xiv Preface describing surfaces, beginning with the traditional representation by equations in three-dimensional space, proceeding to parametric rep- resentation, and introducing the less intuitive, but central for our purposes, representation as factor spaces. It also includes a prelimi- nary discussion of the metric geometry of surfaces. Subsequent chap- ters introduce fundamental mathematical structures: topology, com- binatorial (piecewise-linear) structure, smooth structure, Riemannian metric, and complex structure in the specific context of surfaces. The assumed background is the standard calculus sequence, some linear algebra, and rudiments of ODE and real analysis. All notions are introduced and discussed, and virtually all results proved, based on this background. The focal point of the book is the Euler characteristic, which ap- pears in many different guises and ties together concepts from com- binatorics, algebraic topology, Morse theory, ODE, and Riemannian geometry. The repeated appearance of the Euler characteristic pro- vides both a unifying theme and a powerful illustration of the notion of an invariant in all those theories. A further idea of both the motivations and the material presented in the book may be found in the Table of Contents, which is quite detailed. My plan for teaching the course was somewhat bold and ambi- tious, and could have easily miscarried had I not been blessed with a teaching assistant who became the book’s co-author. I decided to use no text either for my own preparations or as a prop for students. In- stead, I decided to present the material the way I understand it, with not only descriptions and examples, but also proofs, coming directly from my head. A mitigating factor was that, although suﬃciently broadly educated, I am not a professional topologist or geometer. Hence, the stuff I had ready in my head or could easily reconstruct should not have been too obscure or overly challenging. So, this is how the book came about. I prepared each lecture (usually without or with minimal written notes), and my TA, the third year Ph.D. student Vaughn Climenhaga, took notes and within 24 hours, usually less, prepared a very faithful and occasionally even somewhat embellished version typed in TeX. I usually did some very

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