Vector bundles with infinitely many souls

Abstract

Abstract. We construct the first examples of manifolds, the simplest one being S 3 ×S 4 ×R 5 , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.According to the soul theorem of J. Cheeger and D. Gromoll [CG72], a complete open manifold of nonnegative sectional curvature is diffeomorphic to the total space of the normal bundle of a compact totally geodesic submanifold, called a soul. The soul is not unique but any two souls are mapped to each other by an ambient diffeomorphism inducing an isometry on the souls [Sha79]. In this note we show that the homeomorphism type of the soul generally depends on the metric; namely the following is true. Theorem 1. There exist infinitely many complete Riemannian metrics on S3 × S 4 × R 5 with sec ≥ 0 and pairwise nonhomeomorphic souls.The proof applies some classical techniques of geometric topology to recent examples of nonnegatively curved manifolds due to K. Grove and W. Ziller [GZ00]. As we explain below, it is much easier to produce a manifold with finitely many nonnegatively curved metrics having nonhomeomorphic souls, however the full power of [GZ00] is needed to get infinitely many such metrics.Grove and Ziller [GZ00] showed that any principal S 3 ×S 3 -bundle over S 4 admits an S 3 × S 3 -invariant metric with sec ≥ 0. By O'Neill's formula, all associated bundles admit metrics with sec ≥ 0 which gives rise to a rich class of examples, including all sphere bundles over S 4 with structure group SO(4). Note that the souls in Theorem 1 are the total spaces of S 3 -bundles over S 4 with structure group SO(3). Theorem 1 is a particular case of the following. The main topological tool used in this paper is a result of L. Siebenmann [Sie69] that generalizes the famous Masur's theorem: any tangential homotopy equivalence of closed smooth n-manifolds is homotopic to a diffeomorphism after taking the

Moduli spaces of nonnegative sectional curvature and non-unique souls

“…(Without invoking Remark 3.2 we only get an infinite sequence of D(η k,n i,l−i+m )'s that are diffeomorphic to some D(η k,n i 0 ,l−i 0 +m ).) As in [Bel03] results of Grove-Ziller show that each E(η k,n i,l−i+m ) are nonnegatively curved with zero section S(ξ k i ) being a soul, and p 1 (S(ξ k i )) is ±4imultiple of the generator, so assuming i ≥ 0, we get that the souls are pairwise non-homeomorphic.…”

“…This section contains examples of manifolds that admit metrics with infinitely many non-homeomorphic souls of codimension 4. The examples are obtained by modifying arguments in [Bel03,KPT05] and invoking the new topological ingredient, Proposition 3.1 below, which is best stated with the following notation.…”

“…Proof. It is explained in [Bel03] that S(ξ k l ), S(ξ k i ) are fiber homotopy equivalent if l − i is divisible by 12 for k ≥ 4. (In fact, up to fiber homotopy equivalence there are only finitely many oriented S 3 -fibrations over a finite complex Z that admit a section, because their classifying map in [Z, BSG 4 ] factor through BSF 3 and [Z,…”

“…Belegradek in [Bel03] used examples of Grove-Ziller [GZ00] to produce first examples of infinitely many nondiffeomorphic souls for metrics on the same 2000 Mathematics Subject classification. Primary 53C20.…”

“…One motivation for the present work was to construct non-diffeomorphic souls of the smallest possible codimension; of course, multiplying by a Euclidean space then yields examples in any higher codimension. We sharpen examples in [Bel03] by arranging the soul to have codimension 4 and any given dimension ≥ 7.…”