The connects of continuous maps to a space's topology. It introduces the , a topological invariant that provides information about fixed points without explicitly computing them.
This powerful result has applications in and . It generalizes the and relates to the , offering insights into the global topology of spaces and their self-maps.
Lefschetz fixed-point theorem
Powerful result in algebraic topology that relates the fixed points of a continuous mapping to the topology of the space
Provides a method for determining the existence of fixed points without explicitly finding them
Has important applications in various areas of mathematics, including dynamical systems and differential equations
Fixed points of continuous maps
A fixed point of a continuous function f:X→X is a point x∈X such that f(x)=x
The set of fixed points of f is denoted by Fix(f)
Determining the existence and properties of fixed points is a fundamental problem in topology and analysis
Examples of fixed points include the center of rotation for a rigid body and the equilibrium points of a dynamical system
Lefschetz number
The Lefschetz number L(f) is a topological invariant associated with a continuous map f:X→X
Defined as the alternating sum of the traces of the induced homomorphisms on the of X: L(f)=∑i=0n(−1)itr(f∗:Hi(X)→Hi(X))
Provides information about the fixed points of f without explicitly computing them
If L(f)=0, then f has at least one fixed point
Traces of induced homomorphisms
The induced homomorphisms f∗:Hi(X)→Hi(X) are linear maps between the homology groups of X
The trace of a linear map is the sum of its diagonal entries in any matrix representation
Computing the traces of the induced homomorphisms is a key step in determining the Lefschetz number
The traces capture information about the action of f on the homology of X
Connection to Euler characteristic
The Euler characteristic χ(X) is a topological invariant that measures the "shape" of a space X
Defined as the alternating sum of the ranks of the homology groups: χ(X)=∑i=0n(−1)irank(Hi(X))
For a self-map f:X→X, the Lefschetz number L(f) is related to the Euler characteristic by the formula L(f)=∑i=0n(−1)itr(f∗:Hi(X)→Hi(X))
This connection provides a link between the fixed points of f and the global topology of X
Computation using simplicial approximation
is a technique for approximating continuous maps between simplicial complexes by simplicial maps
Allows for the computation of the induced homomorphisms and Lefschetz number in a combinatorial setting
The simplicial approximation theorem ensures that any continuous map can be approximated by a simplicial map, preserving the essential topological features
Simplicial approximation is particularly useful for computations in algebraic topology, including the Lefschetz fixed-point theorem
Applications in topology
The Lefschetz fixed-point theorem has numerous applications in various branches of topology
In dynamical systems, it can be used to study the existence and stability of fixed points and periodic orbits
In differential equations, it provides a tool for analyzing the solutions and bifurcations of nonlinear systems
The theorem also has applications in algebraic geometry, complex analysis, and other areas of mathematics
Brouwer fixed-point theorem vs Lefschetz
The Brouwer fixed-point theorem is a special case of the Lefschetz fixed-point theorem for continuous self-maps of closed, bounded, and convex subsets of Euclidean space
Brouwer's theorem states that any continuous function from a closed, bounded, and convex set to itself has at least one fixed point
The Lefschetz fixed-point theorem generalizes Brouwer's theorem to a broader class of spaces and mappings
While Brouwer's theorem guarantees the existence of a fixed point, the Lefschetz theorem provides additional information about the number and nature of fixed points
Nielsen fixed-point theory
is an extension of the Lefschetz fixed-point theorem that provides a more refined analysis of fixed points
Introduces the concept of Nielsen fixed point classes, which are equivalence classes of fixed points related by homotopies
The N(f) is a lower bound for the number of fixed points of f and is a homotopy invariant
Nielsen theory provides a stronger criterion for the existence of fixed points compared to the Lefschetz number
Holomorphic Lefschetz fixed-point formula
The is a version of the Lefschetz fixed-point theorem for holomorphic maps on complex manifolds
Relates the fixed points of a holomorphic map to the cohomology of the manifold and the local behavior of the map near the fixed points
The formula involves the holomorphic Euler characteristic and the local holomorphic indices of the fixed points
Has important applications in complex geometry and the study of complex dynamical systems
Atiyah-Bott fixed-point theorem
The is a generalization of the Lefschetz fixed-point theorem to elliptic complexes on compact manifolds
Relates the fixed points of a map to the of the manifold and the local behavior of the map near the fixed points
Involves the equivariant Euler class and the equivariant Todd class of the normal bundle to the fixed point set
Has applications in mathematical physics, particularly in the study of gauge theories and string theory
Lefschetz fixed-point theorem for manifolds
The Lefschetz fixed-point theorem can be formulated specifically for continuous self-maps of compact manifolds
In this setting, the theorem relates the fixed points of the map to the intersection of the graph of the map with the diagonal in the product manifold
The intersection number can be computed using the and the of the diagonal
This formulation provides a geometric interpretation of the Lefschetz number and its relation to fixed points
Fixed-point indices
The is a local topological invariant associated with an isolated fixed point of a continuous self-map
Measures the local behavior of the map near the fixed point and provides information about the multiplicity and stability of the fixed point
The sum of the fixed-point indices over all fixed points is equal to the Lefschetz number
Fixed-point indices can be computed using local degree theory or the
Geometric interpretation of theorem
The Lefschetz fixed-point theorem has a geometric interpretation in terms of the intersection of the graph of a map with the diagonal
For a continuous self-map f:X→X, the graph of f is the set Γf={(x,f(x))∣x∈X} in the product space X×X
The diagonal Δ={(x,x)∣x∈X} represents the set of fixed points
The Lefschetz number L(f) can be interpreted as the algebraic intersection number of Γf and Δ
Generalizations of Lefschetz theorem
The Lefschetz fixed-point theorem has been generalized in various directions to encompass a wider range of spaces and mappings
The combines the ideas of Nielsen fixed-point theory with the Lefschetz number
The considers maps that are equivariant with respect to a group action
The extends the theorem to constructible sheaves and their endomorphisms
These generalizations provide more powerful tools for studying fixed points in different mathematical contexts
Converse of Lefschetz fixed-point theorem
The converse of the Lefschetz fixed-point theorem asks whether the existence of a fixed point implies a non-zero Lefschetz number
In general, the converse does not hold, as there exist maps with fixed points but a zero Lefschetz number
However, under certain conditions, such as the map being homotopic to the identity or the space having a specific homological structure, the converse may be true
The converse of the Lefschetz fixed-point theorem is an active area of research in algebraic topology, with connections to other fixed-point theorems and topological invariants