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Register a new userD{e}vissage for Waldhausen K-theory
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Authors
George Raptis
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A devissage--type theorem in algebraic $K$-theory is a statement that identifies the $K$-theory of a Waldhausen category $mathscr{C}$ in terms of the $K$-theories of a collection of Waldhausen subcategories of $mathscr{C}$ when a devissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between devissage theorems of emph{single type} and of emph{multiple type} depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillens original devissage theorem for abelian categories (single type) and Waldhausens theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of devissage--type theorems and prove a general devissage theorem of single type and a general devissage theorem of multiple type.
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We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausens $S_{bullet}$-construction and therefore it defines a model for Waldhausen $K$-theory. As an example, we discuss this model for $A$-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausens $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.
For a group G that splits as an amalgamation of A and B over a common subgroup C, there is an associated Waldhausen Nil-group, measuring the failure of Mayer-Vietoris for algebraic K-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups A,B,G satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of G. We note that a special case of an acylindrical amalgamation includes any amalgamation over a finite group C.
We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.
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 Luries notion of a Poincare $infty$-category, which permits an abstract counterpart of unimodular forms called Poincare objects. We analyse the special cases of hyperbolic and metabolic Poincare objects, and establish a version of Ranickis algebraic Thom construction. For derived $infty$-categories of rings, we classify all Poincare 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 Poincare structures on $infty$-categories of parametrised spectra, recovering the visible signature of a Poincare duality space. We conduct a thorough investigation of the global structural properties of Poincare $infty$-categories, showing in particular that they form a bicomplete, closed symmetric monoidal $infty$-category. We also study the process of tensoring and cotensoring a Poincare $infty$-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 Poincare $infty$-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.
For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.
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