Ramón J. Flores, Richard M. Foote
In this paper we complete the description of the $B\mathbb{Z} /p$-cellularization of the classifying spaces of all finite groups, for all primes $p$. The techniques are based in a careful analysis of the $p$-fusion structure of the groups involved -with special attention to their strongly closed subgroups- and Chachólski's description of the $A$-cellular approximation.
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One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
Cellular approximations of fusion systems
本文研究了融合系统分类空间的细胞化问题，提出了判断分类空间是否为细胞的代数准则，并对特定异常示例进行了计算。
Let A be the classifying space of an abelian p-torsion group. We compute A-cellular approximations (in the sense of Chachólski and Farjoun) of classifying spaces of p-local compact groups, with special emphasis in the cases which arise from honest compact Lie groups.
The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally normal regular cellular stratified space $X$ can be embedded in $X$ as a strong deformation retract. Here we elaborate on this idea and develop the theory of cellular stratified spaces. We introduce the notion of cylindrically normal cellular stratified spaces and associate a top…
We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.
We present a constructive method to compute the cellularization with respect to K(Z/p, m) for any integer m > 0 of a large class of H-spaces, namely all those which have a finite number of non-trivial K(Z/p, m)-homotopy groups (the pointed mapping space map(K(Z/p, m), X) is a Postnikov piece). We prove in particular that the K(Z/p, m)-cellularization of an H-space having a finite number of K(Z/p, m)-homotopy groups is a p-torsion Postnikov piece. Along the way we characterize the BZ/p^r-cellular classifying spaces of nilpotent groups.
By examining commensurators of virtually cyclic groups, we show that for each natural number n, any locally finite-by-virtually cyclic group of cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+3. As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers.
In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p-local finite group in terms of representation rings of subgroups of its Sylow. We also prove a stable elements formula for generalized cohomological invariants of p-local finite groups, which is used to show the existence of unitary embeddings of p-local finite groups. Finally, we show that the augmentation map for the cochains of the classifying space of a p-local finite group is Gorenstein in the sense of Dwyer-Greenlees-Iyengar and obtain some consequences about the co…
We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower z_k X whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections, and that any nilpotent space for which the space of pointed self-maps map_*(X,X) is "canonically" discrete must be aspher…
We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.
Let G be a finite group and p a prime dividing its order. We define new collections of p-subgroups of G. We study the homotopy relations among them and with the standard collections of p-subgroups. We determine their ampleness and sharpness properties.
Realizing wedges of Moore spaces as classifying spaces of finite semigroups
针对Fiedorowicz提出的猜想，证明有限生成阿贝尔群的单连通Moore空间的有限楔和同伦等价于有限半群的分类空间，表明仅同调群无法排除有限单连通CW复形等价于有限半群分类空间的可能。
The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying spa…
The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable p-completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in partic…
Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When $\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we constru…
We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the F-cohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every finitely generated regular branch group has infinite rational cohomological dimension. Moreover, we prove that the first Grigorchuk group G is not contained in Kropholler's class HF .
A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these class…
A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.
Let $G$ be an infinite discrete group. A classifying space for proper actions of $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are contractible for all finite subgroups $H$ of $G$. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper $G$-spectra and study its finiteness properties. We investigate when $G$ admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper $G$-spectra a…
Classifying spaces of finite groups of tame representation type
本文研究有限群中具有tame表示类型的块的p-完成分类空间及其循环空间，重点在于$A_\infty$代数结构和奇点余奇点范畴。
We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino-Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space of G as an object under the stable p-completed classifying space of S via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution to the Martino-Priddy conjecture, we obtain the surprising result that the unstable p-completed homotopy type of BG is determined by the stable p-completed inclus…
We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.
Among the generalizations of Serre's theorem on the homotopy groups of a finite complex we isolate the one proposed by Dwyer and Wilkerson. Even though the spaces they consider must be 2-connected, we show that it can be used to both recover known results and obtain new theorems about p-completed classifying spaces.
Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g greater than 1 admits a (9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean bui…
The classifying diagram was defined by Rezk and is a generalization of the nerve of a category; in contrast to the nerve, the classifying diagram of two categories is equivalent if and only if the categories are equivalent. In this paper we prove that the classifying diagram of any category is characterized in terms of classifying spaces of stabilizers of groups. We also prove explicit decompositions of the classifying diagrams for the categories of finite ordered sets, finite dimensional vector spaces, and finite sets in terms of classifying spaces of groups.
Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in \mathcal{F}$ the fixed point subspace $X^H$ is contractible. The group $G$ is of type $\mathcal{F}\text{-}\mathrm{F}_{n}$ if it admits a model for $E_\mathcal{F} G$ with $n$-skeleton with compact orbit space. The main result of the article provides is a characterization of $\mathcal{F}\text{-}\mathrm{F}_{n}$ analogue to Brown's criterion for $…
For n>2, we prove the mod 2 cohomology of the finite Chevalley group Spin_n(F_q) is isomorphic to that of the classifying space of the loop group of the spin group Spin(n).
The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces.…
The localising subcategories of the derived category of the cochains on the classifying space of a finite group are classified. They are in one to one correspondence with the subsets of the set of homogeneous prime ideals of the cohomology ring $H^*(G,k)$.
Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.
We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model.
Let G be a discrete group which acts properly and isometrically on a complete CAT(0)-space X. Consider an integer d with d=1 or d greater or equal to 3 such that the topological dimension of X is bounded by d. We show the existence of a G-CW-model E_fin(G) for the classifying space for proper G-actions with dim(E_fin(G)) less or equal to d. Provided that the action is also cocompact, we prove the existence of a G-CW-model E_vcyc(G) for the classifying space of the family of virtually cyclic subgroups such that dim(E_vcyc(G)) is less or equal to d+1.
Cellular sheaf Laplacians on the set of simplices of symmetric simplicial set induced by hypergraph
本文将有序有限抽象单纯复形上的胞腔层拉普拉斯算子推广至超图诱导的对称单纯集的单纯集合上，通过构造超图到有限对称单纯集范畴的函子定义了该算子并给出了具体公式。
Universal Assembly and Cellular Loop Spaces on Regular CW Complexes
本文为正则CW复形建立了代数手术理论中链复形的经典装配形式的类比，通过构造组合路径和环路对象，并引入通用装配函子，将局部化同伦范畴等同于导出范畴的Verdier商，证明了局部化理论可通过环路DG代数的DG模进行Morita描述。
We show that every discrete subgroup of $\mathrm{GL}(n,\mathbb{R})$ admits a finite dimensional classifying space with virtually cyclic stabilizers. Applying our methods to $\mathrm{SL}(3,\mathbb{Z})$, we obtain a four dimensional classifying space with virtually cyclic stabilizers and a decomposition of the algebraic $K$-theory of its group ring.
We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic completion of the Grothendieck group of finite principal (G,K)-bundles whose isotropy groups are p-groups. Collecting the result for different primes p, we get a new and simple description of the group of homotopy classes of stable maps between (uncompleted) classifying spaces of groups. This description allows us to determine the kernel of the map from the …
In this paper, we compute the rational cohomology groups of the classifying space of a simply connected Kac-Moody group of infinite type. The fundamental principle is "from finite to infinite". That is, for a Kac-Moody group G(A) of infinite type, the input data for computation are the rational cohomology of classifying spaces of parabolic subgroups of G(A)(which are of finite type), and the homomorphisms induced by inclusions of these subgroups. In some special cases, we can further determine the cohomology rings. Our method also applies to study the mod p cohomology of the classifying spaces…
We construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_\infty$-operad. We use one of these to construct a cellular $E_\infty$-bialgebra structure on the interval and derive from it a natural cellular $E_\infty$-coalgebra structure on the geometric realization of a simplicial set which, passing to cellular chains, recovers up to signs the Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse and McClure-Smith. We use another prop, a quotient of the first…
We extend Björner's characterization of the face poset of finite CW complexes to a certain class of stratified spaces, called cylindrically normal stellar complexes. As a direct consequence, we obtain a discrete analogue of cell decompositions in smooth Morse theory, by using the classifying space model introduced in arXiv:1612.08429. As another application, we show that the exit-path simplicial set $\mathrm{Exit}(X)$ of a finite cylindrically normal CW stellar complex $X$ is a quasicategory.
Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G$. We give the first examples of groups $G$ with a cocompact classifying space for proper actions $\underline{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$-equivariant (virtual) vector bundle over $\underline{E}G$. This implies that the Atiyah-Hirzeburch spectral sequen…
In this paper we explore the possibility of defining p-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question posed by Haynes Miller, whether an equivalent theory can be obtained by studying triples (f,t,X), where X is a p-complete, nilpotent space with finite fundamental group, f is a map from the classifying space of a finite p-group, and t is a stable retraction of f satisfying Frobenius reciprocity at the level of stable homotopy. We refer to t as a retractive transfer of f and to (f,t,X) as a retractive transfer triple ove…
Type Annihilation for Classifying Maps of Rack Spaces
该研究针对有限类型rack的分类映射，证明了其同伦群相关的零化性质，确定了特定同调群，并给出在辛与Alexander结构上的低维应用。
We show that the classifying space of a $p$-local compact group is approximated by a telescope of classifying spaces of $p$-local finite groups. This result has numerous implications, like a Stable Elements Theorem for $p$-local compact groups, or the description of the homotopy type of certain mapping spaces.
Greenlees and Sadofsky showed that the classifying spaces of finite groups are self-dual with respect to Morava K-theory K(n). Their duality map was constructed using a transfer map. We generalize their duality map and prove a K(n)-version of Poincare duality for classifying spaces of orbifolds. By regarding these classifying spaces as the homotopy types of certain differentiable stacks, our construction can be viewed as a stack version of Spanier-Whitehead type construction. Some examples and calculations will be given at the end.
Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology of the classifying space BG up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculat…
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.
Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is isomorphic to the mod ell cohomology of the classifying space of the loop group LG for q=p^{ab}, a>0.
We give the first examples of finite groups G such that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of Q. As a tool, we give several characterizations of the varieties which satisfy Kunneth properties for Chow groups or motivic homology. We define the (compactly supported) motive of a quotient stack in Voevodsky's derived category of motives. This makes it possible to ask when the motive of BG is mixed Tate, which is equivalent to the motivic Kunneth property. We prove that BG is mixed Tate for various "well-behaved" fin…
On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds
本文证明了 Kontsevich 猜想的同调版本，即当 $M$ 为具有非平凡且非球面边界的不可约 $3$-流形连通和时，其相对微分同胚分类空间仅有有限个有限生成的非零同调群。
We consider Γ-equivariant principal G-bundles over proper Γ-CW-complexes with prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups with finite covering dimension Γand G such that G is almost connected.
We define algebraic cobordism of classifying spaces, Ω^*(BG) and G-equivariant algebraic cobordism Ω^*_G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^j(Ω^*(-)), which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We calculate Ω^*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n+1). We also calculate Ω^*(BG) when G is a finite abelian group. A finite non-abelian group for …
We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.
We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory $K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a $G$-Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it…
Picard groups and composition nilpotence for finite cellular isotropic spectra
研究有限胞腔各向同性谱，证明\(Pic(SH(k/k)^c_{cell})\cong\mathbb{Z}^2\)，指出\(E_{**}\)性质及权重结构作用，还表明特定核为合成幂零理想，可检测合成幂零性并给出典范分解。
We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.
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