Related papers: A model for the Whitehead product in rational mapp…

Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $\Gamma$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a…

We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. We also explain how this makes available the entire range…

We investigate the topology of the space of M\"obius conjugacy classes of degree $d$ rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks…

We study a naturally occurring $E_{\infty}$-subalgebra of the full $E_2$-Hochschild cochain complex arising from coherent cochains. For group rings and certain category algebras, these cochains detect $H^*(B {\cal{C}})$, the simplicial…

Let $K$ be a nontrivial knot. For each $n\in \mathbb{N}$, we prove that the rank of its $n$th iterated Whitehead doubled knot group $\pi_1(S^3 \setminus \operatorname{WD}^n(K))$ is bounded below by $n+1$. As an application, we show that…

We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class…

We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen…

Quillen showed how to describe the homotopy theory of simply-connected rational spaces in terms of differential graded Lie algebras. Here we survey a generalization of Quillen's results that describes the $v_n$-periodic localizations of…

We give lower bounds for genera of components of fiber products of holomorphic maps between compact Riemann surfaces, extending results on genera of components of algebraic curves of the form $A(x)-B(y)=0,$ where $A$ and $B$ are rational…

This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of…

We compute the arithmetic ranks of the defining ideals of homogeneous coordinate rings of certain Segre products arising from elliptic curves. The cohomological dimension of these ideals varies with the characteristic of the field, though…

We define and study binary operations for homotopy groups with coefficients. We give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of…

This paper contains two results on how homotopy limits of topological spaces interact with connectivity. The first is a formula for the connectivity of the homotopy limit of diagrams shaped over suitably finite categories, in terms of the…

Let $\mathfrak{G}=\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$ be the $\mathbb{Z}/2$-extension of the product of two symmetric groups $\mathfrak{S}_{q} \times \mathfrak{S}_q$. In this paper, we compute the…

We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are…

We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of…

Consider the topologically enriched category of compact smooth manifolds (possibly with corners), with morphisms given by codimension zero smooth embeddings. Now formally identify any object X with its thickening X x [-1,1]. We prove that…

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for…

The Andr\'e-Quillen cohomology of an algebra with coefficients in a module is defined by deriving a functor based on K\"ahler differential forms. It can be computed using a cofibrant resolution of the algebra in a model category structure…

Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the Andr\'{e}-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$…