Infinity-operadic foundations for embedding calculus

Abstract

Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of $\infty$-categories of truncated right-modules over a unital $\infty$-operad $\mathcal{O}$. We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as $\mathcal{O}$ varies, and generalise these results to the level of Morita $(\infty,2)$-categories. Applied to the ${\rm BO}(d)$-framed $E_d$-operad, this extends Goodwillie-Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the $E_d$-operad, it yields new versions of embedding calculus, such as one for topological embeddings, based on ${\rm BTop}(d)$, or one similar to Boavida de Brito-Weiss' configuration categories, based on ${\rm BAut}(E_d)$. In addition, we prove a delooping result in the context of embedding calculus, establish a convergence result for topological embedding calculus, improve upon the smooth convergence result of Goodwillie, Klein, and Weiss, and deduce an Alexander trick for homology 4-spheres.

Figures (3)

• Figure 1: The directed paths indicate an element $\gamma\in \partial^h C_3(M)$ starting at an element $\gamma(-)\in\mathrm{Sym}_3(M)$, which has been lifted to $\mathsf{B}_1(\mathsf{E}_M,\mathsf{Disc}_{\leq 3},\mathrm{Sym}_3(-))$ as indicated by the coloured discs. To lift $\gamma$ to $\mathsf{B}_1(\mathsf{E}_M,\mathsf{Disc}_{\leq 3},\partial^h C_3(-))$, we shrink the domain of the directed paths until their image lies in the innermost discs.
• Figure 2: A mapping cylinder thickening $W$ of $M$ as in \ref{['def:thickenings']}\ref{['enum:mcg-thickening']}. The grey lines indicate the map $\pi \colon \partial_1 W \to M$. Note that $\pi$ need not be injective.
• Figure 3: By pushing along the interval direction of the mapping cylinder increasingly much as we approach the boundary of $U$, we can isotope $h(\mathrm{cyl}(\pi_U))$ (the dark region) into $U'$ (the green region), with image given by the red region.

Theorems & Definitions (224)

• Theorem A
• Remark
• Remark
• Remark : Configuration categories
• Theorem B
• Remark
• Remark 1.1
• Remark 1.2
• Lemma 1.3
• proof