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.

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] }

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).

- ↑ 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. - ↑ Chronomaths, (in French).
- ↑ Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999 , pp. 61–63.
- ↑ McCleary, John. "Bourbaki and Algebraic Topology" (PDF). gives documentation (translated into English from French originals).
- ↑ 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. - ↑ Chen, Li; Rong, Yongwu (December 8–11, 2008).
*Linear Time Recognition Algorithms for Topological Invariants in 3D*. 19th International Conference on Pattern Recognition (ICPR 2008). Tampa, Florida. pp. 3254–3257. arXiv: 0804.1982 .

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, but not tearing or gluing.

**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, to 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 .

**Amalie Emmy Noether** was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's theorem, which is fundamental in mathematical physics. She invariably used the name "Emmy Noether" in her life and publications. 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 some theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

**Commutative algebra** 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.

**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.

**Heinz Hopf** was a German mathematician who worked on the fields of topology and geometry.

In mathematics, the **simplicial approximation theorem** is a foundational result for algebraic topology, guaranteeing that continuous mappings can be approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices—that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (*affine*-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of the domain, and (ii) replacement of the actual mapping by a homotopic one.

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

In mathematics, an **H-space**, or a **topological unital magma**, is a topological space *X* together with a continuous map μ : *X* × *X* → *X* with an identity element *e* such that μ(*e*, *x*) = μ(*x*, *e*) = *x* for all *x* in *X*. Alternatively, the maps μ(*e*, *x*) and μ(*x*, *e*) are sometimes only required to be homotopic to the identity, sometimes through basepoint preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes. Every topological group is an H-space; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.

In mathematics, **Alexander duality** refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace *X* in Euclidean space, a sphere, or other manifold. It is generalized by Spanier–Whitehead duality.

**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 mathematics, **real algebraic geometry** is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them.

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 it is the case in Euclidean and CW complex. Abstract cell complexes play an important role in image analysis and computer graphics.

- Alexandrov, Pavel S. (1956),
*Combinatorial Topology Vols. I,II,III*, translated by Horace Komm, Graylock Press, MR 1643155 - Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century",
*Mathematics Magazine*, Mathematical Association of America,**60**(5): 282–291, doi:10.1080/0025570X.1988.11977391, JSTOR 2689545 - Teicher, Mina, ed. (1999),
*The Heritage of Emmy Noether*, Israel Mathematical Conference Proceedings, Bar-Ilan University/American Mathematical Society/Oxford University Press, ISBN 978-0-19-851045-1, OCLC 223099225 - Novikov, Sergei P. (2001) [1994], "Combinatorial topology",
*Encyclopedia of Mathematics*, EMS Press

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