Surfaces

Mathematicians have developed the language of topology; but the conventional development leaves until last the discussion and ideas about shapes, so those who have not mastered the initial ideas of analysis and general topology are barred from participation by a rigid formalism. This book, first published in 1976, adopts a different approach, developing a language close to that of ordinary discourse. It is intended to encourage readers of varying backgrounds, but especially students, to think spatially. The text is well illustrated, with toned drawings creating three-dimensional effects where appropriate, and there are abundant exercises. Useful appendices accompany the text, providing hints and solutions to these exercises and also a sketch of how the treatment can be modelled within a conventional topology course for more advanced students. The book can be regarded as an example of the emerging discipline of mathematics education, as well as being about surfaces. This book is an attempt to break that cycle, being designed for students who might later teach children. It has two aspects, one mathematical, and one educational. Mathematically, it expounds the topology of compact surfaces as far as the Classification Theorem, and treats also the Morse Theory of such surfaces. In fact the whole treatment is a 'handle-body' approach. However, a conventional mathematical treatment would begin 4logically' with topological spaces, continuity and homeomorphisms, before settling down to combinatorial ideas. Any student who could not jump the initial conceptual thresholds would then be cut off from the rich intuitive material that underlies the later work. And it is the rich intuitive material that a future teacher will need, for introducing children to 3-dimensional thinking, so that he can stress the meaning rather than the syntax.