Topology: 2nd edition

Product Information

Follows the present-day trend in the teaching of topology which explores the subject much more extensively with one semester devoted to general topology and a second to algebraic topology. Ex.___ For a senior undergraduate or first year graduate-level course in Introduction to Topology. Appropriate for a one-semester course on both general and algebraic topology or separate courses treating each topic separately.

Topology - MIT Mathematics Intro to Topology - MIT Mathematics

James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT [1] and the author of several texts in the area of topology, including Topology (an undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.Advanced topics—Such as metrization and imbedding theorems, function spaces, and dimension theory are covered after connectedness and compactness.

MIT Mathematics

This text is designed to provide instructors with a convenient single text resource for bridging between general and algebraic topology courses. 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. Optional, independent topics and applications can be studied and developed in depth depending on course needs and preferences. Features The reason I've given this long explanation (because I hope it will also help others studying Topology who have similarities), is because the path most Topology students follow is the following Among Munkres' contributions to mathematics is the development of what is sometimes called the Munkres assignment algorithm. A significant contribution in topology is his obstruction theory for the smoothing of homeomorphisms. [3] [4] These developments establish a connection between the John Milnor groups of differentiable structures on spheres and the smoothing methods of classical analysis.Another subfield is geometric topology, which is the study of manifolds, spaces that are locally Euclidean. For example, hollow spheres and tori are 2-dimensional manifolds (or “2-manifolds”). Because of this Euclidean feature, very often (although unfortunately not always), a differentiable structure can be put on manifolds, and geometry (which is the study of local properties) can be used as a tool to study their topology (which is the study of global properties). A very famous example in this field is the Poincaré conjecture, which was proven using (advanced) geometric notions such as Ricci flows. Of course, algebraic tools are still useful for these spaces. This seminar is an introduction to knot theory, and there is often one each year. Like other junior seminars, students are expected to learn and present a topic on their own. Topics covered vary, but typically include tri-colorability of knots and links, numerical knot invariants such as the crossing number, unknotting number and bridge number, and polynomial invariants such as the Jones polynomial and the Alexander-Conway polynomial. More advanced students may learn about homology invariants, such as the Khovanov homology and the Heegaard Floer homology. Carefully guides students through transitions to more advanced topics being careful not to overwhelm them. Motivates students to continue into more challenging areas. Ex.___ Greatly expanded, full-semester coverage of algebraic topology—Extensive treatment of the fundamental group and covering spaces. What follows is a wealth of applications—to the topology of the plane (including the Jordan curve theorem), to the classification of compact surfaces, and to the classification of covering spaces. A final chapter provides an application to group theory itself. Unless one is (and you are not!) planning to write a PhD thesis in General Topology, Munkres is (more than) enough.

Math 131 - Fall 2019 - Harvard University

He was elected to the 2018 class of fellows of the American Mathematical Society. [5] Textbooks [ edit ] Deepen students' understanding of concepts and theorems just presented rather than simply test comprehension. The supplementary exercises can be used by students as a foundation for an independent research project or paper. Ex.___ It is great to study topology at Princeton. Princeton has some of the best topologists in the world; Professors David Gabai, Peter Ozsvath and Zoltan Szabo are all well-known mathematicians in their fields. The junior faculty also includes very promising young topologists. Prof. Gabai has been an important figure in low-dimensional topology, and is especially known for his contributions in the study of hyperbolic 3-manifolds. Profs. Ozsváth and Szabó together invented Heegaard Floer homology, a homology theory for 3-manifolds. After finishing the sequence MAT 365 and MAT 560, topology students can consider taking a junior seminar in knot theory (or some other topic), or, if that is not available, writing a junior paper under the guidance of one of the professors. (Both junior and senior faculty members are probably willing to provide supervision.) 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. The graduate courses are challenging, but not impossible, so interested students are recommended to speak to the respective professors early. It may also be beneficial to learn other related topics well, including basic abstract algebra, Lie theory, algebraic geometry, and, in particular, differential geometry. Courses Each of the text's two parts is suitable for a one-semester course, giving instructors a convenient single text resource for bridging between the courses. The text can also be used where algebraic topology is studied only briefly at the end of a single-semester course. Ex.___Munkres completed his undergraduate education at Nebraska Wesleyan University [2] and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise. Earlier in his career he taught at the University of Michigan and at Princeton University. [2] NEW - Greatly expanded, full-semester coverage of algebraic topology—Extensive treatment of the fundamental group and covering spaces. What follows is a wealth of applications—to the topology of the plane (including the Jordan curve theorem), to the classification of compact surfaces, and to the classification of covering spaces. A final chapter provides an application to group theory itself. The study of 1- and 2-manifolds is arguably complete – as an exercise, you can probably easily list all 1-manifolds without much prior knowledge, and inexplicably, much about manifolds of dimension greater than 4 is known. 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. This is an active area of research, and in recent years has been found to be closely related to quantum field theory in physics. While I certainly have a lot more Differential Topology and Algebraic Topology to learn (and I look forward to it), I also feel like I should learn a bit more of General Topology. Firstly I apologize if this is a bit of a soft question, it's hard for me to ask this quite concretely so I do apologize if this post doesn't seem like I'm asking something immediately.