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Algebraic Topology

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Haynes Miller, course on homotopy theory of the vector field problem, part 1 and part 2, (handwritten notes my Matt Ando).

Sometimes when the definition of a term is supplied (such as for "open simplex"), it will be immediately abused and applied to other things without comment that are not really the same thing (such as happens with "open simplex") -- thus causing countless hours of needless confusion. Three papers that achieve perfect marriages of algebraic topology and differential geometry: Thom's Quelques propriétés des variétés différentiables founded cobordism theory. I'll close this box and throw it open to the floor by sharing what may be the first such textbook available as a massive set of online notes.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. Long story short I think once you have learned the contents of Chapters 0 and 1 in Hatcher from some source, you can just read the rest of it.

This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. 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.

Many of these are available online, but often aren't well advertised or easy to find, since they usually don't get published or make it to arxiv. For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized here much more than in the books of an earlier generation. Maybe it’s just because his book has reached Rudin levels of being the standard text for the subject, but it seems to me that often whenever this book is recommended online, there is at least one person who detests it. He takes care to give you the intuition of whats going on as well as a complete and easy to follow rigorous development.

Often done with simple examples, this gives an opportunity to get comfortable with them first and makes this book about as readable as a book on algebraic topology can be.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. His beautiful concise course is a classic for good reason; we so rarely have an expert give us his "take" on a field.

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