Stable moduli spaces of hermitian forms
Fabian Hebestreit, Wolfgang Steimle
TL;DR
The paper develops a stable ∞-categorical framework for hermitian K-theory by viewing Grothendieck-Witt spaces as group completions of moduli spaces of Poincaré-objects, even without inverting 2. It introduces weight structures and parametrised algebraic surgery to compare GW-spaces with derived K- and L-theoretic data, providing a general proof strategy via an algebraic cobordism category Cob(C,Qoppa). The main contributions include a weight-theorem identifying GW(C,Qoppa) with the group completion of the heart's Poincaré forms, and a parametrised surgery argument that yields equivalences between cobordism subcategories and thus explicit descriptions of GW-spaces in terms of derived categories; these results recover classical GW-spaces for rings and produce fiber sequences relating K- and L-theory in broad settings. The framework also extends to rings, form parameters, and even spectra, offering a robust, 2-agnostic approach with potential applications to the cohomology of orthogonal groups and beyond, thereby unifying classical hermitian K-theory with modern stable ∞-categorical methods.
Abstract
We prove that Grothendieck-Witt spaces of Poincaré categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubi's classical hermitian and quadratic K-groups with the genuine Grothendieck-Witt groups from our joint work with Calmès, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our solution of several conjectures in hermitian K-theory. The method of proof is abstracted from work of Galatius and Randal-Williams on cobordism categories of manifolds using the identification of the Grothendieck-Witt space of a Poincaré category as the homotopy type of the associated cobordism category.