# Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

Abstract : We define Grothendieck-Witt spectra in the setting of Poincar\'e $\infty$-categories, show that they fit into an extension with a K- and an L-theoretic part and deduce localisation sequences for Verdier quotients. As special cases we obtain generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible. A novel feature of our approach is the systematic use of ideas from cobordism theory by interpreting the hermitian Q-construction as an algebraic cobordism category. We also use this to give a new description of the LA-spectra of Weiss and Williams.
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https://hal-univ-artois.archives-ouvertes.fr/hal-02941256
Contributor : Baptiste Calmès <>
Submitted on : Wednesday, September 16, 2020 - 9:17:35 PM
Last modification on : Thursday, September 17, 2020 - 3:02:07 AM

### Identifiers

• HAL Id : hal-02941256, version 1
• ARXIV : 2009.07224

### Citation

Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, et al.. Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity. 2020. ⟨hal-02941256⟩

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