List of algebraic topology topics

This is a list of .

=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 (mathematics) **Riemann-Hurwitz formula *Singular homology *Cellular homology *Relative homology *Mayer-Vietoris sequence *Excision theorem *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 *Mapping cylinder *Mapping cone *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 *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 (mathematics) **Sheaf cohomology **Grothendieck topology *Derived category

=History=

*Combinatorial topology