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In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as "Hermitian K-theory", is important in surgery theory. One can define L-groups for any ring with involution R: the quadratic L-groups (Wall) and the symmetric L-groups (Mishchenko, Ranicki). The even-dimensional L-groups are defined as the Witt groups of ε-quadratic forms over the ring R with . More precisely, is the abelian group of equivalence classes of non-degenerate ε-quadratic forms over R, where the underlying R-modules F are finitely generated free. The equivalence relation is given by stabilization with respect to hyperbolic ε-quadratic forms: The addition in is defined by The zero element is represented by for any . The inverse of is . Defining odd-dimensional L-groups is more complicated; further details and the definition of the odd-dimensional L-groups can be found in the references mentioned below. The L-groups of a group are the L-groups of the group ring . In the applications to topology is the fundamental group of a space . The quadratic L-groups play a central role in the surgery classification of the homotopy types of -dimensional manifolds of dimension , and in the formulation of the Novikov conjecture. The distinction between symmetric L-groups and quadratic L-groups, indicated by upper and lower indices, reflects the usage in group homology and cohomology. The group cohomology of the cyclic group deals with the fixed points of a -action, while the group homology deals with the orbits of a -action; compare (fixed points) and (orbits, quotient) for upper/lower index notation. The quadratic L-groups: and the symmetric L-groups: are related by a symmetrization map which is an isomorphism modulo 2-torsion, and which corresponds to the polarization identities. The quadratic and the symmetric L-groups are 4-fold periodic (the comment of Ranicki, page 12, on the non-periodicity of the symmetric L-groups refers to another type of L-groups, defined using "short complexes").

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Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.

Ε-quadratic form

In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called -quadratic forms, particularly in the context of surgery theory. There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms.

Arf invariant

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.

Let G be a simple linear algebraic group over an algebraically dosed field K of characteristic p >= 0 and let V be an irreducible rational G-module with highest weight A. When is self-dual, a basic question to ask is whether V has a non-degenerate G-invari ...

Elsevier2017

Hermitian Forms over Algebras with Involution and Hermitian Categories

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EPFL2012

Let k be a field of characteristic ≠2, A be a central simple algebra with involution σ over k and W(A,σ) be the associated Witt group of hermitian forms. We prove that for all purely inseparable extensions L of k, the canonical map rL/k:W(A,σ)⟶W(AL,σL) is ...

Springer Verlag2011