What is the best book on algebraic topology?

Another modern textbook is Algebraic Topology by Tammo tom Dieck.

Finally, there’s Algebraic Topology from a Homotopical Viewpoint by Marcelo Aguilar, Samuel Gitler, Carlos Prieto. As the title suggests, it is based on homotopy theory.

The other answer suggests Spanier’s Algebraic topology.

Why is algebraic topology so hard?

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.

What is algebraic topology good for?

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

What should I read after Hatcher algebraic topology?

You should try and learn some homological algebra at some stage (derived functors for example).

Stable Homotopy and Generalised Homology by Adams.

You can try and add to your knowledge of spectral sequences with A User’s Guide to Spectral Sequence by McCleary.

What are the prerequisites for algebraic topology?

Prerequisites: The only formal requirements are some basic algebra, point-set topology, and “mathematical maturity”. However, the more familiarity you have with algebra and topology, the easier this course will be.

What is the hardest math subject?

1. Algebra: Algebra is a branch of mathematics that studies symbols and the rules that control how they are used.

How can I prepare for topology exam?

You should attempt solving problems on the topics of your exam….Some good sources are:

1. Lee’s “Introduction to Topological Manifolds” which covers all of your topics.
2. Hatcher’s point set topology notes (links to the pdf)
3. Hatcher’s “Algebraic Topology” for problems on homotopy, fundamental group and covering spaces.

Is algebraic topology interesting?

In a less direct way, algebraic topology is interesting because of the way we have chosen to study space. By focusing on the global properties of spaces, the developments and constructions in algebraic topology have been very general and abstract.

Who invented algebraic topology?

H. Poincaré
H. Poincaré may be regarded as the father of algebraic topology. The concept of fundamental groups invented by H. Poincaré in 1895 conveys the first transition from topology to algebra by assigning an algebraic structure on the set of relative homotopy classes of loops in a functorial way.

Who invented differential topology?

Modern differential geometry (1900–2000) The subject of modern differential geometry emerged out of the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on the foundations of topology.

What are the prerequisites for learning topology?

How much algebra do you need for algebraic topology?

Therefore, it is fair to say a good knowledge of abstract algebra should suffices to take a first course in algebraic topology.

Do you need algebraic topology for differential geometry?

Having said that, topological theory built on differential forms needs background/experience in Algebraic Topology (and some Homological Algebra). In other words, for a proper study of Differential Topology, Algebraic Topology is a prerequisite.

Who is the father of topology?

Mathematicians associate the emergence of topology as a distinct field of mathematics with the 1895 publication of Analysis Situs by the Frenchman Henri Poincaré, although many topological ideas had found their way into mathematics during the previous century and a half.

Is topology a pure math?

Topology is a relatively new branch of mathematics; most of the research in topology has been done since 1900. The following are some of the subfields of topology. General Topology or Point Set Topology. General topology normally considers local properties of spaces, and is closely related to analysis.

Is topology a hard class?

For ex- ample, it can be difficult to determine if two topological spaces are homeomorphic. The idea of algebraic topology is to transform these questions to questions in alge- bra that may be easier to answer. The slogan for this class is topology is hard and algebra is easy.

Why do people study topology?

Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis. It is also used in string theory in physics, and for describing the space-time structure of universe.