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 $n\equiv3\pmod4$ with $n\geq11$ and $n\not\equiv255\pmod{256}$. 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 $4b+3$, $b\geq2$, where $ρ_b$ is the representation of $S^3$ on $\mathbb{R}^3\oplus\mathbb{H}^b$ 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 $\hat A$- and $L$-genera. We then show that, for every $b\geq2$ with $b\not\equiv63\pmod{64}$, 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 $n\equiv255\pmod{256}$ below $2^{16}$.