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.

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 ...

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 ...

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.

I would like to mention that in An Epsilon of Room, remark 1.1.3, Tao states:. The notion of a measurable space (X, S) (and of a measurable function) is superficially similar to that of a topological space (X, F) (and of a continuous function); the topology F contains ∅ and X just as the σ-algebra S does, but is now closed under arbitrary unions and finite intersections, rather …

Motivation: I am taking my first graduate course in algebraic topology next semester, and, up to this point, I have never taken the time to learn any category theory. I've read that category theory helps one to understand the underlying structure of the subject and that it was developed by those studying algebraic topology.