Algebraic Topology

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So the example of mapping cylinders we saw before will not be an illustration of what we're doing with mapping cylinders here. Coming to the book, this is great starting point for Algebraic Topology if you know Point set Topology ( because he assumes that the reader knows Quotient Topology). For every n, we construct two arcs in the plane that intersect at least n times and do not form spirals. In this book the authors present the technical tools needed for proving rigorously the classification theorem, give a detailed proof using these tools, and also discuss the history of the theorem and its various proofs.

One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales. I'm hoping that will not only expose me to the cutting edge,but allow me to work with one of the greats. There is a recent beautiful textbook that's a very good addition to the literature, Davis and Kirk's Lectures in Algebraic Topology - but most of the material in that book is pre-1980 and focuses on the geometric aspects of the subject.Whether this is a good book or a bad book depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory.

In terms of notation, if A is a subspace of X, Hatcher just assumes in Chapter 0 that you know what X/A is supposed to mean (the cryptic mutterings in the user-hostile language of CW complexes on page 8 don't help).

The only thing that comes to mind that might be controversial is the usage of Delta complexes, which I’ve heard is seldom used elsewhere and doesn’t do much to simplify material.