11.3 Applications to topology and geometry

Hermitian K-theory and L-theory are powerful tools for classifying manifolds and understanding their structure. These theories extend classical quadratic form theory and provide a framework for analyzing surgery obstructions, crucial in high-dimensional topology.

Applications to topology and geometry include the S-cobordism theorem, classification of exotic spheres, and proofs of topological invariance. These theories also play a role in studying singular spaces, intersection homology, and connecting algebraic geometry with topology.

Hermitian K-theory for Manifold Classification

Foundations of Hermitian K-theory and L-theory

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• Hermitian K-theory (KH(R)) studies hermitian forms over a ring R with involution generalizing classical quadratic form theory
• L-theory examines symmetric bilinear forms and quadratic forms on chain complexes providing a framework for surgery obstructions
• Surgery exact sequence connects L-groups to the structure set of a manifold classifying h-cobordism classes of homotopy equivalent manifolds
• Wall's realization theorem links L-groups and manifold classification stating every L-group element realizes as a normal map surgery obstruction

Applications to Manifold Classification

• S-cobordism theorem utilizes Whitehead torsion from K-theory to classify h-cobordisms and diffeomorphisms of manifolds
• Hermitian K-theory and L-theory play crucial roles classifying high-dimensional manifolds (dimensions ≥ 5)
• Novikov conjecture relates homotopy invariance of higher signatures to L-theory and the assembly map in algebraic K-theory
  • Remains a major open problem in topology
  • Has implications for understanding large-scale geometry of manifolds

Examples and Specific Applications

• Classification of exotic spheres uses surgery theory and L-groups (Kervaire-Milnor spheres)
• Topological invariance of rational Pontryagin classes proved using L-theory (Novikov)
• Hermitian K-theory applied to study of quadratic forms over fields (Witt rings)
• L-theory used in proofs of topological rigidity for certain aspherical manifolds (Borel conjecture cases)

Hermitian K-theory of Singular Spaces

Intersection Homology and L-theory

• Intersection homology theory extends Poincaré duality to singular spaces (stratified spaces and pseudomanifolds)
• L-theory applies to intersection homology defining signature invariants for singular spaces generalizing classical manifold signatures
• Cappell-Shaneson supernaturality theorem relates L-groups of stratified spaces to strata L-groups analyzing singular space topology
• Stratified L-theory extends classical L-theory to stratified spaces studying surgery problems on these general objects

Algebraic Approaches to Singular Spaces

• Algebraic cycle theory and associated K-theory studies singular algebraic varieties connecting algebraic geometry and topology
• Hermitian K-theory defines characteristic classes for singular spaces generalizing classical classes (Pontryagin classes)
• Resolution of singularities relates to K-theory and L-theory providing insights into singular space structure and desingularizations

Examples in Singular Space Theory

• Intersection homology applied to study of Schubert varieties in flag manifolds
• Stratified L-theory used to analyze surgeries on pseudomanifolds (Siegel-Sullivan theory)
• K-theory of singular algebraic varieties applied in motivic cohomology (Voevodsky's work)
• Characteristic classes for singular spaces used in studying orbifolds and quotient singularities

Hermitian K-theory Connections

Index Theory and K-theory

• Atiyah-Singer index theorem relates analytical index of elliptic operators to topological invariants deeply connected to K-theory
• Novikov conjecture on higher signatures formulated in L-theory terms relates to Baum-Connes conjecture in operator K-theory
• Farrell-Jones conjecture in algebraic K-theory and L-theory impacts geometric topology and aspherical manifold study

Bridging Algebraic and Geometric Topology

• Algebraic K-theory of spaces bridges algebraic K-theory and homotopy theory applying to diffeomorphism group study of manifolds
• Characteristic class theory (Chern classes, Pontryagin classes) formulated in K-theory terms applies in algebraic and differential topology
• Topological K-theory studying vector bundles over topological spaces connects to index theory and elliptic operator study on manifolds

Applications to Dynamical Systems and Foliations

• Foliation study and characteristic classes involve K-theory and L-theory connecting to dynamical systems and geometric group theory
• K-theory applied to study of C*-algebras associated with dynamical systems (crossed product algebras)
• L-theory used in analyzing leaf spaces of foliations and their singularities

Research in Hermitian K-theory and Topology

Open Problems and Conjectures

• Borel conjecture remains open in many cases relating to Farrell-Jones conjecture in K-theory and L-theory
• Controlled algebra and geometry studies spaces with additional metric structure applying K-theory and L-theory to geometric group theory and coarse geometry
• Assembly map study in K-theory and L-theory connects to Novikov and Baum-Connes conjectures

Emerging Research Directions

• Derived algebraic geometry applications to K-theory and L-theory open new avenues including derived manifold study and invariants
• Equivariant K-theory and L-theory versions yield results in group action study on manifolds and stratified spaces
• Hermitian K-theory, L-theory, and motivic homotopy theory relationships actively researched potentially applying to algebraic geometry and number theory

Higher Categorical Approaches

• Higher categorical structures in K-theory and L-theory (∞-categories, spectral algebraic geometry) grow connecting to homotopy theory and higher algebra
• ∞-categorical methods applied to study of algebraic K-theory spectra (Waldhausen S-construction generalization)
• Spectral algebraic geometry techniques used in formulating new cohomology theories with K-theoretic flavors (topological modular forms)