Euler–Poincaré characteristic formula

E904009

mathematical formula topological invariant relation

The Euler–Poincaré characteristic formula is a fundamental relation in topology and algebraic geometry that expresses a space’s Euler characteristic in terms of alternating sums of dimensions of its cohomology groups.

Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Euler characteristic	1

Statements (48)

Predicate	Object
instanceOf	mathematical formula ⓘ topological invariant relation ⓘ
appliesTo	cochain complexes of finite type ⓘ elliptic complexes ⓘ finite CW-complexes ⓘ finite simplicial complexes ⓘ topological spaces with finite-dimensional cohomology ⓘ
categoryTheoreticFormulation	alternating sum of dimensions of objects in a finite cochain complex equals alternating sum of dimensions of its cohomology ⓘ
equivalentTo	χ(X) = Σ_i (-1)^i b_i(X) ⓘ
expresses	Euler characteristic as alternating sum of Betti numbers ⓘ Euler characteristic as alternating sum of cohomology dimensions ⓘ
field	algebraic geometry ⓘ differential geometry ⓘ homological algebra ⓘ topology ⓘ
generalizes	Euler characteristic formula for polyhedra NERFINISHED ⓘ Euler’s formula V − E + F for convex polyhedra ⓘ
historicalDevelopment	originates from Euler’s work on polyhedra and Poincaré’s development of homology theory ⓘ
namedAfter	Henri Poincaré NERFINISHED ⓘ Leonhard Euler NERFINISHED ⓘ
property	depends only on isomorphism class of cohomology groups ⓘ invariant under homotopy equivalence of spaces ⓘ
relatedResult	Atiyah–Singer index theorem NERFINISHED ⓘ Gauss–Bonnet theorem NERFINISHED ⓘ Lefschetz fixed-point theorem NERFINISHED ⓘ
relatesConcept	Betti numbers NERFINISHED ⓘ Euler characteristic NERFINISHED ⓘ chain complexes ⓘ cohomology ⓘ cohomology groups ⓘ finite CW-complexes ⓘ homology groups ⓘ simplicial complexes ⓘ
requires	finite-dimensional cohomology groups in each degree ⓘ vanishing of cohomology in sufficiently high degrees for convergence of sum ⓘ
statement	For a finite chain complex C, Σ_i (-1)^i dim C_i = Σ_i (-1)^i dim H_i(C) ⓘ For a space X with finite-dimensional cohomology, χ(X) = Σ_i (-1)^i dim H^i(X) ⓘ
usedIn	Hodge theory NERFINISHED ⓘ algebraic geometry ⓘ algebraic topology ⓘ index theory ⓘ representation theory ⓘ sheaf cohomology ⓘ spectral sequence computations ⓘ
usesNotation	Betti numbers b_i NERFINISHED ⓘ H^i(X) ⓘ H_i(X) ⓘ χ(X) ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Morse Theory → keyConcept → Euler–Poincaré characteristic formula ⓘ

subject surface form: Morse theory

this entity surface form: Euler characteristic

Grothendieck–Ogg–Shafarevich formula → typeOf → Euler–Poincaré characteristic formula ⓘ