Modules, Vector Spaces, and Gradings

v v v v v a a b b v b b a a v w w v v v v b b a a Fig. 4.1. Diagrams above and corresponding surfaces. Identifying the boundary of the disk on the left with point v gives us a sphere S 2 . Identifying the opposite edges of the squares, as indicated by the arrows, gives us the torus T 2 , the real projective plane R P 2 , and the Klein bottle K 2 , respectively, from left to right. The projective plane and the Klein bottle are not embeddable in R 3 . Rather, we show Steiner’s Roman surface, one of the famous immersions of the former and the standard immersion of the latter. |x| = 1}. The torus plural tori T 2 is the boundary of a donut. The real pro- jective plane RP 2 may be constructed also by identifying opposite antipodal points on a sphere. S 2 and T 2 can exist in R 3 , as shown in Figures 1.7 and 2.4. Both RP 2 and the Klein bottle K 2 , however, cannot be realized in R 3 without self-intersections. Example 4.1 χ of basic 2-manifolds Let’s calculate the Euler characteristic for our basic 2-manifolds. Recall that the surface of a tetrahedron triangulates a sphere, as shown in Figure 2.13. So, χS 2 = 4−6+4 = 2. To compute the Eu- ler characteristic of the other manifolds, we must build triangulations for them. We simply triangulate the square used for the diagrams in Figure 4.1, as shown in Figure 4.2. This triangulation gives us χT 2 = 9 − 18 + 27 = 0. We may complete the table in Figure 4.2b in a similar fashion. As χT 2 = χK 2 = 0, the Euler characteristic by itself is not powerful enough to differentiate be- tween surfaces. We may connect manifold to form larger manifolds that have complex connec- tivity. Definition 4.4 connected sum The connected sum of two n-manifolds 1 2 6 7 8 6 1 2 3 4 5 3 a A triangulation for the diagram of the torus T 2 2-Manifold χ Sphere S 2 2 Torus T 2 Klein bottle K 2 Projective plane RP 2 1 b The Euler characteristics of our basic 2-manifolds Fig. 4.2. A triangulation of the diagram of the torus T 2 = Fig. 4.3. The connected sum of two tori is a genus 2 torus. M 1 , M 2 is M 1 M 2 = M 1 − ˚ D n 1 ∂ ˚ D n 1 =∂ ˚ D n 2 M 2 − ˚ D n 2 , 4.2 where D n 1 , D n 2 are n-dimensional closed disks in M 1 , M 2 , respectively. In other words, we cut out two disks and glue the manifolds together along the boundary of those disks using a homeomorphism. In Figure 4.3, for example, we connect two tori to form a sum with two handles. Suppose we form the connected sum of two surfaces M 1 , M 2 by removing a single triangle from each and identifying the two boundaries. Clearly, the Euler characteristic should be the sum of the Euler characteristics of the two surfaces minus 2 for the two missing triangles. In fact, this is true for arbitrary shaped disks. Theorem 4.1 For compact surfaces M 1 , M 2 , χM 1 M 2 = χM 1 + χM 2 − 2. 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.