11.1 Hermitian K-theory and its properties

Hermitian K-theory adds symmetric or skew-symmetric bilinear forms to classical K-theory. It's defined for rings with involution and generalizes Witt groups. KH₀(R) is the Grothendieck group of non-degenerate hermitian forms, while higher groups use advanced constructions.

This theory exhibits 4-fold periodicity and connects to classical K-theory through a forgetful map. It satisfies key properties like localization and excision. Computations vary based on ring structure, with fields and basic rings offering simpler examples.

Hermitian K-theory Definition

Fundamental Concepts

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• Hermitian K-theory incorporates additional structure related to symmetric or skew-symmetric bilinear forms
• Defined for rings with involution equipped with an anti-automorphism that squares to the identity
• Generalizes the notion of Witt groups classifying quadratic forms over fields
• KH₀(R) defined as the Grothendieck group of non-degenerate hermitian forms over a ring R with involution
• Higher Hermitian K-groups KHₙ(R) use advanced constructions (Quillen Q-construction, Waldhausen's S•-construction) applied to categories of hermitian forms

Relationship to Classical K-theory

• Natural forgetful map exists from Hermitian K-theory to classical K-theory
• Forgetful map disregards the hermitian structure and only considers the underlying module
• Provides insights into the additional structure captured by Hermitian K-theory compared to classical K-theory
• Allows for comparison and interplay between the two theories (classical and Hermitian K-theory)

Hermitian K-theory Properties

Periodicity and Bott Theorem

• Exhibits 4-fold periodicity known as the Hermitian Bott periodicity theorem
• States that KHₙ(R) ≅ KHₙ₊₄(R) for all n ≥ 0
• Proof involves constructing explicit isomorphisms between relevant K-groups using homotopy theory and algebraic topology techniques
• Closely related to 8-fold periodicity of real topological K-theory
• Difference arises from additional structure of the involution in Hermitian K-theory

Fundamental Theorems and Sequences

• Karoubi-Villamayor sequence relates Hermitian K-theory to classical K-theory and Witt groups
• Fundamental theorem of Hermitian K-theory connects K-theory of ring R to its polynomial ring R[t]
• Analogous to fundamental theorem of algebraic K-theory
• Satisfies localization and excision properties crucial for computations and applications in topology and geometry
• These properties enable systematic study and calculation of Hermitian K-groups in various contexts

Hermitian K-theory for Examples

Fields and Basic Rings

• For field F with trivial involution, KH₀(F) isomorphic to Witt ring W(F) of the field
• Higher Hermitian K-groups of fields computed using Hermitian Bott periodicity theorem and relationship to Witt groups
• Rings of integers in number fields require techniques from algebraic number theory and class field theory
• Hermitian K-theory of finite fields explicitly computed and related to classical K-theory through forgetful map
• Quaternion algebras with standard involution exhibit special properties in their Hermitian K-groups

Advanced Computations

• General rings with involution often require spectral sequence techniques
• Utilization of localization and devissage methods for more complex ring structures
• Computation strategies may vary depending on the specific properties of the ring and its involution
• Examples include complex algebraic varieties, group rings, and topological spaces

Hermitian K-theory of Group Rings

Connections to Surgery Theory

• Closely related to study of surgery theory and classification of manifolds
• L-theory of group ring R[G] crucial in formulation of surgery exact sequence and Novikov conjecture
• Computation involves techniques from homological algebra (spectral sequences, equivariant homology theories)
• Farrell-Jones conjecture formulated in terms of Hermitian K-theory of group rings
• Major open problem in geometric topology with implications for manifold classification

Applications and Insights

• Study of intersection forms on manifolds utilizes Hermitian K-theory of group rings
• Classification of high-dimensional knots benefits from group ring Hermitian K-theory analysis
• Relationship between Hermitian and algebraic K-theory of group rings provides structural insights
• Applications extend to geometric topology, manifold theory, and algebraic topology
• Hermitian K-theory of group rings bridges abstract algebra and geometric topology