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Differentiable classification of 4-manifolds with singular Riemannian foliations

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Abstract

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: \(\mathbb {S}^4\), \(\mathbb {CP}^2\), \(\mathbb {S}^2\times \mathbb {S}^2\), or \(\mathbb {CP}^2\)#\(\pm \mathbb {CP}^2\). As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard \(\mathbb {S}^4\), \(\pm \mathbb {CP}^2\) and \(\mathbb {S}^2\times \mathbb {S}^2\). A classification of singular Riemannian foliations of codimension 1 on all closed simply connected 4-manifolds is obtained as a byproduct. In particular, there are exactly 3 non-homogeneous singular Riemannian foliations of codimension 1, complementing the list of cohomogeneity one 4-manifolds.

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Notes

  1. Throughout this paper, all manifolds considered are connected and smooth.

  2. Here we ignore the specific orientations and identify \(L(\pm m,1)\).

  3. Here orientability refers to that of the total space.

  4. In [44] the second case in \(\mathbb {CP}^2\) was missing, as remarked also in [26, 32]. Note also that it should be \(\mathbb {CP}^2\)#\(-\mathbb {CP}^2\) other than \(\mathbb {CP}^2\)#\(\mathbb {CP}^2\), as the covering space of the manifolds \(43\) and \(49\) in Parker’s list.

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Acknowledgments

The authors are very grateful to Alexander Lytchak for kindly introducing each other, very nice suggestions and useful comments. The authors also thank Zizhou Tang, Gudlaugur Thorbergsson, Burkhard Wilking and Wolfgang Ziller for their support and valuable discussion. Many thanks also to Marcos M. Alexandrino and Fernando Galaz-Garcia for their interest and helpful conversation. The first author would like to thank the Alexander von Humboldt Foundation and the University of Cologne for their support and hospitality during his Humboldt postdoctoral position in 2012–2014, under the supervision of professor Gudlaugur Thorbergsson.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Jianquan Ge

  2. Mathematisches Institut, WWU Münster, Einsteinstr. 62, Münster, Germany

    Marco Radeschi

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  1. Jianquan Ge
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  2. Marco Radeschi
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Correspondence to Jianquan Ge.

Additional information

J. Ge is partially supported by the NSFC (No. 11001016 and No. 11331002), the SRFDP (No. 20100003120003), the Fundamental Research Funds for the Central Universities, and a research fellowship from the Alexander von Humboldt Foundation.

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Ge, J., Radeschi, M. Differentiable classification of 4-manifolds with singular Riemannian foliations. Math. Ann. 363, 525–548 (2015). https://doi.org/10.1007/s00208-015-1172-5

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