We prove new structural results for the rational homotopy type of the classifying space $B\operatorname{aut}(X)$ of fibrations with fiber a simply connected finite CW-complex $X$.
We first study nilpotent covers of $B\operatorname{aut}(X)$ and show that their rational cohomology groups are algebraic representations of the associated transformation groups. For the universal cover, this yields an extension of the Sullivan--Wilkerson theorem to higher homotopy and cohomology groups. For the cover corresponding to the kernel of the homology representation, this proves algebraicity of the cohomology of the homotopy Torelli space.
For the cover that classifies what we call normal unipotent fibrations, we then prove the stronger result that there exists a nilpotent dg Lie algebra $\mathfrak g(X)$ in algebraic representations that models its equivariant rational homotopy type. This leads to an algebraic model for the space $B\operatorname{aut}(X)$ and to a description of its rational cohomology ring as the cohomology of a certain arithmetic group $Γ(X)$ with coefficients in the Chevalley-Eilenberg cohomology of $\mathfrak g(X)$. This has strong structural consequences for the cohomology ring and, in certain cases, allows it to be completely determined using invariant theory and calculations with modular forms. We illustrate these points with concrete examples.
As another application, we significantly improve on certain results on self-homotopy equivalences of highly connected even-dimensional manifolds due to Berglund--Madsen, and we prove parallel new results in odd dimensions.
In this paper, we discuss the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves. One construction proceeds via Brown representability and provides a classification in the pointed model category. The second construction is given by the classifying space of the monoid of homotopy self-equivalences of a simplicial sheaf and provides the unpointed classification.
Let $B{ aut}_1X$ be the Dold-Lashof classifying space of orientable fibrations with fiber $X$. For a rationally weakly trivial map $f:X\to Y$, our strictly induced map $a_f: (Baut_1X)_0\to (Baut_1Y)_0$ induces a natural map from a $X_0$-fibration to a $Y_0$-fibration. It is given by a map between the differential graded Lie algebras of derivations of Sullivan models. We note some conditions that the map $a_f$ admits a section and note some relations with the Halperin conjecture. Furthermore we give the obstruction class for a lifting of a classifying map $h: B\to (Baut_1Y)_0$ and apply it for …
Given $X$ a finite nilpotent simplicial set, consider the classifying fibrations $$ X\to Baut_G^*(X)\to Baut_G(X),\qquad X\to Z\to Baut_π^*(X), $$ where $G$ and $π$ denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of $X$ which act nilpotently on $H_*(X)$ and $π_*(X)$. We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if $L$ is a cdgl model of $X$, there are connected sub cdgl's $Der^G L$ and $Der^π L$ of the Lie algebra $D…
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.
Let $f:E\longrightarrow O$ be a Hurewicz fibration with a fiber space $F_{r_{o}}$ and a lifting function $L_{f}$. The \emph{$Lf-$function} $Θ_{L_{f}}$ of $f$ is defined by the restriction map of $L_{f}$ on the space $Ω(O,r_{o})\times F_{r_{o}}\times \{1\}$. The purpose of this paper is to give some results which show the role of $Lf-$functions in finding a fiber homotopically equivalent relation between two fibrations, over a common polyhedron base. Furthermore we will prove the equivalently between our results and Dold's theorem in fiber bundles, over a common suspension base of polyhedron sp…
Given a spherical fibration $ξ$ over the classifying space $BG$ of a finite group we define a dimension function for the $m-$fold fiber join of $ξ$ where $m$ is some large positive integer. We show that the dimension functions satisfy the Borel-Smith conditions when $m$ is large enough. As an application we prove that there exists no spherical fibration over the classifying space of $\text{Qd}(p)= (\mathbb{Z}/p)^2\rtimes\text{SL}_2(\mathbb{Z}/p)$ with $p-$effective Euler class, generalizing the result of Özgün Ünlü about group actions on finite complexes homotopy equivalent to a sphere. We hav…
For a fibration with the fiber $K(π,n)$-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of $K(π,n)$-complex is given which preserves multiplicative structure as well. In terms of this model the action of the $n$-cohomology of the base with coefficients in $π$ on the homology of fibration is described.
The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.
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.
Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory of principal bundles. Thus, Goerss-Jardine appealed to topological methods for the verification. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use the classifying theory of principal bundles or appeal to topological methods.
We consider certain rational homotopical conditions of simly connected CW complex $X$ such that the rational cohomology of the classifying space $Baut_1X$ for fibrations with two-stage fibre $X$ is (not) free. First, we consider when is $Baut_1X$ a rational factor of $Baut_1(X\times S^n)$ for an odd-integer $n$ and observe for a non-formal elliptic space $X$ of rank 3. Second, we compute the Sullivan minimal models of $Baut_1X$ when $X$ are certain non-formal pure spaces of rank 5.
Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we prove that this is possible for the category of functors indexed a small category I which assign to any morphism in I a weak equivalence in a given model category. We identify the homotopy type of this category with the mapping space out of the nerve of I into the classifying space of the space of weak equivalences of values of these functors. We use it to re…
Given a fibration of simply connected CW complexes of finite type, we study the evaluation subgroup of the fibre inclusion as an invariant of fibre-homotopy type. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the rationalized evaluation subgroup of the fibre inclusion decomposes as the direct sum of the rationalized Gottlieb group of the fibre and the rationalized homotopy group …
Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.
We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension of the cohomology groups of classifying spaces.
Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_\infty \colon UE \to \mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ ω\colon \mathrm{aut}_1(\mathrm{Baut}_1(X_{\mathbb Q})) \to \mathrm{Baut}_1(X_{\mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_\infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $\mathrm{Baut}_1(X_{\mathbb Q})$ as a consequence. We also prove tha…
We study the monodromy representation corresponding to a fibration introduced by G. Denham and A. Suciu, which involves polyhedral products given in Definition 2.2. Algebraic and geometric descriptions for these monodromy representations are given. In particular, we study the case of a product of two finite cyclic groups and obtain representations into $Out(F_n)$ and $SL_n(\mathbb Z)$. We give algebraic descriptions of monodromy for the case of a product of any two finite groups . Finally we give a geometric description for monodromy representations of a product of 2 or more finite groups to $…
Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,ΩC]$ acts on $[X,E]$ with orbit space the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the map of $X$ to the base point of $E$ is also well-known to be the image of $[X, ΩB]$. The isotropy subgroups for other maps $e:X -> E$ can definitely change as $e$ does. The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is a group. If $f$ has a lift to $E$, the set $p_#^{-1}(f)$ is identified with the cokernel of a natural homomorphism from this grou…
The main objective of this paper is to show that the homotopy colimit of a diagram of quasi-categories and indexed by a small category is a localization of Lurie's higher Grothendieck construction of the diagram. We thereby generalize Thomason's classical result which states that the homotopy colimit of a diagram of categories has the homotopy type of (the classifying space of) the Grothendieck construction of the diagram of categories.
We consider the Lyndon-Hochschild-Serre spectral sequence with mod-p coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential. Using this result we deduce a similar formula for the Serre spectral sequence for a principal fibration with fibre the classifying space of a cyclic p-group. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in…
We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. As a new result, we describe the moduli space of M-bundles over a fixed space X, when M is a simplicial monoid. Moreover, the moduli space of M-bundles is classified by the classifying space of the simplicial submonoid generated by homotopy invertible elements of M. We give a gen…
We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky's h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Cech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.
We define a class of symplectic fibrations called symplectic configurations. They are natural generalization of Hamiltonian fibrations. Their geometric and topological properties are investigated. We are mainly concentrated on integral symplectic manifolds. We construct the classifyng space \B of symplectic integral configurations. The properties of the classifying map \B --> BSymp(M,w) are examined. The universal symplectic bundle over \B has a natural connection whose holonomy group is isomorphic to the enlarged Hamiltonian group recently defined by McDuff. The space \B is identified with th…
Using the theory of extensions of L-infinity algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley-Eilenberg and Harrison cohomology. We also investigate the algebraic structure of the Chevalley-Eilenberg complexes of L-infinity algebras and show that they possess, along with the Gerstenhaber bracket, an L-infinity structure that is homotopy abelian.
Given pointed $CW$-complexes $X$ and $Y$, $\rmph(X, Y)$ denotes the set of homotopy classes of phantom maps from $X$ to $Y$ and $\rmsph(X, Y)$ denotes the subset of $\rmph(X, Y)$ consisting of homotopy classes of special phantom maps. In a preceding paper, we gave a sufficient condition such that $\rmph(X, Y)$ and $\rmsph(X, Y)$ have natural group structures and established a formula for calculating the groups $\rmph(X, Y)$ and $\rmsph(X, Y)$ in many cases where the groups $[X,Ω\widehat{Y}]$ are nontrivial. In this paper, we establish a dual version of the formula, in which the target is the t…
The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad $\mathcal{O}$ in symmetric spectra. Our main result is that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total $\mathcal{O}$-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-complet…
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 …
The category of rational O(2)-equivariant spectra splits as a product of cyclic and dihedral parts. Using the classification of rational G-equivariant spectra for finite groups G, we classify the dihedral part of rational O(2)-equivariant spectra in terms of an algebraic model.
We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to a new and unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic K-theory invariant…
A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the framework of knot theory the notion of classifying spaces for families pops out naturally and we provide explicit models for them. The study of their topological properties hints at the existence of arcane connections between knot theory and algebraic number theory. Trying to track these connections, we discovered a behavioural analogy between the fundamental group…
The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's $Γ$-spaces. The formalism of topological operads generalises well to different categories yielding such notions as $\mathbb E_n$-algebras in chain complexes, while the $Γ$-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding "weak" sections of a homotopical Grothendieck fibration. We propose a model for such sections, c…
The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber $M$ depend only on the underlying $τ_M$-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for $τ_M$-fibrations, $Baut(τ_M)$, and its cohomology ring, i.e., the ring of characteristic classes of $τ_M$-fibrations. For a bundle $ξ$ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable $ξ$-fibration together with explicit cocycle representatives for the characteri…
Let F be a fibration on a simply-connected base with symplectic fibre (M, ω). Assume that the fibre is nilpotent and T^{2k}-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston's criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is als…
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…
We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform) fibrations and that satisfy the (functorial) Frobenius condition. As applications, we obtain a new proof that the Quillen model structure for Kan complexes is right proper, avoiding entirely the use of topological realization and minimal fibrations, and we solve an open problem in the study of Voevodsky's simplicial model of type theory, proving a constructive…
We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$ induces an isomorphism in cohomology in degrees $\le K$. This is then applied to show that if $G$ is a connected non-special group there is a $G$-torsor $P \to X$ for which we do not have $[P]=[G][X]$ in the (completion of the) Grothendieck ring of varieties.
A central question in arrangement theory is to determine whether the characteristic polynomial $Δ_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $Δ_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $β_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $…
In 1970s Segal outlined proofs of two theorems relating spaces of Fredholm and self-adjoint Fredholm operators with Quillen's constructions used to define higher algebraic K-theory. In the present paper we provide detailed proofs of these theorems of Segal (different from the ones outlined by Segal) and then transplant some further ideas of Quillen to the real of Fredholm and self-adjoint Fredholm operators. The main results may be considered as partial analogues of the equivalence of Quillen's two definition of higher algebraic K-theory. Along the way we relate spaces of Fredholm and self-adj…
We compute the number of rational points of classifying stacks of Chevalley group schemes using the Lefschetz-Grothendieck trace formula of Behrend for $\ell$-adic cohomology of algebraic stacks. From this we also derive associated zeta functions for these classifying stacks.
Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected points of a space. In other words, we can do abstract homotopy theory, but not algebraic topology. Shulman's Real Cohesive HoTT remedies this issue by introducing a system of modalities that relate the spatial structure of types to their homotopical structure. In this paper, we develop a theory of modal fibrations for a general modality, and apply it in par…
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.
For locally homotopy trivial fibrations, one can define transition functions $$ g\dab : U\da\cap U\db \to H = H(F)$$ where $H$ is the monoid of homotopy equivalences of $F$ to itself but, instead of the cocycle condition, one obtains only that $g\dab g\dbgam$ is homotopic to $g\dagam$ as a map of $U\da\cap U\db\cap U\dgam$ into $H$. Moreover on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis \cite{wirth:diss}. Here we present it using language that has been developed …
