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