arXiv:math/0610349v1 [math.DG] 11 Oct 2006 AN INFINITE FAMILY OF GROMOLL-MEYER SPHERES CARLOS DUR ´ AN, THOMAS P ¨ UTTMANN, AND A. RIGAS Abstract. We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere. 1. Introduction This paper provides a new geometric way to construct all exotic 7-spheres. Ex- otic spheres are differentiable manifolds that are homeomorphic but not diffeomor- phic to standard spheres. The first examples were found by Milnor [Mi1] in 1956 among the S 3 -bundles over S 4 . It turned out that 7 is the smallest dimension where exotic spheres can occur except possibly in the special dimension 4. In any dimension n> 4 the exotic spheres and the standard sphere form a finite abelian group: the group Θ n of (orientation preserving diffeomorphism classes of) homo- topy spheres [KM]. The inverse element in Θ n can be obtained by a change of orientation. In dimension 7 we have Θ 7 ≈ Z 28 . Hence, ignoring orientation there are 14 exotic 7-spheres. From these 14 exotic 7-spheres four (corresponding to 2, 5, 9, 12, 16, 19, 23, 26 ∈ Z 28 ) are not diffeomorphic to an S 3 -bundle over S 4 [EK]. In 1974 Gromoll and Meyer [GM] constructed an exotic 7-sphere, Σ 7 GM , as quo- tient of the compact group Sp(2) by a two-sided S 3 -action. This construction pro- vided Σ 7 GM automatically with a metric of nonnegative sectional curvature (K ≥ 0). The Gromoll-Meyer sphere Σ 7 GM was the only exotic sphere known to admit such a metric until 1999 when Grove and Ziller [GZ] constructed metrics with K ≥ 0 on all Milnor spheres, i.e., on all exotic 7-spheres that are S 3 -bundles over S 4 . In 2002 Totaro [To] and independently Kapovitch and Ziller [KZ] showed that Σ 7 GM is the only exotic sphere that can be modeled by a biquotient of a compact group and thus underlined the singular status of the Gromoll-Meyer sphere among all models for exotic spheres. We nevertheless provide an elementary and direct generalization of the Gromoll- Meyer construction. The essential components in this construction are natural self- maps of S 7 , namely, the n-powers of unit octonions, n ∈ Z. In terms of quaternions these maps are defined by ρ n : S 7 → S 7 , ( cos t+p sin t w sin t ) → ( cos nt+p sin nt w sin nt ) C. Duran and A. Rigas were supported by CNPq. C. Duran was also supported by FAPESP grant 03/016789 and FAEPEX grant 15406. T. P¨ uttmann was supported by a DFG Heisenberg fellowship and by the DFG priority program SPP 1154 “Globale Differentialgeometrie”. 1