The Wild World of 4-Manifolds By: Alexandru Scorpan

Author: Alexandru Scorpan

This book offers a panorama of the topology of simply-connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but too small to have room to undo them. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. The first part of the book puts things in context with a survey of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold – the intersection form – and its interaction with the topology of the manifold. The third part reviews complex surfaces as an important source of examples. The fourth and final part of the book presents gauge theory. This differential-geometric method has brought to light the unwieldy nature of smooth 4-manifolds; and although the method brings new insights, it has raised more questions than answers.