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
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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