List of algebraic topology topics

This is a list of algebraic topology topics.

Homology (mathematics)

• Simplex
• Simplicial complex
• Polytope
• Triangulation
• Barycentric subdivision
• Simplicial approximation theorem
• Abstract simplicial complex
• Simplicial set
• Simplicial category
• Chain (algebraic topology)
• Betti number
• Euler characteristic
• Genus
• Riemann–Hurwitz formula
• Singular homology
• Cellular homology
• Relative homology
• Mayer–Vietoris sequence
• Excision theorem
• Universal coefficient theorem
• Cohomology
• List of cohomology theories
• Cocycle class
• Cup product
• Cohomology ring
• De Rham cohomology
• Čech cohomology
• Alexander–Spanier cohomology
• Intersection cohomology
• Lusternik–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 of a continuous mapping
• 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
• Mapping cylinder
• Mapping cone (topology)
• Wedge sum
• Smash product
• Adjunction space
• 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
• Cofibration
• Homotopy groups of spheres
• Plus construction
• Whitehead theorem
• Weak equivalence
• Hurewicz theorem
• H-space

Further developments

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

Applied topology

Example: DE-9IM

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

See also

• Glossary of algebraic topology
• topology glossary
• List of topology topics
• List of general topology topics
• List of geometric topology topics
• Publications in topology
• Topological property