On the classification of Smale-Barden manifolds with Sasakian structures

2020/02/02 by Aleksy Tralle, Vicente Muñoz, Tralle, Aleksy +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Geometry and complex manifolds #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2002.00457

Abstract

Smale-Barden manifolds M are classified by their second homology H₂(M,\mathbb Z) and the Barden invariant i(M). It is an important and dificult question to decide when M admits a Sasakian structure in terms of these data. In this work we show methods of doing this. In particular we realize all M with H₂(M)=\mathbb Z^k⊕(⊕_i=1^r\mathbb Z_mi^2g_i) and i=0,∞, provided that k≥ 1, m_i≥ 2, g_i≥ 1, m_i are pairwise coprime. Using our methods we also contribute to the problem of the existence of definite Sasakian structures on rational homology spheres. Also, we give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures.

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• Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds

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