In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a spherical 3-manifold ''M'' is a
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
of the form
:
where
is a
finite
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 marke ...
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of
SO(4) acting freely by rotations 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 dimensio ...
. All such manifolds are
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
,
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, and
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds.
Properties
A spherical 3-manifold
has a finite
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to Γ itself. The
elliptization conjecture, proved 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 ...
, states that conversely all compact 3-manifolds with finite fundamental group are spherical manifolds.
The fundamental group is either
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
, or is a central extension of a
dihedral,
tetrahedral
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
,
octahedral
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
, or
icosahedral group by a cyclic group of even order. This divides the set of such manifolds into 5 classes, described in the following sections.
The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of 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-dimens ...
.
Cyclic case (lens spaces)
The manifolds
with Γ
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
are precisely the 3-dimensional
lens spaces. A lens space is not determined by its fundamental group (there are non-
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
lens spaces with
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
fundamental groups); but any other spherical manifold is.
Three-dimensional lens spaces arise as quotients of
by
the action of the group that is generated by elements of the form
:
where
. Such a lens space
has fundamental group
for all
, so spaces with different
are not homotopy equivalent.
Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces
and
are:
#homotopy equivalent if and only if
for some
#homeomorphic if and only if
In particular, the lens spaces ''L''(7,1) and ''L''(7,2) give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic.
The lens space ''L''(1,0) is the 3-sphere, and the lens space ''L''(2,1) is 3 dimensional real projective space.
Lens spaces can be represented as
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s in many ways, usually as fiber spaces over the 2-sphere with at most two exceptional fibers, though the lens space with fundamental group of order 4 also has a representation as a Seifert fiber space over the projective plane with no exceptional fibers.
Dihedral case (prism manifolds)
A prism manifold is a closed
3-dimensional manifold ''M'' whose fundamental group
is a central extension of a dihedral group.
The fundamental group π
1(''M'') of ''M'' is a product of a cyclic group of order ''m'' with a group having presentation
:
for integers ''k'', ''m'', ''n'' with ''k'' ≥ 1, ''m'' ≥ 1, ''n''
≥ 2 and ''m'' coprime to 2''n''.
Alternatively, the fundamental group has presentation
:
for coprime integers ''m'', ''n'' with ''m'' ≥ 1, ''n'' ≥ 2. (The ''n'' here equals the previous ''n'', and the ''m'' here is 2
''k''-1 times the previous ''m''.)
We continue with the latter presentation. This group is a
metacyclic group
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group ''G'' for which there is a short exact sequence
:1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\,
where ''H'' and ''K'' ar ...
of order 4''mn'' with
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
of order 4''m'' (so ''m'' and ''n'' are both determined by this group).
The element ''y'' generates a
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of order 2''n'', and the element ''x'' has order 4''m''. The
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
is cyclic of order 2''m'' and is generated by ''x''
2, and the quotient by the center is the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of order 2''n''.
When ''m'' = 1 this group is a binary dihedral or
dicyclic group
In group theory, a dicyclic group (notation Dic''n'' or Q4''n'', Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST) is a particular kind of non-abelian group of order 4''n'' (''n'' > 1). It is an extension of the ...
. The simplest example is ''m'' = 1, ''n'' = 2, when π
1(''M'') is the
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
of order 8.
Prism manifolds are uniquely determined by their fundamental groups: if a closed 3-manifold has the same fundamental group as a prism manifold ''M'', it is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to ''M''.
Prism manifolds can be represented as
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s in two ways.
Tetrahedral case
The fundamental group is a product of a cyclic group of order ''m'' with a group having presentation
:
for integers ''k'', ''m'' with ''k'' ≥ 1, ''m'' ≥ 1 and ''m'' coprime to 6.
Alternatively, the fundamental group has presentation
:
for an odd integer ''m'' ≥ 1. (The ''m'' here is 3
''k''-1 times the previous ''m''.)
We continue with the latter presentation. This group has order 24''m''. The elements ''x'' and ''y'' generate a normal subgroup isomorphic to the
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
of order 8. The
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
is cyclic of order 2''m''. It is generated by the elements ''z''
3 and ''x''
2 = ''y''
2, and the quotient by the center is the tetrahedral group, equivalently, the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
''A''
4.
When ''m'' = 1 this group is the
binary tetrahedral group
In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of ...
.
These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 3.
Octahedral case
The fundamental group is a product of a cyclic group of order ''m'' coprime to 6 with the
binary octahedral group In mathematics, the binary octahedral group, name as 2O or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group ''O'' or (2, ...
(of order 48) which has the presentation
:
These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 4.
Icosahedral case
The fundamental group is a product of a cyclic group of order ''m'' coprime to 30 with the
binary icosahedral group In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120.
It is an extension of the icosahedral group ''I'' or (2,3,5) of o ...
(order 120) which has the presentation
:
When ''m'' is 1, the manifold is the
Poincaré homology sphere
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luci ...
.
These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 5.
References
*
Peter Orlik
Peter Paul Nikolas Orlik (born 12 November 1938, in Budapest) is an American mathematician, known for his research on topology, algebra, and combinatorics.
Orlik earned in 1961 his bachelor's degree from the Norwegian Institute of Technology in Tr ...
, ''Seifert manifolds'', Lecture Notes in Mathematics, vol. 291,
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
(1972).
*
William Jaco
William "Bus" H. Jaco (born July 14, 1940 in Grafton, West Virginia) is an American mathematician who is known for his role in the Jaco–Shalen–Johannson decomposition theorem and is currently Regents Professor and Grayce B. Kerr Chair at Okl ...
, ''Lectures on 3-manifold topology''
*
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thursto ...
, ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35.
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial ...
,
Princeton, New Jersey
Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of w ...
, 1997. {{ISBN, 0-691-08304-5
Geometric topology
Riemannian geometry
Group theory
3-manifolds