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 $n\equiv 3(\mathrm{mod}4)$ with $n\ge 11$ and $n\not\equiv 255(\mathrm{mod}256)$. 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 $\rho$ of $S^3$ and an $S^3$-conjugation-equivariant map $\beta$, and are denoted ${\Sigma }_{\rho ,\beta }$. By recent constructions, such spheres admit Riemannian metrics of strictly positive sectional curvature. In this paper we consider the spheres ${\Sigma }_{{\rho }_b,\beta }$ of dimension $4b+3$, $b\ge 2$, where ${\rho }_b$ 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 $k(\beta )$ depending only on $\beta$ 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 $b\ge 2$ with $b\not\equiv 63(\mathrm{mod}64)$, there is a map $\beta$ for which the Eells-Kuiper invariant does not vanish, and thus the corresponding sphere is exotic. A computer calculation also gives the dimensions $n\equiv 255(\mathrm{mod}256)$ below $2^{16}$.

arXiv comment36 pages, 3 tables. Programs and outputs archived at Zenodo, doi:10.5281/zenodo.23571372