10.3 K-Theory and fixed point theorems

K-Theory provides powerful tools for studying fixed points of continuous maps on manifolds. The Lefschetz fixed point theorem, formulated using K-theory groups, connects fixed points to traces of induced maps, offering insights into the topology and geometry of manifolds.

K-theoretic methods, including the Atiyah-Bott fixed point theorem, extend these ideas to more general settings. These techniques have applications in dynamical systems, algebraic topology, and gauge theory, showcasing K-Theory's importance in geometry and topology.

Lefschetz Fixed Point Theorem via K-Theory

Formulation and Key Concepts

• The Lefschetz fixed point theorem connects the fixed points of a continuous map on a compact manifold to the traces of induced maps on homology groups
• In K-Theory, the theorem is formulated using the Lefschetz number, which is defined as the alternating sum of traces of the induced maps on the K-theory groups
  • K-theory groups are abelian groups constructed from vector bundles over the manifold (tangent bundle, normal bundle)
  • Induced maps on K-theory groups are obtained by pulling back vector bundles along the given continuous map (pullback bundle, pushforward bundle)
• The theorem states that if the Lefschetz number is nonzero, then the continuous map must have at least one fixed point (Brouwer fixed point theorem, Hairy ball theorem)

Proof and Techniques

• The proof relies on the Atiyah-Singer index theorem, which relates the Fredholm index of an elliptic operator to topological invariants of the manifold
  • The Fredholm index is the difference between the dimensions of the kernel and cokernel of the operator (Fredholm operator, compact operator)
  • The relevant elliptic operator is constructed using the data of the continuous map and the Dolbeault operator on the manifold (Dirac operator, Laplace operator)
• The trace formula for the Lefschetz number is derived using the heat kernel approach and the McKean-Singer formula (heat equation, spectral theory)
• Other techniques used in the proof include Hodge theory, Chern-Weil theory, and the Chern character (De Rham cohomology, characteristic classes)

K-Theory and Fixed Point Theorems

Atiyah-Bott Fixed Point Theorem

• The Atiyah-Bott fixed point theorem generalizes the Lefschetz fixed point theorem to elliptic complexes
• In the context of K-theory, the theorem relates the fixed points of a map to the equivariant K-theory of the manifold
  • Equivariant K-theory is a version of K-theory that takes into account the action of a group on the manifold (group action, orbit space)
  • Equivariant K-theory groups are constructed using G-equivariant vector bundles, where G is the group acting on the manifold (representation theory, group cohomology)
• The theorem expresses the Lefschetz number as an element in the equivariant K-theory of the fixed point set (localization theorem, Weyl group)
• The proof involves the localization theorem in equivariant K-theory, which relates the equivariant K-theory of the manifold to that of the fixed point set (Atiyah-Segal completion theorem, equivariant cohomology)

Applications and Connections

• The Atiyah-Bott fixed point theorem has applications in the study of group actions on manifolds and in mathematical physics (symplectic geometry, gauge theory)
• It is connected to other areas of mathematics, such as representation theory, algebraic geometry, and index theory (character theory, Grothendieck group, Riemann-Roch theorem)
• The theorem has been generalized and extended in various directions, including to infinite-dimensional manifolds and to more general cohomology theories (loop spaces, K-homology)

K-Theoretic Methods for Fixed Points

Detecting and Studying Fixed Points

• K-theory provides a powerful tool for studying fixed points of continuous maps on manifolds
• The Lefschetz fixed point theorem and its generalizations can be used to detect the existence of fixed points (Nielsen fixed point theory, Reidemeister trace)
• The K-theory groups and their induced maps contain information about the fixed point set and its properties
  • The ranks of the K-theory groups provide lower bounds for the number of fixed points (Euler characteristic, Betti numbers)
  • Induced maps on K-theory groups can be used to study the structure of the fixed point set, such as its connectedness and orientability (homology groups, cohomology rings)

Techniques and Applications

• K-theoretic methods can be combined with other topological techniques, such as obstruction theory and characteristic classes, to obtain more refined information about fixed points (vector field, Euler class)
• Applications include the study of dynamical systems, algebraic topology, and gauge theory (Morse theory, Floer homology)
• K-theory has also been used to study fixed points in more general settings, such as for maps between different manifolds or for correspondences (coincidence theory, Lefschetz-Hopf theorem)
• The methods have been extended to study equivariant fixed points, periodic points, and higher-order fixed points (Reidemeister torsion, Lefschetz zeta function)

K-Theory vs Hopf Trace Formula

K-Theoretic Hopf Trace Formula

• The Hopf trace formula relates the Lefschetz number of a map to a trace in the cohomology of the manifold
• In the context of K-theory, the formula can be generalized to express the Lefschetz number in terms of traces in the K-theory groups
• The K-theoretic version of the Hopf trace formula involves the Chern character, a homomorphism from K-theory to cohomology
  • The Chern character maps the K-theory groups to the even-dimensional cohomology groups (Chern classes, Todd class)
  • The formula expresses the Lefschetz number as the trace of the induced map on cohomology, obtained by composing the Chern character with the induced map on K-theory (Chern-Simons form, eta invariant)

Proof and Applications

• The proof of the K-theoretic Hopf trace formula relies on the properties of the Chern character and the Atiyah-Singer index theorem (Grothendieck-Riemann-Roch theorem, local index theorem)
• The formula provides a link between the fixed point theory and the cohomological properties of the manifold (Poincaré duality, intersection theory)
• Applications include the study of Reidemeister torsion, zeta functions, and the Riemann-Roch theorem (analytic torsion, Selberg trace formula)
• The K-theoretic Hopf trace formula has been generalized to other settings, such as equivariant K-theory and algebraic K-theory (Bismut-Lott theorem, Lefschetz-Riemann-Roch theorem)