We prove the existence of a map of spectra τ A : kA → ℓA between connective topological K-theory and connective algebraic L-theory of a complex C *-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence KA[ 1 2 ] ≃ − → LA[ 1 2 ] of periodic K-and L-theory spectra after inverting 2. We show that this equivalence extends to K-and L-theory of real C *-algebras. Using this we give a comparison between the real Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.

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This paper is the first in a series in which we offer a new framework for hermitian $${\text {K}}$$ 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 the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré $$\infty $$ ∞ -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived $$\infty $$ ∞ -categories of rings, we classify all Poincaré 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 Poincaré structures on $$\infty $$ ∞ -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré $$\infty $$ ∞ -categories, showing in particular that they form a bicomplete, closed symmetric monoidal $$\infty $$ ∞ -category. We also study the process of tensoring and cotensoring a Poincaré $$\infty $$ ∞ -category over a finite simplicial complex, a construction featuring prominently in the definition of the $${\text {L}}$$ L - and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré $$\infty $$ ∞ -category using generators and relations. We extract its basic properties, relating it in particular to the 0th $${\text {L}}$$ L - and algebraic $${\text {K}}$$ K -groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

We define Grothendieck-Witt spectra in the setting of Poincaré ∞-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. CONTENTS IntroductionRecollection 1 Poincaré-Verdier sequences and additive functors 1.1 Poincaré-Verdier sequences 1.2 Split Poincaré-Verdier sequences and Poincaré recollements 1.3 Poincaré-Karoubi sequences 1.4 Examples of Poincaré-Verdier sequences 1.5 Additive and localising functors 2 The hermitian Q-construction and algebraic cobordism categories 2.1 The hermitian Q-construction 2.2 The cobordism category of a Poincaré ∞-category 2.3 Algebraic surgery 2.4 The additivity theorem 2.5 Fibrations between cobordism categories 2.6 Additivity in K-Theory 3 Structure theory for additive functors 3.1 Cobordisms of Poincaré functors 3.2 Isotropic decompositions of Poincaré ∞-categories 3.3 The group-completion of an additive functor 3.4 The spectrification of an additive functor 3.5 Bordism invariant functors 3.6 The bordification of an additive functor 3.7 The genuine hyperbolisation of an additive functor 4 Grothendieck-Witt theory 4.1 The Grothendieck-Witt space 4.2 The Grothendieck-Witt spectrum 4.3 The Bott-Genauer sequence and Karoubi's fundamental theorem 4.4 L-theory and the fundamental fibre square 4.5 The real algebraic K-theory spectrum and Karoubi periodicity 4.6 LA-theory after Weiss and Williams Date: September 16, 2020. 144 B.2 Schlichting's Grothendieck-Witt-spectrum of a ring with 2 invertible 146 References 150which we term the fundamental fibre square. Now, in [HM] Hesselholt and Madsen promoted the Grothendieck-Witt spectrum GW s cl ( , ) into the genuine fixed points of what they termed the real algebraic K-theory KR s cl ( , ), a genuine C 2 -spectrum. We similarly produce a functor KR ∶ Cat p ∞ ⟶ Sp gC 2 using the language of spectral Mackey functors, with the property that the isotropy separation square of KR(C, Ϙ) is precisely the fundamental fibre square above, so that in particular KR(C, Ϙ) gC 2 ≃ GW(C, Ϙ) and KR(C, Ϙ) C 2 ≃ L(C, Ϙ); here (−) gC 2 and (−) C 2 ∶ S gC 2 → S denote the genuine and geometric fixed points functors, respectively. Combined with the comparison results of [HS20] this affirms the conjecture of Hesselholt and Madsen, that the geometric fixed points of the real algebraic K-theory spectrum of a discrete ring are a version of Ranicki's L-theory.As the ultimate expression of periodicity, we then enhance our extension of Karoubi's periodicity to the following statement in the language of genuine homotopy theory: Theorem D. The boundary map of the metabolic Poincaré-Verdier sequence provides a...

In this paper we study the index theoretic interpretation of the analytical assembly map that appears in the Baum-Connes conjecture. In its general form it may be constructed using Kasparov's equivariant KK-theory. In the special case of a torsionfree group the domain simplifies to the usual K-homology of the classifying space BG of G and it is frequently used that in this case the analytical assembly map is given by assigning to an operator an equivariant index. We give a precise formulation of this statement and prove it.