Summary of the topics covered in class  

 

  1. October 8th 2014: Review of fundamental topological concepts, like topological spaces (Euclidean topology on R^n, discrete and trivial topology on a nonempty set X), continuous maps, product and quotient topology, Haussdorff property, compactness, connectedness, homeomorphisms. Basic examples: the 2-sphere, the torus T^2 and the 2-dimensional real projective space. Motivation for introducing algebraic objects for distinguishing spaces.

  2. October 9th 2014: Definition of (topological) manifold and surface. Presentation of a surface as the quotient of a 2-gon. Examples covered in details: the sphere, the torus, the projective space. Definition of orientable surface of genus g and Klein bottle. Definition of triangulation.

  3. October 15th 2014: Definition of Euler characteristic of a compact surface (using triangulations). Explicit examples of "good triangulations", including the minimal triangulation of the torus (see the picture below). Orientable and nonorientable surfaces. Proof that the Möbius strip and the 2-dimensional real projective space are nonorientable. Definition of connected sum of two surfaces. Statement of the classification theorem of compact surfaces.

 

 

 

The Figure was done by Alessandro Fasse. It represents the minimal triangulation of the torus, with exactly 7 vertices, 21 edges and 14 faces. The triangles are in white, while the torus comes from identifying the two pairs of parallel black edges in the picture.

4. October 16th 2014: Definition of homotopic maps, homotopy equivalent spaces, contractible space, retraction, deformation retract. Definition of real projective space.

 

5. October 22nd 2014: Paths: composition, inverse path, equivalence of two paths. Path-connectedness. Fundamental group: definition, group structure and independence on the base point within the same path-connected component.

 

6. October 23rd 2014: Functorial properties of the fundamental group: invariance with respect to homeomorphisms and homotopy equivalences. Fundamental group of the circle, part 1: Proof of the path lifting property and homotopy lifting property (sketch)

 

7. October 29th 2014: The fundamental group of the circle and its consequences, among which: Brouwer fixed point theorem in dimension 2, Fundamental Theorem of Algebra. Fundamental group of a product of two spaces.

 

8. October 30th 2014: Van Kampen's Theorem. Computation of the fundamental group of spheres.

 

9. November 5th 2014: More examples. Computation of fundamental group of surfaces. Definition of CW complex. 

 

10. November 6th 2014: Some properties of the fundamental group of CW complexes. Introduction to homology theory - motivation.

 

11. November 12th 2014: Chain complexes. Simplicial homology: definition of delta complexes and of simplicial homology groups. Examples.

 

12. November 13th 2014: Singular homology theory. Properties of singular homology theory. Computation of H_0(X), and of singular homology groups  of a point. Reduced homology groups. Functoriality properties of homology groups. Definition of chain maps and chain homotopic maps. 

 

13. November 19th 2014: Theorem: Homotopic maps induce the same homomorphism in homology. Invariance of homology for homotopy equivalent spaces. Relation between the first fundamental group and the first homology group. 

 

14. November 20th 2014: Computation of the first homology group of spheres and compact surfaces. Exact sequences. Short exact sequence of complexes and `Snake Lemma'.

 

15. November 26th 2014: Relative homology groups. Long exact sequence of the pair. Excision theorem (idea of the proof).

 

16. November 27th 2014: Computation of homology groups of spheres. Brouwer fixed point theorem. Local homology groups. Degree of maps from S^n to S^n and properties.

 

17. December 3rd 2014: Theorem on the existence of nowhere vanishing continuous vector fields on spheres. Cellular homology - definition and main properties. Equivalence of cellular homology and singular homology. Computation of the homology groups of the complex projective space and spheres using cellular homology. 

 

18: December 4th 2014: Analysis of the boundary operator in cellular homology. Computation of the (cellular) homology groups of the following spaces: compact surfaces (orientable and non-orientable) and real projective spaces.

Prof. SILVIA SABATINI

Ph.D.