List of algebraic topology topics - Definition and Overview

This is a list of algebraic topology topics, by Wikipedia page. See also: list of general topology topics.

Homology (mathematics)

Main article:Homology theory

• Simplex
• Simplicial complex
  • Polytope
  • Triangulation
  • Barycentric subdivision
  • Simplicial approximation theorem
  • Abstract simplicial complex
  • Simplicial set
  • Simplicial category
• Chain (mathematics)
• Betti number
• Euler characteristic
  • Genus
  • Riemann-Hurwitz formula
• Singular homology
• Mayer-Vietoris sequence
• Universal coefficient theorem
• Cohomology
  • Cocycle class
  • Cup product
  • Cohomology ring
  • De Rham cohomology
  • Cech cohomology
  • Alexander-Spanier cohomology
  • Intersection cohomology
  • Lyusternik-Schnirelmann category
• Poincaré duality
• Fundamental class
• Applications
  • Jordan curve theorem
  • Brouwer fixed point theorem
  • Invariance of domain
  • Lefschetz fixed-point theorem
  • Hairy ball theorem
  • Degree (mathematics)
  • Borsuk-Ulam Theorem
  • Ham sandwich theorem
  • Homology sphere

Homotopy theory

• Homotopy
• Path (topology)
• Fundamental group
• Homotopy group
• Seifert-van Kampen theorem
• Pointed space
• Winding number
• Simply connected
  • Universal cover
• Monodromy
• Homotopy lifting property
• Wedge sum
• Smash product
• Cohomotopy
• Cohomotopy group
• Brown's representability theorem
• Eilenberg-MacLane space
• Fibre bundle
  • Möbius strip
  • Line bundle
    • Canonical line bundle
  • Vector bundle
  • Associated bundle
  • Fibration
  • Hopf bundle
  • Classifying space
• Homotopy groups of spheres
• Plus construction
• Whitehead theorem
• Weak equivalence
• Hurewicz theorem
• H-space

Further developments

• Künneth theorem
• De Rham cohomology
• Characteristic class
  • Chern class
  • Chern-Simons form
  • Pontryagin class
  • Pontryagin number
  • Stiefel-Whitney class
• Poincaré conjecture
• Cohomology operation
• Bott periodicity theorem
• K-theory
  • Adams operation
  • Algebraic K-theory
  • Whitehead torsion
• Cobordism
• Thom space
• Suspension functor
• Stable homotopy theory
• Spectrum (homotopy theory)
• Morava K-theory
• Hodge conjecture
• Weil conjectures

Homological algebra

• Chain complex
• Commutative diagram
• Exact sequence
  • Five lemma
  • Short five lemma
  • Snake lemma
  • Splitting lemma
  • Extension problem
• Spectral sequence
• Abelian category
• Group cohomology
• Sheaf
  • Sheaf cohomology
  • Grothendieck topology
• Derived category

History

• Combinatorial topology