Editing K-theory (section)

==Developments==
The other historical origin of algebraic K-theory was the work of [[J. H. C. Whitehead]] and others on what later became known as [[Whitehead torsion]].

There followed a period in which there were various partial definitions of ''[[Algebraic K-theory#Higher K-theory|higher K-theory functors]]''. Finally, two useful and equivalent definitions were given by [[Daniel Quillen]] using [[homotopy theory]] in 1969 and 1972. A variant was also given by [[Friedhelm Waldhausen]] in order to study the ''algebraic K-theory of spaces,'' which is related to the study of pseudo-isotopies.  Much modern research on higher K-theory is related to algebraic geometry and the study of [[motivic cohomology]].

The corresponding constructions involving an auxiliary [[quadratic form]] received the general name [[L-theory]]. It is a major tool of [[surgery theory]].

In [[string theory]], the K-theory classification of [[Ramond–Ramond field]] strengths and the charges of stable [[D-branes]] was first proposed in 1997.<ref>by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and [[Greg Moore (physicist)|Gregory Moore]] in [https://arxiv.org/abs/hep-th/9710230 K-theory and Ramond–Ramond Charge].</ref>

In 2022, Russian mathematician Alexander Ivanovich Efimov constructed a significant generalization of algebraic K-theory, particularly applied to dualizable <math>(\infty,1)</math>-categories <ref>{{Citation |last=Efimov |first=Alexander I. |title=K-theory and localizing invariants of large categories |date=2025-02-06 |arxiv=2405.12169 }}</ref>