Combinatorial topology

Last updated

In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour.

Contents

The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, [1] and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, [2] who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology. [3]

A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. [4] This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods.

Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology. The digital forms of the Euler characteristic theorem and the Gauss–Bonnet theorem were obtained by Li Chen and Yongwu Rong. [5] [6] A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935).

See also

Notes

  1. ↑ For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), (in French) note 41, explicitly names Noether as inventing homology groups.
  2. ↑ Chronomaths, (in French).
  3. ↑ Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999 , pp. 61–63.
  4. ↑ McCleary, John. "Bourbaki and Algebraic Topology" (PDF). gives documentation (translated into English from French originals).
  5. ↑ Chen, Li; Rong, Yongwu (2010). "Digital topological method for computing genus and the Betti numbers". Topology and Its Applications . 157 (12): 1931–1936. doi:10.1016/j.topol.2010.04.006. MR   2646425.
  6. ↑ Chen, Li; Rong, Yongwu. Linear Time Recognition Algorithms for Topological Invariants in 3D. 19th International Conference on Pattern Recognition (ICPR 2008). pp. 3254–7. arXiv: 0804.1982 . CiteSeerX   10.1.1.312.6573 . doi:10.1109/ICPR.2008.4761192. ISBN   978-1-4244-2174-9.

Related Research Articles

<span class="mw-page-title-main">Topology</span> Branch of mathematics

In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

<span class="mw-page-title-main">Algebraic topology</span> Branch of mathematics

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

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .

<span class="mw-page-title-main">Emmy Noether</span> German mathematician (1882–1935)

Amalie Emmy Noether was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

<span class="mw-page-title-main">Commutative algebra</span> Branch of algebra that studies commutative rings

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces, the sequence of Betti numbers is 0 from some point onward, and they are all finite.

In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

<span class="mw-page-title-main">Discrete geometry</span> Branch of geometry that studies combinatorial properties and constructive methods

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

<span class="mw-page-title-main">Pavel Alexandrov</span> Soviet mathematician (1896–1982)

Pavel Sergeyevich Alexandrov, sometimes romanized Paul Alexandroff, was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topology, the Alexandroff compactification and the Alexandrov topology are named after him.

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is,

<span class="mw-page-title-main">Triangulation (topology)</span>

In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling.

<span class="mw-page-title-main">Circuit rank</span> Fewest graph edges whose removal breaks all cycles

In graph theory, a branch of mathematics, the circuit rank, cyclomatic number, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It is equal to the number of independent cycles in the graph. Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank r is easily computed using the formula

Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties or topological features of objects.

In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.

In the mathematical field of low-dimensional topology, a clasper is a surface in a 3-manifold on which surgery can be performed.

In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a Hausdorff space as is the case in Euclidean and CW complexes. Abstract cell complexes play an important role in image analysis and computer graphics.

This is a glossary of properties and concepts in algebraic topology in mathematics.

References

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.