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# Topology of Surfaces

### Autumn 2001

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Lecturer: Sigmundur Gudmundsson

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

A topological space $\left(X,T\right)$ 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.

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)