Higher Smooth Surgery Structure Sets of Complex Projective Spaces, Part I

TL;DR

This paper investigates the higher smooth surgery structure sets of complex projective spaces, focusing on the free subgroup and providing information across all dimensions, with a follow-up study on torsion.

Contribution

It determines the higher smooth surgery structure sets of complex projective spaces and analyzes the forgetful map to topological versions, primarily in the free subgroup.

Findings

Determined the structure sets in all dimensions for the free subgroup.

Provided insights into the forgetful map from smooth to topological structures.

Laid groundwork for analyzing torsion in the second part.

Abstract

This is the first of the two articles where we determine the higher smooth surgery structure sets of complex projective spaces (up to some extension problems) and the forgetful map to their topological versions in low dimensions. In this part, we concentrate on the free subgroup, where we obtain information in all dimensions. In the second part, we study the torsion.