Structure Theorem Advanced Structures

For a compact surface M, let gM be the connected sum of g copies of M. If M is a torus, we get a multi-donut surface, as shown in Figure 4.3. Definition 4.5 genus The connected sum of g tori is called a surface with genus g . The genus refers to how many “holes” the donut surface has. We are now ready to give a complete answer to the homeomorphism problem for closed compact 2-manifolds. Combining this theorem with the table in Figure 4.2b, we get the following. Corollary 4.1 χgT 2 = 2 − 2g and χgRP 2 = 2 − g. We are now ready to fully classify all compact closed 2-manifolds as connected sums, using the Euler characteristic and orientability. Theorem 4.2 homeomorphy of 2-manifolds Closed compact surfaces M 1 and M 2 are homeomorphic, M 1 ≈ M 2 , iff a χM 1 = χM 2 and b either both surfaces are orientable or both are nonorientable. Observe that the theorem is “if and only if.” We can easily compute the Euler characteristic of any 2-manifold by triangulating it. Computing orientability is also easy by orienting one triangle and “spreading” the orien- tation throughout the manifold if it is orientable. Together, χ and orientability tell us the genus of the surface if we apply Corollary 4.1 Therefore, we have a full computational method for capturing the topology of 2-manifolds. Our success in classifying all 2-manifolds up to topological type encourages us to seek similar results for higher dimensional manifolds. Unfortunately, Markov showed in 1958 that both the homeomorphism and the homotopy problems are undecidable for n-manifolds, n ≥ 4: There exist no algorithms for classifying manifolds according to topological or homotopy type Markov, 1958. We will sketch his result in an extended example later this section. Markov’s result leaves the homeomorphism problem unsettled for 3-manifolds. Three-manifold topology is currently an active area in topology. Weeks 1985 provides an accessible view, while Thurston 1997 and Fomenko and Matveev 1997 furnish the theoretical and algorithmic results. Table 4.1. Some categories and their morphisms. category morphisms sets arbitrary functions groups homomorphisms topological spaces continuous maps topological spaces homotopy classes of maps

4.1.2 Functors

A more powerful technique for studying topological spaces is to form and study algebraic images of them. This idea forms the crux of algebraic topology. Usually, these “images” are groups, but richer structures such as rings and modules also arise. Our hope is that, in the process of forming these images, we retain enough detail to accurately reconstruct the shapes of spaces. As we are interested in understanding how spaces are structurally related, we also want maps between spaces to be converted into maps between the images. The mechanism we use for forming these images is a functor. To use functors, we need a concept called categories, which may be viewed as an abstraction of abstractions. Definition 4.6 category A category C consists of: a a collection Ob C of objects; b sets Mor X,Y of morphisms for each pair X,Y ∈ ObC; including a distinguished identity morphism 1 = 1 X ∈ MorX, X for each X. c a composition of morphisms function ◦ : MorX,Y × MorY, Z → Mor X, Z for each triple X,Y, Z ∈ ObC, satisfying f ◦ 1 = 1 ◦ f = f , and f ◦ g ◦ h = f ◦ g ◦ h. We have already seen a few examples of categories in the previous chapter, as listed in Table 4.1. Definition 4.7 functor A covariant functor F from a category C to a cate- gory D assigns to each object X ∈ C an object FX ∈ D and to each morphism f ∈ MorX,Y a morphism F f ∈ MorFX, FY such that F1 = 1 and F f ◦ g = F f ◦ Fg. Figure 4.4 gives an intuitive picture of a functor in action.