The smooth structure set of $S^p \times S^q$

TL;DR

This paper computes the smooth structure set of products of spheres, revealing that it generally lacks a group structure and that the forgetful map's image is not always a subgroup, impacting understanding of smooth versus topological structures.

Contribution

It provides explicit calculations of the smooth structure set for products of spheres and demonstrates limitations of group structures and subgroup properties in this context.

Findings

The smooth structure set of $S^p imes S^q$ is explicitly calculated.

In general, $S(4j-1, 4k)$ does not admit a group structure compatible with the surgery sequence.

The image of the forgetful map is not always a subgroup of the topological structure set.

Abstract

We calculate the smooth structure set of $S^p \times S^q$, $S(p, q)$, for $p, q \geq 2$ and $p+q \geq 5$. As a consequence we show that in general $S(4j-1, 4k)$ cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map $F: S(4j, 4k) --> S^{Top}(4j, 4k)$ is not in general a subgroup of the topological structure set.