Searcharxiv⌕ Search

arXiv subjects

Connor Malin

Publications and source records attributed to Connor Malin.

8 recordsLinked to original sources
arXiv2026

Surgery on manifold operads

We study cobordisms of a class of topological operads called ``manifold operads''. These operads are generalizations of the Fulton-MacPherson operad: an operad built from configurations of points in Euclidean space. Cobordism of manifold operads, along with the associated theory of surgery, depends crucially on delicate combinatorial results for trees associated to operadic bimodules. As an application of surgery, we produce infinitely many manifold operads which are left or right ``bimodule cobordant'' to, but not homotopy equivalent to the Fulton-MacPherson operad.

math.AT↗
arXiv2025

Unstable $1$-semiadditivity as classifying Goodwillie towers

A stable $\infty$-category is $1$-semiadditive if the norms for all finite group actions are equivalences. In the presence of $1$-semiadditivity, Goodwillie calculus simplifies drastically. We introduce two variants of $1$-semiadditivity for an $\infty$-category $C$ and study their relation to the Goodwillie calculus of functors $C \rightarrow \s(C)$. We demonstrate that these variations of $1$-semiadditivity are complete obstructions to the problem of endowing $\partial_\ast F$ with either a right module or a divided power right module structure which completely classifies the Goodwillie tower of $F$. We find applications to algebraic localizations of spaces, the Morita theory of operads, and bar-cobar duality of algebras. Along the way, we address several milestones in these areas including: Lie structures in the Goodwillie calculus of spaces, spectral Lie algebra models of $v_h$-periodic homotopy theory, and the Poincar\'e/Koszul duality of $E_d$-algebras.

math.AT↗
arXiv2024

Koszul duality and a classification of stable Weiss towers

We introduce a version of Koszul duality for categories, which extends the Koszul duality of operads and right modules. We demonstrate that the derivatives which appear in Weiss calculus (with values in spectra) form a right module over the Koszul dual of the category of vector spaces and orthogonal surjections, resolving conjectures of Arone--Ching and Espic. Using categorical Fourier transforms, we then classify Weiss towers. In particular, we describe the $n$-th polynomial approximation as a pullback of the $(n-1)$-st polynomial approximation along a ``generalized norm map''.

math.AT↗
arXiv2023

Koszul self duality of manifolds

We show that Koszul duality for operads in $(\mathrm{Top},\times)$ can be expressed via generalized Thom complexes. As an application, we prove the Koszul self duality of the little disk modules $E_M$. We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.

math.AT↗
arXiv2022

The stable embedding tower and operadic structures on configuration spaces

Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $\operatorname{Emb}(M,N)$ via the "embedding tower", which is constructed using the homotopy theory of presheaves on $M$. The same theory allows us to study the stable homotopy type of $\operatorname{Emb}(M,N)$ via the "stable embedding tower". By analyzing cubes of framed configuration spaces, we prove that the layers of the stable embedding tower are tangential homotopy invariants of $N$. If $M$ is framed, the moduli space of disks $E_M$ is intimately connected to both the stable and unstable embedding towers through the $E_n$ operad. The action of $E_n$ on $E_M$ induces an action of the Poisson operad $\mathrm{pois}_n$ on the homology of configuration spaces $H_*(F(M,-))$. In order to study this action, we introduce the notion of Poincare-Koszul operads and modules and show that $E_n$ and $E_M$ are examples. As an application, we compute the induced action of the Lie operad on $H_*(F(M,-))$ and show it is a homotopy invariant of $M^+$.

math.AT↗
arXiv2022

An elementary proof of the homotopy invariance of stabilized configuration spaces

In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual of $\Sigma^\infty_+ F(M,k)$ and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to ``generalized configuration spaces'' which have recently undergone study.

math.AT↗