In mathematics, a spherical 3-manifoldM is a 3-manifold of the form
where is a finite subgroup of O(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds.
A special case of the Bonnet–Myers theorem says that every smooth manifold which has a smooth Riemannian metric which is both geodesically complete and of constant positive curvature must be closed and have finite fundamental group. William Thurston's elliptization conjecture, proven by Grigori Perelman using Richard Hamilton's Ricci flow, states a converse: every closed three-dimensional manifold with finite fundamental group has a smooth Riemannian metric of constant positive curvature. (This converse is special to three dimensions.) As such, the spherical three-manifolds are precisely the closed 3-manifolds with finite fundamental group.
According to Synge's theorem, every spherical 3-manifold is orientable, and in particular must be included in SO(4). The fundamental group is either cyclic, or is a central extension of a dihedral, tetrahedral, octahedral, or icosahedral group by a cyclic group of even order. This divides the set of such manifolds into five classes, described in the following sections.
The spherical manifolds are exactly the manifolds with spherical geometry, one of the eight geometries of Thurston's geometrization conjecture.
The manifolds with Γ cyclic are precisely the 3-dimensional lens spaces. A lens space is not determined by its fundamental group (there are non-homeomorphic lens spaces with isomorphic 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:
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 spaces 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.
A prism manifold is a closed 3-dimensional manifold M whose fundamental group is a central extension of a dihedral group.
The fundamental group π1(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 2n.
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 2k-1 times the previous m.)
We continue with the latter presentation. This group is a metacyclic group of order 4mn with abelianization of order 4m (so m and n are both determined by this group). The element y generates a cyclic normal subgroup of order 2n, and the element x has order 4m. The center is cyclic of order 2m and is generated by x2, and the quotient by the center is the dihedral group of order 2n.
When m = 1 this group is a binary dihedral or dicyclic group. The simplest example is m = 1, n = 2, when π1(M) is the quaternion group 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 to M.
Prism manifolds can be represented as Seifert fiber spaces in two ways.
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 3k-1 times the previous m.)
We continue with the latter presentation. This group has order 24m. The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elements z3 and x2 = y2, and the quotient by the center is the tetrahedral group, equivalently, the alternating group A4.
When m = 1 this group is the binary tetrahedral group.
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 3.
The fundamental group is a product of a cyclic group of order m coprime to 6 with the binary octahedral group (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 spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 4.
The fundamental group is a product of a cyclic group of order m coprime to 30 with the binary icosahedral group (order 120) which has the presentation
When m is 1, the manifold is the Poincaré homology sphere.
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.
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination of finitely many elements of the subset and their inverses.
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
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring and a two-sided ideal in , a new ring, the quotient ring , is constructed, whose elements are the cosets of in subject to special and operations.
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-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.
In the mathematical disciplines of topology and geometry, an orbifold is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group.
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is,
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean 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 and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals. The theorem was first proven by Emanuel Lasker for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether.
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a -bundle over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.
In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.
In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy.
In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order , for a fixed prime number and varying integer exponents . Such groups are briefly called finitep-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups.
In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups.
