Advanced Geometry 2
UNITS/SISSA Laurea Magistrale


“final” version of the notes: notesAG2.pdf (last update Nov. 4 2020)

(this is called “final” version, but there will certainly be many typos and mistakes!)


The exam will consist in an oral exam with questions on the topics from the notes and on the exercises therein.


You can practice by solving the exercises from the notes.

Additional exercises from the book “Introduction to smooth manifolds”, by Lee:

all exercises from chapters 1, 2, 3, 4, 5, 6, 8, 9, 10, 11


There are also some exams from the previous years (unfortunately some of them are in Italian):

07/02/17 28/09/17 07/06/17 21/02/17 03/09/18 25/06/18 14/02/18 31/01/18

28/01/19 26/02/19 07/05/19 23/07/19 26/09/19


Alternatively, the brave student can choose to give a short seminar on one of the topics below:


  1. Solve the “Knot problem” exercise, i.e. Exercise 90 from the notes.

  2. Extend Remark 118 to the case of the Grassmannian of k-planes in R^n.
    At some point you will need to understand the cellular structure of Grassmannians using Schubert cells, as in Chapter 6 of the book “Characteristic classes” by Milnor. (Another valuable source for this is the draft of the book “Vector bundles and k-theory”, of Hatcher, available     here)
    This will require a little work, I am available to give hints and discuss some steps together with interested students.

  3. Study the classification of vector bundles on spheres (Section 3.5 of the notes)

  4. Proof of the Nerve Lemma. Study Section 4.G (“Gluing constructions”) and the proof of Corollary 4G.3 from Hatcher’s book, available here

  5. The Whitney arc. Study the construction from the paper “A function not constant on a connected set of critical points”, by H. Whitney, published on. Duke Math. J. 1, (1935) 514-517. (contact me for the reference if you cannot download the paper)

  6. Poincaré–Hopf theorem
    Study the proof presented in Section 6 of the book “Topology from the differentiable viewpoint” by Milnor (contact me if you do not have access to the reference)