Nonnegatively curved five-manifolds with non-abelian symmetry

Abstract

Known examples of manifolds which admit metrics of positive (sectional) curvature are rare when compared with nonnegatively curved examples. In fact, besides rank one symmetric spaces, compact manifolds with positive curvature are known to exist only in dimensions below 25, while to generate new nonnegatively curved manifolds from known ones it is enough, for example, to take products, quotients or biquotients (see [32] for a survey).By the Soul Theorem any complete non-compact nonnegatively curved manifold is diffeomorphic to a vector bundle over a compact manifold with nonnegative curvature. For compact manifolds of positive curvature Bonnet-Myers implies that the fundamental group is finite and in nonnegative curvature a finite cover is diffeomorphic to the product of a torus with a compact simply-connected manifold of nonnegative curvature (see [5]). We will hence only consider compact simply-connected manifolds.Recently, positively and nonnegatively curved manifolds were studied under the additional assumption of having a "large" isometry group (see the surveys [12] and [30]). The beginning of this subject was the result by Hsiang and Kleiner [17] that a compact simply-connected 4-dimensional Riemannian manifold with positive curvature and S 1 -symmetry must be either S 4 or CP 2 . The possible isometric circle-actions were classified in [10] and [15].The classification of isometric circle actions on positively curved 5-manifolds is a very difficult problem and at the moment seems out of reach. In 2002, Rong [26] showed that a positively curved compact simply-connected 5-dimensional manifold with a 2-torus acting by isometries is diffeomorphic to a 5-sphere. In 2009 Galaz-Garcia and Searle [11], only assuming nonnegative curvature, showed that a simply-connected 5-manifold which admits an isometric action of a 2torus is diffeomorphic to either S 5 , S 3 × S 2 , the nontrivial S 3 -bundle over S 2 , denoted by S 3× S 2 , or the Wu-manifold W = SU(3)/ SO(3). The description of the actions is not yet solved in any of these cases.In this context, a question that naturally arises is which 5-manifolds admit a metric of nonnegative (or positive) curvature with symmetry containing a connected non-abelian group G. In this paper we will classify such manifolds with nonnegative curvature and obtain a partial classification in positive curvature. For this purpose, we first classify all five-dimensional compact simply-connected manifolds which admit an action of a connected non-abelian Lie group without any geometric assumptions. They are either S 5 , S 3 × S 2 , S 3 × S 2 , connected sums of S 3 × S 2 , or connected sums k W # l B of copies of the Wu-manifold W and the Brieskorn variety B of type (2, 3, 3, 3). Since any non-abelian connected Lie group contains SO(3) or SU(2) as a subgroup, it is natural to classify in addition the actions by these two groups up to equivariant diffeomorphisms. This is the content of Theorems C and D below.To describe the actions we introduce the following key construction.

Positive curvature and symmetry in small dimensions

“…[GGR15, GR15,PP03]), while in dimension 5, the best result is due to Rong and Galaz-Garcia-Searle (see [Ron02,GGS14], cf. [Goz15,Sim]). In dimensions 6 and 7, the above results are the best known, however there remains a large gap in our understanding due to the vast number of known positively curved examples (see [Zil07,Dea11,GVZ11,PW]).…”

Proper actions and decompositions in equivariant K-theory

“…Lets look at actions of SU (2) on simply connected 5-dimensional manifolds with only one isotropy type and orbit space S 2 . These manifolds have been classified up to equivariant diffeomorphism in [Sim16]. There can be only three types of orbit types (e), (SU (2)/(Z 2 )), (SU (2)/(Z m )) and we will describe now the possible equivariant diffeomorphic types: Consider for natural numbers l, m with (l, m) = 1,…”