Rational Homology 5-Spheres with Positive Ricci Curvature

TL;DR

This paper constructs infinitely many simply connected rational homology 5-spheres with positive Ricci curvature, characterized by their second homology group order, and establishes their uniqueness based on prime decompositions.

Contribution

It introduces a method to produce rational homology 5-spheres with positive Ricci curvature for any integer k>1, and classifies them by prime factorization.

Findings

Existence of such 5-spheres for all k>1.

Explicit relation between homology group order and prime factorization.

Uniqueness of the manifolds based on prime decomposition.

Abstract

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decomposition of $\scriptstyle{k}$ has the form $\scriptstyle{k=p_1... p_r}$ for distinct primes $\scriptstyle{p_i}$ then $\scriptstyle{M^5_k}$ is uniquely determined.