Related papers: Homotopy nilpotent groups

Let $\mathit{s}\mathcal{L}$ be the $\infty$-category of simplicial restricted Lie algebras over $\mathbf{F} = \overline{\mathbf{F}}_p$, the algebraic closure of a finite field $\mathbf{F}_p$. By the work of A. K. Bousfield et al. on the…

We develop a simple theory of Andr\'e-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences…

For a finite extension $F$ of $\mathbb{Q}_p$ and $n \geq 1$, let $D$ be the division algebra over $F$ of invariant $1/n$ and let $G^0$ be the subgroup of $\text{GL}_n(F)$ of elements with norm $1$ determinant. We show that the action of…

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

We show that the common basis complex of a free group of rank $n$ has the homotopy type of a wedge of spheres of dimension $2n-3$. This establishes an $\mathrm{Aut}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated…

We prove that homotopy invariants of finite degree distinguish homotopy classes of maps of a connected compact CW-complex to a nilpotent connected CW-complex with finitely generated homotopy groups.

We identify the obstructions for the functoriality and the uniqueness of the totalization functor, (partially) defined on the category of simplicial objects in the homotopy category of a stable model category, and we use a result from the…

It is shown that if T is a connected nontrivial graph and X is an arbitrary finite simplicial complex, then there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma.…

For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and…

Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of…

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same…

We consider the category whose objects are filtered, or complete, $L_\infty$-algebras and whose morphisms are $\infty$-morphisms which respect the filtrations. We then discuss the homotopical properties of the Getzler-Hinich simplicial…

We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly to H_*(X_0) when X is 0-connected. The E^1…

We study groups $G$ where the $\varphi$-conjugacy class $[e]_{\varphi}=\{g^{-1}\varphi(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $\varphi$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th…

We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy…

We define unstable $p$-completion in general $\infty$-topoi and the unstable motivic homotopy category, and prove that the $p$-completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the…

Homotopy type theory is a logical setting based on Martin-L\"of type theory in which geometric constructions and proofs can be carried out synthetically. Here, types can be interpreted as spaces up to homotopy, and proofs as…

We introduce the notion of homotopically discrete n-fold category as an n-fold generalization of a groupoid with no non-trivial loops. We give two equivalent descriptions of this structure: in terms of a Segal-type model and in terms of…

If G is a finitely generated powerful pro-p group satisfying a certain law v=1, and if G can be generated by a normal subset T of finite width which satisfies a positive law, we prove that G is nilpotent. Furthermore, the nilpotency class…