Euler–Poincaré characteristic formula
E904009
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Euler characteristic | 1 |
| Euler–Poincaré characteristic formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11086030 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Euler–Poincaré characteristic formula Context triple: [Grothendieck–Ogg–Shafarevich formula, typeOf, Euler–Poincaré characteristic formula]
-
A.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
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B.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
C.
Noether’s formula
Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
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D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
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E.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Euler–Poincaré characteristic formula Target entity description: 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.
-
A.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
-
B.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
C.
Noether’s formula
Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
- F. None of above. chosen
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) ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Euler–Poincaré characteristic formula Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Morse theory
this entity surface form:
Euler characteristic