Topological Classification of Multiaxial U(n)-Actions

TL;DR

This paper develops a homotopy classification framework for multiaxial U(n)-manifolds, extending previous work by relaxing conditions and utilizing classical surgery theory, with specific results for spheres and Sp(n)-manifolds.

Contribution

It introduces a general approach to classify multiaxial U(n)-manifolds via a decomposition into pairs of strata, broadening the scope beyond previous strict conditions.

Findings

Homotopy classification reduces to classical surgery theory computations.

Explicit classification results for standard representation spheres.

Extension of results to multiaxial Sp(n)-manifolds.

Abstract

A U(n)-manifold is multiaxial if the isotropy groups are always conjugate to unitary subgroups. The classification and the concordance of such manifolds have been studied by Davis, Hsiang and Morgan under much more strict conditions. We show that in general, without much extra condition, the homotopy classification of multiaxial manifolds can be split into a direct sum of the classification of pairs of adjacent strata, which can be computed by the classical surgery theory. Moreover, we also compute the homotopy classification for the case of the standard representation sphere. We also present the result for the similar multiaxial Sp(n)-manifolds.