A Note on Smale Manifolds and Lorentzian Sasaki-Einstein Geometry

TL;DR

This paper constructs new Lorentzian Sasaki-Einstein metrics on Smale manifolds, including those with nontrivial torsion in second homology, expanding the known examples and addressing open questions in the field.

Contribution

It introduces new Lorentzian Sasaki-Einstein metrics on a broader class of Smale manifolds, including those with torsion in homology and mixed structures, which were previously unknown.

Findings

Existence of LSE metrics on Smale manifolds with nontrivial torsion in H_2

Most simply-connected positive Sasakian rational homology 5-spheres are negative Sasakian

Construction of LSE Smale manifolds with prescribed torsion and arbitrary second Betti number

Abstract

In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds $M.$ It has already been established in \cite{Gmz2} that such metrics exist on the so-called torsion free Smale manifolds, i.e. the $k$-fold connected sum of $S^{2}\times S^{3}.$ Now, we show that LSE metrics exist on Smale manifolds for which $H_{2}(M,\mathbb{Z})_{tor}$ is nontrivial. In particular, we show that most simply-connected positive Sasakian rational homology 5-spheres are also negative Sasakian (hence Lorentzian Sasaki-Einstein). Moreover, we show that for each pair of positive integers $(n,s)$ with $n,s >1$, there exists a Lorentzian Sasaki-Einstein Smale manifold $M$ such that $H_{2}(M,{\mathbb{Z}})_{tors}=(\mathbb{Z}/n)^{2s}$. Finally, we are able to construct so-called mixed Smale manifolds (connect sum of torsion free Smale manifolds with rational homology spheres) which admit LSE metrics and have arbitrary second Betti number. This gives infinitely many examples which do not admit positive Sasakian structures. These results partially address the open problems