Ph.D. course in geometry and topology VT2019


Georgios Dimitroglou Rizell
office: 14136.


Handouts of lecture notes will be made available here. There is no official course literature, but references to supplementary optional reading for the different topics will be given.


The participants should ideally hand in solutions to 24 exercises of choice from the lectures. In order to pass 50% of the problems should be solved correctly. The deadline is Monday 1 July.

Plan (preliminary)

01.Mon. 28/1Topological spaces, topological manifolds, homotopy groups.[Bre], [May]
02.Fri. 1/2The fundamental group and groupoid, the universal cover, higher homotopy groups.*[Bre], [May]
03.Mon. 4/2Computations, simplicial complexes, triangulations.[Bre], [May], [Arm]
04.Mon. 11/2Principal bundles, long exact sequences of homotoy groups.[Hus], [Bre], [May]
05.Wed. 13/2Homotopy lifting property and examples.[Bre], [May]
06.Fri. 1/3More examples of principal bundles, classifying spaces.[Hus]
07.Tue. 5/3The Poincaré homology sphere, classification of principal bundles.[Hus]
08.Mon. 11/3Introduction to knot theory. Smooth manifolds.[Mil]
09.Thu. 14/3The isotopy extension theorem. Knot projections and knot diagrams.[Kos], [Bur]
10.Mon. 18/3Reidemeister moves, the Quandle invariant.[Bur], [Man]
11.Thu. 21/3Linking numbers, surfaces with and without boundary[Man], [Arm]
12.Mon. 25/3Seifert surfaces, connected sum*[Man]
13.Tor. 28/3Connected sum, Intersection form, Seifert matrix, Alexander polynomial[Arm],[Bur]
14.Fri. 5/4Finitely presented groups, the Braid group[Man]
15.Mon. 8/4Jones polynomial, Seifert van Kampen theorem, the knot group, the Wirtinger representation[Man],[Bre]
16.Tue. 7/5Tangent bundle, Lie bracket, de Rham complex.[Bot]
17.Fri. 10/5Integration of forms, Lie groups, Adjoint representation[Bot],[Kos]
18.Tue. 14/5Cartan one-form, Maurer-Cartan equation, Ehresmann connection[Son]
19.Fri. 24/5S1-bundles, Parallel transport, Monodromy (holonomy)[Son]
20.Tue. 28/5Flatness, curvature, Gauss-Bonnet theorem[Son]
* Lecture notes have been updated.
  • Exercises 18, 19, 25.2 have been clarified.
  • New hint in Exercise 13 and corrected formula just before (L.6).
  • Definition of torus knots in Lecture 12 has been corrected.
  • Sign convention for the Lie bracket has been corrected.