Past exam of the geometry course of the University of Oxford / 2009 / A / 1

The classification of compact surfaces states that every compact connected topological surface without boundary is homeomorphic to exactly one of the sphere, a connected sum of surfaces consisting of $g\ge 1$ tori, or a connected sum of $h\ge 1$ real projective planes. The torus family is orientable and the projective-plane family is nonorientable; their respective genera are unique. Including $g=0$ makes the sphere the genus-zero orientable case. If boundaries are allowed, the number of removed open discs is an additional invariant.

A finite subdivision into vertices, edges and polygonal faces defines the Euler characteristicCount cells after all identifications. Subdividing an edge adds one vertex and one edge; splitting a face adds one edge and one face. Each operation leaves $V-E+F$ unchanged. Comparing subdivisions through a common refinement proves independence of the subdivision.

For the connected sum of surfaces, remove an embedded open disc from each surface and identify the resulting boundary circles by a homeomorphism. The resulting closed connected surface is denoted $M\#N$.

Removing a disc decreases the Euler characteristic by one. Gluing the remaining surfaces along their common boundary subtracts the characteristic of a circle, which is zero. ThusThis can be verified by choosing matching subdivisions of the boundary circles and counting the vertices and edges identified in the gluing.

A square model of the torus has one vertex, two edges and one face, so its Euler characteristic is zero. The connected-sum formula givesThe real projective plane has a bigon model with one vertex, one edge and one face, hence characteristic one. Therefore

Take $M=T^2$, $N={\mathbb{RP}}^2\#{\mathbb{RP}}^2$ and $S={\mathbb{RP}}^2$. The first is orientable, while $N$, the Klein bottle, is not, so $M$ and $N$ are not homeomorphic.

Both $M\#S$ and $N\#S$ are nonorientable and have Euler characteristic $-1$. By the classification of compact surfaces, both are homeomorphic to the connected sum of three projective planes. HenceThis is failure of cancellation for connected sums of surfaces.