The Problems of Dimension Four, and Some Ramifications

Abstract

In this short note, I present a very quick review of the peculiarities of dimension four in geometric topology. I consider, in particular, the role of geometric simple connectivity (which means handle decomposition without handles of index one) for both closed manifolds and open manifolds and for finitely presented groups, together with some of recent developments in geometric group theory.

Keywords

• DIFF category
• geometric simple connectivity
• 4-dimensional topology
• finitely presented groups
• QSF property