geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...

, the spherical space form conjecture (now a theorem) states that a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

acting on the

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

is conjugate to a group of isometries of the 3-sphere.

History

The conjecture was posed by

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...

in 1926 after determining the fundamental groups of three-dimensional spherical space forms as a generalization of the

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

to the non-simply connected case.

Status

The conjecture is implied by Thurston's

geometrization conjecture In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theor ...

, which was proven by

Grigori Perelman Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his ...

in 2003. The conjecture was independently proven for groups whose actions have

fixed points Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fi ...

—this special case is known as the Smith conjecture. It is also proven for various groups acting without fixed points, such as

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

s whose orders are a power of two (George Livesay, Robert Myers) and cyclic groups of order 3 ( J. Hyam Rubinstein).

References

Conjectures that have been proved Geometric topology {{topology-stub

History

Status

See also

References