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Topology studies properties of spaces that are invariant under any continuous deformation. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken. For example, a square can be deformed into a circle without breaking it, but a figure 8 cannot. Hence a square is topologically equivalent to a circle, but different from a figure.

## Defintion

So toplogy is , in general, the study of topological spaces. But there is difference between topology and "A toplogy". Here are some examples of typical questions in topology: How many holes are there in an object? How can you define the holes in a torus or sphere? What is the boundary of an object? Is a space connected? Does every continuous function from the space to itself have a fixed point?

## Neighbourhood

The concept of Neighbourhood is very import in topology and geometric algebra. Intuitively speaking neighbour of a point is the set containing that point where one can move a small amount to any distance without leaving the set.

Topology is a relatively new branch of mathematics; most of the research in topology has been done since 1900. The following are some of the subfields of topology.
General Topology or Point Set Topology: General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense. Combinatorial Topology. Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology, and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic. This is the number (V - E + F), where V, E, and F are the number of vertices, edges, and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any “triangulation” of a sphere will have an Euler characteristic of 2.
Algebraic Topology: Algebraic topology also considers the global properties of spaces, and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups.

Topological space as explained are charaterized in terms of holes or handles. A sphere has zero hole. Toplogy has given birth to a new science to study the shape of the universe. We are three dimensional being who inhabit a four dimensional world ( space-time). We can not leave our universe to see what is the shpae or structure of it. So topological features can tell how our universe looks entirely within itself.

All topological spaces which have indentical number of holes are of same kind. They can be transformed onto each other by continuous mapping or functions.

## The genus of compact Riemann surface

The first stage is to classify the surfaces according to their topology, that is to say, according to that aspect of things preserved by continuous transformations. The topological classification of compact 2-dimensional orientable surfaces is really very simple. It is given by a single natural number called the genus of the surface. Roughly speaking, all we have to do is count the number of 'handles' that the surface has. In the case of the sphere the genus is 0, whereas for the torus it is 1. The surface of an ordinary teacup also has genus 1 (one handle!), so it is topologically the same as a torus. The surface of a normal pretzel has genus 3. See Figure below for several examples.

The genus does not in itself fix the Riemann surface, however, except for genus 0. We also need to know certain complex parameters known as moduli. Let me illustrate this issue in the case of the torus (genus 1). An easy way to construct a Riemann surface of genus 1 is to take a region of the complex plane bounded by a parallelogram, say with vertices 0, 1, 1 þ p, p (described cyclicly). See Figure below. Now we must imagine that opposite edges of the parallelogram are glued together, that is, the edge from 0 to 1 is glued to that from p to 1 + p, and the edge from 0 to p is glued to that from 1 to 1 + p. (We could always Wnd other patches to cover the seams, if we like.) The resulting Riemann surface is indeed topologically a torus. Now, it turns out that, for diVering values of p, the resulting surfaces are generally inequivalent to each other; that is to say, it is not possible to transform one into another by means of a holomorphic mapping. (There are certain discrete equivalences, however, such as those arising when p is replaced by 1 + p, by -p, or by 1/p.

It can be made intuitively plausible that not all Riemann surfaces with the same topology can be equivalent, by considering the two cases illustrated in Figure below.

In one case I have chosen a very tiny value of p, and we have a very stringy looking torus, and in the other case I have chosen p close to i, where the torus is nice and fat. Intuitively, it seems pretty clear that there can be no conformal equivalence between the two, and indeed there is none.
There is just this one complex modulus p in the case of genus 1, but for genus 2 we find that there are three. To construct a Riemann surface of genus 2 by pasting together a shape, in the manner of the parallelogram that we used for genus 1, we could construct the shape from a piece of the hyperbolic plane; see Figure. The same would hold for any higher genus. The number m of complex moduli for genus g, where g>2, is m = 3g - 3

## Riemann Mapping theorem

Riemann mapping theorem applies to topology. Some appreciation of the considerable freedom involved in holomorphic transformations can, however, be obtained from a famous result known as the Riemann mapping theorem. This asserts that if we have some closed region in the complex plane, bounded by a non-self-intersecting closed loop, then there exists a holomorphic map matching this region to the closed unit disc . (There are some mild restrictions on the 'tameness' of the loop, but these do not prevent the loop from having corners or other worse kinds of place where the loop may be not differentiable.

The Riemann mapping theorem asserts that any open region in the complex plane, bounded by a simple closed (not necessarily smooth) loop, can be mapped holomorphically to the interior of the unit circle, the boundary being also mapped accordingly.
One can go further than this and select, in a quite arbitrary way, three distinct points a, b, c on the loop, and insist that they be taken by the map to three speciWed points a`, b`, c` on the unit circle (say a` = 1, b` = ω, c` = ω^2), the only restriction being that the cyclic ordering of the points a, b, c, around the loop agrees with that of a`, b`, c` around the unit circle. Furthermore, the map is then determined uniquely. Another way of specifying the map uniquely would be to choose just one point a on the loop and one additional point j inside it, and then to insist that a maps to a speciWc point a` on the unit circle (say a` = 1) and j maps to a specific point j` inside the unit circle (say j` = 0).

Zhoukowski's transformation w = 1/2 (z + 1/z) takes the exterior of a circle through z= -1 to an aerofoil cross-section, enabling the airflow pattern about the latter to be calculated.
aerofoil transformation, illustrated in Figure above, which can be given explicitly by the eVect of the transformation
w = 1/2 (z + 1/z)
on a suitable circle passing through the point z = -1. This shape indeed closely resembles a cross-section through the wing of an aeroplane of the 1930s, so that the (idealized) airflow around it can be directly obtained from that around a 'wing' of circular cross-section—which, in turn, is obtained by another such holomorphic transformation. (I was once told that the reason that such a shape was so commonly used for aeroplane wings was merely that then one could study it mathematically by just employing the Zhoukowski transformation. I hope that this is not true!) Of course, there are specific assumptions and simplifications involved in applications such as these. Not only are the assumptions of zero viscosity and incompressible, irrotational flow mere convenient simpliWcations, but there is also the very drastic simpliWcation that the flow can be regarded as the same all along the length of the wing, so that an essentially threedimensional problem can be reduced to one entirely in two dimensions. It is clear that for a completely realistic computation of the floow around an aeroplane wing, a far more complicated mathematical treatment would be needed. There is no reason to expect that, in a more realistic treatment, we could get away with anything approaching such a direct and elegant use of holomorphic functions as we have with the Zhoukowski transformation.
It could, indeed, be argued that there is a strong element of good fortune in finding such an attractive application of complex numbers to a problem which had a distinctive importance in the real world. Air, of course, consists of enormous numbers of individual fundamental particles (in fact, about 1020 of them in a cubic centimetre), so airflow is something whose macroscopic description involves a considerable amount of averaging and approximation. There is no reason to expect that the mathematical equations of aerodynamics should reflect a great deal of the mathematics that is deeply involved in the physical laws that govern those individual particles.
I referred to the 'extraordinary and very basic role' that complex numbers actually play at the 'tiniest scales' of physical action, and there is indeed a holomorphic equation governing the behaviour of particles. However, for macroscopic systems, this 'complex structure' generally becomes completely buried, and it would appear that only in exceptional circumstances (such as in the airflow problem considered above) would complex numbers and holomorphic geometry find a natural utility. Yet there are circumstances where a basic underlying complex structure shows through even at the macroscopic level. This can sometimes be seen in Maxwell’s electromagnetic theory and other wave phenomena. There is also a particularly striking example in relativity theory. In the following chapter, we shall see something of the remarkable way in which complex numbers and holomorphic functions can exert their magic from behind the scenes.
A unit circle is the circle in the complex plane having |z| = 1 . That is to say the absolute value of all the points on the circle is unity.

## Knot theory

Knot theory , in topology, is the study of mathematical knots. In our daily live we often encounter knots that are so entangled that we can not unwrap or disantangle those. In knot theory knots are closed curves that end points are joined. Mathematically , A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop. Simply, we can say a knot K is a "simple closed curve" or "(closed) Jordan curve" (see Curve) — that is: a "nearly" injective and continuous function K [0, 1] -> R^3, with the only "non-injectivity" being K(0)=K(1).
Knot diagrams are created from projecting knots onto two dimensional plane. Every knot has crossing numbers which describes how many times there are crossoven between loops or lines. The isotopy of knots can be defined using the Reidemeister moves :

If these three transformations can be made then two knot diagrams always represent the same knot.
Knot ploynomial is a topological invariant which is a polynomial. Consider an oriented link diagram, i.e. one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let L+, L_ , L0 be the oriented link diagrams resulting from changing the diagram as

The original diagram might be either L+, L_-, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial C(z), is recursively defined according to the rules: C(O) = 1 (where O is any diagram of the unknot) C(L_) = C(L_-) + z C(L_0).
These are few basic things about knot theory.

## Brouwer's fixed point theorem

Brouwer's fixed point theorem states that any continuous function that maps a compact convex set onto itself has a fixed point such that f(x0) = x0. A disc is an example of such compact convex set. The proof is somewhat lengthy

Suppose that a function f(x) maps a triangle which is topologically equivalent to a disc onto itself. Then there will be a single point p which must be mapped to itself.

The mapping must be continuous.

## Diffeomorphism and homeomorphism

Each diffeomorphism is a homomorphism but the converse is true.
Morphisms. Let X and Y be topological spaces. The map f : X → Y is called continuous (or a topological morphism) iff the preimage of open sets is again open. The continuous map f : X → Y is called a topological isomorphism iff it is bijective and the inverse map f−1 : Y → X is continuous, too. Topological isomorphisms are also called homeomorphisms.
Standard examples. In order to give the reader a feeling for the universality of the notion of topological space, let us consider the following examples. • For n = 0, 1, . . . , the sets Rn and Cn and their subsets are topological spaces. • Every Hilbert space and Banach space and their subsets are topological spaces. • Every real (resp. complex) finite-dimensional manifold is a topological space. • For n = 1, 2, . . . , the n-dimensional unit sphere Sn := {x ∈ Rn+1 : x_1^2 + x_2^2 + . . . + x_n+1^2 = 1} is an n-dimensional real manifold. •
Define F(ϕ) := eiϕ for all ϕ ∈ R. The continuous map F : R → S1 sends the real line onto the unit circle S1. This is not a homeomorphism. We say that the real line R is a covering space of the unit circle.

## Connectedness

connected, but a torus is not simply connected.
The Jordan curve theorem. A topological space is called a closed Jordan curve iff it is homeomorphic to the unit circle S1. Obviously, the topological space R2 \ S1
consists of two components, namely, the interior and the exterior of the unit circle. In 1887 Camille Jordan (1838-1922) proved the much deeper result that for any closed Jordan curve C living in R2, the topological space R2 \ C
consists of two components; one component is bounded and the other component is unbounded. These two components are called the interior and the exterior of the curve C, respectively.
Euler characterstics of few topolgical figures

## The Poincar´e–Hopf Theorem for Velocity Fields

We want to show that the topology of a manifold restricts the structure of velocity vector fields on the manifold. Consider a velocity vector field
v(x) = v(x, y)i + w(x, y)j
on the Euclidean plane. Suppose that v(x0) = 0 and that the stationary point x0 is regular, that is, we define

and we assume that det v`(x0) != 0. The number ind v(x0) := sgn detv`(x0) is called the index of the stationary point x0. Some typical situations are pictured in Figure below. For example, if

v(x, y) := λx, w(x, y) := μy with real nonzero numbers λ and μ, then ind v`(0) = sgn(λμ). In particular, the index of a sink (or source) is equal to +1, whereas the index of a saddle is equal to −1. Consider a smooth velocity vector field on a real, 2-dimensional, compact, arcwise connected manifold (e.g., a sphere) which has only a finite number of stationary points x1, ..., xN. In addition, suppose that all of the stationary points are regular. Then, the Poincar´e–Hopf theorem tells us that

where χ is the Euler characteristic of the manifold.

## Riemannian geometry of two dimensional sphere

To begin with, let us introduce spherical coordinates
−π < ϕ ≤ π,
−π/2 ≤ ϑ ≤ π/2 .
Here, ϕ and ϑ denote geographic longitude and geographic latitude, respectively. Moreover, we get the following:
• equator: ϑ = 0;
• North Pole: ϑ = π/2 ;
• South Pole: ϑ = −π/2 ;
• meridian: ϕ = const;
• parallel of latitude: ϑ = const.
In terms of Cartesian coordinates x, y, z, the sphere S^2 R can be parametrized in the following way:

In fact, it follows from cos^2 α + sin^2 α = 1 that x^2 + y^2 + z^2 = R2. Now consider a smooth curve

on the sphere. In Cartesian coordinates,

Differentiation with respect to time yields

Similarly, we get y˙ and z˙. Using again cos^2 α + sin^2 α = 1,

This yields the arc length of the curve C,

Setting u1 := ϕ, u2 := ϑ, we get

where we sum over i, j = 1, 2. The functions

are called the components of the metric tensor of the sphere. Set g := det(gij). For the sphere, g = g11g22 = R^4 cos^2 ϑ. The differential form

is called the volume form of the sphere S^2 R. The integral

is equal to the surface area of the sphere. In the 1820s, the significance of the metric tensor was first noticed by Gauss in the context of his surface theory. The theorema egregium (the beautiful theorem) of Gauss tells us that the curvature of a surface can be computed by means of the second partial derivatives of the functions g_ij (see Vol. III). In the 1850s, Riemann introduced the components R_ijkl of the Riemann curvature tensor for n-dimensional Riemannian manifolds. In the special case of the sphere, there is only one essential component of the Riemann curvature tensor, namely, R_1212 = Kg
where K = 1/R2 is the Gaussian curvature of the sphere S2 R. In 1915 Einstein critically used the Riemann curvature tensor of the 4-dimensional space-time manifold. in order to describe the gravitational force in our universe in the framework of the theory of general relativity.

## Equations of our universe

Our universe is , in all respect, mathematical. Bertrand Russell qouted that mathematics is the only language which all universes confront, not just this accidental one. All physical theories are mathematical model which describes the natural phenomena. One such biggest idea was the Higg's Boson. Higg's boson is the God's particle which has been experimentally found in LHC in Switchzerland. For more explanation follow this page of quantum field theory.