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A theorem of O. Forster says that if R is a noetherian ring of Krull dimension d, then every projective R-module of rank n can be generated by d+n elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d+n elements. We view rank-n projective R-modules as R-forms of the non-unital R-algebra R^n where the product of any two elements is 0. The first two authors generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for finite étale algebras. In this paper, we prove new upper and lower bounds on the number of generators of an R-form of a k-algebra, where k is an infinite field and R is a finitely generated k-ring of Krull dimension d. In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack BG, where G is the automorphism group of the algebra in question, by algebraic spaces of a certain type. [ABSTRACT FROM AUTHOR]

In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S 2 and S 6 . The proof described here uses characteristic classes and the Bott periodicity theorem in topological K -theory. This paper originates from the talk “Almost Complex Structures on Spheres” given by the second author at the MAM1 workshop “(Non)-existence of complex structures on S 6 ”, held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition. [ABSTRACT FROM AUTHOR]

This paper provides certain computations with transfer associated with projective bundles of \mathrm {Spin} vector bundles. One aspect is to revise the proof of the main result of [Trans. Amer. Math. Soc.349 (1997), pp. 4385-4399] which says that all fourfold products of the Ray classes are zero in symplectic cobordism. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching μ on a finite regular CW complex X , Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of X. Forman also proved the existence of a CW complex which is homotopy equivalent to X and whose cells are in one-to-one correspondence with the critical cells of μ , but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman's gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of X , while retaining a combinatorial description. The critical difference from Forman's gradient flows is the existence of a partial order on the set of flow paths, from which a 2-category C (μ) is constructed. It is shown that the classifying space of C (μ) is homotopy equivalent to X by using homotopy theory of 2-categories. This result can be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory in early 90's. [ABSTRACT FROM AUTHOR]

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex. [ABSTRACT FROM AUTHOR]

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring for the Hilbert modular group (and its reduced version). We also compute the rational topological K-theory of their reduced C*-algebras. This is done by computing the source of the assembly maps in the Farrell{Jones and the Baum{Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups. [ABSTRACT FROM AUTHOR]

This paper presents a notion of Lusternik–Schnirelmann category for small categories, which is an invariant under homotopy equivalences based on natural transformations. We focus on the relationship between this categorical Lusternik–Schnirelmann category and the classical one via the classifying space. We provide a combinatorial method to calculate the classical Lusternik–Schnirelmann category of the classifying space of a finite acyclic category, taking the barycentric subdivision into account. Moreover, we establish the product inequality for fibered and cofibered categories as an analogue of the inequality of the classical Lusternik–Schnirelmann category for Hurewicz fibrations. [ABSTRACT FROM AUTHOR]

Every principal G-bundle over X is classified up to equivalence by a homotopy class X → BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space ˜ΩX, that is a group. Moreover, the morphisms of topological groups ˜Ω X → G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory. [ABSTRACT FROM AUTHOR]

Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend lower categorical analogues that have been classically used in algebraic topology and algebraic K-theory, such as the homotopy invariance theorem (by Bousfield and Kan), the homotopy colimit theorem (Thomason), Theorems A and B (Quillen), or the homotopy cofinality theorem (Hirschhorn). [ABSTRACT FROM AUTHOR]

In this paper we study algebraic and geometric properties of closed oriented smooth 4-manifolds M with H2(M;Z)≅0. Moreover, we investigate the problem of embedding M in 5-space or other standard simply-connected 5-manifolds according to Barden''s list [Ann. of Math. 82 (1965) 365–385]. These results are related with papers [Invent. Math. 77 (1984) 173–184; Topology 23 (1984) 257–269] of Cochran. [Copyright &y& Elsevier]

Let p be a fixed prime number. The main purpose of this paper is to introduce irreducible p-local compact groups and to describe all p-local compact groups of rank 1 in terms of this concept. [ABSTRACT FROM AUTHOR]

In topology there is a well-known theorem of Atiyah, Hirzebruch, and Segal which states that for a connected compact Lie group G there is an isomorphism R(G) ≅ K0(BG), where BG is the classifying space of G. In the present paper we consider an algebraic analogue of this theorem. For a split reductive group G over a field k, we prove that there is a natural isomorphism ... where KnG is Thomason's G-equivariant K-theory of Spec k, BG is a motivic étale classifying space introduced by Voevodsky and Morel, and IG s the augmentation ideal of K0G (k). [ABSTRACT FROM AUTHOR]

Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-understood classifying spaces, which give geometric meaning to their cells. This paper contains some contributions to the study of the relationship between bicategories and the homotopy types of their classifying spaces. Mainly, generalizations are given of Quillen's Theorems A and B to lax functors between bicategories. [ABSTRACT FROM AUTHOR]

For any prime p, the theory of p-local compact groups is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups and generalises the earlier concept of p-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational p-adic cohomology of the classifying space of a p-local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on p-local compact groups are determined in the appropriate sense by the map they induce on rational cohomology. [ABSTRACT FROM AUTHOR]

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, C R Acad Sci Paris 256:1198-1201, , Ann Sci Ec Norm Super 80:349-425, ) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying a natural 'filling condition'. Any such double groupoid characteristically has associated to it 'homotopy groups', which are defined using only its algebraic structure. Thus arises the notion of 'weak equivalence' between such double groupoids, and a corresponding 'homotopy category' is defined. Our main result in the paper states that the geometric realization functor induces an equivalence between the homotopy category of double groupoids with filling condition and the category of homotopy 2-types (that is, the homotopy category of all topological spaces with the property that the $n{\text{th}}$ homotopy group at any base point vanishes for n ≥ 3). A quasi-inverse functor is explicitly given by means of a new 'homotopy double groupoid' construction for topological spaces. [ABSTRACT FROM AUTHOR]

Abstract: This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well-understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street''s) geometric nerves. [Copyright &y& Elsevier]

Abstract: One major open problem in geometric topology is the Hilbert–Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor''s construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions. [Copyright &y& Elsevier]

Abstract: In this note we correct one argument in the proof of Theorem 1.1 of our paper Martino and Priddy (1995) , given as below. Let , be finite groups and be a prime number. The goal is to give necessary and sufficient algebraic conditions on the-subgroups of and which determine if their -completions and are stably homotopy equivalent. [Copyright &y& Elsevier]