L-theory

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 ${\displaystyle L_{*}(R)}$ (Wall) and the symmetric L-groups ${\displaystyle L^{*}(R)}$ (Mishchenko, Ranicki).

The even-dimensional L-groups ${\displaystyle L_{2k}(R)}$ are defined as the Witt groups of ε-quadratic forms over the ring R with ${\displaystyle \epsilon =(-1)^{k}}$. More precisely,

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