A Smale-Barden manifold admitting K-contact but not Sasakian structure

TL;DR

This paper presents the first example of a simply connected 5-manifold that admits a K-contact structure without supporting any Sasakian structure, resolving a longstanding open problem.

Contribution

It provides the first known example of a Smale-Barden manifold with a K-contact but non-Sasakian structure, answering a key question in contact geometry.

Findings

First example of such a manifold confirming the distinction between K-contact and Sasakian structures.

Demonstrates that not all K-contact structures can be upgraded to Sasakian structures.

Settles a long-standing open question in the field of contact and Sasakian geometry.

Abstract

We give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki.