Back in the day there must have been a movement towards thin sleek books. Of course, there's tradeoffs. On the downside there's a lack of narration and context - the usual what, why, and where we're going type of stuff. The upside is what mathematicians call 'elegance'. For the layman this can be summarized as describing an object or thought in its most minimal form, whatever that may be.

My rating system of five stars is based on how successful the two authors succeed in the thin book paradigm. That is, I think there's enough there to "get it" and what's there is correct (or with a minimum of erratas). I agree with all the reviewers that gave this book 5 stars and 1-2 stars (it sucks). With respect to the 5 star folks, I agree that the authors met their objectives stated in the preface. They succeeded of creating an engineering schematic or wiring diagram to get from A to B. With respect with the 1-2 star folks I concur with their opinion. Yes, this book would not pass an editorial board standards based on modern publishing criteria. Nowadays it's more than just the handwaving. For example, reading chapter 1 section 1 I don't think there's enough there to explain to your grandmother what a manifold is and why should we care about them - a true test of mastery.

My advice to the 1-2 star folks that's not used to reading these thin sleek books is a technique I call pre-reading the book. It's a three-fold process. If you're forced into a quarter sytem (12-14 week) math class covering the whole book I highly recommend doing an 80-hour crash study session prior to the first class. Otherwise it'll seem like you just walked into a middle of a movie. First, read the preface to see where the authors 'think' they're going. Second, map out the key chapters and problems associated with each topical goal. Third, starting from the end of the book and going all the way to the start, build a dependency outline linking the 'big' result at the end with all the preceding. You'll be surprised how quickly you can cover what initially seemed advanced book. Most of the 'filler' can be gotten off the web. There's alot out there since this book has been used for the last 30 plus years.

My excitement from reading this book is how the authors bridge the study of smooth compact objects to topics that previously were in the realm of algebraic topology. These objects can be classified quickly by looking at global properties such as how many holes, described by the genus, how much curvature they have, described by the euclidean characteristic, how many vector zeros on the surfaces, described by how many bald spots would appear if hair sprouted on the surface and someone wanted to comb it and how many times the surface wraps around the interior, described by its degree. The authors do a good job of pretending like you don't have to know anything about algebraic topology but like I stated in the previous paragraph I couldn't resist googling because without getting some precursory knowledge it felt like being in the middle of a movie.

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