DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Euler characteristic of a finite CW complex

Definition

If $X$ has finitely many cells and $c_{n} (X)$ is its number of $n$ -cells, define $χ (X) = \sum_{n} (- 1)^{n} c_{n} (X)$ . The sum is finite by hypothesis.

Used by

Euler characteristic of an orientable compact surface Corollary
Morse Euler characteristic identity Corollary
Morse homology recovers the Morse inequalities Corollary
Orientability and Euler characteristic determine a nonempty compact connected boundaryless surface Corollary
The Euler characteristic does not determine the signature Counterexample
Euler characteristic of a compact manifold Definition
Genus and Euler characteristic of a compact Riemann surface Definition
A torus translation has zero Lefschetz number and no fixed points Example
Genus-two orientable polygon Example
Klein bottle as two crosscaps Example
Projective plane crosscap polygon Example
Real projective space shows coefficient-dependent perfectness Example
Sphere as a polygonal quotient Example
The Lefschetz number of the identity is Euler characteristic Example
The right-angled cube Davis complex and its boundary 2-sphere Example
Torus commutator polygon Example
Two CW structures on the circle have the same Euler characteristic Example
Gauss-Bonnet alone does not classify surfaces False statement
Cellular homology of the one-polygon surface model Lemma
Finite tangent index count and inward boundary sum Lemma
Planar facial graph facts and isomorphism extension for arbitrary arc drawings Lemma
Euler characteristic is additive for finite CW pairs Proposition
Euler characteristic of a finite CW product Proposition
Finiteness and additivity of the Euler characteristic Proposition
Classification of compact connected surfaces Theorem
Euler–Poincare formula for finite CW complexes Theorem
Polygonal normal forms for compact connected surfaces Theorem
Riemann–Hurwitz formula for compact Riemann surfaces Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Allen Hatcher, Algebraic Topology, Section 2.2 (standard reference, not scraped)