2016年6月17日 · Algebraic topology, by it's very nature,is not an easy subject because it's really an uneven mixture of algebra and topology unlike any other subject you've seen before.However,how difficult it can be to me depends on how you present algebraic topology and the chosen level of abstraction.

There are also relationships between more classical algebraic geometry and algebraic topology, e.g. as in Hirzebruch's book and his proof of his version of the Riemann--Roch theorem. These sorts of connections don't seem to be as much in vogue right now (although I am not an expert in algebraic topology by any means, so may be wrong on this).

2015年6月20日 · Algebraic topology gives us better tools to answer these questions (although it is a homotopy-type "tester" a priori). Not only that, but homotopy and homology are clearly related to a notion of "holes" in a given space: a notion that is very geometric.

2010年5月2日 · Tom Dieck's "Algebraic Topology" calls a triad $(Y;X_1,X_2)$ excisive for a given homology theory (where ...

2010年8月13日 · What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free. The deeper the methods used, the better. All the above can be proved with just the fundamental ...

2016年4月2日 · I have also looked at the book Basic Topology by Armstrong, there it looks different. He starts with the definition of n-simplex. This time the set up is not a Topological Space; it is $\mathbb{R}^n$ and first few simplexes are point, line, triangle and tetrahedron then he moves on to define a simplicial complex.

2024年10月20日 · Sadly, most courses on algebraic topology are extremely handwavy, especially those that follow Hatcher's book. It's not just non formal proofs, this book doesn't even contain formal definitions of some important terms. I also strongly prefer Munkres's style. His "topology" book is indeed a great source to learn about the fundamental group.

Algebraic topology. There is an excellent book by Allen Hatcher called Algebraic Topology that is available for free on his website, and also as a hard copy on Amazon. This is an excellent geometrically oriented book on the subject that contains much of what you would learn in a graduate course on the subject plus a large number of additional ...

2023年9月28日 · Topology and Geometry, by Breadon is book primarily on algebraic topology but it treats the subject using differential manifolds, so you probably enjoy reading its 2-3 chapters. Here is the link . Foundations of Differentiable Manifolds and Lie Groups, by Warner.

2015年5月25日 · A major and important area of algebraic topology. The categorical requirements for appreciating the Quillen machinery are not modest, and the book does a great job presenting all that is needed. Having said all that, if you find a text that you like but which avoids the category theoretic language, then you should be able to quite easily fill ...