Classification of $SO(3)$-Actions on Five Manifolds

Nakanishi, Aiko

doi:10.2977/prims/1195188834

Publ. Res. Inst. Math. Sci.

1978

DOI: 10.2977/prims/1195188834

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Classification of $SO(3)$-Actions on Five Manifolds

Aiko Nakanishi¹

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1981

2015

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Cited by 2 publications

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“…The case of G = SO(3) was studied in [18], although the author missed the equivariant connected sums of S 3 × S 2 with the above SO(3)-action, and did not describe some of the actions explicitly. For partial results about differentiable classifications, see [22], [23], [24] and [25]. In [20] the SO (3) and SU(2)-actions on compact simply-connected 4-manifolds are classified.…”

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Nonnegatively curved five-manifolds with non-abelian symmetry

Simas

2015

Geom Dedicata

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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.

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Nonnegatively curved five-manifolds with non-abelian symmetry

Simas

2015

Geom Dedicata

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Simply connected closed smooth $5$-manifolds with effective smooth ${\rm U}(2)$-actions

Ôike¹

1981

Tohoku Math. J. (2)

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Classification of $SO(3)$-Actions on Five Manifolds

Cited by 2 publications

References 6 publications

Nonnegatively curved five-manifolds with non-abelian symmetry

Nonnegatively curved five-manifolds with non-abelian symmetry

Simply connected closed smooth $5$-manifolds with effective smooth ${\rm U}(2)$-actions

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