Betti numbers

E790522
algebraic topology concept homological invariant topological invariant

Betti numbers are topological invariants that count the number of independent cycles or holes in each dimension of a topological space, reflecting its underlying shape and structure.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (3)

Surface form Occurrences
Betti group 2
Betti number 1
Betti numbers conjecture 1

Statements (52)

Predicate Object
instanceOf algebraic topology concept ⓘ
homological invariant ⓘ
topological invariant ⓘ
appearsIn Poincaré’s work on homology ⓘ
appliesTo CW-complex ⓘ
simplicial complex ⓘ
topological manifold ⓘ
b0Meaning number of path-connected components ⓘ
b1Meaning number of independent 1-dimensional cycles ⓘ
b2Meaning number of independent 2-dimensional voids ⓘ
characterizes number of holes in each dimension ⓘ
number of independent cycles in each dimension ⓘ
definedOver abelian group ⓘ
vector space ⓘ
describes topological space ⓘ
field algebraic topology ⓘ
homological algebra ⓘ
topology ⓘ
generalMeaning rank of kth homology group ⓘ
hasComponent b0 ⓘ
b1 ⓘ
b2 ⓘ
bk ⓘ
introducedIn 19th century ⓘ
invariantUnder homeomorphism ⓘ
homotopy equivalence ⓘ
namedAfter Enrico Betti NERFINISHED ⓘ
property can be computed algorithmically for finite simplicial complexes ⓘ
finite for finite CW-complexes ⓘ
form a sequence indexed by nonnegative integers ⓘ
relatedTo Euler characteristic NERFINISHED ⓘ
Künneth formula NERFINISHED ⓘ
Poincaré polynomial NERFINISHED ⓘ
cellular homology ⓘ
homology group ⓘ
simplicial homology ⓘ
singular homology ⓘ
universal coefficient theorem NERFINISHED ⓘ
usedIn Hodge theory NERFINISHED ⓘ
Morse theory NERFINISHED ⓘ
algebraic geometry ⓘ
classification of topological spaces ⓘ
computational topology ⓘ
differential topology ⓘ
image analysis ⓘ
manifold classification ⓘ
materials science microstructure analysis ⓘ
network topology analysis ⓘ
persistent homology ⓘ
sensor network coverage analysis ⓘ
shape analysis ⓘ
topological data analysis ⓘ

Referenced by (6)

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

Castelnuovo–Mumford regularity → relatedTo → Betti numbers ⓘ
Weil conjectures → hasPart → Betti numbers ⓘ
this entity surface form: Betti numbers conjecture
Enrico Betti → notableWork → Betti numbers ⓘ
Enrico Betti → notableWork → Betti numbers ⓘ
this entity surface form: Betti group
Enrico Betti → notableConcept → Betti numbers ⓘ
this entity surface form: Betti number
Enrico Betti → notableConcept → Betti numbers ⓘ
this entity surface form: Betti group