Branches of Mathematics
Algebraic Topology
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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.
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 Cross-Tradition Connections
Sources
1. Wikipedia: Algebraic Topology
Wikimedia FoundationOpening paragraphQuote, Opening paragraph
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
1. Wikipedia: Algebraic Topology
Wikimedia FoundationMethod of algebraic invariants sectionQuote, 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.
1. Wikipedia: Algebraic Topology
Wikimedia FoundationSetting in category theory sectionQuote, 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.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead sectionQuote, 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.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaApplications sectionQuote, Applications section
The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.
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
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