Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG.
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB 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),BB) 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 varian…
The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory from Tonks' thesis, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equ…
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.
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.
Every principal G-bundle is classified up to equivalence by a homotopy class of maps into the classifying space of G. On the other hand, for every nice topological space Milnor constructed a strict model of loop space, that is a group. Moreover the morphisms of topological groups defined on the loop space of X generate all the bundles over X up to equivalence. In this paper, we show that the relationship 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 …
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…
We show that the cohomology group of the equivariant simplicial de Rham complex is isomorphic to the cohomology group of the classifying space of a semi-direct product group.
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…
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…
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.
We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying s…
We study the structure of the $E_2$-term of the Rothenberg-Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the compact, connected, simply connected, exceptional Lie group of rank 6.
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).
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
We consider a problem on the conditions of a compact Lie group G that the loop space of the p-completed classifying space be a p-compact group for a set of primes. In particular, we discuss the classifying spaces BG that are p-compact for all primes when the groups are certain subgroups of simple Lie groups. A survey of the p-compactness of BG for a single prime is included.
We are interested in the relationship between the virtual cohomological dimension (or vcd) of a discrete group Gamma and the smallest possible dimension of a model for the classifying space of Gamma relative to its family of virtually cyclic subgroups. In this paper we construct a model for the virtually cyclic classifying space of the Heisenberg group. This model has dimension 3, which equals the vcd of the Heisenberg group. We also prove that there exists no model of dimension less than 3.
In this paper we describe a classifying theory for families of simplicial topological groups. If $B$ is a topological space and $G$ is a simplicial topological group, then we can consider the non-abelian cohomology $H(B,G)$ of $B$ with coefficients in $G$. If $G$ is a topological group, thought of as a constant simplicial group, then the set $H(B,G)$ is the set of isomorphism classes of principal $G$ bundles, or $G$ torsors, on $B$. For more general simplicial groups $G$, the set $H(B,G)$ parametrizes the set of equivalence classes of higher $G$ torsors on $B$. In this paper we consider a more…
This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac-Moody group and source the classifying space of either a p-compact group or a connected Kac-Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac-Moody group can be detected by restricti…
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 …
This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen's Theorems A and B for non-unital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the Group-Completion Theorem.
For a prime knot group, the classifying space for the family of the subgroups generated by the meridians can be seen as an abstract analogue of the ambient manifold in which the knot lives. An explicit model of this ambient classifying space is constructed as a branched covering space of the 3-sphere branched over the knot; more general branched covering spaces, obtained quotienting the ambient classifying space by finite-index normal subgroups, are studied. Various homological properties of said spaces are established, some of which have parallels in algebraic number theory. In particular, pr…
This paper presents a commutative complex-oriented cohomology theory that realizes the Buchstaber formal group law localized away from 2. Also, the restriction of the classifying map of FB on special unitary cobordism ring localized away from 2 defines a four parameter genus, studied by Hoehn and Totaro.
We show that the Rothenberg--Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the exceptional Lie group E_8 does not collapse at the E_2-level.
In this paper we construct a model for the classifying space, BVCG, of a crystallographic group G of rank n relative to the family VC of virtually-cyclic subgroups of G. The model is used to show that there exists no other model for the virtually-cyclic classifying space of G with dimension less than vcd(G)+1, where vcd(G) denotes the virtual cohomological dimension of G. In addition, the dimension of our construction realizes this limit.
We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain paracompactness condition on \X, we show that X is actually well-defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks.
In this paper, we give a new proof of a well-known theorem due to tom Dieck that the fat realization and Segal's classifying space of an internal category in the category of topological spaces are homotopy equivalent.
We give a bound for the geometric dimension for the family of virtually cyclic groups in mapping class groups of a compact surface with punctures, possibly with nonempty boundary and negative Euler characteristic.
We construct classifying spaces for discrete and compact Lie groups, with the property that they are topological groups and complete metric spaces in a natural way. We sketch a program in view of extending these constructions.
We prove that, for $n\geq 3$, the minimal dimension of a model of the classifying space of the full braid group $B_n$, and of the pure braid group $P_n$, with respect to the family of virtually cyclic groups is $n$.
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 M be a closed simply connected 2n-dimensional manifold. The present paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.
We study the classifying space of a twisted loop group $L_σG$ where $G$ is a compact Lie group and $σ$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $σ$-twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^*(BL_σG)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.
Let $G$ be a simply-connected, simple compact Lie group of type $\{n_{1},\ldots,n_{\ell}\}$, where $n_{1}\le\cdots \le n_{\ell}$. Let $\mathcal{G}_k$ be the gauge group of the principal $G$-bundle (namedright{P}{}{S^{4}}) whose isomorphism class is determined by the the second Chern class having value $k\in\mathbb{Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal{G}_k$ provided that $n_{\ell}<p-1$ and $p$ does not divide $k$.
We compute the cotorsion product of the mod 2 cohomology of spinor group spin(n), which is the E_2-term of the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of the classifying space of the spinor group spin(n). As a consequence of this computation, we show the non-collapsing of the Rothenberg-Steenrod spectral sequence for n > 16.
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.
We extend the formalism of Topological T-duality to spaces which are the total space of a principal $S^1$-bundle $p:E \to W$ with an $H$-flux in $H^3(E,Z)$ together the together with an automorphism of the continuous-trace algebra on $E$ determined by $H$. The automorphism is a `topological approximation' to a gerby gauge transformation of spacetime. We motivate this physically from Buscher's Rules for T-duality. Using the Equivariant Brauer Group, we connect this problem to the $C^{\ast}$-algebraic formalism of Topological T-duality of Mathai and Rosenberg. We show that the study of this prob…
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.
For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and $\mathbb{F}_2$-cohomology rings of $B_\text{com} G$ (the classifying space for commutativity of $G$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of $E_\text{com} G$ (the homotopy fiber of the inclusion $B_\text{com} G \to BG$), and some low-dimensional homotopy groups of $B_\text{com} G$.
For a compact Lie group G we show that if the representing spectrum for Borel cohomology generates its category of modules if G is connected. For a closed subgroup H of G we consider the map C^*(BG)--->C^*(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.
We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.
In this note we show that the second homotopy group of $B(2,G)$, the classifying space for commutativity for a compact Lie group $G$, contains a direct summand isomorphic to $π_1(G)\oplusπ_1([G,G])$, where $[G,G]$ is the commutator subgroup of $G$. It follows from a similar statement for $E(2,G)$, the homotopy fiber of the canonical inclusion $B(2,G)\hookrightarrow BG$. As a consequence of our main result we obtain that if $E(2,G)$ is 2-connected, then $[G,G]$ is simply-connected. This last result completes how the higher connectivity of $E(2,G)$ resembles the higher connectivity of $[G,G]$ fo…
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.
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.
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 describe a cocompact model for the classifying space for proper actions of the mapping class group of a surface with punctures and boundary components. Our construction relies on a known model for the case of a closed surface and uses an induction on the number of punctures and boundary components.
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…
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.
For certain contractible G-CW-complexes and F a family of subgroups of G, we construct a spectral sequence converging to the F-Bredon cohomology of G with E1-terms given by the F-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space for the family F. We also introduce a hierarchically defined class of groups which contains all countable elementary amenable groups and countable linear groups of characteristic zero, and show that if a group G is in this class, then G has finite …
We describe the K-ring of the classifying space of the generalized quaternion group in terms of generators and the minimal set of relations. We also compute the order of the main generator in the truncated rings.
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 $H^*(G,k)$.
There is the problem, whether for a given virtually torsionfree discrete group $Γ$ there exists a cocompact proper topological $Γ$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. It is related to Nielsen's Realization and to the problem of Brown, whether there is a d-dimensional model for the classifying space for proper actions, if the underlying group has virtually cohomological dimension d. Assuming that the expected manifold model has a zero-dimensional singular set, we solve the problem in the Poincaré category and obtain new results about…
