arXiv subjects
Publications and source records attributed to Pedro Boavida de Brito.
The little $n$-disks operad is $SO(n)$ and $O(n)$-equivariantly formal over the rationals. Equivalently, the oriented and unoriented framed little disks operads are rationally formal as $\infty$-operads.
We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie-Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime $p$. Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the $(n+1)$-st Goodwillie-Weiss approximation is a $p$-local universal Vassiliev invariant of degree $\leq n$ for every $n \leq p + 1$.
Using a variant of the Boardman-Vogt tensor product, we construct an action of the Grothendieck-Teichmüller group on the completion of the little n-disks operad $E_n$. This action is used to establish a partial formality theorem for $E_n$ with mod $p$ coefficients and to give a new proof of the formality theorem in characteristic zero.
We prove that the set of concordance classes of sections of an infinity-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss. This is reminiscent of an h-principle in which the role of isotopy is played by concordance. As an application, we offer an answer to the question: what does the classifying space of a Segal space classify?
Using log-geometry, we construct a model for the configuration category of a smooth algebraic variety. As an application, we prove the formality of certain configuration spaces.
We show that in codimension at least 3, spaces of locally flat topological embeddings of manifolds are correctly modelled by derived spaces of maps between their configuration categories (under mild smoothability conditions). That general claim was reduced in an earlier paper to the special cases where the manifolds in question are euclidean spaces. We deal with these special cases by comparing to other special cases where the manifolds have the form "torus" and "torus times euclidean space" respectively, and by setting up a torus trick for configuration categories.
We investigate the relationship between the configuration category of a manifold and the configuration category of a covering space of that manifold.
The configuration category of a manifold is a topological category which we view as a Segal space, via the nerve construction. Our main result is that the unordered configuration category, suitably truncated, admits a finite presentation as a complete Segal space if the manifold in question is the interior of a compact manifold.
We discuss right fibrations in the $\infty$-categorical context of Segal objects in a category V and prove some basic results about these.
A construction related to the Boardman-Vogt tensor product of operads allows us to describe the configuration category of a product manifold $M\times N$ in terms of the configuration categories of the factors $M$ and $N$.
In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.
We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichmüller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal.
We study the covariant model structure on dendroidal spaces and establish direct relations to the homotopy theory of algebras over a simplicial operad as well as to the homotopy theory of special $Γ$-spaces. As an important tool in the latter comparison, we present a sharpening of the classical Barratt-Priddy-Quillen theorem.
The functor that takes a manifold to its configuration category exhibits a type of full faithfulness in some cases.
Manifold calculus is a form of functor calculus concerned with functors from some category of manifolds to spaces. A weakness in the original formulation is that it is not continuous in the sense that it does not handle well the natural enrichments. In this paper, we correct this by defining an enriched version of manifold calculus which essentially extends the discrete setting. Along the way, we recast the Taylor tower as a tower of homotopy sheafifications. As a spin-off we obtain a natural connection to operads: the limit of the Taylor tower is a certain (derived) space of right module maps over the framed little discs operad.