Chromatic Homotopy Theory
《Chromatic Homotopy Theory》 · 第 272 页 · Section 9.1.2，Homology spheres
By Poincaré duality, a closed 3-manifold \(M\) is a homology sphere if and only if \(H_1(M)\) vanishes. Since \(H_1(M)\) is the abelianization of \(\pi_1(M)\), this happens precisely when \(\pi_1(M)\) is a perfect group, that is a group with trivial abelianization.