chain

From Middle English cheyne, chaine, from Old French chaine, chaene (“chain”), from Latin catēna (“chain”), from Proto-Indo-European *kat- (“to braid, twist; hut, shed”). Doublet of catena.

der

a chain is only as strong as its weakest link

(algebraic topology, originally) A formal sum of cells in a CW complex of a certain dimension k (in which case the formal sums are called k'''-chains); a formal sum of simplices or cubes of a certain dimension in a simplical complex or cubical complex (respectively).

(algebraic topology, homological algebra, more generally) An element of a group (or module) in a chain complex.


cycle

→ Japanese: サイクル (saikuru)

From Middle English cicle (“fixed length period of years”), from Late Latin cyclus, from Ancient Greek κύκλος (kúklos, “circle”), from Proto-Hellenic *kúklos, *kʷókʷlos, from Proto-Indo-European *kʷékʷlos (“circle, wheel”).
Doublet of chakra, chakram, charkha, chukker, cyclus, and wheel (see there for more).

pyörä

bicycle, tricycle, motorcycle etc. — see also bicycle

(topology, algebraic topology) A chain whose boundary is zero.


simplex

en-noun

simplex (plural simplexes or simplices or simplicia)

simplexes (plural)
simplices (plural)
simplicia (plural)

Henri Poincaré
Pieter Hendrik Schoute
Simplex (disambiguation)
simplex

From Latin simplex (“simple”).
In the mathematical sense, apparently first used (in German) in 1902, Pieter Hendrik Schoute, Mehrdimensionale Geometrie, where Schoute first suggests the term Simplicissimum, but then from the next page decides to use simplex "for short". In his pioneering works on algebraic topology, Henri Poincaré had previously introduced the concept, but not the actual term simplex.

linearly independent
linearly independent

(geometry, algebraic topology) An analogue in a space of arbitrary dimensionality of the triangle or tetrahedron; the convex hull of n+1 linearly independent points in n-dimensional space; in other words, the triangle, tetrahedron etc., generalized to an arbitrary number of dimensions.

单纯形

an analogue in any dimension of the triangle or tetrahedron


pointed

en-adj

pointed (comparative more pointed, superlative most pointed)

more pointed (comparative)
most pointed (superlative)

The class of all pointed spaces forms a category - Top_• - with basepoint preserving continuous maps as morphisms.

(topology, algebraic topology, of a topological space) That has a named, but otherwise arbitrary, point (called the basepoint) that remains unchanged during subsequent discussion and is kept track of during all operations.


homology

en-noun

homology (countable and uncountable, plural homologies)

LL-Q1860 (eng)-Flame, not lame-homology.wav

Henri Poincaré
Homology
MathWorld
singular homology

From homo- + -logy.
In topology, first used by French polymath Henri Poincaré, in the sense (close to what is now called a bordism) of a relation between manifolds mapped into a reference manifold: that is, the property of such manifolds that they form the boundary of a higher-dimensional manifold inside the reference manifold. Poincaré's version was eventually replaced by the more general singular homology, which is what mathematicians now mean by homology.

homologia

biology, psychology: relationship of having a common evolutionary or developmental origin

One encounters a similar situation in homology theory. Beside singular homology, which is a homotopy invariant, and Čech homology, which is a shape invariant, there exists strong homology, which is a strong shape invariant. In the special case of metric compacta, this homology was introduced by N.E. Steenrod in 1940 and is often referred to as the Steenrod homology.

2000, Sibe Mardešić, Strong Shape and Homology, Springer, page v:

Brieskorn homology spheres are a special case of Siefert fibered homology spheres.

2002, Nikolai Saveliev, Invariants of Homology 3-Spheres, Springer, page 2:

The relationship of being homologous; a homologous relationship.

(topology, algebraic topology) A general way of associating a sequence of algebraic objects, such as abelian groups or modules, to a sequence of topological spaces; also used attributively: see Usage notes below.

同调

algebraic topology: way of associating a sequence of groups to a sequence of topological spaces


Betti number

A calque of French nombre de Betti, coined in 1892 by Henri Poincaré; named after Italian mathematician Enrico Betti in recognition of an 1871 paper.

Poincaré proved that Betti numbers are invariants and used them to extend Euler's polyhedral formula to higher dimensional spaces.

1979 [W. H. Freeman & Company], Michael Henle, A Combinatorial Introduction to Topology, 1994, Dover, page 163,
Prove that, for compact surfaces, the zeroth Betti number is the number of components of the surface, where a component is a connected subset of the surface, such that any larger containing subset is not connected.

2007, Oscar García-Prada, Peter Beier Gothen, Vicente Muñoz, Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles, American Mathematical Society, page 7,
PROPOSITION 2.1. Fix the rank r. For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers.

The goal is to compute Betti numbers in 2D and 3D image partitions using the practical definition of Betti numbers. Thus, depending on the dimension of the topological map, we count the number of connected components, the number of tunnels, and the number of cavities to obtain the Betti numbers.[…]The number of connected components of region r in a 2D image partition is equal to the first Betti number b#x5F;0(r).

2012, Guillaume Damiand, Alexandre Dupas, “12: Combinatorial Maps for Image Segmentation”, in Valentin E. Brimkov, Reneta P. Barneva, editors, Digital Geometry Algorithms, Springer, page 380:

(topology, algebraic topology) Any of a sequence of numbers, denoted bₙ, which characterise a given topological space K by giving, for each dimension, the number of holes in K of said dimension; (formally) the rank of the nth homology group, Hₙ, of K.

Betti number
Enrico Betti
Henri Poincaré

Bettital

number that characterises a topological space with respect to a given dimension


fibration

(algebraic topology) A continuous mapping satisfying the homotopy lifting property with respect to any space.


chain complex

boundary homomorphism
boundary homomorphism

(algebraic topology) A sequence of Abelian groups (or modules), together with a sequence of homomorphisms (called boundary homomorphisms) between consecutive groups, such that the image of one homomorphism is contained in the kernel of the next.


K-theory

From circa 1960. The K stands for German Klasse (“class”).
The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

algebraic K-theory

study of rings generated by vector bundles

K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.

1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

K'''-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K'''-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

topological K-theory
topological K-theory

(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;

The study of rings R generated by the set of vector bundles over some topological space or scheme;
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.


單純

/tän⁵⁵ ʈ͡ʂʰu̯ən³⁵/

/tän⁵⁵ ʈ͡ʂʰu̯ən³⁵/

(geometry, algebraic topology) simplicial


fibrational

en-adj

fibrational (comparative more fibrational, superlative most fibrational)

more fibrational (comparative)
most fibrational (superlative)

Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory.

2015, Emily Riehl, Dominic Verity, “Kan extensions and the calculus of modules for ∞-categories”, in arXiv:

(algebraic topology) Of or pertaining to a fibration.


單純形

ㄉㄢ ㄔㄨㄣˊ ㄒㄧㄥˊ

/tän⁵⁵ ʈ͡ʂʰu̯ən³⁵ ɕiŋ³⁵/

/tän⁵⁵ ʈ͡ʂʰu̯ən³⁵ ɕiŋ³⁵/


algebraic analysis

The area of study in algebraic geometry began as a research programme, instigated in 1959 by Japanese mathematician Mikio Sato.

1986, Goro Kato (translator), Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of Algebraic Analysis, Princeton Universty Press, https://books.google.com.au/books?id=qYUrDgAAQBAJ&printsec=frontcover&dq=%22algebraic+analysis%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiK5MSlvsLsAhWXbn0KHbPuBPQQ6AEwAnoECAYQAg#v=onepage&q=%22algebraic%20analysis%22&f=false.

In their classical work [123], Kashiwara, Kawai, and Kimura pointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we may add that what we mean by algebraic microlocal analysis is the successful attempt to adapt the methods of abstract algebraic geometry to the non-commutative setting in which the base ring is now the ring of variable coefficients^([sic]) partial differential operators.

1999, Goro Kato, Daniele C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, page v:

2008, Yoshitsugu Takei, The work of T. Kawai on exact WKB analysis, T. Aoki, H. Majima, Y. Takei, N. Tose (editors), Algebraic Analysis of Differential Equations, Springer, page 19,
Thus Kawai has been engaged mainly in the research of exact WKB analysis or algebraic analysis of singular perturbation theory since 1989.

partial differential equation
partial differential equation

(algebraic geometry, algebraic topology) The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions and certain generalisations of functions (such as hyperfunctions and microfunctions).


chain group

kth chain group (canonical)
chain groups (plural)

(algebraic topology) A free abelian group generated by all the k-dimensional oriented simplices of a given simplicial complex.


homology group

en-noun

kth homology group (plural homology groups)

kth homology group (canonical)
homology groups (plural)

(algebraic topology) The quotient group of the kth cycle group modulo the kth boundary group (derived from the chain complex of, e.g., a given simplicial complex).


boundary group

kth boundary group (plural boundary groups)

kth boundary group (canonical)
boundary groups (plural)

(algebraic topology) The image of the (k + 1)ᵗʰ boundary homomorphism (for a given chain complex).


cycle group

kth cycle group (canonical)
cycle groups (plural)

(algebraic topology) The kernel of the kᵗʰ boundary homomorphism (of a given chain complex).


boundary homomorphism

kth boundary homomorphism (plural boundary homomorphisms)

kth boundary homomorphism (canonical)
boundary homomorphisms (plural)

(algebraic topology, in a simplicial chain complex) A group homomorphism on the group of simplicial k-chains induced by mapping each oriented k-simplex to a particular formal sum of its faces.

(algebraic topology, in greater generality) A homomorphism that operates on the kth element in a chain complex.


connecting homomorphism

connecting homomorphism (plural connecting homomorphisms)

(homological algebra) The homomorphism constructed by the snake lemma, leading to a particular exact sequence of kernels and cokernels.

(algebraic topology) Such a homomorphism in the context of exact sequences of chain complexes, leading to a long exact sequence of homology groups.