Homotopy and the Fundamental Group

Abstract

The results of the preceding chapter left a serious gap in our attempt to classify compact 2-manifolds up to homeomorphism: although we have exhibited a list of surfaces and shown that every compact connected surface is homeomorphic to one on the list, we still have no way of knowing when two surfaces are not homeomorphic. For all we know, all of the surfaces on our list might be homeomorphic to the sphere! (Think, for example, of the unexpected homeomorphism between ${{\mathbb{P}}^{2} \#}{{\mathbb{P}}^{2} \#}{{\mathbb{P}}^{2}} \ {\rm and} \ {{\mathbb{T}}^{2} \#}{{\mathbb{P}}^{2}}$.)