🔢Algebraic K-Theory Unit 11 – Hermitian K–Theory and L–Theory

Key Concepts and Definitions

• Hermitian K-Theory extends algebraic K-theory to rings with involution, incorporating the structure of hermitian forms
• L-Theory studies quadratic forms and symmetric bilinear forms over rings with involution
• Involution is an anti-automorphism of a ring $R$ of order 2, denoted by $x \mapsto \bar{x}$, satisfying $\overline{x+y} = \bar{x} + \bar{y}$ and $\overline{xy} = \bar{y}\bar{x}$
  • Examples include complex conjugation on $\mathbb{C}$ and the transpose map on matrix rings
• Hermitian form over a ring with involution $(R, -)$ is a map $h: M \times M \to R$ satisfying:
  • $h(x+y, z) = h(x,z) + h(y,z)$
  • $h(ax, y) = ah(x,y)$
  • $h(x,y) = \overline{h(y,x)}$
• Quadratic form is a map $q: M \to R/\{r - \bar{r} \mid r \in R\}$ such that $q(ax) = a\bar{a}q(x)$ and the map $(x,y) \mapsto q(x+y) - q(x) - q(y)$ is bilinear
• Witt group $W(R)$ consists of equivalence classes of non-degenerate hermitian forms over $(R, -)$, with addition given by orthogonal sum
• $L$-groups $L^n(R)$ are defined using quadratic forms and symmetric bilinear forms over $(R, -)$, and are 4-periodic: $L^n(R) \cong L^{n+4}(R)$

Historical Context and Development

• Hermitian K-Theory and L-Theory emerged as important tools in surgery theory and the classification of manifolds in the 1960s and 1970s
• Milnor, Novikov, and Wall pioneered the use of quadratic forms and hermitian forms in the study of manifolds
• Mishchenko and Ranicki developed the foundations of symmetric L-theory in the 1970s
• Karoubi and Villamayor introduced Hermitian K-theory in the early 1970s as a generalization of algebraic K-theory
• Hermitian K-theory and L-theory have since found applications in diverse areas of mathematics, including:
  • Surgery theory and geometric topology
  • Algebraic topology and homotopy theory
  • Algebraic geometry and number theory
• Recent developments include the study of equivariant and motivic versions of Hermitian K-theory and L-theory, as well as connections to other areas such as C*-algebras and noncommutative geometry

Hermitian Forms and Quadratic Forms

• Hermitian forms generalize symmetric bilinear forms to rings with involution
  • Over $\mathbb{R}$ with trivial involution, hermitian forms are symmetric bilinear forms
  • Over $\mathbb{C}$ with complex conjugation, hermitian forms are classical hermitian forms on complex vector spaces
• Quadratic forms can be studied over any commutative ring, but the theory is particularly rich over rings with involution
• Metabolic forms play a key role in Hermitian K-theory and L-theory
  • A hermitian form $(M, h)$ is metabolic if there exists a submodule $N \subseteq M$ such that $N = N^\perp$ with respect to $h$
  • Metabolic forms are trivial in the Witt group $W(R)$
• Hyperbolic forms are another important class of hermitian forms
  • The hyperbolic form $H(M)$ on $M \oplus M^*$ is defined by $((x,f), (y,g)) \mapsto f(y) + \overline{g(x)}$
  • Hyperbolic forms are metabolic and play a role in defining the Grothendieck-Witt group $GW(R)$
• Witt cancellation theorem states that if $(M, h) \perp H(N) \cong (M', h') \perp H(N')$, then $(M, h) \cong (M', h')$

Hermitian K-Theory: Foundations

• Hermitian K-theory extends algebraic K-theory to rings with involution, taking into account the structure of hermitian forms
• Grothendieck-Witt group $GW(R)$ is the zeroth Hermitian K-group, analogous to the zeroth algebraic K-group $K_0(R)$
  • $GW(R)$ consists of equivalence classes of hermitian forms over $(R, -)$, with addition given by orthogonal sum and the hyperbolic form $H(M)$ considered trivial
• Higher Hermitian K-groups $GW_n(R)$ are defined using the Grothendieck-Witt space $GW(R)$, which is an infinite loop space
  • $GW_n(R) = \pi_n(GW(R))$, the homotopy groups of the Grothendieck-Witt space
• Karoubi's fundamental theorem relates Hermitian K-theory to algebraic K-theory via an exact sequence:
  • $K_n(R) \to GW_n(R) \to W_n(R) \to 0$
  • Here, $W_n(R)$ is the higher Witt group, defined as the cokernel of the hyperbolic map $K_n(R) \to GW_n(R)$
• Hermitian K-theory satisfies various functorial properties and has a product structure compatible with the product in algebraic K-theory

L-Theory: Basics and Applications

• L-theory studies quadratic forms and symmetric bilinear forms over rings with involution
• $L$-groups $L^n(R)$ are defined using quadratic forms and symmetric bilinear forms over $(R, -)$, and are 4-periodic: $L^n(R) \cong L^{n+4}(R)$
  • $L^0(R)$ is the Witt group of non-degenerate quadratic forms over $(R, -)$
  • $L^1(R)$ is the Witt group of non-degenerate symmetric bilinear forms over $(R, -)$
  • $L^2(R)$ and $L^3(R)$ are defined using formations and linking forms, respectively
• L-theory plays a central role in surgery theory and the classification of manifolds
  • Surgery obstruction for a normal map $(f, b): M \to X$ lives in the L-group $L^n(\mathbb{Z}[\pi_1(X)])$
  • The obstruction vanishes if and only if $f$ is normally cobordant to a homotopy equivalence
• Assembly maps in L-theory, such as the symmetric signature over $\mathbb{Z}$, have been extensively studied
  • The Novikov conjecture on the homotopy invariance of higher signatures is closely related to the assembly map in L-theory
• Visible L-theory, introduced by Weiss, is a variant that incorporates the visibility of quadratic forms and has applications in the study of stratified spaces

Connections to Algebraic K-Theory

• Hermitian K-theory and L-theory are closely related to algebraic K-theory, with various connecting maps and exact sequences
• Karoubi's fundamental theorem relates Hermitian K-theory to algebraic K-theory via an exact sequence:
  • $K_n(R) \to GW_n(R) \to W_n(R) \to 0$
• The Grothendieck-Witt group $GW(R)$ can be viewed as a hermitian analogue of the zeroth algebraic K-group $K_0(R)$
• Ranicki's algebraic surgery exact sequence connects L-theory to algebraic K-theory:
  • $\cdots \to L^n(R) \to \mathcal{S}_n(R) \to H_n(R) \to L^{n-1}(R) \to \cdots$
  • Here, $\mathcal{S}_n(R)$ is the $n$th symmetric L-group, and $H_n(R)$ is the $n$th hyperquadratic L-group
• The assembly maps in Hermitian K-theory and L-theory are analogous to the assembly maps in algebraic K-theory, and their study is central to the Novikov and Borel conjectures
• Waldhausen's A-theory, a variant of algebraic K-theory for spaces with a notion of cofibrations and weak equivalences, has a hermitian analogue called Hermitian A-theory

Computational Techniques and Examples

• Computing Hermitian K-groups and L-groups can be challenging, but various techniques have been developed
• For finite fields $\mathbb{F}_q$, the Grothendieck-Witt group $GW(\mathbb{F}_q)$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, generated by the trivial form and the norm form
• For the ring of integers $\mathbb{Z}$ with trivial involution, the Grothendieck-Witt group $GW(\mathbb{Z})$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, generated by the trivial form and the form $\langle 1, -1 \rangle$
  • The higher Hermitian K-groups $GW_n(\mathbb{Z})$ are related to the classical Witt groups of quadratic forms over $\mathbb{Q}$ and the finite fields $\mathbb{F}_p$
• For the complex numbers $\mathbb{C}$ with complex conjugation, the Grothendieck-Witt group $GW(\mathbb{C})$ is isomorphic to $\mathbb{Z}$, generated by the trivial form
  • The higher Hermitian K-groups $GW_n(\mathbb{C})$ are isomorphic to the topological K-groups $KU_n$, the K-theory of complex vector bundles
• Quillen's localization theorem for algebraic K-theory has analogues in Hermitian K-theory and L-theory, allowing for computations using long exact sequences
• Spectral sequences, such as the Atiyah-Hirzebruch spectral sequence, can be used to compute Hermitian K-groups and L-groups in certain cases

Advanced Topics and Current Research

• Equivariant Hermitian K-theory and L-theory, which take into account the action of a group on the ring with involution, have been developed and have applications in equivariant surgery theory
• Motivic Hermitian K-theory and L-theory are versions that incorporate the motivic homotopy theory of Morel and Voevodsky, allowing for the study of hermitian forms over schemes
  • Motivic Hermitian K-theory is related to the study of quadratic forms over fields and the Milnor conjecture on quadratic forms
• Hermitian K-theory and L-theory have been extended to the setting of C*-algebras and noncommutative geometry
  • The L-theory of C*-algebras is closely related to the Baum-Connes conjecture and the study of positive scalar curvature on manifolds
• Assembly maps in Hermitian K-theory and L-theory, and their relationship to the Novikov and Borel conjectures, remain active areas of research
  • The Farrell-Jones conjecture predicts that certain assembly maps in L-theory are isomorphisms for a wide class of groups
• Trace methods, which relate Hermitian K-theory and L-theory to cyclic homology and noncommutative differential forms, have been developed and have applications in geometry and physics
• Connections between Hermitian K-theory, L-theory, and other areas such as homotopy theory, algebraic geometry, and number theory continue to be explored and have led to new insights and conjectures