How to know if you are "tough enough" to study Algebraic Topology
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.
Applications of algebraic topology? - Mathematics Stack Exchange
2015年6月20日 · In short, algebraic topology gives certainty and clarity of understanding of global topology in higher-dimensional manifolds where the intuition cannot go. That's just my opinion anyway. Of course, then you would have to ask why anyone would want to know what the global topology of a topological space looks like. But that's actually a different ...
big picture - Advanced algebraic topology topics overview
2012年1月8日 · If you want a more modern perspective on algebraic topology, Peter May's "A concise course in algebraic topology" phrases a lot of things in terms of fiber/cofiber sequences, takes an axiomatic approach to (co)homology, and discusses the bare bones of some more advanced topics like K-theory.
Modern Research in Algebraic Topology - Mathematics Stack …
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).
algebraic topology - Definition of n-cell and cell complex ...
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.
reference request - Learning Roadmap for Algebraic Topology ...
2021年4月16日 · These are very good and comprehensive books which have stood the test of time; books that present algebraic topology properly and algebraically. Many great algebraic topologists grew up on these books. That having been said, I am also a fan of Munkres Elements of Algebraic Topology which works out examples very nicely using simplicial ...
How much algebra and how much topology is there in "algebraic …
2019年11月24日 · In modern (or more advanced) algebraic topology, you will also start to need some more advanced notions and nowadays algebraic topology heavily relies on category theory. This will of course be introduced along the way but you should be prepared to deal with more abstract objects and heavy machinery.
How to study algebraic topology? - Mathematics Stack Exchange
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.
Why algebraic topology is also called combinatorial topology?
Combinatorial topology is the older name for algebraic topology when all topological problems were expressed, set up and solved in Euclidean space of dimensions 1,2 and 3. In such spaces, all topological invariants-such as the fundamental group-can be expressed combinatorially via simplexes and related objects.
Applications of algebraic topology - Mathematics Stack Exchange
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 ...