structural:reformatting, linguistic:paraphrasing, structural:summarization
Topological properties

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

The 3-sphere is homeomorphic to the one-point compactification of R. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere.

The homology groups of the 3-sphere are as follows: H (S, Z) and H (S, Z) are both infinite cyclic, while H (S, Z) = {} for all other indices i. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1 / n on any knot in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the homotopy groups, we have π (S) = π (S) = {} and π (S) is infinite cyclic. The higher-homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres.