Please take this page in conjunction with the Part III Guide to Courses Geometry and Topology section.
Prerequisite areas
Differential Geometry
The course generally starts from scratch, and since it is taken by people with a variety of interests [including topology, analysis and physics] it is usually fairly accessible. It is an important stepping stone for many other geometry courses.
You will find this helpful for the following Part III courses:
- Complex Manifolds
- [Algebraic Topology]
- Other geometry and geometric analysis courses which change from year to year [eg Riemannian Geometry]
- Theoretical Physics courses [eg General Relativity, Symmetries, Fields and Particles, Applications of Differential Geometry to Physics]
Relevant undergraduate courses are:
- Differential Geometry
- Riemann Surfaces
- Algebraic Topology
- Geometry 1B
First level prerequisites
Linear algebra: abstract vector spaces and linear maps, bilinear forms. See e.g. Ib Linear Algebra.
Multi-variable calculus: derivatives of functions as linear maps, the chain rule, partial derivatives, Taylor's theorem in several variables. See e.g.Ib Analysis II. You can check if you are at the required level by doing the following exercises: Analysis II 2015-16 Sheet 4 [Questions 4, 5, 11].
Solutions to first-order differential equations [Picard's theorem]
Elementary point-set topology: topological spaces, continuity, compactness etc. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by doing the following exercises: Met & Top 2015-16 Example Sheet 1
Second level prerequisites
Some exposure to ideas of classical differential geometry, e.g. Riemannian metrics on surfaces, curvature, geodesics.
Useful books and resources
Notes from the Part II Course.
Milnor's classic book "Topology from the Differentiable Viewpoint" is a terrific introduction to differential topology as covered in Chapter 1 of the Part II course. It is quite different in feel from the Part III course but would be great to look at in preparation.
Nakahara "Geometry, Topology and Physics". This is not a pure maths book, so comes with a warning that it is not always completely precise and rigorous. It also covers lots of material outside the Part III course. However, it is excellent for giving an intuitive picture of the concepts, and may be particularly helpful to physicists taking the course.
Algebraic Topology
Relevant undergraduate courses are:
- Part II Algebraic Topology
First level prerequisites
Second level prerequisites
Some experience of some version of homology in algebraic topology. For example you should know about:
- Homotopic maps and homotopy equivalence of spaces.
- Chain complexes and exact sequences.
- Simplicial homology. [Or another type of homology.]
Useful books and resources
Chapter 1, Algebraic Topology, Allen Hatcher, CUP, 2009
Part II notes for Algebraic Topology on Oscar Randal-Williams’ teaching page
Typical Scheduling:
The differential topology of smooth manifolds.
Course Text:
At the level of A Comprehensive Introduction to Differential Geometry, Vol. 1, Spivak or Differential Topology Guillemin and Pollack
Topic Outline:
- Definition of differential manifolds
- Vectors bundles
- Tangent vectors, vectors fields and flows
- Smooth functions on manifolds, derivatives
- Regular values, Morse functions, transversality, degree theory
- Tensors and forms
- Integration on manifolds, Stokes theorem and de Rham cohomology
Topology is the study of properties of geometric objects which do not involve distance or angle; allowing consideration of such additional properties leads to Geometry.
- 18.900 [Geometry and Topology in the Plane] provides an introduction to concepts and constructs in geometry and topology using examples accessible with only 18.0x preparation. Prerequisite: 18.03 or 18.06.
- 18.901 [Introduction to Topology] explores the notion of continuity at its deepest level. Prerequisite: 18.100.
- 18.904 [Seminar in Topology] is an introduction to algebraic topology: the fundamental group, covering spaces, classification of surfaces. Prerequisite: 18.901.
- 18.950 [Differential Geometry] explores concepts of curvature, focusing on curves and surfaces. Prerequisites: 18.100 and Linear Algebra.
- 18.952 [Theory of Differential Forms] gives a modern theoretical treatment of Stokes' theorem using differential forms. Prerequisites: 18.101 and either 18.700 or 18.701.
- 18.994 [Seminar in Geometry] Prerequisites: 18.100 and Linear Algebra.
I would say, it depends on how much Differential Topology you are interested in. Generally speaking, Differential Topology makes use of Algebraic Topology at various places, but there are also books like Hirsch' that introduce Differential Topology without [almost] any references to Algebraic Topology. 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.
Addendum [book recommendations]:
1] For a general introduction to Geometry and Topology:
- Bredon "Topology and Geometry": I can wholeheartedly recommend it! First part covers all the necessary and important general topology, then moves on to Differentiable Manifolds, after which it goes to Algebraic Topology [fundamental groups, [co]homology,homotopy theory]. I think it is a good first book because it is self-contained, has exercises and gives a taste of different basic parts of modern geometry and topology without leaving the impression that they are isolated from each other.
2] For Algebraic Topology:
Hatcher "Algebraic Topology": personally, I find his book overrated and quite annoying. It made me hate algebraic topology in my undergraduate years! YMMV.
Spanier "Algebraic Topology": probably not a good first book, a little outdated, but much less annoying than Hatcher's. There is a lot to learn from it!
Switzer "Algebraic Topology": great second book for Algebraic Topology, covering various topics of modern topology.
Rotman "Introduction to Algebraic Topology": good introduction to very basic Algebraic Topology.
tom Dieck "Algebraic Topology": good introduction to Algebraic Topology, but it contains several typos, so it's probably not so great for beginners. On the bright side, that keeps you on your toes to make sure you are paying attention :P It covers a good amount of topics too.
Greenberg and Harper "Algebraic Topology... a first course": "reader-friendly" introduction to basic Algebraic Topology, but some prior experience with the language of category might be helpful.
Davis and Kirk "Lecture notes on algebraic topology": since they are lecture notes, the material is "compressed" without filling in between [which is nice in my opinion] and covers various important topics of Algebraic Topology. The notes can be downloaded from their homepage .
May "A concise course in algebraic topology": in my opinion, not suitable for readers without prior experience with Algebraic Topology [unless they are very gifted students].
3] For Algebraic Topology with homotopical focus:
Selick "Introduction To Homotopy Theory": it is suited for readers who already have some experience with the basic concepts of Algebraic Topology, Category theory and Homological Algebra. Although the first part deals with all these prerequisites, the material would get dense for readers without prior exposure to them.
Gray "Homotopy Theory - An Introducton to Algebraic Topology": nice self-contained introductory exposition with exercises, suitable for beginners in Algebraic Topology [knowledge of general point-set topology is assumed]
Fomenko, Fuchs "Homotopic Topology" - it is one of those books where you have to work your way through it [i.e. lots of stuff left to the reader as exercises].
Aguilar, Gitler, Prieto "Algebraic Topology from a Homotopical Viewpoint": only requires solid understanding of basic general topology. Prior experience with Algebraic Topology and Category Theory might be helpful, but not necessary.
Strom "Modern Classical Homotopy Theory": little gem that only assumes solid understanding of basic general topology. All the theory is presented as mini-problems to work through. What better way to learn something than to work it out on your own. Thus some experience with Algebraic Topology might be helpful, but not strictly necessary. And it is self-contained in the sense that it takes care of the necessary category theory.
Whitehead "Elements of homotopy theory": requires a first course in algebraic topology.
Warner "Topics in Topology and Homotopy Theory": not suitable for beginners, it is more of an encyclopedia [mere 900+ pages].
4] For Differential Topology:
Hirsch' "Differential Topology": self-contained, in particualar requires no prior knowledge of Algebraic Topology;
Milnor's "Topology from Differentiable Viewpoint": self-contained, in particular requires no prior knowledge of Algebraic Topology;
Milnor's "Morse Theory": the classic book on Morse theory;
Guillemin and Pollack's "Differential Topology": self-contained, the last chapter introduces some cohomology theory, thus it does not omit this important tool.
5] For Differential Algebraic Topology:
Milnor and Stasheff's "Characteristic Classes": A first course in smooth manifolds might be helpful, but not necessary.
Bott and Tu's "Differential Forms in Algebraic Topology": another classic text, the title is self-explanatory;
Kreck's "Differential algebraic topology - from stratifolds to exotic spheres": advanced text.