Round handle decompositions of 1-connected 5-manifolds

An nn-dimensional round kk-handle is the product of a circle and an ordinary (n−1)(n−1)-dimensional kk-handle. This exposition gives an introduction to the topic of round handles, exploring several aspects like the existence of round handle decompositions and their manipulation as well as some applications. As a driving example we show how to explicitly construct minimal round handle decompositions of closed, oriented, simply-connected 5-manifolds, also presenting more specialised results of Shkol’nikov and techniques of Fomenko and Sharko along the way.