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### Sol geometry

The geometry of the Lie group semidirect product with , where acts on by .

### Knot diagram

A knot diagram is a picture of a projection of a knot onto a plane. Usually, only double points are allowed (no more than two points are allowed to be superposed), and the double or crossing points must be "genuine crossings" which transverse in the plane. This means that double points must look like the above left diagram, and not the above right one. Also, it is usually demanded that a knot diagram contain the information if the crossings are overcrossings or undercrossings so that the original knot can be reconstructed.The knot diagram of the trefoil knot is illustratedabove.Knot polynomials can be computed from knot diagrams. Such polynomials often (but not always) allow the knots corresponding to given diagrams to be uniquely identified.Rolfsen (1976) gives a table of knot diagrams for knots up to 10 crossings and links up to four components and 9 crossings. Adams (1994) gives a smaller table of knots diagrams up to 9 crossings, two-component..

### Hyperplane

Let , , ..., be scalars not all equal to 0. Then the set consisting of all vectorsin such thatfor a constant is a subspace of called a hyperplane.More generally, a hyperplane is any codimension-1 vector subspace of a vector space. Equivalently, a hyperplane in a vector space is any subspace such that is one-dimensional. Equivalently, a hyperplane is the linear transformation kernel of any nonzero linear map from the vector space to the underlying field.

### Rank

The word "rank" refers to several related concepts in mathematics involving graphs, groups, matrices, quadratic forms, sequences, set theory, statistics, and tensors.In graph theory, the graph rank of a graph is defined as , where is the number of vertices on and is the number of connected components (Biggs 1993, p. 25).In set theory, rank is a (class) function from sets to ordinal numbers. The rank of a set is the least ordinal number greater than the rank of any member of the set (Mirimanoff 1917; Moore 1982, pp. 261-262; Rubin 1967, p. 214). The proof that rank is well-defined uses the axiom of foundation.For example, the empty set has rank 0 (since it has no members and 0 is the least ordinal number), has rank 1 (since , its only member, has rank 0), has rank 2, and has rank . Every ordinal number has itself as its rank.Mirimanoff (1917) showed that, assuming the class of urelements is a set, for any ordinal number , the class..

### Refinement

A refinement of a cover is a cover such that every element is a subset of an element .

### Gauss's theorema egregium

Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. "Residents" of the surface may observe the Gaussian curvature of the surface without ever venturing into full three-dimensional space; they can observe the curvature of the surface they live in without even knowing about the three-dimensional space in which they are embedded.In particular, Gaussian curvature can be measured by checking how closely the arc lengths of circles of small radii correspond to what they should be in Euclidean space, . If the arc length of circles tends to be smaller than what is expected in Euclidean space, then the space is positively curved; if larger, negatively; if the same, 0 Gaussian curvature.Gauss (effectively) expressed the theorema egregium by saying that the Gaussian curvature at a point is given by where is the Riemann tensor, and and are an orthonormal..

### Direct sum

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.The matrix direct sum is defined by(1)(2)(Ayres 1962, pp. 13-14).The direct sum of two subspaces and is the sum of subspaces in which and have only the zero vector in common (Rosen 2000, p. 357).The significant property of the direct sum is that it is the coproduct in the category of modules (i.e., a module direct sum). This general definition gives as a consequence the definition of the direct sum of Abelian groups and (since they are -modules, i.e., modules over the integers) and the direct sum of vector spaces (since they are modules over a field). Note that the direct sum of Abelian groups is the same as the group direct product, but that the term direct sum is not used for groups which are non-Abelian.Note that direct products and direct sums differ for infinite indices. An element of the direct sum is..

### Cartesian product

The Cartesian product of two sets and (also called the product set, set direct product, or cross product) is defined to be the set of all points where and . It is denoted , and is called the Cartesian product since it originated in Descartes' formulation of analytic geometry. In the Cartesian view, points in the plane are specified by their vertical and horizontal coordinates, with points on a line being specified by just one coordinate. The main examples of direct products are Euclidean three-space (, where are the real numbers), and the plane ().The graph product is sometimes called the Cartesianproduct (Vizing 1963, Clark and Suen 2000).

### Group torsion

If is a group, then the torsion elements of (also called the torsion of ) are defined to be the set of elements in such that for some natural number , where is the identity element of the group .In the case that is Abelian, is a subgroup and is called the torsion subgroup of . If consists only of the identity element, the group is called torsion-free.

### Group rank

For any prime number and any positive integer , the -rank of a finitely generated Abelian group is the number of copies of the cyclic group appearing in the Kronecker decomposition of (Schenkman 1965). The free (or torsion-free) rank of is the number of copies of appearing in the same decomposition. It can be characterized as the maximal number of elements of which are linearly independent over . Since it is also equal to the dimension of as a vector space over , it is often called the rational rank of . Munkres (1984) calls it the Betti number of .Most authors refer to simply as the "rank" of (Kargapolov and Merzljakov 1979), whereas others (Griffith 1970) use the word "rank" to denote the sum . In this latter meaning, the rank of is the number of direct summands appearing in the Kronecker decomposition of ...

### Homotopy group

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The th homotopy group of a topological space is the set of homotopy classes of maps from the n-sphere to , with a group structure, and is denoted . The fundamental group is , and, as in the case of , the maps must pass through a basepoint . For , the homotopy group is an Abelian group.The group operations are not as simple as those for the fundamental group. Consider two maps and , which pass through . The product is given by mapping the equator to the basepoint . Then the northern hemisphere is mapped to the sphere by collapsing the equator to a point, and then it is mapped to by . The southern hemisphere is similarly mapped to by . The diagram above shows the product of two spheres.The identity element is represented by the constant map . The choice of direction of a loop in the fundamental group corresponds to a manifold orientation of in a homotopy..

### Bordism group

There are two types of bordism groups: bordism groups, also called cobordism groups or cobordism rings, and there are singular bordism groups. The bordism groups give a framework for getting a grip on the question, "When is a compact boundaryless manifold the boundary of another manifold?" The answer is, precisely when all its Stiefel-Whitney numbers are zero. Singular bordism groups give insight into Steenrod's realization problem: "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too.The machinery of the bordism group winds up being important for homotopytheory as well.

### Homotopy

A continuous transformation from one function to another. A homotopy between two functions and from a space to a space is a continuous map from such that and , where denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping space from the first function to the second.Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. The concept of homotopy was first formulated by Poincaré around 1900 (Collins 2004).

### Closed set

There are several equivalent definitions of a closed set. Let be a subset of a metric space. A set is closed if 1. The complement of is an open set, 2. is its own set closure, 3. Sequences/nets/filters in that converge do so within , 4. Every point outside has a neighborhood disjoint from . The point-set topological definition of a closed set is a set which contains all of its limit points. Therefore, a closed set is one for which, whatever point is picked outside of , can always be isolated in some open set which doesn't touch .The most commonly encountered closed sets are the closed interval, closed path, closed disk, interior of a closed path together with the path itself, and closed ball. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points (and is nowhere dense, so it has Lebesgue measure 0).It is possible for a set to be neither open nor closed, e.g., the half-closed interval ...

### Dimension axiom

One of the Eilenberg-Steenrod axioms. Let be a single point space. unless , in which case where are some groups. The are called the coefficients of the homology .

### Exponential map

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp() is defined to be , where is the unique Lie group homeomorphism from the real numbers to the Lie group such that its velocity at time 0 is .On a Riemannian manifold, exp is a map from the tangent bundle of the manifold to the manifold, and exp() is defined to be , where is the unique geodesic traveling through the base-point of such that its velocity at time 0 is .The three notions of exp (exp from complex analysis, exp from Lie groups, and exp from Riemannian geometry) are all linked together, the strongest link being between the Lie groups and Riemannian geometry definition. If is a compact Lie group, it admits a left and right invariant Riemannian metric. With respect to that metric, the two exp maps agree on their common domain. In other words, one-parameter subgroups are geodesics. In the case of the manifold..

### Chern class

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The th Chern class is an obstruction to the existence of everywhere complex linearly independent vector fields on that vector bundle. The th Chern class is in the th cohomology group of the base space.

### Characteristic class

Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the existence of everywhere linearly independent vector fields on the vector bundle. The most common examples of characteristic classes are the Chern, Pontryagin, and Stiefel-Whitney classes.

### Excision axiom

One of the Eilenberg-Steenrod axioms which states that, if is a space with subspaces and such that the set closure of is contained in the interior of , then the inclusion map induces an isomorphism .

### Parallel postulate

Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended. This statement is equivalent to the fifth of Euclid's postulates, which Euclid himself avoided using until proposition 29 in the Elements. For centuries, many mathematicians believed that this statement was not a true postulate, but rather a theorem which could be derived from the first four of Euclid's postulates. (That part of geometry which could be derived using only postulates 1-4 came to be known as absolute geometry.)Over the years, many purported proofs of the parallel postulate were published. However, none were correct, including the 28 "proofs" G. S. Klügel analyzed in his dissertation of 1763 (Hofstadter 1989). The main motivation for all of this effort was that Euclid's parallel postulate did..

### Long exact sequence of a pair axiom

One of the Eilenberg-Steenrod axioms. It states that, for every pair , there is a natural long exact sequencewhere the map is induced by the inclusion map and is induced by the inclusion map . The map is called the boundary map.

### Compact manifold

A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold.For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold.For any positive integer , a distinct nonorientable..

### Cohomology

Cohomology is an invariant of a topological space, formally "dual" to homology, and so it detects "holes" in a space. Cohomology has more algebraic structure than homology, making it into a graded ring (with multiplication given by the so-called "cup product"), whereas homology is just a graded Abelian group invariant of a space.A generalized homology or cohomology theory must satisfy all of the Eilenberg-Steenrodaxioms with the exception of the dimension axiom.

### Lorentz group

The Lorentz group is the group of time-preserving linear isometries of Minkowski space with the Minkowski metric(where the convention is used). It is also the group of isometries of three-dimensional hyperbolic geometry. It is time-preserving in the sense that the unit time vector is sent to another vector such that .A consequence of the definition of the Lorentz group is that the full group of time-preserving isometries of Minkowski space is the group direct product of the group of translations of (i.e., itself, with addition as the group operation), with the Lorentz group, and that the full isometry group of the Minkowski is a group extension of by the product .The Lorentz group is invariant under space rotations and Lorentztransformations.

### Poincar&eacute; hyperbolic disk

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk , with hyperbolic metric(1)The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. The illustration above shows a hyperbolic tessellation similar to M. C. Escher's Circle Limit IV (Heaven and Hell) (Trott 1999, pp. 10 and 83).The endpoints of any arc can be specified by two angles around the disk and . Define(2)(3)Then trigonometry shows that in the above diagram,(4)(5)so the radius of the circle forming the arc is and its center is located at , where(6)The half-angle subtended by the arc is then(7)so(8)The..

### Disk

An -dimensional disk (sometimes spelled "disc") of radius is the collection of points of distance (closed disk) or (open disk) from a fixed point in Euclidean -space. A disk is the shadow of a ball on a plane perpendicular to the ball-radiant point line.The -disk for is called a ball, and the boundary of the -disk is a -hypersphere. The standard -disk, denoted (or ), has its center at the origin and has radius .

### Stiefel manifold

The Stiefel manifold of orthonormal -frames in is the collection of vectors (, ..., ) where is in for all , and the -tuple (, ..., ) is orthonormal. This is a submanifold of , having dimension .Sometimes the "orthonormal" condition is dropped in favor of the mildly weaker condition that the -tuple (, ..., ) is linearly independent. Usually, this does not affect the applications since Stiefel manifolds are usually considered only during homotopy theoretic considerations. With respect to homotopy theory, the two definitions are more or less equivalent since Gram-Schmidt orthonormalization gives rise to a smooth deformation retraction of the second type of Stiefel manifold onto the first.

### Smooth manifold

Another word for a (infinitely differentiable) manifold, also called a differentiable manifold. A smooth manifold is a topological manifold together with its "functional structure" (Bredon 1995) and so differs from a topological manifold because the notion of differentiability exists on it. Every smooth manifold is a topological manifold, but not necessarily vice versa. (The first nonsmooth topological manifold occurs in four dimensions.) Milnor (1956) showed that a seven-dimensional hypersphere can be made into a smooth manifold in 28 ways.

### Brouwer degree

Let be a map between two compact, connected, oriented -dimensional manifolds without boundary. Then induces a homomorphism from the homology groups to , both canonically isomorphic to the integers, and so can be thought of as a homomorphism of the integers. The integer to which the number 1 gets sent is called the degree of the map .There is an easy way to compute if the manifolds involved are smooth. Let , and approximate by a smooth map homotopic to such that is a "regular value" of (which exist and are everywhere dense by Sard's theorem). By the implicit function theorem, each point in has a neighborhood such that restricted to it is a diffeomorphism. If the diffeomorphism is orientation preserving, assign it the number , and if it is orientation reversing, assign it the number . Add up all the numbers for all the points in , and that is the , the Brouwer degree of . One reason why the degree of a map is important is because it is a homotopy invariant...

### Bordism

A relation between compact boundaryless manifolds (also called closed manifolds). Two closed manifolds are bordant iff their disjoint union is the boundary of a compact -manifold. Roughly, two manifolds are bordant if together they form the boundary of a manifold. The word bordism is now used in place of the original term cobordism.

### Riemann sphere

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold (C-star) which is the one-point compactification of the complex numbers , together with two charts. (Here denotes complex infinity.) The notation is also used (Krantz 1999, p. 82; Lorentzen, and Waadeland 2008, p. 3).For all points in the complex plane, the chart is the identity map from the sphere (with infinity removed) to the complex plane. For the point at infinity, the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending infinity to 0 and all other points to .

### Grassmann manifold

A special case of a flag manifold. A Grassmann manifold is a certain collection of vector subspaces of a vector space. In particular, is the Grassmann manifold of -dimensional subspaces of the vector space . It has a natural manifold structure as an orbit-space of the Stiefel manifold of orthonormal -frames in . One of the main things about Grassmann manifolds is that they are classifying spaces for vector bundles.

### Ambient isotopy

An ambient isotopy from an embedding of a manifold in to another is a homotopy of self diffeomorphisms (or isomorphisms, or piecewise-linear transformations, etc.) of , starting at the identity map, such that the "last" diffeomorphism compounded with the first embedding of is the second embedding of . In other words, an ambient isotopy is like an isotopy except that instead of distorting the embedding, the whole ambient space is being stretched and distorted and the embedding is just "coming along for the ride." For smooth manifolds, a map is isotopic iff it is ambiently isotopic.For knots, the equivalence of manifolds under continuous deformation is independent of the embedding space. Knots of opposite chirality have ambient isotopy, but not regular isotopy.

### Flag manifold

For any sequence of integers , there is a flag manifold of type (, ..., ) which is the collection of ordered sets of vector subspaces of (, ..., ) with and a subspace of . There are also complex flag manifolds with complex subspaces of instead of real subspaces of a real -space.These flag manifolds admit the structure of manifoldsin a natural way and are used in the theory of Lie groups.

### Module

A module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of the rules of manipulating vectors hold. A module is abstractly very similar to a vector space, although in modules, coefficients are taken in rings that are much more general algebraic objects than the fields used in vector spaces. A module taking its coefficients in a ring is called a module over , or a R-module.Modules are the basic tool of homological algebra. Examples of modules include the set of integers , the cubic lattice in dimensions , and the group ring of a group. is a module over itself. It is closed under addition and subtraction (although it is sufficient to require closure under subtraction). Numbers of the form for and a fixed integer form a submodule since, for all ,and is still in .Given two integers and , the smallest module containing and is the module for their greatest common divisor, ...

### Homology

Homology is a concept that is used in many branches of algebra and topology. Historically, the term "homology" was first used in a topological sense by Poincaré. To him, it meant pretty much what is now called a bordism, meaning that a homology was thought of as a relation between manifolds mapped into a manifold. Such manifolds form a homology when they form the boundary of a higher-dimensional manifold inside the manifold in question.To simplify the definition of homology, Poincaré simplified the spaces he dealt with. He assumed that all the spaces he dealt with had a triangulation (i.e., they were "simplicial complexes"). Then instead of talking about general "objects" in these spaces, he restricted himself to subcomplexes, i.e., objects in the space made up only on the simplices in the triangulation of the space. Eventually, Poincaré's version of homology was dispensed with and..

### Determinant

Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. As shown by Cramer's rule, a nonhomogeneous system of linear equations has a unique solution iff the determinant of the system's matrix is nonzero (i.e., the matrix is nonsingular). For example, eliminating , , and from the equations(1)(2)(3)gives the expression(4)which is called the determinant for this system of equation. Determinants are definedonly for square matrices.If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, the matrix is said to be unimodular.The determinant of a matrix ,(5)is commonly denoted , , or in component notation as , , or (Muir 1960, p. 17). Note that the notation may be more convenient when indicating the absolute value of a determinant, i.e., instead of . The determinant is implemented in the Wolfram Language as Det[m].A determinant is defined..

### Thurston's geometrization conjecture

Thurston's conjecture proposed a complete characterization of geometric structureson three-dimensional manifolds.Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."Thurston's conjecture is that, after you split a three-manifold into its connected sum and..

### Hole

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a disconnectivity is interpreted as a hole in the space. Examples of holes are things like the "donut hole" in the center of the torus, a domain removed from a plane, and the portion missing from Euclidean space after cutting a knot out from it.Singular homology groups form a measure of the hole structure of a space, but they are one particular measure and they don't always detect all holes. homotopy groups of a space are another measure of holes in a space, as well as bordism groups, K-theory, cohomotopy groups, and so on.There are many ways to measure holes in a space. Some holes are picked up by homotopy groups that are not detected by homology groups, and some holes are detected by homology groups that are not picked up by homotopy groups. (For example, in the torus, homotopy..

### Handle

A handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a zip around each puncture travelling in opposite directions, pulling the edges of the zips together, and then zipping up.Handles are to manifolds as cells are to CW-complexes. If is a manifold together with a -sphere embedded in its boundary with a trivial tubular neighborhood, we attach a -handle to by gluing the tubular neighborhood of the -sphere to the tubular neighborhood of the standard -sphere in the dim()-dimensional disk. In this way, attaching a -handle is essentially just the process of attaching a fattened-up -disk to along the -sphere . The embedded disk in this new manifold is called the -handle in the union of and the handle.Dyck's theorem states that handles and cross-handlesare equivalent in the presence of a cross-cap...

### Exotic sphere

Milnor (1956) found more than one smooth structure on the seven-dimensional hypersphere. Generalizations have subsequently been found in other dimensions. Using surgery theory, it is possible to relate the number of diffeomorphism classes of exotic spheres to higher homotopy groups of spheres (Kosinski 1992).Kervaire and Milnor (1963) computed a list of the number of distinct (up to diffeomorphism) differential structures on spheres indexed by the dimension of the sphere. For , 2, ..., assuming the Poincaré conjecture, they are 1, 1, 1, , 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, ... (OEIS A001676). The status of is still unresolved, and it is not known whether there is 1, more than 1, or infinitely many smooth structures on the 4-sphere (Scorpan 2005). The claim that there is exactly one is known as the smooth Poincaré conjecture for .The only exotic Euclidean spaces are a continuum ofexotic R4 structures...

### Surgery

In the process of attaching a -handle to a manifold , the boundary of is modified by a process called -surgery. Surgery consists of the removal of a tubular neighborhood of a -sphere from the boundaries of and the standard sphere, and the gluing together of these two scarred-up objects along their common boundaries.

### Sphere eversion

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. This result is so counterintuitive and the proof so technical that the result remained controversial for a number of years.In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin's eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979).The..

### Quotient space

The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in . Quotient spaces are also called factor spaces.This can be stated in terms of maps as follows: if denotes the map that sends each point to its equivalence class in , the topology on can be specified by prescribing that a subset of is open iff is open.In general, quotient spaces are not well behaved, and little is known about them. However, it is known that any compact metrizable space is a quotient of the Cantor set, any compact connected -dimensional manifold for is a quotient of any other, and a function out of a quotient space is continuous iff the function is continuous.Let be the closed -dimensional disk and its boundary, the -dimensional sphere. Then (which is homeomorphic to ), provides an example of..

### Projective space

A projective space is a space that is invariant under the group of all general linear homogeneous transformation in the space concerned, but not under all the transformations of any group containing as a subgroup.A projective space is the space of one-dimensional vector subspaces of a given vector space. For real vector spaces, the notation or denotes the real projective space of dimension (i.e., the space of one-dimensional vector subspaces of ) and denotes the complex projective space of complex dimension (i.e., the space of one-dimensional complex vector subspaces of ). can also be viewed as the set consisting of together with its points at infinity.

### Free

When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.In algebraic topology, a free abstract mathematical object is generated by elements in a "free manner" ("freely"), i.e., such that the elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic gadget is freely generated by a subset if, for any function where is any other algebraic gadget, there exists a unique homomorphism (which has different meanings depending on what kind of gadgets you're dealing with) such that restricted to is .If the algebraic gadgets are vector spaces, then freely generates iff is a basis for . If the algebraic gadgets are..

### Lifting problem

Given a map from a space to a space and another map from a space to a space , does there exist a map from to such that ? If such a map exists, then is called a lift of .

### Lift

Given a map from a space to a space and another map from a space to a space , a lift is a map from to such that . In other words, a lift of is a map such that the diagram (shown below) commutes. If is the identity from to , a manifold, and if is the bundle projection from the tangent bundle to , the lifts are precisely vector fields. If is a bundle projection from any fiber bundle to , then lifts are precisely sections. If is the identity from to , a manifold, and a projection from the orientation double cover of , then lifts exist iff is an orientable manifold.If is a map from a circle to , an -manifold, and the bundle projection from the fiber bundle of alternating n-forms on , then lifts always exist iff is orientable. If is a map from a region in the complex plane to the complex plane (complex analytic), and if is the exponential map, lifts of are precisely logarithms of ...

### Immersion

A special nonsingular map from one manifold to another such that at every point in the domain of the map, the derivative is an injective linear transformation. This is equivalent to saying that every point in the domain has a neighborhood such that, up to diffeomorphisms of the tangent space, the map looks like the inclusion map from a lower-dimensional Euclidean space to a higher-dimensional Euclidean space.

### Hopf map

The first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic. Its discovery was a shock to the mathematical community, since it was believed at the time that all such maps were null-homotopic, by analogy with homology groups.The Hopf map arises in many contexts, and can be generalized to a map . For any point in the sphere, its preimage is a circle in . There are several descriptions of the Hopf map, also called the Hopf fibration.As a submanifold of , the 3-sphere is(1)and the 2-sphere is a submanifold of ,(2)The Hopf map takes points (, , , ) on a 3-sphere to points on a 2-sphere (, , )(3)(4)(5)Every point on the 2-sphere corresponds to a circlecalled the Hopf circle on the 3-sphere.By stereographic projection, the 3-sphere can be mapped to , where the point at infinity corresponds to the north pole. As a map, from , the Hopf map can be pretty complicated. The diagram above shows some of..

### Chern number

The Chern number is defined in terms of the Chern class of a manifold as follows. For any collection Chern classes such that their cup product has the same dimension as the manifold, this cup product can be evaluated on the manifold's fundamental class. The resulting number is called the Chern number for that combination of Chern classes. The most important aspect of Chern numbers is that they are bordism invariant.

### Map

A map is a way of associating unique objects to every element in a given set. So a map from to is a function such that for every , there is a unique object . The terms function and mapping are synonymous for map.

### Laurent polynomial

A Laurent polynomial with coefficients in the field is an algebraic object that is typically expressed in the formwhere the are elements of , and only finitely many of the are nonzero. A Laurent polynomial is an algebraic object in the sense that it is treated as a polynomial except that the indeterminant "" can also have negative powers.Expressed more precisely, the collection of Laurent polynomials with coefficients in a field form a ring, denoted , with ring operations given by componentwise addition and multiplication according to the relationfor all and in the integers. Formally, this is equivalent to saying that is the group ring of the integers and the field . This corresponds to (the polynomial ring in one variable for ) being the group ring or monoid ring for the monoid of natural numbers and the field ...

### Identity function

The identity function is the function which assigns every real number to the same real number . It is identical to the identity map.The identity function is trivially idempotent, i.e., .The identity function in the complex plane is illustrated above.A function that approximates the identity function for small to terms of order is given by(OEIS A115183 and A115184).This function leads to some nice pi approximations.

### Ricci curvature tensor

The Ricci curvature tensor, also simply known as the Ricci tensor (Parker and Christensen 1994), is defined bywhere is the Riemann tensor.Geometrically, the Ricci curvature is the mathematical object that controls the growthrate of the volume of metric balls in a manifold.

### Contour winding number

The winding number of a contour about a point , denoted , is defined byand gives the number of times curve passes (counterclockwise) around a point. Counterclockwise winding is assigned a positive winding number, while clockwise winding is assigned a negative winding number. The winding number is also called the index, and denoted .The contour winding number was part of the inspiration for the idea of the Brouwer degree between two compact, oriented manifolds of the same dimension. In the language of the degree of a map, if is a closed curve (i.e., ), then it can be considered as a function from to . In that context, the winding number of around a point in is given by the degree of the mapfrom the circle to the circle.

### Topology

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.The definition of topology leads to the following mathematical..

### Metric tensor

Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements in a generalized Pythagorean theorem:(1)In Euclidean space, where is the Kronecker delta (which is 0 for and 1 for ), reproducing the usual form of the Pythagorean theorem(2)In this way, the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be "arithmetized" by way of introducing a sort of generalized coordinate system (Borisenko and Tarapov 1979).In the above simplification, the space in question is most often a smooth manifold , whereby a metric tensor is essentially a geometrical object taking two vector inputs and calculating either the squared length of a single vector or a scalar product of two different vectors (Misner et al. 1978). In this..

### Dynamical system

A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system. If is any continuous function, then the evolution of a variable can be given by the formula(1)This equation can also be viewed as a difference equation(2)so defining(3)gives(4)which can be read "as changes by 1 unit, changes by ." This is the discrete analog of the differential equation(5)

### Meromorphic function

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function is a function of the formwhere and are entire functions with (Krantz 1999, p. 64).A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. A meromorphic function with an infinite number of poles is exemplified by on the punctured disk , where is the open unit disk.An equivalent definition of a meromorphic function is a complex analytic mapto the Riemann sphere.The word derives from the Greek (meros), meaning "part," and (morphe), meaning "form" or "appearance."..

### Curvature

In general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.After the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the shape operator. Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature.Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface,..

Relations in the definition of a Steenrod algebra which state that, for ,where denotes function composition and is the floor function.

### Dimension

The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object. For example, a rectangle is two-dimensional, while a cube is three-dimensional. The dimension of an object is sometimes also called its "dimensionality."The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane.The notion of dimension is important in mathematics because it gives a precise parameterization of the conceptual or visual complexity of any geometric object. In fact, the concept can even be applied to abstract objects which cannot be directly visualized. For example, the notion of time can be considered as one-dimensional, since it can be thought of as consisting of only "now," "before" and "after."..

### Hairy ball theorem

There does not exist an everywhere nonzero tangent vector field on the 2-sphere . This implies that somewhere on the surface of the Earth, there is a point with zero horizontal wind velocity. The theorem can be generalized to the statement that the -sphere has a nonzero tangent vector field iff is odd.

### Steenrod algebra

The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every and there are natural transformations of functors(1)satisfying: 1. for . 2. for all and all pairs . 3. . 4. The maps commute with the coboundary maps in the long exact sequence of a pair. In other words,(2)is a degree transformation of cohomology theories. 5. (Cartan relation)(3)6. (Adem relations) For ,(4)7. where is the cohomology suspension isomorphism. The existence of these cohomology operations endows the cohomology ring with the structure of a module over the Steenrod algebra , defined to be , where is the free module functor that takes any set and sends it to the free module over that set. We think of as being a graded module, where the th gradation is given by . This makes the tensor algebra into a graded algebra over . is the ideal generated by the elements and for . This makes into a graded algebra.By the definition of..

### De rham cohomology

de Rham cohomology is a formal set-up for the analytic problem: If you have a differential k-form on a manifold , is it the exterior derivative of another differential k-form ? Formally, if then . This is more commonly stated as , meaning that if is to be the exterior derivative of a differential k-form, a necessary condition that must satisfy is that its exterior derivative is zero.de Rham cohomology gives a formalism that aims to answer the question, "Are all differential -forms on a manifold with zero exterior derivative the exterior derivatives of -forms?" In particular, the th de Rham cohomology vector space is defined to be the space of all -forms with exterior derivative 0, modulo the space of all boundaries of -forms. This is the trivial vector space iff the answer to our question is yes.The fundamental result about de Rham cohomology is that it is a topological invariant of the manifold, namely: the th de Rham cohomology vector space..

### Normal bundle

The normal bundle of a submanifold is the vector bundle over that consists of all pairs , where is in and is a vector in the vector quotient space . Provided has a Riemann metric, can be thought of as the orthogonal complement to .

### Vector bundle

A vector bundle is special class of fiber bundle in which the fiber is a vector space . Technically, a little more is required; namely, if is a bundle with fiber , to be a vector bundle, all of the fibers for need to have a coherent vector space structure. One way to say this is that the "trivializations" , are fiber-for-fiber vector space isomorphisms.A vector bundle is a total space along with a surjective map to a base manifold . Any fiber is a vector space isomorphic to .The simplest nontrivial vector bundle is a line bundleon the circle, and is analogous to the Möbius strip.One use for vector bundles is a generalization of vector functions. For instance, the tangent vectors of an -dimensional manifold are isomorphic to at a point in a coordinate chart. But the isomorphism with depends on the choice of coordinate chart. Nearby , the vector fields look like functions. To define vector fields on the whole manifold requires the tangent bundle,..

### Tangent map

If , then the tangent map associated to is a vector bundle homeomorphism (i.e., a map between the tangent bundles of and respectively). The tangent map corresponds to differentiation by the formula(1)where (i.e., is a curve passing through the base point to in at time 0 with velocity ). In this case, if and , then the chain rule is expressed as(2)In other words, with this way of formalizing differentiation, the chain rule can be remembered by saying that "the process of taking the tangent map of a map is functorial." To a topologist, the form(3)for all , is more intuitive than the usual form of the chain rule.

### Fibration

If is a fiber bundle with a paracompact topological space, then satisfies the homotopy lifting property with respect to all topological spaces. In other words, if is a homotopy from to , and if is a lift of the map with respect to , then has a lift to a map with respect to . Therefore, if you have a homotopy of a map into , and if the beginning of it has a lift, then that lift can be extended to a lift of the homotopy itself.A fibration is a map between topological spaces such that it satisfies the homotopy lifting property.

### Fiber bundle

A fiber bundle (also called simply a bundle) with fiber is a map where is called the total space of the fiber bundle and the base space of the fiber bundle. The main condition for the map to be a fiber bundle is that every point in the base space has a neighborhood such that is homeomorphic to in a special way. Namely, ifis the homeomorphism, thenwhere the map means projection onto the component. The homeomorphisms which "commute with projection" are called local trivializations for the fiber bundle . In other words, looks like the product (at least locally), except that the fibers for may be a bit "twisted."A fiber bundle is the most general kind of bundle. Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., vector bundles and principal bundles.Examples of fiber bundles include any product (which is a bundle over with fiber ), the Möbius strip (which..

### Whitney sum

An operation that takes two vector bundles over a fixed space and produces a new vector bundle over the same space. If and are vector bundles over , then the Whitney sum is the vector bundle over such that each fiber over is naturally the direct sum of the and fibers over .The Whitney sum is therefore the fiber for fiber direct sum of the two bundles and . An easy formal definition of the Whitney sum is that is the pull-back bundle of the diagonal map from to , where the bundle over is .

### Pontryagin number

The Pontryagin number is defined in terms of the Pontryagin class of a manifold as follows. For any collection of Pontryagin classes such that their cup product has the same dimension as the manifold, this cup product can be evaluated on the manifold's fundamental class. The resulting number is called the Pontryagin number for that combination of Pontryagin classes. The most important aspect of Pontryagin numbers is that they are bordism invariant. Together, Pontryagin and Stiefel-Whitney numbers determine an oriented manifold's oriented bordism class.

### Pontryagin class

The th Pontryagin class of a vector bundle is times the th Chern class of the complexification of the vector bundle. It is also in the th cohomology group of the base space involved.

### Tubular neighborhood

A tubular neighborhood of a submanifold is an embedding of the normal bundle () of into , i.e., , where the image of the zero section of the normal bundle is equal to .

### Extension problem

Given a subspace of a space and a map from to a space , is it possible to extend that map to a map from to ?

### Net

The word net has several meanings in mathematics. It refers to a plane diagram in which the polyhedron edges of a polyhedron are shown, a point set satisfying certain uniformity of distribution conditions, and a topological generalization of a sequence.The net of a polyhedron is also known as a development, pattern, or planar net (Buekenhout and Parker 1998). The illustrations above show polyhedron nets for the cube and tetrahedron.In his classic Treatise on Measurement with the Compass and Ruler, Dürer(1525) made one of the first presentations of a net (Livio 2002, p. 138).The net of a polyhedron must in general also specify which edges are to be joined since there might be ambiguity as to which of several possible polyhedra a net might fold into. For simple symmetrical polyhedra, the folding procedure can only be done one way, so edges need not be labeled. However, for the net shown above, two different solids can be constructed from..

### Multiply connected

A set which is connected but not simply connected is called multiply connected. A space is -multiply connected if it is -connected and if every map from the -sphere into it extends continuously over the -diskA theorem of Whitehead says that a space is infinitelyconnected iff it is contractible.

### Connected space

A space is connected if any two points in can be connected by a curve lying wholly within .A space is 0-connected (a.k.a. pathwise-connected) if every map from a 0-sphere to the space extends continuously to the 1-disk. Since the 0-sphere is the two endpoints of an interval (1-disk), every two points have a path between them. A space is 1-connected (a.k.a. simply connected) if it is 0-connected and if every map from the 1-sphere to it extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible. A space is -multiply connected if it is -connected and if every map from the -sphere into it extends continuously over the -disk.A theorem of Whitehead says that a space is infinitelyconnected iff it is contractible.

### Topologically transitive

A function is topologically transitive if, given any two intervals and , there is some positive integer such that . Vaguely, this means that neighborhoods of points eventually get flung out to "big" sets so that they don't necessarily stick together in one localized clump.

### Metric space

A metric space is a set with a global distance function (the metric ) that, for every two points in , gives the distance between them as a nonnegative real number . A metric space must also satisfy 1. iff , 2. , 3. The triangle inequality .

### Simply connected

A pathwise-connected domain is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the domain is connected but not simply, it is said to be multiply connected. In particular, a bounded subset of is said to be simply connected if both and , where denotes a set difference, are connected.A space is simply connected if it is pathwise-connected and if every map from the 1-sphere to extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible.

### Set closure

A set and a binary operator are said to exhibit closure if applying the binary operator to two elements returns a value which is itself a member of .The closure of a set is the smallest closed set containing . Closed sets are closed under arbitrary intersection, so it is also the intersection of all closed sets containing . Typically, it is just with all of its accumulation points.The term "closure" is also used to refer to a "closed" version of a given set. The closure of a set can be defined in several equivalent ways, including 1. The set plus its limit points, also called "boundary" points, the union of which is also called the "frontier." 2. The unique smallest closed set containing the givenset. 3. The complement of the interior of the complementof the set. 4. The collection of all points such that every neighborhood of these points intersects the original set in a nonempty set. In topologies where the T2-separation..

### Cantor set

The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the interval whose ternary expansions do not contain 1, illustrated above.The th iteration of the Cantor is implemented in the Wolfram Language as CantorMesh[n].Iterating the process 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101, .... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ... (OEIS A088917) whose th term is amazingly given by (mod 3), where is a (central) Delannoy number and is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006). The recurrence plot for this..

### Stokes' theorem

For a differential (k-1)-form with compact support on an oriented -dimensional manifold with boundary ,(1)where is the exterior derivative of the differential form . When is a compact manifold without boundary, then the formula holds with the right hand side zero.Stokes' theorem connects to the "standard" gradient, curl, and divergence theorems by the following relations. If is a function on ,(2)where (the dual space) is the duality isomorphism between a vector space and its dual, given by the Euclidean inner product on . If is a vector field on a ,(3)where is the Hodge star operator. If is a vector field on ,(4)With these three identities in mind, the above Stokes' theorem in the three instances is transformed into the gradient, curl, and divergence theorems respectively as follows. If is a function on and is a curve in , then(5)which is the gradient theorem. If is a vector field and an embedded compact 3-manifold with boundary in , then(6)which..

### Galois extension field

The following are equivalent definitions for a Galois extension field (also simply known as a Galois extension) of .1. is the splitting field for a collection of separable polynomials. When is a finite extension, then only one separable polynomial is necessary. 2. The field automorphisms of that fix do not fix any intermediate fields , i.e., . 3. Every irreducible polynomial over which has a root in factors into linear factors in . Also, must be a separable extension. 4. A field automorphism of the algebraic closure of for which must fix . That is to say that must be a field automorphism of fixing . Also, must be a separable extension. A Galois extension has all of the above properties. For example, consider , the rationals adjoined by the imaginary number , over , which is a Galois extension. Note that contains all of the roots of , and is generated by them, so it is the splitting field of . Of course, there are two distinct roots in so it is separable. The only..

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