Definition 3.27 vector space Let F be a field and V be an Abelian group. A vector space over F
is a unitary F-module, where V is the associated Abelian group. The elements of F are called scalars and the elements of V are called
vectors . We often refer to V as the vector space.
We briefly quickly recall some familiar properties of vector spaces.
Theorem 3.18 basis, dimension If we can write any vector in a vector space
V as a linear combination of the vectors in a finite linearly independent subset B
= {α
i
| i ∈ I} of V , B forms a basis for V and V is finite-dimensional with dimension
|B|. As for free Abelian groups, the dimension is invariant over the set of bases for
the vector space. Our final new concept for this section is that of gradings. Given a ring, we
may be able to decompose the structure into a direct sum decomposition, such that multiplication has a nice form with respect to this decomposition.
Definition 3.28 graded ring A graded
ring is
a ring
R, +, ⊗ equipped with a direct sum decomposition of Abelian groups R ∼
=
i
R
i
, i ∈ Z,
so that multiplication is defined by bilinear pairings R
n
⊗ R
m
→ R
n +m
. El- ements in a single R
i
are homogeneous and have degree i, deg e = i, for all
e ∈ R
i
. If a module is defined over a graded ring as just defined, we may also seek a
similar decomposition for the module.
Definition 3.29 graded module A graded module M over a graded ring R is a module equipped with a direct sum decomposition, M ∼
=
i
M
i
, i ∈ Z, so that the action of R on M is defined by bilinear pairings R
n
⊗ M
m
→ M
n +m
. Our decomposition may be infinite in size. We will be interested, however, in
those gradings that are bounded from below.
Definition 3.30 non-negatively graded A graded ring module is non- negatively graded
if R
i
= 0 M
i
= 0, respectively for all i 0.
Example 3.10 standard grading Let R [t] be the ring of polynomials with
indeterminate t. We may grade R [t] non-negatively with t
n
= t
n
· R[t], n ≥ 0. This is called the standard grading for R
[t].
3.3.4 Structure Theorem
Building upon the concept of a group, we have defined a number of richer structures. A natural question that arises is the classification of these structures.
The fundamental theorem Theorem 3.10 gave a full description of finitely generated Abelian groups in terms of a direct sum of cyclic groups. Amazingly,
the theorem generalizes to any PID or graded PID.
Theorem 3.19 Structure Theorem If
D is
a PID,
then every
finitely generated D-module is isomorphic to a direct sum of cyclic D-modules. That is, it decomposes uniquely into the form
D
β
⊕
m i
=1
D d
i
D ,
3.1 for d
i
∈ D, β ∈ Z, such that d
i
|d
i +1
. Similarly, every graded module M over a graded PID D decomposes uniquely into the form
n i
=1
Σ
α
i
D ⊕
m j
=1
Σ
γ
j
D d
j
D ,
3.2 where d
j
∈ D are homogeneous elements so that d
j
|d
j +1
, α
i
, γ
j
∈ Z, and Σ
α
denotes an α-shift upward in grading.
In both cases, the theorem decomposes the structures into free left and torsion right parts. In the latter case, the torsional elements are also homogeneous.
In the statement of the theorem, there is some new notation. For exam- ple, we write the free part of the module with a a power notation. That
is, D
β
is the direct sum of β copies of D, where β is the Betti number for the
PID. The shift operator Σ
α
simply moves an element in grading i to grading i
+ α. Note that if D = Z, we get Theorem 3.10. If D = F, where F is a field, then the D-module is a finite-dimensional vector space V over F, and we see
that V is isomorphic to a direct sum of vector spaces of dimension 1 over F. These are two of the cases that will concern us in our discussion of homology
in the next chapter.
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
