Embedding calculus and smooth structures

TL;DR

This paper investigates how embedding calculus detects smooth structures, showing it cannot distinguish exotic smooth structures in dimension 4 but can in higher dimensions, and introduces an isotopy extension theorem.

Contribution

It demonstrates the limitations of embedding calculus in dimension 4 and its ability to distinguish exotic spheres in higher dimensions, along with a new isotopy extension theorem.

Findings

Embedding calculus does not distinguish exotic smooth structures in dimension 4.

Embedding calculus can distinguish certain exotic spheres in higher dimensions.

An isotopy extension theorem for the embedding calculus limit is proved.

Abstract

We study the dependence of the embedding calculus Taylor tower on the smooth structures of the source and target. We prove that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro. In contrast, we show that embedding calculus does distinguish certain exotic spheres in higher dimensions. As a technical tool of independent interest, we prove an isotopy extension theorem for the limit of the embedding calculus tower, which we use to investigate several further examples.