4
Homology
The goal of this chapter is to identify and describe a feasible combinatorial method for computing topology. I use the word “feasible” in a computational
sense: We need a method that will provide us with fast implementable al- gorithms. Our method of choice will be simplicial homology, which com-
plements our representation of spaces in simplicial form. Homology utilizes finitely generated Abelian groups for describing the topology of spaces. For-
tunately, we fully understand the structure of these groups from Chapter 3. We may now define homology easily, and even venture confidently into some
advanced topics.
But first, I need to justify the choice of homology, which is weaker than both forms of topological classification we have seen earlier. I do so in the first
section of this chapter. I devote the next section to the definition of simplicial homology, a quick history of the proof of its invariance, and the relationship
of homology and the Euler characteristic. In the final section, I examine the Universal Coefficient Theorem in order to develop a faster procedure for com-
puting the topology of subcomplexes of R
3
. I borrow heavily from Hatcher 2001 and Munkres 1984 for the content
of this chapter. I am also influenced by great introductory books in algebraic topology, including Giblin 1981, Henle 1997, and, my first encounter with
the subject, Massey 1991.
4.1 Justification
The primary goal of topology is to classify spaces according to their connectiv- ity. We have seen that there are different meanings of the word “connectivity,”
corresponding to finer and coarser levels of classifications. In this section, we examine homeomorphy and homotopy and see how they are not suitable for
our purposes. In addition, we look at the powerful framework of categories
60
and functors. A classic functor, the fundamental group, motivates the defini- tion of homology.
A common tool for differentiating between spaces is an invariant.
Definition 4.1 invariant A topological invariant is a map that assigns the same object to spaces of the same topological type.
Note that an invariant may assign the same object to spaces of different topological types. In other words, an invariant need not be complete.
All that is required by the definition is that if the spaces have the same type, they are mapped to the same object. Generally, this characteristic of invariants
implies their utility in contrapositives: If two spaces are assigned different objects, they have different topological types. On the other hand, if two spaces
are assigned the same object, we usually cannot say anything about them. A good invariant, however, will have enough differentiating power to be useful
through contrapositives.
Rather than classifying all topological spaces, we could focus on interesting subsets of spaces with special structure. One such subset is the set of mani-
folds, as defined in Section 2.2. Here, we use a famous invariant, the Euler characteristic, defined first for graphs by Euler.
Definition 4.2 Euler characteristic Let K be a simplicial complex and s
i
= card
{σ ∈ K | dim σ = i}. The Euler characteristic χK is χK =
dim K
∑
i =0
−1
i
s
i
=
∑
σ
∈
K
−{∅}
−1
dim σ
. 4.1
While it is defined for a simplicial complex, the Euler characteristic is an in- teger invariant for
|K|, the underlying space of K. Given any triangulation of a space M, we always will get the same integer, which we will call the Euler
characteristic of that space χM.
4.1.1 Surface Topology
One of the achievements of topology in the nineteenth century was the classi- fication of all closed compact 2-manifolds using the Euler characteristic. We
examine this classification by first looking at a few basic 2-manifolds.
Definition 4.3 basic 2-manifolds Figure 4.1 gives the basic 2-manifolds us- ing diagrams. We may also define the sphere geometrically by S
2
= {x ∈ R
3
|
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
