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Answer : Maybe this is the picture you are looking for. I refer to a path tracing the oriented edge as a a a , and its inverse as a − 1 a^{-1} a − 1 . Note that every continuous path on the dunce cap (any CW-complex really) can be continuously deformed to trace only edges (of the given CW-structure). This should be somewhat visually intuitive, by straightening your path along edges, vertex to vertex. Thus if the path a a a is trivial, so is any other path (do you see why?). A comment as to what deformations are allowed - you are free to push your path anyway you want, and over edges (reemerging on any of the copies of the edge, noting orientations). This is what happens between the first and the second image. The one important rule is that you CANNOT ever move the basepoint (=initial and final point of your path). Note that I did not move the basepoint passing from the first to the second picture, as the three corners of the triangle are literally the same point of the dunce cap. ...

Answer : It is definitely for someone who already knows the subject and is looking for a different perspective. It is not advanced and it is not introductory, more a supplement. It is also sloppy and very hard to follow for someone who does not know the subject. The praises are from people who know the subject and like the presentation and a few things not easily found elsewhere. Overall, it is not a good book in my opinion. I think Differential Forms in Algebraic Topology is well written. However, it is written in a way a master teaches a student: There are no loss of detail, no fancy language, and explicit examples are everywhere. The student is supposed to emulate Bott's proof writing style, to criticize it, to tease it and understand every detail of it. When I read it I often find questions I would ask myself was discussed by him in very detail. But I am also aware that this is my second or third time reading the book; when I read it the first time my feeling was quite ...