positive sectional curvature on homotopy nine-spheres
BibTeX
@misc{galazgarcia2026positive,
title = {{Positive sectional curvature on exotic spheres in infinitely many dimensions}},
author = {Fernando Galaz-García},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31528},
eprint = {2610.31528},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31528},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive sectional curvature on exotic spheres in infinitely many dimensions
AU - Fernando Galaz-García
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31528
UR - https://arxiv.org/abs/2610.31528
AB - We show that there exist exotic spheres with metrics of strictly positive sectional curvature in infinitely many dimensions: in every dimension n≡3pmod4 with n≥11 and nnot≡255pmod256. Previously, every known closed simply connected manifold of positive sectional curvature in a dimension above $24$ was diffeomorphic to a compact rank-one symmetric space. The spheres we consider are certain homotopy spheres arising as quotients of special S^3-S^3 bundles introduced by Sperança. These homotopy spheres are determined by an orthogonal representation ρ of S^3 and an S^3-conjugation-equivariant map β, and are denoted Σ_ρ,β. By recent constructions, such spheres admit Riemannian metrics of strictly positive sectional curvature. In this paper we consider the spheres Σ_ρ_b,β of dimension 4b+3, b≥2, where ρ_b is the representation of S^3 on ℝ^3⊕ℍ^b by left multiplication on the second summand. For these spheres, we show that the Eells-Kuiper invariant is the product of an integer k(β) depending only on β and a rational number r_b given by one coefficient of a power series built from the Â- and L-genera. We then show that, for every b≥2 with bnot≡63pmod64, there is a map β for which the Eells-Kuiper invariant does not vanish, and thus the corresponding sphere is exotic. A computer calculation also gives the dimensions n≡255pmod256 below 2^16.
ER -