We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups and obtains a bijection with the set of classifying spaces of compact connected Lie groups topologically localised away from the characteristic. We also study the representations of perfectly reductive groups. We establish a highest weight classification of simple modules, the decomposition into blocks, and relate extension groups to those of the underlying abstract group.

Given a group $G$ and an integer $n\geq 0$, we consider the family ${\mathcal F}_n$ of all virtually abelian subgroups of $G$ of $\textrm{rank}$ at most $n$. In this article, we prove that for each $n\ge 2$ the Bredon cohomology, with respect to the family ${\mathcal F}_n$, of a free abelian group with $\textrm{rank}$ $k \gt n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook et al. (Homology Homotopy Appl. 19(2) (2017), 83–87, Question 2.7). As an application, we compute the minimal dimension of a classifying space for the family ${\mathcal F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge 2$. The main tools that we use are the Mayer–Vietoris sequence for Bredon cohomology, Bass–Serre theory, and the Lück–Weiermann construction.

Let BG be the classifying space of an algebraic group G over the field ${\mathbb C}$ of complex numbers. There are smooth projective approximations X of $BG\times {\mathbb P}^{\infty}$, by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X, by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different.

We state a sufficient condition for a fusion system to be saturated. This is then used to investigate localities with kernels: that is, localities that are (in a particular way) extensions of groups by localities. As an application of these results, we define and study certain products in fusion systems and localities, thus giving a new method to construct saturated subsystems of fusion systems.

We study the $E_2$ -algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\Omega B\Lambda \mathfrak {M}_{*,1}$ : it is the product of $\Omega ^{\infty }\mathbf {MTSO}(2)$ with a certain free $\Omega ^{\infty }$ -space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $\Gamma _{g,n}$ with $g\geqslant 0$ and $n\geqslant 1$ .

We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal {F}$ depends only on $\mathcal {F}$ as long as the object set of the linking system is $\mathrm {Aut}(\mathcal {F})$ -invariant. This was known to be true for linking systems in Oliver’s definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this article. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

We consider the Birman–Hilden inclusion $\phi\colon\Br_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\phi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\phi^*(H_1(\Sigma_{g,1}))$ has only 4-torsion.

Generalising the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown–Peterson tower. We construct non-torsion non-algebraic elements in the Brown–Peterson tower for the prime number 2.

We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$. We give explicit calculations when $\Gamma$ 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 $q\geq 2$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.

For almost any compact connected Lie group $G$ and any field $\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra $H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over $\mathbb{F}_{2}$ , such an isomorphism of Batalin–Vilkovisky algebras does not hold when $G=\text{SO}(3)$ or $G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

Let $G$ be a group hyperbolic relative to a finite collection of subgroups ${\mathcal{P}}$. Let ${\mathcal{F}}$ be the family of subgroups consisting of all the conjugates of subgroups in ${\mathcal{P}}$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{{\mathcal{F}}}G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^{\ast }(M)$ of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $S^{n}\times S^{n}$ for $n$ odd.

We show that the Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16 is true.

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.

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 examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system $\mathcal{F}$ over a finite $p$-group $S$ is saturated can be determined by just looking at smaller classes of subgroups of $S$. We also prove that the homotopy type of the classifying space of a given $p$-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of $\mathcal{F}$-centric $\mathcal{F}$-radical subgroups (at a minimum) to the set of $\mathcal{F}$-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to $p$-constrained finite groups, and prove that they in fact all arise from groups.

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use ${{S}^{1}}$ -equivariant minimal models to give a number of examples of ${{S}^{1}}$-spaces illustrating the properties of the various definitions.