On the classification of Smale-Barden manifolds with Sasakian structures

Abstract

Smale-Barden manifolds $M$ are classified by their second homology $H_2(M,\mathbb{Z})$ 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 $H_2(M)={\mathbb{Z}}^k\oplus ({\oplus }_{i=1}^r{\mathbb{Z}}_{m_i}^{2g_i})$ and $i=0,\infty$, provided that $k\ge 1$, $m_i\ge 2$, $g_i\ge 1$, $m_i$ 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.

16 pages, no figures

Cited by in corpus (2)

• Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds
• Negative Sasakian structures on simply-connected 5-manifolds