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Witt groups. (English) Zbl 1115.19004
Friedlander, Eric M. (ed.) et al., Handbook of $K$-theory. Vol. 1 and 2. Berlin: Springer (ISBN 3-540-23019-X/hbk). 539-579 (2005).

In this survey article the author first describes the (almost) classical theory of the Witt group of an exact category with duality, a straightforward generalization of Knebusch’s definition of the Witt group of a scheme [cf. M. Knebusch, “Symmetric bilinear forms over algebraic varieties”, Proc. Conf. Quadratic Forms, Kingston 1976, Queen’s Pap. pure appl. Math. 46, 103-283 (1976; Zbl 0408.15019)]. Many results and examples are listed ranging from Witt groups of elliptic curves to calculations of Witt groups of surfaces over $ℝ$ and $ℂ$, connections to real algebraic geometry and to other similar theories (e.g., Grothendieck-Witt groups, $L$-theory).

In the second part the author then describes many of his own contributions to the theory of Witt groups, starting with the definition of triangular Witt groups, i.e., Witt groups of triangulated categories with duality [cf. P. Balmer, “Triangular Witt groups. I: The 12-term localization exact sequence”, $K$-Theory 19, 311–363 (2000; Zbl 0953.18003)] and the relation to the classical theory. Many of the more recent results (periodicity, dévissage, Gersten conjecture) and examples are discussed, and the survey ends with a brief account of the recently discovered relationship between Witt groups and ${𝔸}^{1}$-homotopy theory [cf. F. Morel, “An introduction to ${𝔸}^{1}$-homotopy theory”, Trieste: ICTP Lecture Notes 15, 361-441 (2003; Zbl 1081.14029)].

##### MSC:
 19G12 Witt groups of rings 11E81 Algebraic theory of quadratic forms 18E30 Derived categories, triangulated categories
##### Keywords:
Witt groups; triangulated category; Gersten conjecture