Connected sum decompositions of high-dimensional manifolds

TL;DR

This paper explores the existence and uniqueness of connected sum decompositions in high-dimensional manifolds, revealing that unlike in 3D, such decompositions often lack uniqueness in higher dimensions.

Contribution

It extends the classical Kneser-Milnor theorem to higher dimensions, demonstrating the failure of uniqueness in connected sum decompositions.

Findings

Uniqueness of decompositions fails in many high-dimensional cases

The classical theorem does not generalize straightforwardly to higher dimensions

Multiple non-unique decompositions can exist for the same high-dimensional manifold

Abstract

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.