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From Wikipedia, the free encyclopedia

 Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.[1]

Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation.[2] Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

 A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot
A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot

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Transcription

I've got several fun things for you this video. An unsolved problem, a very elegant solution to a weaker version of the problem, and a little bit about what topology is, and why people care. But before we jump into it, it's worth saying a few words on why I'm excited to share the solution. When I was a kid since I loved math and sought out various mathy things, I would occasionally find myself in some talk or a seminar where people wanted to get the youth excited about things that mathematicians care about. A very common go-to topic to excite our imaginations was "topology". We might be shown something like a Mobius strip, maybe building it out of construction paper by twisting a rectangle and gluing its ends. "Look!", we'd be told as we were asked to draw a line along the surface. "It's a surface with just one side!" Or we might be told that topologists view coffee mugs and donuts as the same thing since each has just one hole. But these kinds of demos always left a lurking question. "How is this math?" "How does any of this actually help to solve problems?" it wasn't until I saw the problem that I'm about to show you with its elegant and surprising solution that I started to understand why mathematicians actually care about some of these shapes and the properties they have. So there's this unsolved problem called "the inscribed square problem". If you have a closed-loop meaning you squiggle some line through space in a potentially crazy way and you end up back where you started. The question is whether or not you'll always be able to find four points on this loop that make up a square. If your closed loop was a circle, for example, it's quite easy to find an inscribed square. Infinitely many, in fact. If your loop was, instead, an ellipse, it's still pretty easy to find an inscribed square. The question is whether or not every possible closed-loop, no matter how crazy, has at least one inscribed square. Pretty interesting, right? I mean, just the fact that this is unsolved is interesting that the current tools of math can neither confirm nor deny that there's some loop with no inscribe square in it. Now, if we weaken the question of it and ask about inscribed rectangles instead of inscribed squares it's still pretty hard. But there is a beautiful video-worthy solution that might actually be my favorite piece of math. The idea is to shift the focus away from individual points on the loop and, instead, onto pairs of points. We'll use the following facts about rectangles. Let's label the vertices of some rectangle a, b, c, d. Then the pair of points a, c has a few things in common with the pair of points b, d. The distance between a and c equals the distance between b and d and the midpoint of a and c is the same as the midpoint of b and d. In fact, anytime you have two separate pairs of points in space a, c and b, d if you can guarantee that they share a midpoint and that the distance between a, c equals the distance between b and d it's enough to guarantee that those four points make up a rectangle. So what we're going to do is try to prove that for any closed loop it's always possible to find two distinct pairs of points on that loop that share a midpoint and which are the same distance apart. Take a moment to make sure that's clear. We're finding two distinct pairs of points that share a common midpoint and which are the same distance apart. The way we'll go about this is to define a function that takes in pairs of points on the loop and outputs a single point in 3d space which kind of encodes the midpoint and distance information. It will be sort of like a graph. Consider the closed-loop to be sitting on the xy plane in 3d space. For a given pair of points, label the midpoint M which will be some point on the xy plane and label the distance between them, d. Plot the point which is exactly d units above that midpoint M in the z direction. As you do this for many possible pairs of points you'll effectively be drawing through 3d space and if you do it for all possible pairs of points on the loop you'll draw out some kind of surface above the plane. Now look at the surface and notice how it seems to hug the loop itself. This is actually going to be important later. So let's think about why it happens. As the pair of points on the loop gets closer and closer the plotted point gets lower since its height is, by definition, equal to the distance between the points. Also the midpoint gets closer and closer to the loop as the points approach each other. Once the pair of points coincides meaning the input of our function looks like (X, X) for some point X on the loop the plotted point of the surface will be exactly on the loop at the point X. OK, so remember that. Another important fact is that this function is continuous and all that really means is that if you slightly adjust a given pair of points then the corresponding output in 3d space is only slightly adjusted as well. There's never a sudden discontinuous jump. Our goal, then, is to show that this function has a collision. The two distinct pairs of points each map to the same spot in 3d space. Because the only way for that to happen is if they share a common midpoint and if their distance d apart from each other is the same. So in some sense, finding an inscribed rectangle comes down to showing that this surface has to intersect itself. To move forward from here we need to build up a relationship with the idea of pairs of points on a loop. Think about how we represent pairs of real numbers using a two-dimensional coordinate plane. Analogous to this, we're going to seek out a certain 2d surface which naturally represents all pairs of points on the loop. Understanding the properties of this surface will help to show why the graph that we just defined has to intersect itself. Now, when I say a pair of points there are two things that I could be talking about. The first is "ordered" pairs of points which would mean a pair like (a, b) would be considered distinct from the pair (b, a). That is there some notion of which point is the first one. The second idea is "unordered" points where {a, b} and {b, a} would be considered the same thing, where all that really matters is what the points are and there's no meaning to which one is first. Ultimately, we want to understand “unordered” pairs of points. But to get there, we need to take a path of thought through “ordered” pairs. We'll start out by straightening out the loop cutting it at some point and deforming it into an interval. For the sake of having some labels let's say that this is the interval on the number line from 0 to 1. By following where each point ends up, every point on the loop corresponds with a unique number on this interval. Except for the point where the cut happened which corresponds simultaneously to both endpoints of the interval meaning the number 0 and 1. Now the benefit of straightening out this loop like this is that we can start thinking about pairs of points the same way we think about pairs of numbers. Make a y-axis using a second interval then associate each pair of values on the interval with a single point in this 1x1 square that they span out. Every individual point of this square naturally corresponds to a pair of points on the loop since its x and y coordinates are each numbers between 0 and 1, which are in turn associated to some unique point on the loop. Remember, we're trying to find a surface that naturally represents the set of all pairs of points on the loop and this square is the first step to doing that. The problem is that there's some ambiguity when it comes to the edges of the square. Remember, the endpoints 0 and 1 on the interval really correspond to the same point of the loop as if to say that those endpoints need to be glued together if we're going to faithfully map back to the loop. So, all of the points on the left edge of the square like (0, 0.1), (0, 0.2) on and on and on really represent the same pair of points on the loop as the corresponding coordinates on the right edge of the square. (1, 0.1), (1, 0.2) on and on and on. So for this square to represent the pairs of points on the loop in a unique way we need to glue this left edge to the right edge. I'll mark each edge with some arrows to remember how the edges need to be lined up. Likewise, the bottom edge needs to be glued to the top edge since y coordinates of 0 and 1 really represent the same second point in a given pair of points on the loop. If you bend the square to perform the gluing, first rolling it into a cylinder to glue the left and right edges then gluing the ends of that cylinder which represent the top and bottom edges we get a "torus", better known as the surface of a donut. Every individual point on this torus corresponds to a unique pair of points on the loop. And likewise, every pair of points on the loop corresponds to some unique point on this torus. The torus is to pairs of points on the loop what the xy plane is to pairs of points on the real number line. The key property of this association is that it's continuous both ways meaning if you nudge any point on the torus by just a tiny amount it corresponds to only a very slight nudge to the pair of points on the loop and vice versa. So if the torus is the natural shape for ordered pairs of points on the loop, what's the natural shape for unordered pairs? After all, the whole reason we're doing this is to show the two distinct pairs of points on the loop share a midpoint and are the same distance apart. But if we consider a pair (a, b) to be distinct from (b, a) then that would trivially give us two separate pairs which have the same midpoint and distance apart. That's like saying you can always find a rectangle so long as you consider any pair of points to be a rectangle. Not helpful! So let's think about this. Let's think about how to represent unordered pairs of points. looking back at our unit square. We need to say that the coordinates (0.2, 0.3) represent the same pair as (0.3, 0.2) or the (0.5, 0.7) really represents the same thing as (0.7, 0.5) and in general any coordinates (x, y) has to represent the same thing as (y, x). Once again, we capture this idea by gluing points together when they're supposed to represent the same pair. Which, in this case, requires folding the square over diagonally. Now notice this diagonal line, the crease of the fold it represents all pairs of points that look like (x, x) meaning the pairs which are really just a single point written twice. Right now, it's marked with a red line and you should remember it it will become important to know where all of these pairs like (x, x) live. But we still have some arrows to glue together here. We need to glue that bottom edge to the right edge and the orientation with which we do this is going to be important. Points towards the left of the bottom edge have to be glued to points towards the bottom of the right edge. And points towards the right of the bottom edge have to be glued to points towards the top of the right edge. It's weird to think about, right? Go ahead. Pause and ponder this for a moment. The trick which is kind of clever is to make a diagonal cut which we need to remember to glue back in just a moment. After that, we can glue the bottom and the right like so. But notice the orientation of the arrows here. To glue back what we just cut we don't simply connect the edges of this rectangle to get a cylinder. We have to make a "twist". Doing this in 3d space the shape we get is a "Mobius strip"! Isn't that awesome? Evidently the surface which represents all pairs of unordered points on the loop is the Mobius strip! And notice the edge of the strip shown here in red represents the pairs of points that look like (x, x), those which are really just a single point listed twice. The Mobius strip is to unordered pairs of points on the loop what the xy plane is to pairs of real numbers. That totally blew my mind when I first saw it! Now, with this fact that there is a continuous one-to-one association between unordered pairs of points on the loop and individual points on this Mobius strip, we can solve the inscribed rectangle problem. Remember, we had defined the special kind of graph in 3d space where the loop was sitting in the xy plane. For each pair of points you consider their midpoint M which lives on the xy plane and their distance d apart and you plot a point which is exactly d units above M. Because of the continuous one-to-one association between pairs of points on the loop and the Mobius strip this gives us a natural map from the Mobius strip onto this surface in 3d space. For every point on the Mobius strip, consider the pair of points on the loop that it represents then plug that pair of points into the special function. And here's the key point. When pairs of points on the loop are extremely close together the output of the function is right above the loop itself and in the extreme case of pairs of points like (x, x) the output of the function is exactly on the loop since points on this red edge of the Mobius strip correspond to pairs like (x, x). When the Mobius strip is mapped onto the surface it must be done in such a way that the edge of the strip gets mapped right onto that loop in the xy plane. But if you stand back and think about it for a moment considering the strange shape of the Mobius strip there is no way to glue its edge to something two-dimensional without forcing the strip to intersect itself. Since points of the Mobius strip represent pairs of points on the loop. If the strip intersect itself during this mapping it means that there are at least two distinct pairs of points that correspond to the same output on this surface. Which means they share a midpoint and are the same distance apart. Which, in turn, means that they form a rectangle. And that's the proof! Or at least if you're willing to trust me and saying that you can't glue the edge of a Mobius strip to a plane without forcing it to intersect itself. Then that's the proof! This fact is intuitively clear looking at the Mobius strip. But in order to make it rigorous you basically need to start developing the field of topology. In fact, for any of you who have a topology class in your future going through the exercise of trying to justify this is a good way to gain an appreciation for why topologists choose to make certain definitions and I want you to take note of something here. The reason for talking about the torus and the Mobius strip was not because we were playing around with construction paper or because we were daydreaming about deforming a coffee mug. They came up as a natural way to understand pairs of points on a loop and that's something that we needed to solve a concrete problem. Alright, I have one final animation for you all. But first I'd like to tell you a little about what's making this and future videos possible. First, I want to say a huge thanks to everybody who supported on Patreon. I launched this only last week and have been absolutely blown away by people's enthusiasm for helping make more of these videos. Right now, I'm working on an "Essence of Calculus" series and those supporting on Patreon are getting early access to those videos before I publish the full set in a few months. I also want to thank "hover.com" for supporting this video. I'm sure a lot of you watching have some idea for a website that you want to start. You know that idea that's always been in the back of your mind. But you just haven't gotten to it yet. Maybe it's that cool interactive explanation or a store or just a blog where you want to share the cool things that you learn. Anyway, point is when you want to register a good domain name "hover" is the place to do that doesn't suck. They're just straight to the point. You get your domain name or your email address without having to opt out of a whole bunch of things. They also don't make you pay for things that should obviously be included with your domain. I don't know if you guys know this but when you register a domain name your email address, phone number, home address, all that stuff is published online in a way that marketers, spammers, hackers or anyone can find in what's called a WHOIS database. And unlike most other domain providers "hover" includes free WHOIS privacy with all their supported domains and that keeps your information confidential. If you go there now and use the promo code "TOPOLOGY" you can get 10% off your first purchase. That also lets them know that you came from me which encourages them to support more videos like this one so I can keep making them. And with that, here's the final animation I promised. It shows how that beautiful surface that we defined earlier changes while the loop changes.

Contents

History

 The Seven Bridges of Königsberg was a problem solved by Euler.
The Seven Bridges of Königsberg was a problem solved by Euler.

Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.[3] Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology.[3] On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron. This led to his polyhedron formula, VE + F = 2 (where V, E and F respectively indicate the number of vertices, edges and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signalling the birth of topology.[4][5]

Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.[6] Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print.[7] The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".[8] The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.[citation needed]

Their work was corrected, consolidated and greatly extended by Henri Poincaré. In 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology.[6]

Topological characteristics of closed 2-manifolds[6]
Manifold Euler no.
χ
Orientability Betti numbers Torsion coefficient
(1-dimensional)
b0 b1 b2
Sphere   2 Orientable 1 0 1 none
Torus   0 Orientable 1 2 1 none
2-holed torus −2 Orientable 1 4 1 none
g-holed torus (Genus = g) 2 − 2g Orientable 1 2g 1 none
Projective plane   1 Non-orientable 1 0 0 2
Klein bottle   0 Non-orientable 1 1 0 2
Sphere with c cross-caps 2 − c Non-orientable 1 c − 1 0 2
2-Manifold with g holes
and c cross-caps (c > 0)
2 − (2g + c) Non-orientable 1 (2g + c) − 1 0 2

Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906.[9] A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space.[10] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.[11]

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series. For further developments, see point-set topology and algebraic topology.

Introduction

Topology can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under a certain kind of transformation (called a continuous map), especially those properties that are invariant under a certain kind of invertible transformation (called homeomorphism)."

Topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes' the set X as a topological space by taking proper care of properties such as convergence, connectedness and continuity, upon transformation.

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.

In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This problem in introductory mathematics called Seven Bridges of Königsberg led to the branch of mathematics known as graph theory.

 A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back
A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.

To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.

Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.

Equivalence classes of the English (i.e., Latin) alphabet (sans-serif)
Homeomorphism Homotopy equivalence
{A,R} {B} {C,G,I,J,L,M,N,S,U,V,W,Z}, {D,O} {E,F,T,Y} {H,K}, {P,Q} {X}
{A,R,D,O,P,Q} {B}, {C,E,F,G,H,I,J,K,L,M,N,S,T,U,V,W,X,Y,Z}

An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use the sans-serif Myriad font. Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the "hole" part.

Homeomorphism classes are: [clarification needed]

  • no holes,
  • no holes three tails,
  • no holes four tails,
  • one hole no tail,
  • one hole one tail,
  • one hole two tails,
  • two holes no tail, and
  • a bar with four tails (the "bar" on the K is almost too short to see).

Homotopy classes are larger, because the tails can be squished down to a point. They are:

  • one hole,
  • two holes, and
  • no holes.

To classify the letters correctly, we must show that two letters in the same class are equivalent and two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.

Letter topology has practical relevance in stencil typography. For instance, Braggadocio font stencils are made of one connected piece of material.

Concepts

Topologies on sets

The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology tells how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane (which is a 1-dimensional complex vector space), and the Cantor set can be thought of as the same set with different topologies.

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

  1. Both the empty set and X are elements of τ.
  2. Any union of elements of τ is an element of τ.
  3. Any intersection of finitely many elements of τ is an element of τ.

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open. An open set containing a point x is called a 'neighborhood' of x.

Continuous functions and homeomorphisms

A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the Standard Topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.

Manifolds

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, but also the Klein bottle and real projective plane, which cannot.

Topics

General topology

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.[12][13] It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

The fundamental concepts in point-set topology are continuity, compactness, and connectedness. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby, arbitrarily small, and far apart can all be made precise by using open sets. If we change the definition of open set, we change what continuous functions, compact sets, and connected sets are. Each choice of definition for open set is called a topology. A set with a topology is called a topological space.

Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.[14] The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

The most important of these invariants are homotopy groups, homology, and cohomology.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Differential topology

Differential topology is the field dealing with differentiable functions on differentiable manifolds.[15] It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Geometric topology

Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (i.e. dimensions 2,3 and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.[16] [17] Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem.

In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.

2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.

Generalizations

Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory,[18] while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology theories.[19]

Applications

Biology

Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.[20] Topology is also used in evolutionary biology to represent the relationship between phenotype and genotype.[21] Phenotypic forms that appear quite different can be separated by only a few mutations depending on how genetic changes map to phenotypic changes during development.

Computer science

Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is:

  1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
  2. Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homology.[22]
  3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode.[22]

Physics

In physics, topology is used in several areas such as condensed matter physics,[23] quantum field theory and physical cosmology.

The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials.[24] The compressive strength of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space.[25] Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

The topological classification of Calabi-Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.[26]

In cosmology, topology can be used to describe the overall shape of the universe.[27] This area of research is commonly known as spacetime topology.

Robotics

The various possible positions of a robot can be described by a manifold called configuration space.[28] In the area of motion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired location and pose.[29]

Games and puzzles

Tanglement puzzles are based on topological aspects of the puzzle's shapes and components.[30][31][32][33]

See also

References

Citations

  1. ^ "the definition of topology". 
  2. ^ Bruner, Robert (2000). "What is Topology? A short and idiosyncratic answer". 
  3. ^ a b Croom 1989, p. 7
  4. ^ Richeson 2008, p. 63
  5. ^ Aleksandrov 1969, p. 204
  6. ^ a b c Richeson (2008)
  7. ^ Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
  8. ^ Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature 27, 1 February 1883, pp. 316–317
  9. ^ Fréchet, Maurice (1906). Sur quelques points du calcul fonctionnel. PhD dissertation. OCLC 8897542. 
  10. ^ Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
  11. ^ Croom 1989, p. 129
  12. ^ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
  13. ^ Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
  14. ^ Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0.
  15. ^ Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6. 
  16. ^ Budney, Ryan (2011). "What is geometric topology?". mathoverflow.net. Retrieved 29 December 2013. 
  17. ^ R.B. Sher and R.J. Daverman (2002), Handbook of Geometric Topology, North-Holland. ISBN 0-444-82432-4
  18. ^ Johnstone, Peter T. (1983). "The point of pointless topology". Bulletin of the American Mathematical Society. 8 (1): 41–53. doi:10.1090/s0273-0979-1983-15080-2. 
  19. ^ Artin, Michael (1962). Grothendieck topologies. Cambridge, MA: Harvard University, Dept. of Mathematics. Zbl 0208.48701. 
  20. ^ Adams, Colin (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society. ISBN 0-8218-3678-1 
  21. ^ Barble M R Stadler; et al. "The Topology of the Possible: Formal Spaces Underlying Patterns of Evolutionary Change". Journal of Theoretical Biology. 213: 241–274. doi:10.1006/jtbi.2001.2423. PMID 11894994. 
  22. ^ a b Gunnar Carlsson (April 2009). "Topology and data" (PDF). Bulletin (New Series) of the American Mathematical Society. 46 (2): 255–308. doi:10.1090/S0273-0979-09-01249-X. 
  23. ^ "The Nobel Prize in Physics 2016". Nobel Foundation. 4 October 2016. Retrieved 12 October 2016. 
  24. ^ Stephenson, C.; et., al. (2017). "Topological properties of a self-assembled electrical network via ab initio calculation". Sci. Rep. doi:10.1038/srep41621. 
  25. ^ Cambou, Anne Dominique; Narayanan, Menon (2011). "Three-dimensional structure of a sheet crumpled into a ball". Proceedings of the National Academy of Sciences. 108 (36): 14741–14745. doi:10.1073/pnas.1019192108. 
  26. ^ Yau, S. & Nadis, S.; The Shape of Inner Space, Basic Books, 2010.
  27. ^ The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds 2nd ed (Marcel Dekker, 1985, ISBN 0-8247-7437-X)
  28. ^ John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004
  29. ^ Michael Farber, Invitation to Topological Robotics, European Mathematical Society, 2008
  30. ^ https://math.stackexchange.com How to reason about disentanglement puzzles.
  31. ^ https://www.jstor.org/stable/27642974 Disentangling Topological Puzzles by Using Knot Theory, Mathew Horak; Mathematics Magazine, December 2006.
  32. ^ http://sma.epfl.ch/Notes.pdf A Topological Puzzle, Inta Bertuccioni, December 2003.
  33. ^ https://www.futilitycloset.com/the-figure-8-puzzle The Figure Eight Puzzle, Science and Math, June 2012.

Bibliography

  • Aleksandrov, P. S. (1969) [1956], "Chapter XVIII Topology", in Aleksandrov, A.D.; Kolmogorov, A.N.; Lavrent'ev, M.A., Mathematics / Its Content, Methods and Meaning (2nd ed.), The M.I.T. Press 
  • Croom, Fred H. (1989), Principles of Topology, Saunders College Publishing, ISBN 0-03-029804-0 
  • Richeson, D. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press 

Further reading

External links

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