essential manifold
compact manifold M whose fundamental class [M] defines a nonzero element under the natural homomorphism Hₙ(M) → Hₙ(K(π₁(M),1))
lens space
3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)
compact manifold M whose fundamental class [M] defines a nonzero element under the natural homomorphism Hₙ(M) → Hₙ(K(π₁(M),1))