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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]

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]

Our main purpose in this paper is to resolve, in a rational homotopy theory context, the following open question asked by S. Theriaul: given a topological space X, what one may say about the nilpotency of aut1.(X) when the cocategory of its classifying space Baut1.(X) is finite? Here aut1.(X) denotes the path component of the identity map in the set of self homotopy equivalences of X. More precisely, we prove that HnilQ.aut1.(X) 6 cocatQ.Baut1.(X); when X is a simply connected CW-complex of finite type and that the equality holds when Baut1.(X) is coformal.Many intersections with other popular open questions will be discussed. [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]

Let G be a finite group. A natural place to study the Chow ring of the classifying space BG is Voevodsky's triangulated category of motives, inside which Morel-Voevodsky and Totaro have defined motives M.BG/ and Mc.BG/, respectively. We show that, for any group G of order p³ over a field of characteristic not equal to p which contains a primitive p³-th root of unity, the motive M.BG/ is a mixed Tate motive. We also show that, for a finite group G over a field of characteristic zero, M.BG/ is a mixed Tate motive if and only if Mc.BG/ is a mixed Tate motive. [ABSTRACT FROM AUTHOR]

Let M be a closed simply connected 2n-dimensional manifold. The paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M. [ABSTRACT FROM PUBLISHER]

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]

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]

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]

Recent work of Mishchenko and Morales Meléndez ( [math.AT], ) has shed new light on the classical exact sequence of Conner and Floyd in G-equivariant bordism for a finite group G. This paper is primarily an exposition of the results obtained there as specialized to finite groups. For a countable discrete group G, there is a brief discussion of the generalization to the bordism theory of proper, cocompact G-manifolds. [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]

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]

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens–Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group. [ABSTRACT FROM PUBLISHER]

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 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 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]

We prove that the D'ıaz-Park sharpness conjecture holds for saturated fusion systems defined on a Sylow p-subgroup of the group G2(p), for p 5. [ABSTRACT FROM AUTHOR]

For a finite group G, we define the G-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of G. Also, we find an isomorphism of its fundamental group onto the direct sum Z OSO 2 (BG), whereOSO2 (BG) is the free oriented G-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product G/[G,G] × S1 × XG, where p1(XG) = OSO 2 (BG). Finally, we present some results about the classification of Gtopological quantum field theories in dimension two. [ABSTRACT FROM AUTHOR]

For each n ≥ 3 we give examples of infinitesimally rigid projective manifolds of general type of dimension n with non-contractible universal cover. We provide examples with projective and examples with non-projective universal cover. [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]

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]

For every one-relator monoid M = ⟨ A ∣ u = v ⟩ with u , v ∈ A ∗ we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type FP ∞ , answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property F ∞ , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups. [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]

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]

A gauge group is the topological group of automorphisms of a principal bundle.We compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension. [ABSTRACT FROM AUTHOR]

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces. [ABSTRACT FROM AUTHOR]

We compute the de Rham cohomology rings of BG2 and BSpin(n) for 7 ≤ n ≤ 11 over base fields of characteristic 2 [ABSTRACT FROM AUTHOR]

We present an Eilenberg-MacLane-type description for the first, second and third spaces of the spectrum defined by a symmetric monoidal category. [ABSTRACT FROM AUTHOR]

We determine the mod 2 cohomology ring of the classifying space of the exceptional Lie group E6 and the action of the Steenrod algebra on it. [ABSTRACT FROM AUTHOR]

The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure. [ABSTRACT FROM AUTHOR]

Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra A over a ring represents a 2-torsion class in the Brauer group if and only if there is an algebra A′ in the Brauer class of A admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose A′ such that degA′ = 2 degA. We show that 2 degA is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra A of degree 4 and period 2 such that the degree of any algebra A′ in the Brauer class of A admitting an involution is divisible by 8. Separately, we provide examples of split and nonsplit Azumaya algebras of degree 2 admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa. [ABSTRACT FROM AUTHOR]

We give a new construction of a classifying space for the fibre of the double suspension and an elementary proof of Gray's conjecture at odd primes. [ABSTRACT FROM AUTHOR]

We extend Oprea's result that the Gottlieb group G1(핊2n+1/H) is ZH (the center of H) and show that for a map f : A → 핊2n+1/H, under some conditions on A, we have Gf1 (핊2n+1/H) =ZHf*(π1(A)), the centralizer of the image f*(π1(A)) in H. Then, we compute or estimate the groups Gf m(핊2n+1/H) and Pfm (핊2n+1/H) for certain m > 1. [ABSTRACT FROM AUTHOR]

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group PBn. Elements of Thompson's group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk's thesis, a space CF was considered of configurations of points on the real line allowing for splitting and merging, and a sketch of a proof was given that CF is a classifying space for F. The idea there was to build the universal cover and construct an explicit contraction to a point. However, this was never written up rigorously. Here we start with an established CAT(0) cube complex X on which F acts freely, and construct an explicit homotopy equivalence between X/F and CF, proving that CF is indeed a K(F, 1). [ABSTRACT FROM AUTHOR]

We study the degree 3 cohomological invariants with coefficients in Q/Z(2) of a semisimple group over an arbitrary field. A list of all invariants of adjoint groups of inner type is given. [ABSTRACT FROM AUTHOR]

We develop the fundamentals of Mackey functors in the setup of fusion systems including an acyclicity condition as well as a parametrization and an explicit description of simple Mackey functors. Using this machinery we extend Dwyer’s sharpness results to exotic fusion systems F on a finite p-group S with an abelian subgroup of index p. [ABSTRACT FROM AUTHOR]

Let p: E . B be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut1 (p) denote the identity component of the space of all fibre-homotopy self-equivalences of p. Let Baut1 (p) denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut1(p), connecting the results of recent work by the authors and others. We use this model to give classification results for the rational homotopy types represented by Baut11(p) and also to obtain conditions under which the monoid aut11(p) is a double loop-space after rationalization. [ABSTRACT FROM AUTHOR]

It is shown that the orbit space of universal (in the sense of Palais) G-spaces classifies G-spaces. Theorems on the extension of covering homotopy for G-spaces and on a homotopy representation of the isovariant category ISOV are proved. [ABSTRACT FROM AUTHOR]

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]

For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos ßB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X), ßB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization ∣NB∣ of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, B¨okstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties. [ABSTRACT FROM AUTHOR]

The problems of classifying Hurewicz fibrations whose fibres have just two non-zero homotopy groups and classifying 3-stage Postnikov towers are substantially equivalent. We investigate the case where the fibres have the homotopy type of K(G,m) × K(H, n), for 1 < m < n. Our solution uses a classifying space M∞, i.e. a mapping space whose underlying set consists of all null homotopic maps from individual fibres of the path fibration PK(G,m + 1) → K(G,m + 1) to the space K(H, n + 1), and the group ϵ (K(G,m) × K(H, n)) of based homotopy classes of based self-homotopy equivalences of K(G,m) × K(H, n). If B is a given space, then a group action ϵ(K(G,m) × K(H, n)) × [B,M∞]0 → [B,M∞]0 is defined, and the orbit set [B,M∞]0 / ϵ(K(G,m) × K(H, n)) is shown to classify the above fibrations over B up to fibrewise homotopy type. Our explicit definitions of the classifying spaces, together with our computationally effective group actions, are advantageous for computations and further developments. Two stable range simplifications are given here, together with a classification result for cases where B is a product of spheres. [ABSTRACT FROM AUTHOR]

By using Quillen's S[SUP-1] S construction, we construct a category &script;G with classifying space Ω[SUP&infin] S[SUP&infin], which has a pictorial description by pseudo-tangles. This leads to an elementary interpretation of the generators of the stable stems π[SUPS,SUBn] = π[SUBn](&ohm[SUP&infin] S[SUP&infin]) for n = 0, 1, 2. In addition, we investigate the homotopy fibre of the inclusion &scriptG; ↪ &scritpF;[SUBP,D], where &scriptF;[SUBP,D] denotes the free permutative category with duality, and compute the homotopy groups of &scriptF;[SUBP,D] for n = 0, 1, 2. [ABSTRACT FROM AUTHOR]

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group ( G,⊗) over a small category ℬ. We show a natural bijection between the set of equivalence classes of such torsors and [ B({ℬ}), B( G,⊗)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with π i( Y)=0 for all i≠1,2. [ABSTRACT FROM AUTHOR]

Let X be a simply connected space of LS category two. We show that the LS category of B aut X is not finite when X is not coformal or satisfies Poincaré duality. [ABSTRACT FROM AUTHOR]