11.2 Lefschetz fixed-point theorem

The Lefschetz fixed-point theorem connects fixed points of continuous maps to a space's topology. It introduces the Lefschetz number, a topological invariant that provides information about fixed points without explicitly computing them.

This powerful result has applications in dynamical systems and differential equations. It generalizes the Brouwer fixed-point theorem and relates to the Euler characteristic, 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 $f: X \to X$ is a point $x \in X$ such that $f(x) = x$
• The set of fixed points of $f$ is denoted by $\operatorname{Fix}(f)$
• 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 $f: X \to X$
• Defined as the alternating sum of the traces of the induced homomorphisms on the homology groups of $X$: $L(f) = \sum_{i=0}^n (-1)^i \operatorname{tr}(f_*: H_i(X) \to H_i(X))$
• Provides information about the fixed points of $f$ without explicitly computing them
• If $L(f) \neq 0$, then $f$ has at least one fixed point

Traces of induced homomorphisms

• The induced homomorphisms $f_*: H_i(X) \to H_i(X)$ 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 $\chi(X)$ is a topological invariant that measures the "shape" of a space $X$
• Defined as the alternating sum of the ranks of the homology groups: $\chi(X) = \sum_{i=0}^n (-1)^i \operatorname{rank}(H_i(X))$
• For a self-map $f: X \to X$, the Lefschetz number $L(f)$ is related to the Euler characteristic by the formula $L(f) = \sum_{i=0}^n (-1)^i \operatorname{tr}(f_*: H_i(X) \to H_i(X))$
• This connection provides a link between the fixed points of $f$ and the global topology of $X$

Computation using simplicial approximation

• 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

• 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 Nielsen number $N(f)$ 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 holomorphic Lefschetz fixed-point formula 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 Atiyah-Bott fixed-point theorem is a generalization of the Lefschetz fixed-point theorem to elliptic complexes on compact manifolds
• Relates the fixed points of a map to the equivariant cohomology 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 cup product and the Thom class of the diagonal
• This formulation provides a geometric interpretation of the Lefschetz number and its relation to fixed points

Fixed-point indices

• The fixed-point index 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 Lefschetz-Hopf theorem

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 $f: X \to X$, the graph of $f$ is the set $\Gamma_f = \{(x, f(x)) | x \in X\}$ in the product space $X \times X$
• The diagonal $\Delta = \{(x, x) | x \in X\}$ represents the set of fixed points
• The Lefschetz number $L(f)$ can be interpreted as the algebraic intersection number of $\Gamma_f$ and $\Delta$

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 Nielsen-Lefschetz fixed-point theorem combines the ideas of Nielsen fixed-point theory with the Lefschetz number
• The equivariant Lefschetz fixed-point theorem considers maps that are equivariant with respect to a group action
• The Lefschetz-Verdier fixed-point formula 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