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 Eilenberg-Ganea conjecture (Wikipedia)
Source Eilenberg Zilber theorem (Wikipedia)
Source Hurewicz theorem, Wikipedia
Source Mayer-Vietoris Theorem (Wikipedia)
Source Homology sphere (Wikipedia)
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.
View the Source Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead 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 section
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
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)
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