В математиката, многообразие е пространство ,което "отблизо" прилича на пространствата описани в евклидовата геометрия, но което глобално може да има много по-сложна структура. (Евклидовите пространства, обаче, също са многообразия.). Сферата е пример за многообразие - погледната отблизо тя изглежда плоска, но като цяло е кръгла. Многообразие може да се конструира чрез "залепване" на евклидови прострранства, например карта на Земята може да се получи чрез залепване на няколко карти на нейни части (за да се залепят правилно, обаче, ще трябва отделните карти да са от разтеглив материал).

Друг пример за многообразие е повърхността на тора. За разлика сферата, тя може да се построи като се залепят противоположните страни на една единствена правоъгълна (разтеглива) карта. Сферата и повърхността на тора са примери за двумерни многообразия, но е възможно да се построят многообразия и от по-ниска, и по-висока размерност. Многообразията са важни обекти в математиката и физиката защото позволяват сложни пространства да се изразяват и изследват използвайки сравнително по-добре изучените свойства на по-прости пространства.

Често се дефинират допълнителни структури върху многообразия. Примери за многообразия със допълнителна структура са диференцируемите многообразия, върху които може да се използва диференциално и интегрално смятане, римановите многообразия върху които могат да се дефинират понятията дължина и ъгъл, симплектичните многообразия които служат за фазови пространства в класическата механика, и четиримерните псевдориманови многообразия които моделират пространство-времето в общата теория на относителността.

Окръжността е най-простия пример за топологично многообразие след евклидовото пространство. Нека е зададена окръжност с радиус 1 и център съвпадащ с центъра на координатната система. Ако x и y са координатите на точките от окръжността, то за тях ще е изпълнено x² + y² = 1.

Локално, окръжността прилича на права линия, която е едномерна. Иначе казано локално е нужна само една координата за описание на точките от окръжността. Например, точките от горната част на окръжността, за които y-координатата е положителна (жълтата част във Фигура 1), могат да се опишат чрез x-координатата си. Тоест е зададена непрекъсната биекция χ_top, която изобразява жълтата част от окръжността в отворения интервал (−1, 1) чрез проекция по първата координата

${\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,}$

Такава функция се нарича карта. Аналогично могат да се дефинират карти за долната (червена), лявата (синя), и дясната (зелена) части на окръжността. Заедно тези части покриват цялата окръжност, а четирите карти образуват атлас на многообразието.

Горната и дясната карта се препокриват: тяхното сечение представлява четвъртината от окръжността за която x- и y-координатите са едновременно положителни. Двете карти χ_top и χ_right изобразяват тази част биективно в интервала (0, 1). Следователно може да се конструира функция T от (0, 1) в себе си, която първо обръща жълтата карта и изпраща точката в окръжността, и след това прослвдява зелената карта и се връща пак в интервала:

${\displaystyle T(a)=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}(a)\right)=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2))}\right)={\sqrt {1-a^{2))}.}$

Такава функция се нарича функция на прехода.

Горната, долната, лявата и дясната карти показват, че окръжността е многообразие, но те не образуват единствения възможен атлас. Картите не е нужно да бъдат геометрични проекции, и броят им е въпрос на избор. Например могат да се изберат следните карти

${\displaystyle \chi _{\mathrm {minus} }(x,y)=s={y \over {1+x))}$

и

${\displaystyle \chi _{\mathrm {plus} }(x,y)=t={y \over {1-x)).}$

Тук s е ъгловия коефициент на правата минаваща през произволна точка с координати (x,y) и фиксираната точка (−1,0); t е аналогичното изображение с фиксирана точка (+1,0). Обратното изображение от s в (x,y) се дава чрез

${\displaystyle x=((1-s^{2)) \over {1+s^{2))},\qquad y=((2s} \over {1+s^{2))};}$

Лесно може да се провери, че x²+y² = 1 за всички стойности на s. Тези две карти дават друг атлас на кръга, за който

${\displaystyle t={1 \over s}.}$

Нито една от двете карти не покрива цялата окръжност: s изпуска точката (−1,0), а t - (+1,0). Може да се покаже, че не е възможно една единствена карта да покрива цялата окръжност откъдето се вижда, че дори и простите примери се нуждаят от гъвкавостта която дават на многообразията и многото карти.

Многообразията не е нужно да са свързани (състоящи се от едно парче): двойка отделни окръжности също е топологично многообразие. Не е и нужно те да са затворени: отсечка без краищата си е многообразие. Многообразията не е нужно да са ограничени: параболата е топологично многообразие. Други примери за топологични многообразия са хиперболата и множеството от точките, които са решение на кубичното уравнение y² - x³ + x = 0, което не е нито свързано, нито затворено, нито ограничено.

Обаче примери като две допиращи се окъжности, които образуват 8 не са многообразия, защото не може да се конструира задоволителна карта изпращаща околност на общата точка в отворен интервал.

Oт гледна точка на диференциалното смятане, функцията на прехода T е функция между два отворени интервала, която е диференцируема. Същото е вярно и за другите функции на прехода в атласа. Следователно с този атлас, окръжността се превръща в диференцируемо многообразие. Всъщност тя е още гладко и аналитично.

Окръжността също притежава свойства, които позволяват тя да се разглежда като по-особен тип многообразие. По нея могат да се мерят растояния между точки: дължината на дъгата между две точки. Следователно тя е и риманово многообразие.

The study of manifolds combines many important areas of mathematics: it generalises concepts like curves and surfaces and notions from linear algebra, and incorporates ideas from topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance.

Before the modern concept there were several important results.

Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold.

Non-Euclidean geometry considers spaces where Euclid's parallel postulate (parallel lines never meet) fails. These were first studied in 1733 by Saccheri and developed a hundred years later by Lobachevsky, Bolyai, and Riemann. Their research uncovered two additional types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to manifolds with negative and positive curvature, respectively.

The Euler characteristic is an example of a topological property of a manifold. For a convex polyhedron with V vertices (or corners), E edges and F faces Euler showed that V-E+F=2. This characteristic can be generalised to cover smooth surfaces and higher dimensional spaces by using Betti numbers. The study of other topological properties (or invariants) of manifolds is one of the central themes of topology.

Bernhard Riemann was the first to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit. He distinguishes between stetige Mannigfaltigkeit and discrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n fach ausgedehnte Mannigfaltigkeit (n times extended or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldness. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann.

In the study of complex variables, the process of analytic continuation leads to the construction of manifolds.

Abelian varieties were already implicitly known in Riemann's time as complex manifolds. Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are also naturally manifold theories. All these use the notion of several characteristic axes or dimensions, but these dimensions do not lie along the physical dimensions of width, height, and breadth.

Henri Poincaré studied three-dimensional manifolds and raised a question, today known as the Poincaré conjecture. As of 2006, a consensus among experts is that recent work by Grigori Perelman may have answered this question, after nearly a century of effort by many mathematicians.

Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. The foundational aspects of the subject were clarified during the 1930s by Hassler Whitney and others, making precise intuitions dating back to the latter half of the 19th century, and developed through differential geometry and Lie group theory.

Многообразие в математиката е топологично пространство всяка точка на което има околност хомеоморфна на n-мерно кълбо:

${\displaystyle \mathbf {B} ^{n}=\{(x_{1},x_{2},...,x_{n})|x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}<1\}.}$

Има много различни видове многообразия. Най-простите са топологичните многообразия, които локално изглеждат като евклидови пространства. Всъщност горната дефиниция е дефиниция точно на топологично многообразие. Други типове многообразия имат допълнителна структура. Едни от най-важните са диференцируемите многообразия, чиято структура позволява използването на средства от анализа за изучаването им.

The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can properly represent the entire Earth. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure.

Charts

A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rⁿ and interest focused on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions.

In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane R² minus the negative x-axis and the origin. Another example of a chart is the map χ_top mentioned in the section above, a chart for the circle.

Atlases

The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of charts.

The atlas containing all possible charts consistent with a given atlas is called the maximal atlas. Unlike an ordinary atlas, the maximal atlas of a given atlas is unique. Though it is useful for definitions, it is a very abstract object and not used directly (e.g. in calculations).

Transition maps

Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in Rⁿ to the manifold and then back to another (or perhaps the same) open ball in Rⁿ. The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map.

Additional structure

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all the transition maps are compatible with this structure, the structure transfers to the manifold.

This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of Rⁿ (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold.

In general the structure on the manifold depends on the atlas, but sometimes different atlases give rise to the same structure. Such atlases are called compatible.

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R² is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:

The surface of the sphere can be treated in almost the same way as the circle. It can be viewed as a subset of R³:

${\displaystyle S=\{(x,y,z)\in \mathbf {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}.}$

The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R². Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ, defined by

${\displaystyle \chi (x,y,z)=(x,y),}$

maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z) plane, an atlas of six charts is obtained which covers the entire sphere.

This can be easily generalized to higher-dimensional spheres.

A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold.

The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold.

This can be illustrated with the transition map t = ¹⁄_s from the second half of the circle example. Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together by identifying the point t on the second copy with the point ¹⁄_s on the first copy (the point t = 0 is not identified with any point on the first copy). This gives a circle.

The first construction and this construction are very similar, but they represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point.

The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vector might be.

The n-sphere Sⁿ is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sⁿ can constructed by gluing together two copies of Rⁿ. The transition map between them is defined as

${\displaystyle \mathbf {R} ^{n}\setminus \{0\}\to \mathbf {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}$

This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier.

It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved.

One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. Two points are identified if one is moved onto the other by some group element. If M is the manifold and G is the group, the resulting quotient space is denoted by M / G (or G \ M).

Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively).

The Cartesian product of manifolds is also a manifold. Not every manifold can be written as a product.

The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite cylinders, for example, as S¹ × S¹ and S¹ × [0, 1], respectively.

A manifold with boundary is a manifold with an edge. For example a sheet of paper with rounded corners is a 2-manifold with a 1-dimensional boundary. The edge of the n-manifold will be an (n-1)-manifold. A solid circle, that is a circle with the interior filled in, is called a disk, is a 2-manifold with boundary, its boundary being a circle. A solid sphere is called a ball. This is a 3-manifold with a 2-dimensional boundary, the sphere. (See also Boundary (topology)).

Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together.

Formally, the gluing is defined by a bijection between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly for a differentiable manifold it has to be a diffeomorphism. For other manifolds other structures should be preserved.

A finite cylinder may be constructed as a manifold by starting with a strip R × [0, 1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries.

Шаблон:Seesubarticle

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rⁿ. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rⁿ. These homeomorphisms are the charts of the manifold.

Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable.

The dimension of the manifold at a certain point is the dimension of the Euclidean space charts at that point map to (number n in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to the same Euclidean space. In that case every topological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topological manifolds with differing dimensions.

Шаблон:See details

For most applications a special kind of topological manifold, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point.

Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (a technical definition which loosely means that Taylor's theorem holds). The sphere can be given analytic structure, as can most familiar curves and surfaces.

Шаблон:See details

To measure distances and angles on manifolds, the manifold must be Riemannian. A Riemannian manifold is an analytic manifold in which each tangent space is equipped with an inner product < , > in a manner which varies smoothly from point to point. Given two tangent vectors u and v the inner product <u,v> gives a real number. The dot (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

Most familiar curves and surfaces, including n-spheres and Euclidean space, can be given the structure of a Riemannian manifold.

Шаблон:See details

A Finsler manifold allows the definition of distance, but not of angle; it is an analytic manifold in which each tangent space is equipped with a norm || || in a manner which varies smoothly from point to point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product.

Any Riemannian manifold is a Finsler manifold.

Шаблон:See details

Lie groups are a particularly important class of manifolds. They were named after Sophus Lie (last name pronounced Lee). As well as having an inner product they also have the structure of a topological group, allowing a notion of multiplication of points on the manifold. Any compact Lie group can be given a Riemannian manifold structure. The circle can be given the structure of a Lie group — the circle group. The group structure is then the multiplicative group of all complex numbers with modulus 1.

A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. Other examples of Lie groups include special groups of matrices, which are all subgroups of general linear group, the group of n by n matrices with non-zero determinant. If the matrix entries are real numbers, this will be an n²-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are n(n-1)/2 dimensional manifolds, where n-1 is the dimension of the sphere. Further examples can be found in the table of Lie groups.

• A complex manifold is a manifold modeled on Cⁿ with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. (Note that an n-dimensional complex manifold has dimension 2n as a differentiable manifold.)
• Infinite dimensional manifolds: to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces.
• A symplectic manifold is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contact manifold.

Consider a topological manifold with charts mapping to Rⁿ. Given an ordered basis for Rⁿ, a chart causes its piece of the manifold to itself acquire a sense of ordering, which can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are orientable manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable.

Some illustrative examples are: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in 3-space, and (3) the real projective plane, which arises naturally in geometry.

Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a single side.

Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. Note that in three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space.

Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called antipodes. Although there is no way to do so physically, it is possible to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin projects to the same "point" on this "plane".

• Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (e.g. Euclidean space) by the actions of various finite groups. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense.
• Algebraic varieties and schemes: An algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed using sheaves instead of atlases. Because of singular points one cannot assume a variety is a manifold (even though linguistically the French variété, German Mannigfaltigkeit and English manifold are much the same thing).
• CW-complexes: A CW complex is a topological space formed by gluing objects of different dimensionality together; for this reason they generally are not manifolds. However, they are of central interest in algebraic topology, especially in homotopy theory, where such dimensional defects are acceptable.

• List of manifolds
• Surface (2-manifold)
• 3-manifold
• 4-manifold

• Freedman, Michael H and Quinn, Frank, Topology of 4-Manifolds, Princeton University Press (1990).
• Guillemin, Victor and Pollack, Alan, Differential Topology, Prentice-Hall (1974), ISBN 0132126052. This text was inspired by Milnor, and is commonly used in undergraduate courses.
• Hempel, John, 3-Manifolds, Princeton University Press (1976).
• Hirsch, Morris, Differential Topology, Springer (1997), ISBN 0387901485. Hirsch provides the most complete account with historical insights and excellent, but difficult, problems. This is the standard reference for those wishing to have a deep understanding of the subject.
• Kirby, Robion C. and Siebenmann, Laurence C., Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton, New Jersey: Princeton University Press (1977), ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
• Lee, John M., Introduction to Topological Manifolds, Springer-Verlag, New York (2000), ISBN 0-387-98759-2. Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.
• Massey, William S., Algebraic Topology: An Introduction, Harcourt, Brace & World, 1967.
• Milnor, John, Topology from the Differentiable Viewpoint, Princeton University Press, (revised, 1997), ISBN 0691048339.
• Munkres, James R., Topology, Prentice Hall, (2000) ISBN 0131816292.
• Neuwirth, L. P., editor, Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox, Princeton University Press, (1975).
• Spivak, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers (1965), ISBN 0805390219. This is the standard text used in most graduate courses.
• Riemann, Bernhard, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears.
• Riemann, Bernhard, On the Hypotheses which lie at the Bases of Geometry. The famous Göttingen inaugural lecture (Habilitationsschrift) of 1854.