Decomposing manifolds into Cartesian products

TL;DR

This paper investigates when complex manifolds can be broken down into simpler product structures, providing new results for 3-manifolds and highlighting limitations in higher dimensions.

Contribution

It extends the understanding of manifold decomposability by establishing analogs for 3-manifolds and identifying restrictions needed in higher dimensions.

Findings

3-manifolds decomposability analogous to Borsuk's surface result

Higher-dimensional decomposability requires additional restrictions

Nondecomposable manifolds generally do not split into simpler products

Abstract

The decomposability of a Cartesian product of two nondecomposable manifolds into products of lower dimensional manifolds is studied. For 3-manifolds we obtain an analog of a result due to Borsuk for surfaces, and in higher dimensions we show that similar analogs do not exist unless one imposes further restrictions such as simple connectivity.