A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra

This book will bring the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts.

The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the Poincaré-Hopf theorem on critical points of vector fields on surfaces and the Gauss-Bonnet theorem on the relation between curvature and topology (the Euler characteristic). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincaré approach. Also addressed is the structure of three-dimensional manifolds. In particular, it is proved that the three-dimensional sphere is the union of two doughnuts.

This is the first of three volumes originating from a series of lectures given by the authors at Kyoto University (Japan).