nLab Hermitian K-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Where algebraic K-theory concerns (in degree zero) the Grothendieck groups of (finitely generated projective) modules/vector spaces, Hermitian K-theory is concerned with Grothendieck groups of modules equipped with Hermitian quadratic forms.

Since the resulting groups were originally denoted (in Wall 1969) by the letter “L” (just the next letter in the alphabet after “K” for “class”), this is also known as “L-theory” (cf. Karoubi 1973, p. 306).

References

General

Precursor discussion:

  • Hyman Bass (notes by Amit Roy), §5 in: Lectures on Topics in Algebraic K-Theory, Tata Institute of Fundamental Research (1965) [pdf]

  • C. T. C. Wall (ed. Andrew Ranicki), Surgery on compact Manifolds, Math. Surveys and Monographs 69 (1969) [pdf]

Introduction of Hermtian K-theory as such:

Further early discussion:

For more see:

Further developments:

Topological Hermitian K-theory

The case of topological Hermitian K-theory:

  • Max Karoubi, §III in: Périodicité de la K-théorie hermitienne, in: Hyman Bass (ed.), Algebraic K-Theory III – Hermitian K-Theory and Geometric Applications, Lecture Notes in Mathematics 343 (1973) 301-411 [doi:10.1007/BFb0061366]

and its relation to KR-theory via the hyperbolic functor:

See also:

Last revised on November 16, 2023 at 06:58:03. See the history of this page for a list of all contributions to it.