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Nielsen-Thurston Classification

Nielsen-Thurston classification

In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact surface. William Thurston's theorem completes the work initiated by Nielsen in the 1930s. Given a homeomorphism f : S → S, there is a map g isotopic to f such that either:
- g is periodic;
- g preserves some multi-curve on S (in this case, g is called reducible); or
- g is pseudo-Anosov. The case where S is a torus (i.e., a surface whose genus is one) is handled separately and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic, and the tools of Teichmüller theory become useful. In what follows, we assume S has genus at least two, as this is the case Thurston considered. The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved multi-curve Γ. Each of the resulting compact surfaces with boundary is acted upon by some power (i.e. iterated composition) of g, and the classification can again be applied to this homeomorphism.

The mapping class group

Thurston's classification applies to homeomorphisms of surfaces, but the type of a homeomorphism only depends on its associated element of the mapping class group Mod(S). In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good geometric properties. For example:
- When g is periodic, there is a element of its mapping class that is an isometry of a hyperbolic structure on S.
- When g is pseudo-Anosov, there is a element of its mapping class that preserves a pair of transverse singular foliations of S, stretching the leaves of one (the stable foliation) while contracting the leaves of the other (the unstable foliation).

Mapping tori

Thurston's original motivation for developing this classification was to find geometric structures on mapping tori of the type predicted by the Geometrization conjecture. The mapping torus Mg of a homeomorphism g of a surface S is the 3-manifold obtained from S × [0,1] by gluing S × to S × using g. The geometric structure of Mg is related to the type of g in the classification as follows:
- If g is periodic, then Mg has an H2 × R structure;
- If g is reducible, then Mg has incompressible tori, and should be cut along these tori to yield pieces that each have geometric structures (the JSJ decomposition);
- If g is pseudo-Anosov, then Mg has a hyperbolic (i.e. H3) structure. The first two cases are comparatively easy, while the existence of a hyperbolic structure on the mapping torus of a pseudo-Anosov homeomorphism is a deep and difficult theorem (also due to Thurston). The hyperbolic 3-manifolds that arise in this way are called fibered because they are surface bundles over the circle, and these manifolds are treated separately in the proof of Thurston's geometrization theorem for Haken manifolds. Fibered hyperbolic 3-manifolds have a number of interesting and pathological properties; for example, Cannon and Thurston showed that the limit set of the kleinian group that arises from a fibered hyperbolic 3-manifold is a sphere-filling curve.

Fixed point classification

The three types of surface homeomorphisms are also related to the dynamics of the mapping class group Mod(S) on the Teichmüller space T(S). Thurston introduced a compactification of T(S) that is homeomorphic to a closed ball, and to which the action of Mod(S) extends naturally. The type of an element g of the mapping class group in the Thurston classification is related to its fixed points when acting on the compactification of T(S):
- If g is periodic, then there is a fixed point within T(S); this point corresponds to a hyperbolic structure on S whose isometry group contains an element isotopic to g;
- If g is pseudo-Anosov, then g has no fixed points in T(S) but has a pair of fixed points on the Thurston boundary; these fixed points correspond to the stable and unstable foliations of S preserved by g.
- For some reducible mapping classes g, there is a single fixed point on the Thurston boundary; an example is a Dehn twist along a simple closed curve Γ, in which case the fixed point of g on the Thurston boundary corresponds to Γ. This is reminiscent of the classification of hyperbolic isometries into elliptic, parabolic, and hyperbolic types (which have fixed point structures similar to the periodic, Dehn twist, and pseudo-Anosov types listed above).

References


- Travaux de Thurston sur les surfaces, Astérisque, 66-67, Soc. Math. France, Paris, 1979
- M. Handel and W. P. Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985), no. 2, pp. 173-191
- W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S. 19 (1988), pp. 417-431
- M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995), no. 1, pp. 109-140 Category:Geometric topology Category:Mathematical theorems

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Compact surface

In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. In contexts where manifold includes manifolds with boundary a closed manifold is defined a compact manifold without boundary (whereas a compact manifold may have a boundary). These manifolds are those that are, in an intuitive sense, finite. The simplest example in one dimension is a circle, which is closed while the real line is not. By the basic properties of compactness, a closed manifold is the disjoint union of a finite number of connected closed manifolds. One of the most basic objectives of geometric topology is to understand what the supply of possible closed manifolds is. Category:Geometric topology
-

Jakob Nielsen (mathematician)

For the software usability expert, see Jakob Nielsen (usability consultant). Jakob Nielsen (October 15, 1890 – August 3, 1959) was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium. In 1907 he was expelled for membership to an illicit student club. Nevertheless, he matriculated at the University of Kiel in 1908. Nielsen completed his doctoral dissertation in 1913. Soon thereafter, he was drafted into the German Imperial Navy. He was assigned to coastal defense. In 1915 he was sent to Constantinople as a military adviser to the Turkish Government. After the war, in the spring of 1919, Nielsen married Carola von Pieverling, a German medical doctor. In 1920 Nielsen took a position at the Technical University of Breslau. The next year he published a paper in Mathematisk Tidsskrift in which he proved that any subgroup of a finitely generated free group is free. In 1926 Otto Schreier would generalize this result by removing the condition that the free group be finitely generated; this result is now known as the Nielsen-Schreier theorem. Also in 1921 Nielsen moved to the Royal Veterinary and Agricultural University in Copenhagen, where he would stay until 1925, when he moved to the Technical University in Copenhagen. During World War II some efforts were made to bring Nielsen to the United States as it was feared that he would be assaulted by the Nazis. Nielsen would, in fact, stay in Denmark during the war without being harassed by the Nazis. In 1951 Nielsen became professor of mathematics at the University of Copenhagen, taking the position vacated by the death of Harald Bohr. He resigned this position in 1955 because of his international obligations, in particular with UNESCO, where he served on the executive board from 1952 to 1958. Nielsen, Jakob Nielsen, Jakob Nielsen, Jakob Nielsen, Jakob

Isotopy

In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

Formal definitions

Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × [0,1] → Y from the product of the space X with the unit interval [0,1] to Y such that, for all points x in X, H(x,0)=f(x) and H(x,1)=g(x). If we think of the second parameter of H as "time", then H describes a "continuous deformation" of f into g: at time 0 we have the function f, at time 1 we have the function g.

Properties

Being homotopic is an equivalence relation on the set of all continuous functions from X to Y. This homotopy relation is compatible with function composition in the following sense: if f1, g1 : X → Y are homotopic, and f2, g2 : Y → Z are homotopic, then their compositions f2 o f1 and g2 o g1 : X → Z are homotopic as well. If f and g from X to Y are homotopic, then the group homomorphisms induced by f and g on the level of homology groups are the same: Hn(f) = Hn(g) : Hn(X) → Hn(Y) for all n. If, in addition, X and Y are path-connected, then the group homomorphisms induced by f and g on the level of homotopy groups are also the same: πn(f) = πn(g) : πn(X) → πn(Y). These latter statements are the reason that algebraic topology generally can distinguish spaces only up to homotopy equivalence, to be described next.

Homotopy equivalence and null-homotopy

Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map idX and f o g is homotopic to idY. The maps f and g are called homotopy equivalences in this case. Clearly, every homeomorphism is a homotopy equivalence, but the converse is not true: a solid disk is not homeomorphic to a single point. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and R2 - is homotopy equivalent to the unit circle S1. Those spaces that are homotopy equivalent to a point are called contractible. A function f is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from f to a constant function is then sometimes called a null-homotopy.) For example, it is simple to show that a map from the circle S^1 is null-homotopic precisely when it can be extended to a map of the disc D^2. It follows from these definitions that a space X is contractible if and only if the identity map from X to itself—which is always a homotopy equivalence— is null-homotopic.

Homotopy-invariant properties

Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then
- if X is path-connected, then so is Y
- if X is simply connected, then so is Y
- the (singular) homology and cohomology groups of X and Y are isomorphic
- if X and Y are path-connected, then the fundamental groups of X and Y are isomorphic, and so are the higher homotopy groups An example of an algebraic invariant of topological spaces which is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification, and compactification is not homotopy-invariant).

Homotopy category and homotopy invariants

More abstractly, one can appeal to category theory concepts. One can define the homotopy category, whose objects are topological spaces, and whose morphisms are homotopy classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. A homotopy invariant is any function on spaces, (or on mappings), that respects the relation of homotopy equivalence (resp. homotopy); such invariants are constitutive of homotopy theory. Of course one could have foundational objection to a function whose domain is the collection of all topological spaces. An example of a homotopy invariant is the fundamental group of a space, already mentioned earlier. In practice homotopy theory is carried out by working with CW complexes, for technical convenience; or in some other reasonable category.

Relative homotopy

Especially in order to define the fundamental group, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative to K if there exists a homotopy H : X × [0,1] → Y between f and g such that H(k,t) = f(k) = g(k) for all k∈K and t∈[0,1].

Isotopy

In case the two given continuous functions f and g from the topological space X to the topological space Y are homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of isotopy, which is a homotopy H in the notation used before, such that for each fixed t, H(x,t) gives a homeomorphism. Requiring that two homeomorphisms be isotopic really is a stronger requirement than that they be homotopic. For example, the map of the unit disc in R2 defined by f(x,y) = (-x,-y) is equivalent to a 180 degree rotation around the origin, and so the identity map and f are isotopic because they can be connected by rotations. However, the map on the interval [-1,1] in R defined by f(x) = -x is not isotopic to the identity. Loosely speaking, any homotopy from f to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, f has changed the orientation of the interval, hence it cannot be isotopic to the identity. In geometric topology - for example in knot theory - the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots K1 and K2 in three-dimensional space. The intuitive idea of deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism h such that h moves K1 to K2. Category:Algebraic topologyCategory:Homotopy theory ja:ホモトピー

Pseudo-Anosov map

In mathematics, specifically in topology, a pseudo-Anosov map is a type of homeomorphism of a surface to itself. Thurston coined the term when he provided his classification of homeomorphisms of a surface.

See also


- Anosov flow Category:Dynamical systems Category:Geometric topology

Torus

Geometry

In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle. If the axis of rotation does not intersect the circle, the torus has a hole in the middle and resembles a ring doughnut, a hula hoop or an inflated tire. The other case, when the axis of rotation is a chord of the circle, produces a sort of squashed sphere resembling a round cushion. Torus was the Latin word for a cushion of this shape. A torus can be defined parametrically by: :x(u, v) = (R + r\cos) \cos \, :y(u, v) = (R + r \cos) \sin \, :z(u, v) = r \sin \, where :u, v ∈ [0, 2π], :R is the distance from the center of the tube to the center of the torus, :r is the radius of the tube. The equation in Cartesian coordinates for a torus azimuthally symmetric about the z-axis is :\left(R - \sqrt\right)^2 + z^2 = r^2 The surface area and interior volume of this torus are given by :A = 4\pi^2 Rr = \left( 2\pi r \right) \left( 2 \pi R \right) \, :V = 2\pi^2R r^2 = \left( \pi r^2 \right) \left( 2\pi R \right). \, According to a broader definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.

Topology

conic section Topologically, a torus is a closed surface defined as product of two circles: S1 × S1. The surface described above, given the relative topology from R3, is homeomorphic to a topological torus as long as it does not intersect its own axis. The torus can also be described as a quotient of the Euclidean plane under the identifications :(x,y) ~ (x+1,y) ~ (x,y+1) Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA^B^. The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: :\pi_1(\mathbb^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathbb \times \mathbb Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. The first homology group of the torus is isomorphic to the fundamental group (since the fundamental group is abelian).

The n-torus

One can easily generalize the torus to arbitrary dimensions. An n-torus is defined as a product of n circles: :\mathbb^n = S^1 \times S^1 \times \cdots \times S^1 The torus discussed above is the 2-torus. The 1-torus is just the circle. The 3-torus is rather difficult to visualize. Just as for the 2-torus, the n-torus can be described as a quotient of Rn under integral shifts in any coordinate. That is, the n-torus is Rn modulo the action of the integer lattice Zn (with the action being taken as vector addition). Equivalently, the n-torus is obtained from the n-cube by gluing the opposite faces together. An n-torus is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H•(Tn,Z) can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles.

See also


- algebraic torus
- annulus
- elliptic curve
- maximal torus
- period lattice
- sphere
- surface
- toroid
- torus (nuclear physics)
- Villarceau circles

External links


- [http://www.cut-the-knot.org/shortcut.shtml#torus Creation of a torus] at cut-the-knot
- [http://www.mathsisfun.com/geometry/torus.html More Torus Images] (from [http://www.mathsisfun.com/ Math is Fun])
- Eric W. Weisstein. "Torus." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Torus.html
- [http://www.bubblerings.com/bubblerings/media.cfm Images and movies of bubble rings] from David Whiteis' [http://www.bubblerings.com BubbleRings.com] Category:Surfaces ja:トーラス

Hyperbolic geometry

Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry doesn't hold. Specifically, in Euclidean geometry, given a line L and a point P not on L, there is a unique line through P that does not intersect L. In hyperbolic geometry, there are infinitely many such lines parallel to L.

History

Hyperbolic geometry was initially explored by Giovanni Gerolamo Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by János Bolyai, Karl Friedrich Gauss, and Nikolai Ivanovich Lobachevsky, after whom it is sometimes named. (See article on non-Euclidean geometry for more history.) In Hyperbolic geometry (also called saddle geometry or Lobachevskian geometry) the term parallel only applies to lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Lines that neither intersect in the hyperbolic plane nor the circle at infinity are called ultraparallel. One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem). Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.

Models of the hyperbolic plane

There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, the Poincaré half-plane model and the Lorentz model. The Klein model, also known as the projective disc model and Beltrami-Klein model, uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted. The Poincaré disc model, also known as the conformal disc model, also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle. The Poincaré half-plane model takes one-half of the Euclidean plane, as determined by a Euclidean line B, to be the hyperbolic plane (B itself is not included). Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B. Both Poincaré models preserve hyperbolic angles, and are thereby conformal. All isometries within these models are therefore Möbius transformations. A third model is the Lorentz model or hyperboloid model, which employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds (see below) says that Wilhelm Killing and Karl Weierstrass used this model from 1872. One can take the hyperboloid to represent the events that various moving observers radiating outward from a single point will reach in a fixed proper time. There are alternative ways to set up a physical model of hyperbolic geometry in Einstein's Special Theory of Relativity. For example, set up a polar coordinate system on the hyperbolic plane. Then any point can be identified with a uniform motion on a relativistic plane. The hyperbolic point (2, 30 degrees), for example, could represent an object travelling on a plane with a uniform rapidity of 2 in the direction of 30 degrees north of the polar axis. (The rapidity of an object is the arctanh of its speed as a proportion of the speed of light. An object travelling with rapidity 2, for example, would be going tanh(2) = 96.4% of the speed of light.) The hyperbolic distance between two points of the hyperbolic plane can be identified with the relative speed between two objects travelling in the corresponding uniform motion on the relativistic plane. So every theorem in hyperbolic geometry can be translated into a true statement in special relativity.

Visualizing hyperbolic geometry

The famous circle limit III [http://www.mcescher.com/Gallery/recogn-bmp/LW434.jpg] and IV [http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg] drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can see the geodesics (in III the white lines are not geodesics, but they run alongside them). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares. For example, in III every vertex is the intersection of three triangles and three squares. In normal Euclidean plane, this would sum up to 450°, leading to a contradiction. Hence we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is the fact that the hyperbolic plane has exponential growth. In IV, for example, one can see that the number of angles with a distance of n from the center rises exponentially. The angles have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n. Another fun thing to do, when things are slow at the office, is to cut a couple of sheets of paper into a few dozen identically sized squares, and tape them together putting five squares at each corner. Then note how two "rows" of squares which are next to each other at on point will diverge until they are arbitrarily far apart.

Relationship to Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group \pi_1=\Gamma, known as the Fuchsian group. The quotient space H/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces. The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

See also


- Angle of parallelism
- Hyperbolic space
- Hyperbolic structure
- Fuchsian group
- Fuchsian model
- Hyperbolic 3-manifold
- Kleinian group
- Kleinian model
- Poincaré metric
- Riemann surface
- Special Relativity

References


- Reynolds, William F. (1993) "Hyperbolic Geometry on a Hyperboloid", American Mathematical Monthly 100:442-455.
- Stillwell, John (1996) Sources in Hyperbolic Geometry, volume 10 in AMS/LMS series History of Mathematics. Category:Geometry
- Category:Riemann surfaces

Teichmüller space

In mathematics, given a Riemann surface X, the Teichmüller space of X, notated TX or Teich(X), is a complex manifold whose points represent all complex structures of Riemann surfaces whose underlying topological structure is the same as that of X.

Relation to moduli space

The Teichmüller space of a surface is related to its moduli space, but preserves more information about the surface. More precisely, the surface X (or its underlying topological structure) provides a marking X → Y of each Riemann surface Y represented in TX: whereas moduli space identifies all surfaces which are isomorphic, TX only identifies those surfaces which are isomorphic via a biholomorphic map f that is isotopic to the identity (with respect to the marking, hence its need). The automorphisms of X, up to isotopy, form a discrete group (the Teichmüller modular group, or mapping class group of X) that acts on TX. The action is as follows: if [g] is an element of the mapping class group of X, then [g] sends the point represented by the marking h: X → Y to the point with the marking hg : X → X → Y. The quotient of TX by this action is precisely the moduli space of X.

Properties of TX

The Teichmüller space of X is a complex manifold. Its complex dimension depends on topological properties of X. If X is obtained from a compact surface of genus g (take g greater than 1) by removing n points, then the dimension of TX is 3g-3+n. These are the cases of "finite type". Note that, even though a compact surface with a point removed and the same surface with a disc removed are topologically the same, a complex structure on the surface behaves very differently around a point and around a removed disc. In particular, the boundary of the removed disc becomes an "ideal boundary" for the Riemann surface, and isomorphisms between surfaces with non-empty ideal boundary must take this ideal boundary into account. Varying the structure quasiconformally along the ideal boundary shows that the Teichmüller space of a Riemann surface with nonempty ideal boundary must be infinite-dimensional.

Teichmüller metric

There is, in general, no isomorphism from one Riemann surface to another of the same topological type that is isotopic to the identity. There is, however, always a quasiconformal map from one to the other that is isotopic to the identity, and the measure of how far such a map is from being conformal (i.e., holomorphic) gives a metric on TX, called the Teichmüller metric.

Examples of Teichmüller spaces

A sphere with four points removed and a torus both have Teichmüller spaces of complex dimension 1. Category:Riemann surfaces Category:Moduli theory



Function composition

In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. The notation g o f is read as "g circle f" or "g composed with f". function As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x). Then (c o h)(t) describes the oxygen concentration around the plane at time t. In the mid-20th century, some mathematicians decided that writing "g o f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They wrote "xf" for "f(x)" and "xfg" for "g(f(x))". However, this movement never caught on, and nowadays this notation is found only in old books. The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f o (g o h) = (f o g) o h. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. As a result the set of bijective functions f: X → X form a group with respect to the composition operator. The functions g and f commute with each other if g o f = f o g. In general, composition of functions will not be commutative. Only the simplest functions will be commutative under composition. Derivatives of compositions involving differentiable functions can be found using the chain rule. "Higher" derivatives of such functions are given by Faà di Bruno's formula.

Functional powers

If Y⊂X then f may compose with itself; this is sometimes denoted f 2. Thus: :(f o f)(x) = f(f(x)) = f 2(x) :(f o f o f)(x) = f(f(f(x))) = f 3(x) Repeated composition of a function with itself is sometimes called function iteration. The functional powers f o f n = f n o f = f n+1 for natural n follow immediately.
- By convention, f 0 = idD(f) (the identity map on the domain of f).
- If f:X→X admits an inverse function, negative functional powers f -k (k > 0) are defined as the opposite power of the inverse function, (f −1)k. Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x). (For usual numerical functions, usually the latter is meant, at least for positive exponents. For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan (≠ 1/tan). In some cases, an expression for f in g(x) = f r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration. Iterated functions occur naturally in the study of fractals and dynamical systems.

Composition operator

Given a function g, the composition operator C_g is defined as that operator which maps functions to functions as :C_g f = f \circ g Composition operators are studied in the field of operator theory. :See main article composition operator for more details.

See also


- combinator
- lambda calculus
- higher-order function
- Function composition (computer science) Category:Set theory

Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space. To be more precise, suppose that X is a topological space. Let :(X) be the homeomorphism group of X. Let :_0(X) be the subgroup of (X) consisting of all homeomorphisms isotopic to the identity map on X. It is easy to verify that _0(X) is in fact a subgroup and is normal. The factor group :(X) = (X) / _0(X) is then the mapping class group of X. Thus there is a natural short exact sequence: :1 \rightarrow _0(X) \rightarrow (X) \rightarrow (X) \rightarrow 1 As usual, there is interest in the spaces where this sequence splits. If the mapping class group of X is finite then X is sometimes called rigid. Some mathematicians, when X is an orientable manifold, restrict attention to orientation-preserving homeomorphisms ^+(X). Here convention dictates that the group defined in the second paragraph be called the extended mapping class group.

Examples

An easy exercise is to show that: : (S^1) = /2. For manifolds of dimension two or higher the mapping class group is often infinite. Generalizing the above, for the n-torus we have: : (T^n) = (n, ). The mapping class groups of surfaces have been heavily studied. (Note the special case of (T^2) above.) This is perhaps due to their strange similarity to higher rank linear groups as well as many applications, via surface bundles, in Thurston's theory of geometric three-manifolds. We note that the mapping class group of any closed, orientable surface can be generated by Dehn twists. A few mapping class groups of non-orientable surface have simple presentations: In the projective plane ^2, every self-homeomorphism is isotopic to the identity so : (^2) = 1. The mapping class group of the Klein bottle K is: : (K)=/2+/2. The four elements are the identity, a Dehn twist on the two-sided curve which does not bound a Mobius band, the y-homeomorphism of Lickorish, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity. We also remark that the closed genus three non-orientable surface N_3 also has equal to : (2, ). This is because the surface has a unique one-sided curve that, when cut open, yields a once-holed torus. This is discussed in a paper of Martin Scharlemann.

References

For more information relating to the mapping class groups of surface one should consult the book by Andrew Casson and Steve Bleiler entitled Automorphisms of surfaces after Nielsen and Thurston. The special case where X is a punctured disk is discussed by Joan Birman in Braids, Links, and Mapping Class Groups. Category:Geometric topology

Isometry

:For the mechanical engineering and architecture usage, see isometric projection. In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent. Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

Definitions

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and one should guess from context which one is intended. Let X and Y be metric spaces with metrics d_X and d_Y. A map f:X\to Y is called distance preserving if for any x,y\in X one has d_Y(f(x),f(y))=d_X(x,y). A distance preserving map is automatically injective. A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective). Two metric spaces X and Y are called isometric if there is an isometry from X to Y. The set of isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.

Examples


- Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group.
- The map R\toR defined by x\mapsto |x| is a path isometry but not a global isometry.
- The isometric linear maps from Cn to itself are the unitary matrices.

Generalizations


- Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map f:X\to Y between metric spaces such that
- # for x,x'\in X one has |d_Y(f(x),f(x'))-d_X(x,x')|<\epsilon, and
- # for any point y\in Y there exists a point x\in X with d_Y(y,f(x))<\epsilon. :That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.
- Quasi-isometry is yet an other useful generalization.

See also


- Euclidean group
- Congruence (geometry)
- Euclidean plane isometry
- 3D isometries which leave the origin fixed
- space group
- involution Category:Geometry Category:Metric geometry Category:Symmetry ja:等長写像

Pseudo-Anosov

In mathematics, specifically in topology, a pseudo-Anosov map is a type of homeomorphism of a surface to itself. Thurston coined the term when he provided his classification of homeomorphisms of a surface.

See also


- Anosov flow Category:Dynamical systems Category:Geometric topology

Transverse

The term transverse means "side-to-side", as opposed to longitudinal, which means "front-to-back".
- In automotive engineering, the term transverse refers to an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle. See transverse engine.
- transverse wave See also: velocity, acceleration.

Foliation

In mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. On each sufficiently small piece of the manifold, these stripes give the manifold a local product structure. This product structure does not have to be consistent outside local patches (i. e. well-defined globally): a stripe followed around long enough might return to a different, nearby stripe. More formally, a codimension p foliation F of an n-dimensional manifold M is a covering by charts U_i together with maps :\phi_i:U_i \to \R^n such that on the overlaps U_i \cap U_j the transition functions \varphi_ defined by :\varphi_ =\phi_j \phi_i^ take the form :\varphi_(x,y) = (\varphi_^1(x),\varphi_^2(x,y)) where x denotes the first n-p co-ordinates, and y denotes the last p co-ordinates. In the chart U_i, the stripes x=constant match up with the stripes on other charts U_j. Technically, these stripes are called plaques of the foliation. In each chart, the plaques are n-p dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected submanifolds called the leaves of the foliation. Example: n-dimensional space, foliated as a product by subspaces consisting of points whose first n-p co-ordinates are constant. This can be covered with a single chart. Example: If M \to N is a covering between manifolds, and F is a foliation on N, then it pulls back to a foliation on M. More generally, if the map is merely a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back. Example: If G is a Lie group, and H is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of G, then G is foliated by cosets of H. There is a close relationship, assuming everything is smooth, with vector fields: given a vector field X on M that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension n-1 foliation). This observation generalises to a theorem of Ferdinand Georg Frobenius (the Frobenius theorem), saying that the necessary and sufficient conditions for a distribution (i.e. an n-p dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, are that the set of vector fields tangent to the distribution are closed under Lie bracket. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from GL(n) to a reducible subgroup. The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist. There is a global foliation theory, because topological constraints exist. For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré-Hopf index theorem, which shows the Euler characteristic will have to be 0.

See also


- Frobenius theorem
- G-structure
- Classifying space for foliations
- Reeb foliation
- Taut foliation
- Category:Structures on manifolds

Geometrization conjecture

The geometrization conjecture, also known as Thurston's geometrization conjecture, concerns the geometric structure of compact 3-manifolds. It was proposed by William Thurston in the late 1970s. It 'includes' other conjectures, such as the Poincaré conjecture and the Thurston elliptization conjecture. Here are some essential concepts used in the conjecture: 3-manifolds exhibit a standard two-level decomposition: # the prime decomposition, where every compact 3-manifold is the connected sum of an essentially unique collection of prime three-manifolds, and # the JSJ decomposition. For a three-manifold to have a JSJ decomposition, it must be prime. Here is a formulation of Thurston's conjecture: :Separate a closed 3-manifold into its prime decomposition (capping off spherical boundaries with 3-balls), and then each irreducible summand is reduced by its JSJ decomposition. The interior of each of the resulting manifolds is covered by a simply-connected homogeneous space such that the group of covering transformations are isometries of the homogeneous space, thus endowing the manifold with the local geometry of the homogeneous space. In addition, each manifold (with its induced Riemannian metric) has finite volume. There are exactly eight such simply-connected homogeneous spaces that admit finite volume quotients; they are called Thurston model geometries. The following is a list of the eight geometries: # Euclidean geometry # Hyperbolic geometry # Spherical geometry # The geometry of S2 × R # The geometry of H2 × R # The geometry of SL2R # Nil geometry, i.e. geometry of group of normed upper triangular 3-by-3 matrices. # Sol geometry, i.e. geometry of group of upper triangular 2-by-2 matrices. In the list of geometries above, S2 is the 2-sphere (in a topological sense) and H2 is the hyperbolic plane. Six of the eight geometries above (all except hyperbolic and spherical) are now clearly understood and known to correspond to Seifert manifolds and certain torus bundles. Using information about Seifert manifolds, we can restate the conjecture more tersely as: Every irreducible, compact 3-manifold falls into exactly one of the following categories: #it has a spherical geometry #it has a hyperbolic geometry #The fundamental group contains a subgroup isomorphic to the free abelian group on two generators (this is the fundamental group of a torus). If Thurston's conjecture is correct, then so is the Poincaré Conjecture (via Thurston elliptization conjecture). The Fields Medal was awarded to Thurston in 1982 partially for his proof of the conjecture for Haken manifolds. Progress has been made in proving that 3-manifolds that should be hyperbolic are in fact so. Mainly this progress has been limited to checking examples, although there are some notable results. The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would smooth out any bumps in the metric, resulting in a metric of constant positive curvature, i.e. a spherical metric. He later developed a program to prove the Geometrization Conjecture by Ricci flow. Grigori Perelman may have now solved the Geometrization conjecture (and thus also the Poincaré Conjecture) and there seems to be a consensus among experts that the proof is correct, at least in the case of 3-manifolds with finite fundamental group. Note that Perelman is eligible for a million dollar prize, which he seems uninterested in, but his work will need to survive two years of systematic scrutiny after publication, before the Clay Mathematics Institute can deem the conjecture to have been solved. Category:Geometric topology Category:Riemannian geometry Category:3-manifolds Category:Conjectures

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field t