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I am wondering which chapters in Lee's "Introduction to Topological Manifolds" are most important for further studies in differential geometry. Here is the table of contents.

I intended to read through the first four chapters along with chapters 7, 11, 12 and 13 since I am aware that the topics covered in those chapters feature in various differential geometry texts.

  1. Can I safely skip the other chapters without negatively impacting my studies of differential geometry?

  2. Do chapters 7,11,12 and 13 require any knowledge from the skipped chapters?

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  • $\begingroup$ I've read both books (albeit not that thoroughly) and I would say that if you already took a course in topology (e.g. Munkres) you could probably skip "Topological Manifolds" and begin directly with "Smooth Manifolds". In particular, the chapters on covering spaces and homology in "Topological Manifolds" aren't really used at all in "Smooth Manifolds" except for a couple of arguments in the later chapters (17+ or so). The rest of "Topological Manifolds" is basically a normal treatment of topology emphasizing aspects which are usually neglected in a normal course $\endgroup$ – Chill2Macht Mar 2 '17 at 10:37
  • $\begingroup$ (in particular, an emphasis on proper maps and exhaustion by compact sets come to mind). "Topological Manifolds" is also a good elementary introduction to CW complexes (assuming I remember correctly), but the theory of CW complexes isn't used at all in "Smooth Manifolds", although it is useful knowledge for a study of Differential Topology. $\endgroup$ – Chill2Macht Mar 2 '17 at 10:39
  • $\begingroup$ It is probably worth mentioning that "Smooth Manifolds" has an appendix covering briefly all of the topology prerequisites for the book -- I don't remember whether the appendix gives explicit references to the corresponding chapters in "Topological Manifolds", but even if not hopefully they are not difficult to infer using the Table of Contents and Index of "Topological Manifolds". Thus, in short, if you are only interested in differential geometry right now, I would recommend skipping "Topological Manifolds" entirely, reading the topology appendix of "Smooth Manifolds", looking up gaps. $\endgroup$ – Chill2Macht Mar 2 '17 at 10:42

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