I did as many exercises as I could out of this textbook as an undergraduate one summer, and I believe that doing so took my mathematical maturity to the next level. However, for a long time, many aspects of 3- and 4-manifolds had evaded study; thus developed the subfield of low-dimensional topology, the study of manifolds of dimension 4 or below.

Topology: Readings and Homework - Harvard University Topology: Readings and Homework - Harvard University

Web icon An illustration of a computer application window Wayback Machine Texts icon An illustration of an open book. We build and maintain all our own systems, but we don’t charge for access, sell user information, or run ads. Verð þó að viðurkenna að ég las ekki alla kaflana, komst ekki í gegnum seinasta þriðjunginn um það bil.Deepen students' understanding of concepts and theorems just presented rather than simply test comprehension. In particular, I think the section on coverings and fundamental group to be quite good as a reference (or to learn about for the first time).

Topology - MyMathsCloud Elements Of Algebraic Topology - MyMathsCloud

Two separate, distinct sections (one on general, point set topology, the other on algebraic topology) are each suitable for a one-semester course and are based around the same set of basic, core topics. A couple of cheap, but good, books are Point Set Topology by Gaal and Topology for Analysis by Wilansky. I don't know if this is because Munkres is an especially great textbook, or because the material is just naturally easy, but I certainly didn't see any flaws in the book. Truth be told, this is more of an advanced analysis book than a Topology book, since that subject began with Poincare's Analysis Situs (which introduced (in a sense) and dealt with the two functors: homology and homotopy). It is also a good idea to learn Morse theory, which is an extremely beautiful theory that decomposes a manifold into a CW structure by studying smooth functions on that manifold.Yes, General Topology is fun and there are many neat old theorems that you will learn by studying it in more detail, but you have to prioratize: Life is short and your time in graduate school is even shorter.

Math 131 - Fall 2019 - Harvard University

It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.

The book itself might not be the best for you, depending on things like what level of rigor (or lack thereof) you're comfortable with, but the exercises are excellent, and the book is freely available from Hatcher's own home page.