The Disc-structure space

TL;DR

This paper investigates the Disc-structure space of compact smooth manifolds, revealing its dependence on tangential 2-type, its infinite loop space structure, and nontriviality for spin manifolds, with implications for embedding calculus and automorphism groups.

Contribution

It establishes that in high dimensions, the Disc-structure space depends only on the tangential 2-type, forms an infinite loop space, and is nontrivial for spin manifolds, advancing understanding of manifold automorphisms.

Findings

Disc-structure space depends only on tangential 2-type in high dimensions.

Disc-structure space is an infinite loop space.

Nontrivial for spin manifolds.

Abstract

We study the Disc-structure space $S^{\rm Disc}_\partial(M)$ of a compact smooth manifold $M$. Informally speaking, this space measures the difference between $M$, together with its diffeomorphisms, and the diagram of ordered framed configuration spaces of $M$ with point-forgetting and point-splitting maps between them, together with its derived automorphisms. As the main results, we show that in high dimensions, the Disc-structure space a) only depends on the tangential 2-type of $M$, b) is an infinite loop space, and c) is nontrivial as long as $M$ is spin. The proofs involve intermediate results that may be of independent interest, including an enhancement of embedding calculus to the level of bordism categories, results on the behaviour of derived mapping spaces between operads under rationalisation, and an answer to a question of Dwyer and Hess in that we show that the map ${\rm BTop}(d)\to {\rm BAut}(E_d)$ is an equivalence if and only if $d$ is at most $2$.