# Category: Geometric topology

Gieseking manifold
In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approx
Fully irreducible automorphism
In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group Fn is an element of Out(Fn) which has no periodic conjugacy classes of proper free factors in Fn
Hsiang–Lawson's conjecture
In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian Mathematical Society Gazette a
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unkno
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting (Danish: [ˈhe̝ˀˌkɒˀ]) is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Loop theorem
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma a
Torus bundle
A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds.
Casson invariant
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 Cas
One-relator group
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
Piecewise linear manifold
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pas
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 function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in th
Thurston boundary
In mathematics, the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Th
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadam
Y-homeomorphism
In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces. It can be constructed by sliding a Möbius band included on the surface aroun
Closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.
Boundary parallel
In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M.
Connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
Steenrod problem
In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.
Milnor conjecture (topology)
In knot theory, the Milnor conjecture says that the slice genus of the torus knot is It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer
Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topolo
Milnor–Wood inequality
In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named
Cellular decomposition
In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells (spaces homeomorphic to n-balls Bn). The quotient space M/G has points that cor
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective a
Unknotting problem
In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algori
Handle decomposition
In mathematics, a handle decomposition of an m-manifold M is a union where each is obtained from by the attaching of -handles. A handle decomposition is to a manifold what a CW-decomposition is to a t
Bing shrinking
In geometric topology, a branch of mathematics, the Bing shrinking criterion, introduced by , is a method for showing that a quotient of a space is homeomorphic to the space.
Grushko theorem
In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free pro
Alexander's trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.
Kirby–Siebenmann class
In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.
Dogbone space
In geometric topology, the dogbone space, constructed by R. H. Bing, is a quotient space of three-dimensional Euclidean space such that all inverse images of points are points or , yet it is not homeo
Cannon–Thurston map
In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group
Clutching construction
In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
Lantern relation
In geometric topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The most general version of the rel
Fox derivative
In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related co
Tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple
Crumpled cube
In geometric topology, a branch of mathematics, a crumpled cube is any space in R3 homeomorphic to a 2-sphere together with its interior. Lininger showed in 1965 that the union of a crumpled cube and
Microbundle
In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situat
Round function
In topology and in calculus, a round function is a scalar function , over a manifold , whose critical points form one or several connected components, each homeomorphic to the circle , also called cri
List of geometric topology topics
This is a list of geometric topology topics, by Wikipedia page. See also: * topology glossary * List of topology topics * List of general topology topics * List of algebraic topology topics * Pub
McShane's identity
In geometric topology, McShane's identity for a once punctured torus with a complete, finite-volume hyperbolic structure is given by where * the sum is over all simple closed geodesics γ on the torus
Slam-dunk
In the mathematical field of low-dimensional topology, the slam-dunk is a particular modification of a given surgery diagram in the 3-sphere for a 3-manifold. The name, but not the move, is due to Tim
Surface bundle
In mathematics, a surface bundle is a bundle in which the fiber is a surface. When the base space is a circle the total space is three-dimensional and is often called a surface bundle over the circle.
Free factor complex
In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface.The free factor comp
Pachner moves
In topology, a branch of mathematics, Pachner moves, named after Udo Pachner, are ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homeomorphic manifo
Mapping torus
In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed in
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 projecti
Geospatial topology
Geospatial topology is the study and application of qualitative spatial relationships between geographic features, or between representations of such features in geographic information, such as in geo
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class vanishes), t
Clasper (mathematics)
In the mathematical field of low-dimensional topology, a clasper is a surface (with extra structure) in a 3-manifold on which surgery can be performed.
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. Using four-dime
Whitney disk
In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that
E8 manifold
In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of on
Novikov conjecture
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homot
Nielsen realization problem
The Nielsen realization problem is a question asked by Jakob Nielsen about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by Steven Kerckhoff .
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 -bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are
Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often refe
Annulus theorem
In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomor
Line of greatest slope
In topography, the line of greatest slope is a curve following the steepest slope. In mountain biking and skiing, the line of greatest slope is sometimes called the fall line.
Seifert surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary
Semi-s-cobordism
In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the
In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in
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 h
Spherical 3-manifold
In mathematics, a spherical 3-manifold M is a 3-manifold of the form where is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and clos
Trigenus
In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple . It is obtained by minimizing the genera of three orientable handle bodies — with no i
I-bundle
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can
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 t
Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is th
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to J. H. C. Whitehead discovered this puzzling object while he was trying to prove the Poincaré
Donaldson theory
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (19
Fundamental polygon
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also d
Barycentric subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the
Handlebody
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surge
Surgery structure set
In mathematics, the surgery structure set is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether
De Rham invariant
In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of – either 0 or 1. It can be thought of as the simply-connected symmetric L-gro
PDIFF
In geometric topology, PDIFF, for piecewise differentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth mani
Blaschke selection theorem
The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence of convex sets contained in a bounded set, the theorem guarant
Convergence group
In mathematics, a convergence group or a discrete convergence group is a group acting by homeomorphisms on a compact metrizable space in a way that generalizes the properties of the action of Kleinian
JTS Topology Suite
JTS Topology Suite (Java Topology Suite) is an open-source Java software library that provides an object model for Euclidean planar linear geometry together with a set of fundamental geometric functio
Generalized Poincaré conjecture
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topo
Hauptvermutung
The Hauptvermutung of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdi
Ropelength
In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot
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. Representative topics are the structure th
Mapping class group of a surface
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface vie
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. It cannot be embedded in standard three-dimensional s
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic
Branched surface
In mathematics, a branched surface is a generalization of both surfaces and train tracks.
Double suspension theorem
In geometric topology, the double suspension theorem of James W. Cannon and Robert D. Edwards states that the double suspension S2X of a homology sphere X is a topological sphere. If X is a piecewise-
Orientation character
In algebraic topology, a branch of mathematics, an orientation character on a group is a group homomorphism . This notion is of particular significance in surgery theory.
Pseudo-Anosov map
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its de
Relative dimension
In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map , the difference dim V − dim Q is the relative d
5-manifold
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is hard
Mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group
DE-9IM
The Dimensionally Extended 9-Intersection Model (DE-9IM) is a topological model and a standard used to describe the spatial relations of two regions (two geometries in two-dimensions, R2), in geometry
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-
Fake 4-ball
In mathematics, a fake 4-ball is a compact contractible topological 4-manifold. Michael Freedman proved that every three-dimensional homology sphere bounds a fake 4-ball. His construction involves the
Alexander horned sphere
The Alexander horned sphere is a pathological object in topology discovered by J. W. Alexander.
String topology
String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by Moira Chas and Dennis Sullivan.
Nielsen–Thurston classification
In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen. Given a homeom
Wild arc
In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc
Casson handle
In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them
Cyclic surgery theorem
In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is
Klein bottle
In topology, a branch of mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector can
Signature (topology)
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four. This invariant of a manifold has been studied in detail, sta
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
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles an
Introduction to 3-Manifolds
Introduction to 3-Manifolds is a mathematics book on low-dimensional topology. It was written by Jennifer Schultens and published by the American Mathematical Society in 2014 as volume 151 of their bo
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. When doing so, one must specify both what those pieces are and how they a
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
Side-approximation theorem
In geometric topology, the side-approximation theorem was proved by . It implies that a 2-sphere in R3 can be approximated by polyhedral 2-spheres.
Spherical space form conjecture
In geometric topology, the spherical space form conjecture (now a theorem) states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.
Intersection form of a 4-manifold
In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-
2-sided
In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold of a manifold is said to be 2-sided in when there is an embedding with for each and . In other words, if it
Local flatness
In topology, a branch of mathematics, local flatness is smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a
Train track map
In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice c
Zeeman conjecture
In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex , the space is collapsible. The conjecture, due to Christo
Wall's finiteness obstruction
In geometric topology, a field within mathematics, the obstruction to a finitely dominated space X being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction w(X) which is an
Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1a and S1b (in the same sense that the surface of a cylinder is "fl
Outer space (mathematics)
In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free group Fn is a topological space consisting of the so-called "marked metric graph st
Bing's recognition theorem
In topology, a branch of mathematics, Bing's recognition theorem, named for R. H. Bing, asserts that a necessary and sufficient condition for a 3-manifold M to be homeomorphic to the 3-sphere is that
Borel conjecture
In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. I
Sphere theorem (3-manifolds)
In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedde
Acylindrically hyperbolic group
In the mathematical subject of geometric group theory, an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space
L-theory
In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as "Hermitian K-the
Symplectic filling
In mathematics, a filling of a manifold X is a cobordism W between X and the empty set. More to the point, the n-dimensional topological manifold X is the boundary of an (n + 1)-dimensional manifold W
Moise's theorem
In geometric topology, a branch of mathematics, Moise's theorem, proved by Edwin E. Moise in , states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth st
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topologi