Homology Sphere - Constructions and Examples

Constructions and Examples

• Surgery on a knot in the 3-sphere S3 with framing +1 or − 1 gives a homology sphere.

• More generally, surgery on a link gives a homology sphere whenever the matrix given by intersection numbers (off the diagonal) and framings (on the diagonal) has determinant +1 or −1.

• If p, q, and r are pairwise relatively prime positive integers then the link of the singularity xp + yq + zr = 0 (in other words, the intersection of a small 5-sphere around 0 with this complex surface) is a homology 3-sphere, called a Brieskorn 3-sphere Σ(p, q, r). It is homeomorphic to the standard 3-sphere if one of p, q, and r is 1, and Σ(2, 3, 5) is the Poincaré sphere.

• The connected sum of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere that cannot be written as a connected sum of two homology 3-spheres is called irreducible or prime, and every homology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially unique way. (See Prime decomposition (3-manifold).)

• Suppose that a₁, ..., a_r are integers all at least 2 such that any two are coprime. Then the Seifert fiber space

over the sphere with exceptional fibers of degrees a₁, ..., a_r is a homology sphere, where the b's are chosen so that

(There is always a way to choose the b′s, and the homology sphere does not depend (up to isomorphism) on the choice of b′s.) If r is at most 2 this is just the usual 3-sphere; otherwise they are distinct non-trivial homology spheres. If the a′s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 a′s, not 2, 3, 5, then this is an acyclic homology 3-sphere with infinite fundamental group that has a Thurston geometry modeled on the universal cover of SL₂(R).