Topology
Topology (from the
A
The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. 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.
Motivation
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 or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory.
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
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.[1]
Homeomorphism can be considered the most basic
History
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.
Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.[4] 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.[5] 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".[6]
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.[4]
Manifold | Euler num | Orientability | Betti numbers | Torsion coefficient (1-dim) | ||
---|---|---|---|---|---|---|
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 (c > 0) |
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
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
The 2022 Abel Prize was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".[10]
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 describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the
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:
- Both the empty set and X are elements of τ.
- Any union of elements of τ is an element of τ.
- 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. By definition, every topology is a π-system.
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 (that is, its complement is open). A subset of X may be open, closed, both (a
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. However, the sphere 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
Topics
General topology
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.[11][12] 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 basic object of study is
Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces.[13] 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.[14] 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
Geometric topology
Geometric topology is a branch of topology that primarily focuses on low-dimensional
In high-dimensional topology,
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, and 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
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,[16] 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.[17]
Applications
Biology
Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. 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.[18] In neuroscience, topological quantities like the Euler characteristic and Betti number have been used to measure the complexity of patterns of activity in neural networks.[citation needed]
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 to:
- Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
- Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology.[19]
- Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode.[19]
Several branches of
Physics
Topology is relevant to physics in areas such as condensed matter physics,[21] quantum field theory and physical cosmology.
The topological dependence of mechanical properties in solids is of interest in disciplines of
A
Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things,
The topological classification of Calabi–Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.[24]
In cosmology, topology can be used to describe the overall shape of the universe.[25] This area of research is commonly known as spacetime topology.
In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect, and then generalized in other areas of physics, for instance in photonics[26] by F.D.M Haldane.
Robotics
The possible positions of a robot can be described by a manifold called configuration space.[27] 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 pose.[28]
Games and puzzles
Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components.[29][30][31]
Fiber art
In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path.[32]
Resources and Research
Major Journals
- Geometry & Topology- a mathematic research journal focused on geometry and topology, and their applications, published by Mathematical Sciences Publishers.
- Journal of Topology- a scientific journal which publishes papers of high quality and significance in topology, geometry, and adjacent areas of mathematics.
Major Books
- Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-181629-9
- Willard, Stephen (2016). General topology. Dover books on mathematics. Mineola, N.Y: Dover publications. ISBN 978-0-486-43479-7
- Armstrong, M. A. (1983). Basic topology. Undergraduate texts in mathematics. New York: Springer-Verlag. ISBN 978-0-387-90839-7
See also
- Characterizations of the category of topological spaces
- Equivariant topology
- List of algebraic topology topics
- List of examples in general topology
- List of general topology topics
- List of geometric topology topics
- List of topology topics
- Publications in topology
- Topoisomer
- Topology glossary
- Topological Galois theory
- Topological geometry
- Topological order
References
Citations
- ISBN 978-0-387-94377-0.
- ^ a b Croom 1989, p. 7
- ^ Richeson 2008, p. 63; Aleksandrov 1969, p. 204
- ^ a b c Richeson (2008)
- ^ Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
- doi:10.1038/027316a0.
- OCLC 8897542.
- ^ Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
- ^ Croom 1989, p. 129
- ^ "Prize winner 2022". The Norwegian Academy of Science and Letters. Retrieved 23 March 2022.
- ^ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
- ^ Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
- ISBN 0-521-79160-X, 0-521-79540-0.
- ISBN 978-0-387-95448-6.
- ISBN 0-444-82432-4
- .
- Zbl 0208.48701.
- ISBN 978-0-8218-3678-1.
- ^ .
- ISBN 978-0521576512.
- ^ "The Nobel Prize in Physics 2016". Nobel Foundation. 4 October 2016. Retrieved 12 October 2016.
- PMID 28155863.
- PMID 21873249.
- ^ Yau, S. & Nadis, S.; The Shape of Inner Space, Basic Books, 2010.
- ISBN 0-8247-7437-X)
- S2CID 44745453.
- ^ John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004
- ISBN 978-3037190548.
- JSTOR 27642974..
- ^ http://sma.epfl.ch/Notes.pdf Archived 1 November 2022 at the Wayback Machine A Topological Puzzle, Inta Bertuccioni, December 2003.
- ^ https://www.futilitycloset.com/the-figure-8-puzzle Archived 25 May 2017 at the Wayback Machine The Figure Eight Puzzle, Science and Math, June 2012.
- ISBN 978-1603429733.
Bibliography
- Aleksandrov, P.S. (1969) [1956], "Chapter XVIII Topology", in Aleksandrov, A.D.; Kolmogorov, A.N.; Lavrent'ev, M.A. (eds.), Mathematics / Its Content, Methods and Meaning (2nd ed.), The M.I.T. Press
- Croom, Fred H. (1989), Principles of Topology, Saunders College Publishing, ISBN 978-0-03-029804-2
- Richeson, D. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press
Further reading
- ISBN 3-88538-006-4.
- Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
- Breitenberger, E. (2006). "Johann Benedict Listing". In James, I.M. (ed.). History of Topology. North Holland. ISBN 978-0-444-82375-5.
- ISBN 978-0-387-90125-1.
- orbit spaces.)
- ISBN 0-486-41148-6
- ISBN 978-1-56025-826-1. (Provides a popular introduction to topology and geometry)
- Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover Publications Inc., ISBN 978-0-486-66522-1
External links
- "Topology, general", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov.
- Topology at Curlie
- The Topological Zoo at The Geometry Center.
- Topology Atlas
- Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas.
- Topology Glossary
- Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.