This emphasis also illustrates the book's general slant towards geometric, rather than algebraic, aspects of the subject. However, the passage of the intervening years has helped clarify what are the most important results and techniques. In the example where mapping cylinders were introduced, we were taking a map from the boundary of a "thick" letter to the underlying "thin" letter.

I'll list a couple I know about (I attended some of these courses), and I'll wiki this answer so people can add more. But still, at the end of the day, even though it's often the case that when I add the details to a one page proof by Hatcher it becomes a five page proof (such as for Theorem 2.

If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art. With this criticism the book presents tons really cool example and intuition behind less-straightforward constructions. He covers much more information than any of the other introductory textbooks I have perused, and with tons of explicitly worked out examples. The construction is in three stages: we first exhibit two closed curves on the torus that do not form double spirals, then two arcs on the torus that do not form spirals, and finally two arcs in the plane that do not form spirals.

You can think of that discussion of Hatcher as a "prerequisite quiz" which tests whether you have learned what you need to learn about homotopies. So put on your big boy pants and stop wasting your time looking into lesser books hoping they will be easier to understand. It has a lot of stuff, including one of the nicer introductions to spectral sequences (although I don't know a single book that does this well.I'm hoping that will not only expose me to the cutting edge,but allow me to work with one of the greats. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. Serre's thesis is nice, Hatcher's notes are OK, but this seems to be a topic best learned in a good class).

Homologie singulière des éspaces fibrés has as clear and economical an account of spectral sequences as I've seen anywhere. In the case of non-Abelian Chern-Simons theories, the relevant one-form symmetry is non-invertible, and its “gauging” corresponds to the condensation of a Lagrangian anyon. Like I said earlier: one year of algebra won't necessarily prepare you for these routine abuses by the pros; you'll need two, or else tons of free time. Revisions and Additions: I have made a number of small changes in the text itself in the years since the book was first published. Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach that can deal with various data modalities of varying complexities.Should I be spending a lot more time trying to fill in the gaps or am I supposed to gloss over the details?