nLab Hermitian K-theory

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).

Related concepts

• hyperbolic functor

• KR-theory

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:

• Max Karoubi, Orlando Villamayor, K-théorie algébrique et K-théorie topologique II, Math. Scand. 32 (1973) 57-86 [jstor:24490565]

Further early discussion:

• Hyman Bass (ed.), Algebraic K-Theory III – Hermitian K-Theory and Geometric Applications, Lecture Notes in Mathematics 343 (1973) [doi:10.1007/BFb0061366]

• Alexandr S. Mishchenko, Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis, Uspeki Mat. Nauk 31 2 (1976) 69-134, Russ. Math. Surv. 31 71 (1976) [doi:10.1070/RM1976v031n02ABEH001478, pdf]

• Anthony Bak, Grothendieck Groups of Modules and Forms Over Commutative Orders, American Journal of Mathematics, 99 1 (1977) 107-120 [jstor:2374010, doi:10.2307/2374010]

For more see:

• Max Karoubi, K-théorie hermitienne [webpage]

Further developments:

• Damjan Kobal, K-Theory, Hermitian K-Theory and the Karoubi Tower, K-Theory 17 113–140 (1999) [doi:10.1023/A:1007799508729]

• Max Karoubi, Le théorème de périodicité en K-théorie hermitienne, in Quanta of Maths, Clay Mathematics Proceesings 11, AMS and Clay Math Institute Publications (2010) 257-282 [arXiv:0810.4707, pdf]

• Max Karoubi, Mariusz Wodzicki, Algebraic and Hermitian K-theory of $\mathcal{K}$-rings, The Quarterly Journal of Mathematics 64 3 (2013) 903–940 [arXiv:1310.4123, doi:10.1093/qmath/hat030]

• Marco Schlichting, Higher $K$-Theory of Forms I. From Rings to Exact Categories, Journal of the Institute of Mathematics of Jussieu 20 4 (2021) 1205-1273 [doi:10.1017/S1474748019000410]

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:

• Michael C. Crabb, §9 of: $\mathbb{Z}/2$-Homotopy Theory, Lond. Math. Soc. Lecture Notes 44, Cambridge University Press (1980) [ark:/13960/t3jx7bg4w, pdf]

• Po Hu, Igor Kriz, Appendix of: Topological Hermitian cobordism, Journal of Homotopy and Related Structures 11 (2016) 173–197 [doi:10.1007/s40062-014-0100-9]

• Max Karoubi, Charles Weibel, Thm. 3.5 in: Twisted $K$-theory, Real $A$-bundles and Grothendieck–Witt groups, Journal of Pure and Applied Algebra 221 7 (2017) 1629-1640 [doi:10.1016/j.jpaa.2016.12.020]

See also:

• Max Karoubi, Marco Schlichting, Charles Weibel, The Witt group of real algebraic varieties, Journal of Topology 9 4 (2016) 1257-1302 [arXiv:1506.03862, doi:10.1112/jtopol/jtw024]