Topological properties of Eschenburg spaces and 3-Sasakian manifolds

Abstract

Riemannian manifolds with positive sectional curvature have been a frequent topic of global Riemannian geometry for over 40 years. Nevertheless, there are relatively few known examples of such manifolds. The purpose of this article is to study the topological properties of some of these examples, the so-called Eschenburg spaces, in detail.In addition to positively curved metrics, some Eschenburg spaces also carry another special geometric structure, namely a 3-Sasakian metric, i.e. a metric whose Euclidean cone is Hyperkähler [BG]. 3-Sasakian spaces are interesting since they are Einstein manifolds and are connected to several other geometries: They admit an almost free, isometric action by SU(2) whose quotient is a quaternionic Kähler orbifold. The twistor space of this orbifold, which can be viewed as an S 1 -quotient of the 3-Sasakian manifold, carries a natural Kähler-Einstein orbifold metric with positive scalar curvature.3-Sasakian structures are rare and rigid, in fact the moduli space of such metrics on a fixed manifold consists of at most isolated points. This motivated C. Boyer and K. Galicki to pose the question in [BG][Question 9.9, p. 52] whether a manifold can admit more than one 3-Sasakian structure. Natural candidates for such examples are the 3-Sasakian metrics discovered in [BGM]. They are defined on the Eschenburg biquotients E a,b,c = diag(z a , z b , z c )\SU(3)/ diag(z a+b+c , 1, 1), where a, b, c are positive, pairwise relatively prime integers. The simplest topological invariant of these spaces is the order of the fourth cohomology group, which is a finite cyclic group of order r = ab + ac + bc. By studying further topological invariants of these manifolds we show:Theorem A. For r ≤ 10 7 , there is a unique pair of 3-Sasakian Eschenburg spaces E a,b,c which are diffeomorphic to each other, but not isometric. It is given by (a, b, c) = (2279, 1603, 384) and (2528, 939, 799) with r = 5143925.The two 3-Sasakian metrics are non-isometric since the isometric action by SU(2) has cyclic isotropy groups of order a + b, a + c and b + c. In [BG] they also asked whether two 3-Sasakian manifolds can be homeomorphic to each other but not diffeomorphic. This happens frequently among the 3-Sasakian Eschenburg spaces. There are 96 such pairs for r ≤ 10 7 , the first one of which is given by (a, b, c) = (171, 164, 1) and (223, 60, 53) for r = 28379. See Table 4.5 for the next 4 such pairs.The manifolds E a,b,c also carry a metric of positive sectional curvature, although the 3-Sasakian metric never has positive curvature. They are special cases of the more general Date: December 27, 2017.

Orbifold fibrations of Eschenburg spaces

“…Moreover, the integer h must be odd for Eschenburg manifolds (see [13], Remark 1.4). Notice also that, if E p,q is positively curved, we can assume (1) < 0, and hence there are only finitely many positively curved Eschenburg manifolds for a given order h ( [6]). Finally, we introduce notations for a few orbifolds that we will need.…”

A note on totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces

“…The integral cohomology rings of Eschenburg spaces and Bazaikin spaces are well-known (see [6,3,2,8]). In particular, if σ i (x) := σ i (x 1 , .…”

Open manifolds with non-homeomorphic positively curved souls

“…unabridged versions of the lists of pairs of Eschenburg spaces published in [CEZ07] with us, and Igor Belegradek and Anand Dessai for helpful comments on a preliminary version of this manuscript. Numerous improvements where moreover suggested by an extremely diligent anonymous referee.…”

“…Finally, the equivalence of the notions of homotopy equivalence and tangential homotopy equivalence follows from Proposition 11 since p 1 = 0 for Aloff-Wallach spaces (see Table 2). Table 1, we followed the basic strategy outlined in [CEZ07]. That is, we employed a computer program that first generates all positively curved Eschenburg spaces satisfying ( †) with r bounded by some upper bound R, and then looks for families of spaces whose invariants agree.…”