Topology, in broad terms, is the study of those qualities of an object that are invariant under certain deformations. Such deformations include stretching but not tearing or gluing; in laymen’s terms, one is allowed to play with a sheet of paper without poking holes in it or joining two separate parts together. (A popular joke is that for topologists, a doughnut and a coffee mug are the same thing, because one can be continuously transformed into the other.)
A topology on an object is a structure that determines which subsets of the object are open sets; such a structure is what gives the object properties such as compactness, connectedness, or even convergence of sequences. For example, when we say that [0,1] is compact, what we really mean is that with the usual topology on the real line R, the subset [0,1] is compact. We could easily give R a different topology (e.g., the lower limit topology), such that the subset [0,1] is no longer compact. Point-set topology is the subfield of topology that is concerned with constructing topologies on objects and developing useful notions such as separability and countability; it is closely related to set theory.
There are other subfields of topology. One subfield is algebraic topology, which uses algebraic tools to rigorously express intuitions such as “holes.” For example, how is a hollow sphere different from a hollow torus? One may say that the torus has a “hole” in it while the sphere does not. This intuition is captured by the notion of the fundamental group, which, (very) loosely speaking, is an algebraic object that counts the number of “holes” of a topological space. There are other useful algebraic tools, including various homology and cohomology theories. These can all be viewed as a mapping from the category of topological spaces to algebraic objects, and are very good examples of functors in the language of category theory; it is for this reason that many algebraic topologists are also interested in category theory.
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.
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.
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.
MAT 365: Topology
This is the first course in topology that Princeton offers, and has been taught by Professor Zoltan Szabo for the last many years. The course, following the classic textbook by Munkres, is a careful study of point-set topology. It begins with examining different topologies one can put on familiar spaces, and constructions such as product, metric and quotient topologies. The second part of the course is concerned with developing notions of “desirable” properties, including the separability axioms (T0 , T1 and the Hausdorff condition), as well as countability axioms (first- and second-countability). The final part of the course is an introduction to the fundamental group π1; after some initial calculations (including for the circle), more general tools such as covering spaces and the Seifert-van Kampen theorem are used for more complicated spaces. The latter quarter of the course covers basic notions in algebraic topology (in Munkres, but significantly overlapping with the earliest parts of Hatcher/MAT 560). This course is a prerequisite for all other topology courses at Princeton. Not only should all students interested in topology take this course, but since it deals with so many basic notions that one will certainly meet in the future, almost every mathematics student should take this course. As a bonus, this course satisfies the geometry requirement of the department.
Junior Seminar in Knot Theory
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.
MAT 560: Algebraic Topology
Even though this course is a 500-level, it is aimed at both undergraduate and graduate students. This course is an introduction to algebraic topology, and has been taught by Professor Peter Ozsvath for the last few years. It typically covers the bulk of the classic textbook by Hatcher, including CW complexes, the fundamental group, simplicial and singular homology, and tools to compute these homologies. The dual notion of singular cohomology may also be covered. This course is designed for more serious students of topology and this subject is essentially a prerequisite for any more advanced study in topology. In addition, MAT 345 or equivalent comfort with group theory is strongly recommended before enrolling in this course. After having taken MAT 365, students should have some idea about their interest and comfort with topology, so it is advisable for those considering a research career in the field to immediately follow up with this course.
MAT 56x: Topics in Topology
These graduate courses vary on a semester-by-semester basis and are taught by Professors Gabai, Ozsvath and Szabo.
Chuen Ming Mike Wong ’12 (cw2688[at]columbia[dot]edu)