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# List of geometric topology topics

This is a list of geometric topology topics, by Wikipedia page. 

97 relations: Alexander horned sphere, Alexander polynomial, Alternating knot, Analytic torsion, Annulus (mathematics), Borromean rings, Boy's surface, Braid group, Braid theory, Branched surface, Compression body, Connected sum, Cross-cap, Crossing number (knot theory), Dehn twist, Dehn's lemma, Disk (mathematics), Embedding, Euler characteristic, Exotic sphere, Figure-eight knot (mathematics), Flat manifold, Fundamental group, Genus (mathematics), Geometric topology, Geometrization conjecture, Glossary of topology, Graph manifold, H-cobordism, Haken manifold, Handle decomposition, Handlebody, Hauptvermutung, Heegaard splitting, Hilbert–Smith conjecture, Homology sphere, Hyperbolic 3-manifold, Hyperbolic link, I-bundle, Immersion (mathematics), Incompressible surface, Invariant (mathematics), Jones polynomial, JSJ decomposition, Kirby calculus, Klein bottle, Knot (mathematics), Knot complement, Knot group, Knot invariant, ... Expand index (47 more) »

## Alexander horned sphere

The Alexander horned sphere is a pathological object in topology discovered by.

## Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type.

## Alternating knot

In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link.

## Analytic torsion

In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and.

## Annulus (mathematics)

In mathematics, an annulus (the Latin word for "little ring" is anulus/annulus, with plural anuli/annuli) is a ring-shaped object, a region bounded by two concentric circles.

## Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings).

## Boy's surface

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3-space).

## Braid group

In mathematics, the braid group on strands (denoted), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry.

## Braid theory

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations.

## Branched surface

In mathematics, a branched surface is a generalization of both surfaces and train tracks.

## Compression body

In the theory of 3-manifolds, a compression body is a kind of generalized handlebody.

## Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds.

## Cross-cap

In mathematics, a cross-cap is a two-dimensional surface in 3-space that is one-sided and the continuous image of a Möbius strip that intersects itself in an interval.

## Crossing number (knot theory)

In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot.

## Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).

## Dehn's lemma

In mathematics Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is identical to the original on the boundary of the disk.

## Disk (mathematics)

In geometry, a disk (also spelled disc).

## Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

## Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler&ndash;Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

## Exotic sphere

In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere.

## Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four.

## Flat manifold

In mathematics, a Riemannian manifold is said to be flat if its curvature is everywhere zero.

## Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

## Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

## Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

## Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.

## Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology.

## Graph manifold

In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles.

## H-cobordism

In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps are homotopy equivalences.

## Haken manifold

In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface.

## Handle decomposition

In mathematics, a handle decomposition of an m-manifold M is a union where each M_i is obtained from M_ by the attaching of i-handles.

## Handlebody

In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.

## Hauptvermutung

The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them.

## Heegaard splitting

In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

## Hilbert–Smith conjecture

In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M, it states that G must be a Lie group.

## Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1.

## Hyperbolic 3-manifold

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3&ndash;manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1.

In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry.

## I-bundle

In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold.

## Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.

## Incompressible surface

In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold.

## Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

## Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.

## JSJ decomposition

In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson.

## Kirby calculus

In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves.

## Klein bottle

In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.

## Knot (mathematics)

In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).

## Knot complement

In mathematics, the knot complement of a tame knot K is the three-dimensional space surrounding the knot.

## Knot group

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space.

## Knot invariant

In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots.

## Knot polynomial

In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.

## Knot theory

In topology, knot theory is the study of mathematical knots.

## Lens space

A lens space is an example of a topological space, considered in mathematics.

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space.

## Loop theorem

In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma.

## Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Manifold decomposition

In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces.

## Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space.

## Möbius strip

The Möbius strip or Möbius band, also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary.

## Nielsen–Thurston classification

In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface.

## Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

## Orientability

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

## Poincaré conjecture

In mathematics, 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.

## Pretzel

A Pretzel (Breze(l)) is a type of baked bread product made from dough most commonly shaped into a twisted knot.

## Prime knot

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.

## Racks and quandles

In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams.

## Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.

## Roman surface

The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry.

## Schoenflies problem

In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies.

## Seifert fiber space

A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles.

## Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is a surface whose boundary is a given knot or link.

## Signature (topology)

In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d.

## Skein relation

Skein relations are a mathematical tool used to study knots.

## Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Sphere theorem

In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound.

## Spherical 3-manifold

In mathematics, a spherical 3-manifold M is a 3-manifold of the form where \Gamma is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere S^3.

## Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

## Surface bundle over the circle

In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface.

## Thurston elliptization conjecture

William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

## Torus bundle

In mathematics, in the sub-field of geometric topology, a torus bundle is a kind of surface bundle over the circle, which in turn are a class of three-manifolds.

## Torus knot

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3.

## Train track (mathematics)

In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions.

## Trefoil knot

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot.

## Trigenus

In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple (g_1,g_2,g_3).

## Unknot

The unknot arises in the mathematical theory of knots.

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3.

## Wild knot

In the mathematical theory of knots, a knot is tame if it can be "thickened up", that is, if there exists an extension to an embedding of the solid torus S 1 &times; D 2 into the 3-sphere.

## Writhe

In knot theory, there are several competing notions of the quantity writhe, or Wr.

## 3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

## 3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

## References

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