We study the fibered isomorphism conjecture of Farrell and Jones in $L$-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in 2. of Theorem 1.2, we prove the $L$-theory version of Theorem 1.2 in the 2003 paper by Farrell and Linnell.

Thomason’s Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can also be represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams B of braided monoidal categories, by means of a simplicial set nerve of the diagram. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplicial sets, obtained from composing B with the geometric nerve functor of braided monoidal categories.

Using universal constructions of topological groups, one can endow the fundamental group of a space with a topology and obtain a topological group. Additionally, the fundamental groupoid of a space becomes enriched over Top when the homsets are endowed with similar topologies. This paper is devoted to a generalization of classical covering theory in the context of these constructions.

We prove that for an inclusion of unital associative but not necessarily commutative $\mathbb{k}$-algebras $\mathcal{B}\subseteq \mathcal{A}$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $\mathcal{B}$-module $\mathcal{A}/\mathcal{B}$ is flat. We also prove that for an arbitrary $r$-flat morphism $\varphi\colon\mathcal{B}\to\mathcal{A}$ with an H-unital kernel, one can express the Wodzicki excision sequence and our Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.

Let $M$ be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in $M$. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normalized chain complex functor.

We provide several explicit classes of examples of homotopynormal and of homotopy-conormal maps, when $M$ is the category of simplicial sets or the category of chain complexes over a commutative ring.

In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even in the singular case) in terms of combinatorial data coming from the fan of the toric variety.

We construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors and prove that these cubical derived functors are naturally isomorphic to their simplicial counterparts. We also show that this result generalizes the well-known fact that the simplicial and cubical singular homologies of a topological space are naturally isomorphic.

An algorithm is designed to write down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given connected graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and is quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.

Scissors congruence groups have traditionally been expressed algebraically in terms of group homology. We give an alternate construction of these groups by producing them as the $0$-level in the algebraic $K$-theory of a Waldhausen category.

For a cyclic A-infinity algebra, a potential recording the structure constants can be defined. We define an analogous potential for a homotopy cyclic A-infinity algebra and prove its properties. On the other hand, we find another different potential for a homotopy cyclic A-infinity algebra, which is related to the algebraic analogue of generalized holonomy map of Abbaspour, Tradler and Zeinalian.

Discovering and constructing the topological structure in data has attracted the attention within the community of data analysis. However, most methods developed so far are unsuitable for very large sets of data because of their computational difficulties. This paper presents a fast algorithm for constructing the inherent topological structure in large sets of data that might be noisy in order to enhance the MAPPER algorithm introduced by Singh, Mémoli and Carlsson. The limitation of our method, as shown by our experiments, lies with the storage in the main memory rather than the computing time.

We define and study transversal weight and $t$-structures (for triangulated categories); if a weight structure is transversal to a $t$-one, then it defines certain 'weights' for its heart. Our results axiomatize and describe in detail the relations between the Chow weight structure $w_{Chow}$ for Voevodsky's motives (introduced in a preceding paper), the (conjectural) motivic $t$-structure, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of Deligne's 1-motives (over a smooth base) and of Artin-Tate motives over number fields. In particular, we prove that the 'weights' for Voevodsky's motives (that are given by $w_{Chow}$) are compatible with those for 1-motives (that were 'classically' defined using a quite distinct method); this result is new. Weight structures transversal to the canonical $t$-structures also exist for the Beilinson's $D^b_{\bar{H}_p}$ (the derived category of graded polarizable mixed Hodge complexes) and for the derived category of (Saito's) mixed Hodge modules.

We also study weight filtrations for the heart of $t$ and (the degeneration of) weight spectral sequences. The corresponding relation between $t$ and $w$ is strictly weaker than transversality; yet it is easier to check, and we still obtain a certain filtration for (objects of) the heart of $t$ that is strictly respected by morphisms.

In a succeeding paper we apply the results obtained in order to reduce the existence of Beilinson’s mixed motivic sheaves (over a base scheme $S$) and 'weights' for them to (certain) standard motivic conjectures over a universal domain $K$.

We develop an extension of the framework of combinatorial model categories. The category of small presheaves over large indexing categories and ind-categories are the basic examples of non-combinatorial model categories embraced by the new machinery called class-combinatorial model categories.

The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper, where we introduced the concepts of class-locally presentable and class-accessible categories.

In this work we prove that the category of weak equivalences of a nice class-combinatorial model category is class-accessible. Our extension of J. Smith’s localization theorem depends on the verification of a cosolution-set condition. The deepest result is that the (left Bousfield) localization of a class-combinatorial model category with respect to a strongly class-accessible localization functor is class-combinatorial again.

One of the well-known problems in algebraic $K$-theory is the comparison of higher Chow groups and $K$-groups. In this paper, using the motivic complex defined by Voevodsky– Suslin–Friedlander, we prove the comparison theorem for equi-characteristic regular local rings.

In this note, we correct an error in one of our approaches to encode a group structure by a diagram. We show that we instead obtain the structure of a monoid with involution.