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On the classification of Smale–Barden manifolds with Sasakian structures

Vicente Muñoz

Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Campus de Teatinos, s/n, 29071 Málaga, Spain

E-mail Address: vicente.munoz@uma.es

corresponding author.

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and

Aleksy Tralle

Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, Poland

E-mail Address: tralle@matman.uwm.edu.pl

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https://doi.org/10.1142/S0219199721500772Cited by:2 (Source: Crossref)

Abstract

Smale–Barden manifolds $M$ are classified by their second homology $H_{2} (M, ℤ)$ and the Barden invariant $i (M)$ . It is an important and difficult 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 $H_{2} (M, ℤ) = ℤ^{k} \oplus (\oplus_{i = 1}^{r} ℤ_{m_{i}}^{2 g_{i}})$ and $i (M) = 0, \infty,$ provided that $k \geq 1$ , $m_{i} \geq 2, g_{i} \geq 1$ , $m_{i}$ are pairwise coprime. 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. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.

Keywords:

AMSC: 53C25, 57K50, 57R18

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