Mathematics, Faculty of Science

Sigmundur Gudmundsson

Topology of Surfaces

Autumn 2001

 

Lecturer: Sigmundur Gudmundsson

Literature: James R. Munkres, Topology, Prentice Hall (2000) (approx. sections 51-60 and 70-78)

A topological space (X,T) is said to be a surfaces if it is locally homeomorphic to an open subset of the standard R2. Examples of compact surfaces are the sphere S2, the torus T2, the projective plane RP2 and the celebrated Klein bottle K2, seen above.

The main aim of this course is to construct and classify compact surfaces. They can be divided into two families consisting of those which are orientable and those which are not. The sphere S2 and the torus T2 are orientable but the projective plane RP2 and the Klein bottle K2 are not.

To distinguish between surfaces in the same family we use the genus which is a natural number and a topological invariant of a surfaces. For the sphere S2 the genus is 0 but 1 the projective plane RP2 the torus T2 and 2 for the Klein bottle K2.

Keywords homotopy, the fundamental group, covering spaces, the Brouwer fixed point theorem, the Borsuk-Ulam theorem, deformation retracts, fundamental groups and homology of surfaces, cutting and pasting, the construction and classification of compact surfaces.

Course Programme: Exercises

Examination: Oral exam only.

History of Topology

VIPs: Euler (1707-1883), Möbius (1790-1868), Betti (1823-1892), Jordan (1838-1922), Klein (1849-1925), Poincaré (1854-1912), Brouwer (1881-1966), Hopf (1894-1971), Borsuk (1905-1982), Ulam (1909-1984)

The MacTutor History of Mathematics Archive