Complementing and extending the Inventiones work of Benson, Grodal, Henke [Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507] we give criteria for a space to have cohomology (strongly) F-isomorphic in the sense of Quillen to the stable elements. We extend results about groups models to fusion systems over discrete $p$-toral groups and profinite groups and provide various applications to the Kan-Thurston Theorem.
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…
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 study the mod-$\ell$ homotopy type of classifying spaces for commutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that the mod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces a mod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an …
In this paper we discuss the isomorphism types of parabolic subgroups in Kac-Moody groups. The results have applications in the study of topology of Kac-Moody groups and their classifying spaces.
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 $\mathbb{Z}\oplus Ω_2^{SO}(BG)$, where $Ω_2^{SO}(BG)$ 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 $G/[G,G]\times S^1\times X^G$, where $π_1(X^G)=Ω_2^{SO}(BG)$. Fin…
This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra $W(A)$ associated to any Lie algebroid $A$. We then show that this Weil algebra is related to the Bott-Shulman-Stasheff complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual proof of the main result of Bursztyn et.al. on the reconstructions of multiplicative forms a…
Let $G$ be a finite 2-group of maximal nilpotency class, and let $BG$ be its classifying space. We prove that iterated Massey products in $H^*(BG;\F_2)$ do characterize the homotopy type of $BG$ among 2-complete spaces with the same cohomological structure. As a consequence we get an alternative proof of the modular isomorphism problem for 2-groups of maximal nilpotency class.
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.
For almost any compact connected Lie group $G$ and any field $\mathbb{F}\_p$, we compute the Batalin-Vilkoviskyalgebra $H^{*+\text{dim }G}(LBG;\mathbb{F}\_p)$ on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra is isomorphicto the Hochschild cohomology $HH^*(H\_*(G),H\_*(G))$. Over $\mathbb{F}\_2$, such isomorphism of Batalin-Vilkovisky algebrasdoes not hold when $G=SO(3)$ or $G=G\_2$.
Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let ${\Bbbk}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{*+d}(LM)$ is a Batalin-Vilkovisky algebra. Let $G$ be a topological group such that $M$ is a classifying space of $G$. Denote by $S_*(G)$ the (normalized) singular chains on $G$. Suppose that $G$ is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism $$ HH^{-p}(S_*(G),S_*(G))\cong HH_{p+d}(S_*(G),S_*(G)). $$ Therefore, the Gerstenhaber algebra $HH^{*}(S_*(G),S_*(G))$ is a …
In a previous paper, we have constructed, for an arbitrary Lie group G and any of the fields F=R or C, a good equivariant cohomology theory KF_G^*(-) on the category of proper $G$-CW-complex and have justified why it deserved the label ``equivariant K-theory". It was shown in particular how this theory was a logical extension of the construction of Lück and Oliver for discrete groups and coincided with Segal's classical K-theory when G is a compact group and only finite G-CW-complexes are considered. Here, we compare our new equivariant K-theory with that of N.C. Phillips: it is shown how a na…
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 use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part…
Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in this case the cohomology of the homogeneous space $G/K$ with coefficients in $k$ and the torsion product of $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ are isomorphic as $k$-modules. We show that this isomorphism is multiplicative and natural in the pair $(G,K)$ provided that 2 is invertible in $k$. The proof uses homotopy Gerstenhaber algebras in an essential way. I…
We describe the connected components of the space $\text{Hom}(Γ,SU(2))$ of homomorphisms for a discrete nilpotent group $Γ$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb{RP}^3$. We give explicit calculations when $Γ$ is a finitely generated free nilpotent group. In the second part of the paper we study the filtration $B_{\text{com}}SU(2) = B(2,SU(2))\subset\cdots \subset B(q,SU(2))\subset\cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq2$, the inclusions ind…
Let $F$ be a finite group. We consider the lamplighter group $L=F\wr\mathbb{Z}$ over $F$. We prove that $L$ has a classifying space for proper actions $\underline{E} L$ which is a complex of dimension two. We use this to give an explicit proof of the Baum-Connes conjecture (without coefficients), that states that the assembly map $μ_i^L:K_i^L(\underline{E} L)\rightarrow K_i(C^*L)\;(i=0,1)$ is an isomorphism. Actually, $K_0(C^*L)$ is free abelian of countable rank, with an explicit basis consisting of projections in $C^*L$, while $K_1(C^*L)$ is infinite cyclic, generated by the unitary of $C^*L…
We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum build from the canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i--this is the non-oriented analogue of the Mumford conj…
This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG_m-bundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BG_m. The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theor…
We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and G' are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie type H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of the p-adic units. Our proof uses homotopy theoretic properties of the p-completed classifying spaces of G and G', and we know of no purely algebraic proof of this result.
Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups $H_i(X,G)$ by considering cycles in the simplicial scheme $BG\times X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups $H_i(Δ^j,G)$, which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. <p> We also compute some examples, focusing in particular on the c…
