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 . 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: and are both infinite cyclic, while for all other indices . 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 , 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 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 and is infinite cyclic. The higher-homotopy groups () are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres. Geometric properties The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of . The Euclidean metric on induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to where is the radius. Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group). Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. This implies that the 3-sphere is parallelizable. It follows that the tangent bundle of the 3-sphere is trivial. For a general discussion of the number of linear independent vector fields on a -sphere, see the article vector fields on spheres. There is an interesting action of the circle group on giving the 3-sphere the structure of a principal circle bundle known as