In
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
, the spherical space form conjecture (now a theorem) states that a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
acting on the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
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 topology and geometry.
Early life and education
Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
in 1926 after determining the fundamental groups of three-dimensional spherical space forms as a generalization of the
Poincaré conjecture
In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
to the non-simply connected case.
Status
The conjecture is implied by
Thurston's
geometrization conjecture
In mathematics, Thurston's geometrization conjecture 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 theorem for two-dimensi ...
, which was proven by
Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
in 2003. The conjecture was independently proven for groups whose actions have
fixed points—this special case is known as the
Smith conjecture
In mathematics, the Smith conjecture states that if ''f'' is a diffeomorphism of the 3-sphere of Order (group theory), finite order, then the fixed point set of ''f'' cannot be a nontrivial knot (mathematics), knot.
showed that a non-trivial or ...
. It is also proven for various groups acting without fixed points, such as
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s whose orders are a power of two (George Livesay, Robert Myers) and cyclic groups of order 3 (
J. Hyam Rubinstein).
See also
*
Killing–Hopf theorem
In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously ...
References
Conjectures that have been proved
Geometric topology
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