Algebraic topology is the branch of mathematics that studies topological spaces by attaching algebraic invariants, such as groups, to them, so that continuous shape can be probed by computation. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhich algebraic invariants classify topological spaces up to homeomorphism, or at least up to the coarser equivalence of homotopy, and how much of a space's shape survives translation into algebra. 1 Key DebateWhether purely algebraic reasoning can settle questions that geometric construction alone cannot. L. E. J. Brouwer's fixed-point theorem, proving that every continuous map of a disk to itself fixes some point, was an early demonstration that tracking how a map acts on a space's algebraic invariants could decide a question no direct construction had settled. 1 Classification
Pure or Applied Connections
Associated With
Includes
Additional Source Brouwer Fixed-Point Theorem (Wikipedia)First proofs subsection
Source Brown's representability theorem (Wikipedia)
Source Wikipedia: Edgar H. Brown
Source Eilenberg-Ganea conjecture (Wikipedia)
Source Eilenberg Zilber theorem (Wikipedia)
Source Wolfram MathWorld: Freudenthal Suspension
Theorem
Source Wikipedia: Hans Freudenthal
Source Hurewicz theorem, Wikipedia
Source Wikipedia: J. H. C. Whitehead
Source MacTutor biography of Jean-Pierre Serre, St Andrews
Source Leopold Vietoris, Biography (MacTutor History of Mathematics)
Source Mayer-Vietoris Theorem (Wikipedia)
Source Homology sphere (Wikipedia)
Source Riemann-Hurwitz formula, Wikipedia
Source Wikipedia: Solomon Lefschetz
Source Whitehead Conjecture (Wikipedia)
Sources
1. Wikipedia: Algebraic Topology
Wikimedia FoundationOpening paragraph
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
Method of algebraic invariants section
In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy) of spaces.
Setting in category theory section
They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.
Lead section
The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Applications
The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.
View the Source Brouwer Fixed-Point Theorem (Wikipedia)
Wikimedia FoundationIncludes: Brouwer Fixed-Point Theorem, First proofs subsectionQuote, Includes: Brouwer Fixed-Point Theorem, First proofs subsection
one of the early achievements of algebraic topology
Brown's representability theorem (Wikipedia)
Includes: Brown Representability Theorem, LeadQuote, Includes: Brown Representability Theorem, Lead
Brown's representability theorem in homotopy theory gives necessary and sufficient conditions
Homology sphere (Wikipedia)
Includes: Poincare Homology Sphere, Lead sentenceQuote, Includes: Poincare Homology Sphere, Lead sentence
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 .
Hurewicz theorem, Wikipedia
Includes: Hurewicz Theorem, Lead sentenceQuote, Includes: Hurewicz Theorem, Lead sentence
matics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map kn
Eilenberg-Ganea conjecture (Wikipedia)
Whitehead Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Whitehead Conjecture, Lead sentenceQuote, Includes: Whitehead Conjecture, Lead sentence
e Whitehead asphericity conjecture) is a claim in algebraic topology.
Mayer-Vietoris Theorem (Wikipedia)
Wikimedia FoundationIncludes: Mayer-Vietoris Theorem, Lead sentenceView the Source Eilenberg Zilber theorem (Wikipedia)
Wikipedia: J. H. C. Whitehead
Includes: J. H. C. Whitehead, LeadQuote, Includes: J. H. C. Whitehead, Lead
John Henry Constantine Whitehead (11 November 1904-8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory.
Wikipedia: Hans Freudenthal
Includes: Hans Freudenthal, LeadQuote, Includes: Hans Freudenthal, Lead
He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education.
Wikipedia: Edgar H. Brown
Includes: Edgar H. Brown, LeadQuote, Includes: Edgar H. Brown, Lead
Edgar Henry Brown, Jr. (December 27, 1926, December 22, 2021) was an American mathematician specializing in algebraic topology, and for many years a professor at Brandeis University. == Life == Brown was born in Oak Park, Illinois.
Wikipedia: Solomon Lefschetz
Includes: Solomon Lefschetz, LeadQuote, Includes: Solomon Lefschetz, Lead
Соломо́н Ле́фшец; 3 September 1884-5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. == Life == He was born in Moscow, the son of Alexander
Riemann-Hurwitz formula, Wikipedia
Includes: Riemann-Hurwitz Formula, Lead paragraphQuote, Includes: Riemann-Hurwitz Formula, Lead paragraph
It therefore connects ramification with algebraic topology, in this case.
Wolfram MathWorld: Freudenthal Suspension
Theorem
Wolfram MathWorldIncludes: Freudenthal Suspension Theorem, Subject classificationsQuote, Includes: Freudenthal Suspension Theorem, Subject classifications
Topology Algebraic Topology
MacTutor biography of Jean-Pierre Serre, St Andrews
Includes: Jean-Pierre Serre, SummaryQuote, Includes: Jean-Pierre Serre, Summary
Jean-Pierre Serre is a French mathematician who has made important contributions to algebraic topology, algebraic geometry, and algebraic number theory.
Leopold Vietoris, Biography (MacTutor History of Mathematics)
Includes: Leopold Vietoris, SummaryQuote, Includes: Leopold Vietoris, Summary
Leopold Vietoris was a long-lived Austrian mathematician best known for his contributions to topology, especially the Mayer-Vietoris sequence. He was also interested in mathematical history.
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