We put forward an alternative approach in which, rather than summing over bulk geometries, one gauges a one-form global symmetry of the bulk theory. 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. A couple of notes from courses by Mike Hopkins on elliptic cohomology and related stuff: 1995, 1999. 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.

The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.To clarify, this is not a rant against the book for the heck of it: I really want to try and read this book. But I wanted to know if anyone might have any advice for someone that wants to learn most efficiently/effectively when reading this book. If you want to see the full abstract nonsense type picture you will have to use a book like Spanier's. This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales.

I just wanted to add that I learned a big part of what I know about homology, cohomology and homotopy theory from Hatchers book and I think that that part of the book is a great source for that. This is the first place I've found explanations (that I understand) of things like Mayer-Vietoris sequences of homotopy groups, homotopy pushout and pullback squares etc. But be warned that if you're having difficulty understanding Hatcher, you have zero chance with Spanier. Now, there are usually a lot of examples in each section of the text, but only a small minority of them actually help illuminate the central concepts.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.

which is flimsy because it begins with the wisdom of the burning bush: "We can decompose the Klein bottle as the union of two Mobius bands glued together by a homeomorphism between their boundary circles. So put on your big boy pants and stop wasting your time looking into lesser books hoping they will be easier to understand. Haynes Miller, course on homotopy theory of the vector field problem, part 1 and part 2, (handwritten notes my Matt Ando). This introductory textbook is suitable for use in a first-year graduate course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises.This introductory text is suitable for use in a course on the subject or for self study, featuring broad coverage and a readable exposition, with many examples and exercises. The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides. BibTeX key MR1867354 entry type book address Cambridge year 2002 pages xii+544 publisher Cambridge University Press mrreviewer Donald W.