John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the four mathematicians to have won the Fields Medal, the Wolf Prize, the Abel Prize. Milnor was born on February 1931 in Orange, New Jersey, his father was J. Willard Milnor and his mother was Emily Cox Milnor; as an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and proved the Fary–Milnor theorem. He continued on to graduate school at Princeton under the direction of Ralph Fox and submitted his dissertation, entitled "Isotopy of Links", which concerned link groups and their associated link structure, in 1954. Upon completing his doctorate he went on to work at Princeton, he was a professor at the Institute for Advanced Study from 1970 to 1990. His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, his wife, Dusa McDuff, is a professor of mathematics at Barnard College.
One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. With Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures. An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to mature to this day. In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes which are homeomorphic but combinatorially distinct.
In 1984 Milnor introduced a definition of attractor. The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnor's current interest is dynamics holomorphic dynamics, his work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics: It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the beginning, looking at the simplest nontrivial families of maps; the first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. The case of a unimodal map, that is, one with a single critical point, turns out to be rich; this work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems.
Milnor's work has opened several new directions in this field, has given us many basic concepts, challenging problems and nice theorems. He was an editor of the Annals of Mathematics for a number of years after 1962, he has written a number of books. In 1962 Milnor was awarded the Fields Medal for his work in differential topology, he went on to win the National Medal of Science, the Lester R. Ford Award in 1970 and again in 1984, the Leroy P Steele Prize for "Seminal Contribution to Research", the Wolf Prize in Mathematics, the Leroy P Steele Prize for Mathematical Exposition, the Leroy P Steele Prize for Lifetime Achievement "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64, 399–405". In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology and algebra." Reacting to the award, Milnor told the New Scientist "It feels good," adding that "ne is always surprised by a call at 6 o'clock in the morning."
In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology and dynamical systems". Milnor, John W.. Morse theory. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. ——. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324. ——. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press. ISBN 0-691-08065-8. ——. Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4. Husemoller, Dale. Symmetric bilinear forms. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5. Milnor, John W.. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press. ISBN 0-691-08122-0. Milnor, John W..
Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9. —— (
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 to a small enough observer, all 3-manifolds look like our universe does to a small enough observer; this is made more precise in the definition below. A topological space X is a 3-manifold if it is a second-countable Hausdorff space and if every point in X has a neighbourhood, homeomorphic to Euclidean 3-space; the topological, piecewise-linear, smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, so there is a prevalence of specialized techniques that do not generalize to dimensions greater than three; this special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, partial differential equations.
3-manifold theory is considered a part of low-dimensional topology or geometric topology. A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful in the non-Haken case. Thurston's contributions to the theory allow one to consider, in many cases, the additional structure given by a particular Thurston model geometry; the most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is fruitful; the fundamental groups of 3-manifolds reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group topological methods. Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it.
This is just the standard 3-dimensional vector space over the real numbers. A 3-sphere is a higher-dimensional analogue of a sphere, it consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Real projective 3-space, or RP3, is the topological space of lines passing through the origin 0 in R4, it is a compact, smooth manifold of dimension 3, is a special case Gr of a Grassmannian space. RP3 is SO, hence admits a group structure; the 3-dimensional torus is the product of 3 circles. That is: T 3 = S 1 × S 1 × S 1; the 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer lattice Z3. Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.
A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is an example of a compact abelian Lie group; this follows from the fact. Group multiplication on the torus is defined by coordinate-wise multiplication. Hyperbolic space is a homogeneous space, it is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, models of elliptic geometry that have a constant positive curvature; when embedded to a Euclidean space, every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially; the Poincaré homology sphere is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere with a finite fundamental group, its fundamental group is known as the binary icosahedral group and has order 120.
This shows. In 2003, lack of structure on the largest scales in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. However, there is no strong support for the correctness of the model, as yet. In mathematics, Seifert–Weber space is a closed hyperbolic 3-manifold, it is known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space
Raoul Bott was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, the Borel–Bott–Weil theorem. Bott was born in Budapest, the son of Margit Kovács and Rudolph Bott, his father was of Austrian descent, his mother was of Hungarian Jewish descent. Bott spent his working life in the United States, his family emigrated to Canada in 1938, subsequently he served in the Canadian Army in Europe during World War II. Bott went to college at McGill University in Montreal, where he studied electrical engineering, he earned a Ph. D. in mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled Electrical Network Theory, was written under the direction of Richard Duffin. Afterward, he began teaching at the University of Michigan in Ann Arbor. Bott continued his study at the Institute for Advanced Study in Princeton, he was a professor at Harvard University from 1959 to 1999.
In 2005 Bott died of cancer in San Diego. With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive-real functions. In 1949 they proved a fundamental theorem of filter synthesis. Duffin and Bott extended earlier work by Otto Brune that requisite functions of complex frequency s could be realized by a passive network of inductors and capacitors; the proof, relying on induction on the sum of the degrees of the polynomials in the numerator and denominator of the rational function, was published in Journal of Applied Physics, volume 20, page 816. In his 2000 interview with Allyn Jackson of the American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with network synthesis and electronic filter topology introduced him to algebraic topology. Bott met Arnold S. Shapiro at the IAS and they worked together, he studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem.
In the course of this work, he introduced Morse–Bott functions, an important generalization of Morse functions. This led to his role as collaborator over many years with Michael Atiyah via the part played by periodicity in K-theory. Bott made important contributions towards the index theorem in formulating related fixed-point theorems, in particular the so-called'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem; the major Atiyah–Bott papers on what is now the Atiyah–Bott fixed-point theorem were written in the years up to 1968. In the 1980s, Atiyah and Bott investigated gauge theory, using the Yang–Mills equations on a Riemann surface to obtain topological information about the moduli spaces of stable bundles on Riemann surfaces. In 1983 he spoke to the Canadian Mathematical Society in a talk he called "A topologist marvels at Physics", he is well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups.
He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds and the Bott cannibalistic class. In 1964, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. In 1987, he was awarded the National Medal of Science. In 2000, he received the Wolf Prize. In 2005, he was elected an Overseas Fellow of the Royal Society of London. Bott had 35 Ph. D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, András Szenes, Kevin Corlette. 1995: Collected Papers. Vol. 4. Mathematics Related to Physics. Edited by Robert MacPherson. Contemporary Mathematicians. Birkhäuser Boston, xx+485 pp. ISBN 0-8176-3648-X MR1321890 1995: Collected Papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxii+610 pp. ISBN 0-8176-3647-1 MR1321886 1994: Collected Papers.
Vol. 2. Differential Operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxiv+802 pp. ISBN 0-8176-3646-3 MR1290361 1994: Collected Papers. Vol. 1. Topology and Lie Groups. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xii+584 pp. ISBN 0-8176-3613-7 MR1280032 1982: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics #82. Springer-Verlag, New York-Berlin. Xiv+331 pp. ISBN 0-387-90613-4 doi:10.1007/978-1-4757-3951-0 MR0658304 1969: Lectures on K. Mathematics Lecture Note New York-Amsterdam x +203 pp. MR0258020 Raoul Bott at the Mathematics Genealogy Project Commemorative website at Harvard Math Department "The Life and Works of Raoul Bott", by Loring Tu. "Raoul Bott, an Innovator in Mathematics, Dies at 82", The New York Times, January 8, 2006
In mathematics, topology, a fiber bundle is a space, locally a product space, but globally may have a different topological structure. The similarity between a space E and a product space B × F is defined using a continuous surjective map π: E → B that in small regions of E behaves just like a projection from corresponding regions of B × F to B; the map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, F the fiber. In the trivial case, E is just B × F, the map π is just the projection from the product space to the first factor; this is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself to E is called a section of E. Fiber bundles can be specialized in a number of ways, the most common of, requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber F. In topology, the terms fiber and fiber space appeared for the first time in a paper by Herbert Seifert in 1933, but his definitions are limited to a special case; the main difference from the present day conception of a fiber space, was that for Seifert what is now called the base space of a fiber space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space was given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau, Norman Steenrod, Charles Ehresmann, Jean-Pierre Serre, others. Fiber bundles became their own object of study in the period 1935–1940; the first general definition appeared in the works of Whitney. Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, a fiber bundle whose fiber is a sphere of arbitrary dimension. A fiber bundle is a structure, where E, B, F are topological spaces and π: E → B is a continuous surjection satisfying a local triviality condition outlined below; the space B is called the base space of the bundle, E the total space, F the fiber. The map π is called the projection map. We shall assume in. We require that for every x in E, there is an open neighborhood U ⊂ B of π such that there is a homeomorphism φ: π−1 → U × F in such a way that π agrees with the projection onto the first factor.
That is, the following diagram should commute: where proj1: U × F → U is the natural projection and φ: π−1 → U × F is a homeomorphism. The set of all is called a local trivialization of the bundle, thus for any p in B, the preimage π−1 is homeomorphic to F and is called the fiber over p. Every fiber bundle π: E → B is an open map, since projections of products are open maps; therefore B carries the quotient topology determined by the map π. A fiber bundle is denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category of smooth manifolds; that is, E, B, F are required to be smooth manifolds and all the functions above are required to be smooth maps. Let E = B × F and let π: E → B be the projection onto the first factor. E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle.
Any fiber bundle over a contractible CW-complex is trivial. The simplest example of a nontrivial bundle E is the Möbius strip, it has the circle that runs lengthwise along the center of the strip as a base B and a line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood U of π ∈ B is an arc; the preimage π − 1 in the picture is a slice of the strip one long. A homeomorphism exists that maps the preimage of U to a slice of a cylinder: curved, but not twisted; this pair locally trivializes the strip. The corresponding trivial bundle B × F would be a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point; the elements of differential and integral calculus extend to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, this physical intuition leads to notions such as the divergence and curl. In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain.
This representation of a vector field depends on the coordinate system, there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are discussed on open subsets of Euclidean space, but make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point. More vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold. Vector fields are one kind of tensor field. Given a subset S in Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian coordinates. If each component of V is continuous V is a continuous vector field, more V is a Ck vector field if each component of V is k times continuously differentiable. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space.
Given two Ck-vector fields V, W defined on S and a real valued Ck-function f defined on S, the two operations scalar multiplication and vector addition:= f V:= V + W define the module of Ck-vector fields over the ring of Ck-functions where the multiplication of the functions is defined pointwise. In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system; the transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector. Thus, suppose, a choice of Cartesian coordinates, in terms of which the components of the vector V are V x = and suppose that are n functions of the xi defining a different coordinate system; the components of the vector V in the new coordinates are required to satisfy the transformation law Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: a vector field is a specification of n functions in each coordinate system subject to the transformation law relating the different coordinate systems.
Vector fields are thus contrasted with scalar fields, which associate a number or scalar to every point in space, are contrasted with simple lists of scalar fields, which do not transform under coordinate changes. Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More a vector field F is a mapping from M into the tangent bundle TM so that p ∘ F is the identity mapping where p denotes the projection from TM to M. In other words, a vector field is a section of the tangent bundle. An alternative definition: A smooth vector field X on a manifold M is a linear map X: C ∞ → C ∞ such that X is a derivation: X = f X + X g for all f, g ∈ C ∞. If the manifold M is smooth or analytic—that is, the change of coordinates is smooth —then one can make sense of the notion of smooth vector fields; the c
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions are represented in the form: a + b i + c j + d k where a, b, c, d are real numbers, i, j, k are the fundamental quaternion units. Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, therefore a domain.
In fact, the quaternions were the first noncommutative division algebra. The algebra of quaternions is denoted by H, or in blackboard bold by H, it can be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. The algebra ℍ holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers; these rings are Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, the last normed division algebra over the reals; the unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin, isomorphic to SU and to the universal cover of SO. Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity and Olinde Rodrigues' parameterization of general rotations by four parameters, but neither of these writers treated the four-parameter rotations as an algebra.
Carl Friedrich Gauss had discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the complex numbers could be interpreted as points in a plane, he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of division for a long time, he could not figure out. The great breakthrough in quaternions came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting; as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i 2 = j 2 = k 2 = i j k = − 1 into the stone of Brougham Bridge as he paused on it.
Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery; this letter was published in a letter to a science magazine. An electric circuit seemed to close, a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties, he founded a school of "quaternionists", he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long. After Hamilton's death, his student Peter Tait continued promoting quaternions.
At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described in terms of quaternions. There was a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems. From the mid-1880s, quaternions began to be displaced by vector analysis, developed by Josiah Willard Gibbs, Oliver Heaviside, Hermann von Helmholtz. Vector analys
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.
In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.
In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co