The euler characteristic

Abstract

In Chapter 4 we proved that all surfaces can be classified into three types: the sphere, connected sums of tori, and connected sums of projective planes. We claimed, but did not really prove, the seemingly obvious fact that these are really different types. As you may have noticed, the obviousness of a fact never stops a mathematician from proving it. Thus, of course, our next mission is to prove that these are truly different shapes. Theoretically, there might be some way of continuously deforming the sphere so that it suddenly acquires a handle, or some way of distorting a torus so that a Ms band appears inside of it. In the language of Theorem 4.14, there might be a way of cutting and pasting the planar diagram for the Klein bottle so that it becomes the planar diagram for the torus. Lest you lose all faith, I should reassure you that none of these eventualities ever occur, but we never claimed to have investigated all possible cuttings and pastings. Two figures are topologically equivalent (Definition 2.19) if there is a homeomorphism between them. Are we then to consider all possible homeomorphisms of a space to see if one of them could possibly give the second space?

“The study of mathematics is, if an unprofitable, a perfectly innocent and harmless occupation.”

G.H. Hardy