Combinatorial topology
Encyclopedia
In mathematics
, combinatorial topology was an older name for algebraic topology
, dating from the time when topological invariants of spaces (for example the Betti number
s) were regarded as derived from combinatorial decompositions such as simplicial complex
es. 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 group
s. This point of view is often attributed to Emmy Noether
, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf
, who was influenced by Noether, and to Leopold Vietoris
and Walther Mayer
, who independently defined homology.
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.
Rosenfeld (1979) proposed digital topology
for image processing that can be considered as a new development of combinatorial topology. The digital forms of Euler characteristic
theorem and Gauss–Bonnet theorem
were obtained by Chen et al. (See digital topology.) In history, a 2D grid cell topology
was appeared in Alexandrov-Hopf's book Topologie I (1935).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, combinatorial topology was an older name for algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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.Although algebraic topology...
, dating from the time when topological invariants of spaces (for example the Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s) were regarded as derived from combinatorial decompositions such as simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es. After the proof of the simplicial approximation theorem
Simplicial approximation theorem
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,...
this approach provided rigour.
The change of name reflected the move to organise topological classes such as cycles modulo boundaries explicitly into abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s. This point of view is often attributed to Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
, who was influenced by Noether, and to Leopold Vietoris
Leopold Vietoris
Leopold Vietoris was an Austrian mathematician and a World War I veteran who gained additional fame by becoming a supercentenarian...
and Walther Mayer
Walther Mayer
Walter Mayer was an Austrian mathematician, born 1887 in Graz, Austria.Mayer, who was Jewish, studied at the Federal Institute of Technology in Zürich and the University of Paris before receiving his doctorate in 1912 from the University of Vienna...
, who independently defined homology.
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.
Rosenfeld (1979) proposed digital topology
Digital topology
Digital topology deals with properties and features of two-dimensional or three-dimensional digital imagesthat correspond to topological properties or topological features of objects....
for image processing that can be considered as a new development of combinatorial topology. The digital forms of Euler characteristic
Euler characteristic
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...
theorem and Gauss–Bonnet theorem
Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
were obtained by Chen et al. (See digital topology.) In history, a 2D grid cell topology
Grid cell topology
The grid cell topology is studied in digital topology as part of the theoretical basis for algorithms in computer image analysis or computer graphics....
was appeared in Alexandrov-Hopf's book Topologie I (1935).