Lefschetz number

E904006

fixed point invariant homotopy invariant topological invariant

The Lefschetz number is a topological invariant, computed from the traces of induced maps on homology, that predicts the existence and number of fixed points of a continuous self-map on a topological space.

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Observed surface forms (1)

Surface form	Occurrences
Lefschetz number L(f)	1

Statements (47)

Predicate	Object
instanceOf	fixed point invariant ⓘ homotopy invariant ⓘ topological invariant ⓘ
appliesTo	continuous self-map ⓘ topological space ⓘ
assumes	suitable compactness or finiteness conditions on X ⓘ
canBeDefinedUsing	cohomology groups H^k(X) ⓘ trace of f^* on cohomology ⓘ
definedFor	map f : X → X ⓘ
definitionFormula	L(f) = Σ_k (-1)^k tr(f_* \| H_k(X)) ⓘ
dependsOn	induced maps on homology ⓘ traces of linear maps ⓘ
equals	Euler characteristic χ(X) when f is identity map on X ⓘ sum of fixed point indices for isolated fixed points ⓘ
field	algebraic topology ⓘ fixed point theory ⓘ
generalizes	Euler characteristic of a space ⓘ
hasVariant	Lefschetz number in cohomology ⓘ Lefschetz number with local coefficients ⓘ equivariant Lefschetz number ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	existence of fixed point if nonzero under Lefschetz fixed point theorem ⓘ
interpretsAs	algebraic count of fixed points under suitable hypotheses ⓘ
namedAfter	Solomon Lefschetz NERFINISHED ⓘ
notation	L(f) ⓘ
property	additive with respect to decomposition of space under suitable conditions ⓘ depends only on homotopy class of f ⓘ independent of choice of basis on homology ⓘ invariant under homotopy of maps ⓘ multiplicative under product of maps on product spaces ⓘ
relatedTo	Lefschetz fixed point theorem NERFINISHED ⓘ Lefschetz zeta function ⓘ Lefschetz–Hopf theorem NERFINISHED ⓘ Nielsen fixed point theory NERFINISHED ⓘ Poincaré–Hopf index theorem NERFINISHED ⓘ
requires	finite-dimensional homology groups for standard definition ⓘ
specialCaseOf	Reidemeister trace in some contexts ⓘ
sumsOver	all homological degrees k ⓘ
usedIn	Morse theory NERFINISHED ⓘ algebraic geometry ⓘ differential topology ⓘ dynamical systems ⓘ equivariant topology ⓘ
usedToStudy	fixed points of iterates of a map ⓘ periodic points in dynamical systems ⓘ
uses	homology with coefficients in a field or ring ⓘ singular homology ⓘ

Referenced by (3)

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

Lefschetz fixed-point theorem → defines → Lefschetz number ⓘ

Lefschetz fixed-point theorem → involves → Lefschetz number ⓘ

this entity surface form: Lefschetz number L(f)

Lefschetz → notableFor → Lefschetz number ⓘ

subject surface form: Solomon Lefschetz