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We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi., Comment: 63 pages v2: minor improvements, updated references. v3: slightly reorganised the discussions of quadratic surgery for ring spectra, and the devissage sequence; otherwise minor improvements following a referee report

This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $\infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In this article we lay the foundations of our approach by considering Lurie's notion of a Poincar\'e $\infty$-category, which permits an abstract counterpart of unimodular forms called Poincar\'e objects. We analyse the special cases of hyperbolic and metabolic Poincar\'e objects, and establish a version of Ranicki's algebraic Thom construction. For derived $\infty$-categories of rings, we classify all Poincar\'e structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincar\'e structures on $\infty$-categories of parametrised spectra, recovering the visible signature of a Poincar\'e duality space. We conduct a thorough investigation of the global structural properties of Poincar\'e $\infty$-categories, showing in particular that they form a bicomplete, closed symmetric monoidal $\infty$-category. We also study the process of tensoring and cotensoring a Poincar\'e $\infty$-category over a finite simplicial complex, a construction featuring prominently in the definition of the L- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0-th Grothendieck-Witt group of a Poincar\'e $\infty$-category using generators and relations. We extract its basic properties, relating it in particular to the 0-th L- and algebraic K-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications., Comment: 178 pages v2: minor improvements, updated references. v3: slightly reorganised the discussion of periodicity; otherwise minor improvements following a referee report

We show that quadratic and symmetric L-theory of the integers are related by Anderson duality and show that both spectra split integrally into the L-theory of the real numbers and a generalised Eilenberg-Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space G/Top. Finally, we prove analogous results for the genuine L-spectra recently devised for the study of Grothendieck--Witt theory., Comment: v4: Corrected an oversight in the third comment following Theorem A; v3: Removed an erroneous (but for the paper irrelevant) remark in the appendix. An erratum is available from the webpages of the authors