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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a 3-manifold is a 3-dimensional manifold. ...

In the 3-manifold case, a picturesque description of a lens space is that of a space resulting from gluing two solid tori together by a homeomorphism of their boundaries. Of course, to be consistent, we should exclude the 3-sphere and , both of which can be obtained as just described; some mathematicians include these two manifolds in the class of lens spaces. In mathematics, a solid torus is a topological space homeomorphic to , i. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ...

Three-dimensional lens spaces were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone. J.W. Alexander in 1919 showed that the lens spaces L(5;1) and L(5;2) were not homeomorphic even though they have isomorphic fundamental groups and the same homology. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ...

There is a complete classification of three-dimensional lens spaces.

Definition

Sit the 2n − 1-sphere S^{2n − 1} inside as the set of all n-tuples of unit absolute value. Let ω be a primitive pth root of unity and let be integers coprime to p. Let the set of powers act on the sphere by A sphere is a perfectly symmetrical geometrical object. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... The word unit means any of several things: Physical unit, a fundamental quantity of measurement in science or engineering Units (computer program), a popular program that does unit conversion. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Coprime - Wikipedia /**/ @import /skins-1. ...

The resulting orbit space is a lens space, written as . In mathematics, groups are often used to describe symmetries of objects. ...

We can also define the infinite-dimensional lens spaces as follows. These are the spaces formed from the union of the increasing sequence of spaces for . As before, the must be coprime to p. In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Example in three dimensions

The three dimensional Lens spaces L(p;q) arise as quotients of by the action of the group that is generated by elements of the form .

The Lens space L(p;q) has fundamental group for all q, so spaces with different p are not homotopy equivalent. Moreover, L(p;q₁) and L(p;q₂) are homotopy equivalent if and only if .

See also

• spherical 3-manifold

In mathematics, a spherical 3-manifold M is a prime, orientable, closed 3-manifold of the form where Γ is a finite subgroup of SO(4) acting freely by rotations on NaodW29-math613ae1ab9eaa10f00000002. ...

References

• G. Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993.
• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.