## 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 $$**R**^{2}. Examples of compact surfaces are the sphere $S2$, the torus $T2$, the (real) projective plane $$**R**P^{2} 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 $$**R**P^{2} and the Klein bottle $K2$ are not.

To distinguish orientable surfaces we use the genus which is a natural number and a topological invariant. For the sphere $S2$ the genus is 0 but 1 for the torus $T2$. For the non-oriaentable surfaces we use the so called non-orientable genus. This is 1 for the projective plane $$**R**P^{2} 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)

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