The Euler-Poincaré Formula Homology Groups
While the results from the universal coefficient theorem are theoretically beautiful, our motivation in examining them has a computational nature. We
have seen that some rings of coefficients, such as R, are unable to capture tor- sion. If a space does not have torsion, then we may be able to craft faster algo-
rithms for computing topology by using such rings. The field of real numbers, R
, is not an option, because we do not have infinite precision on computers. The field of rational numbers, Q, does not provide any advantage, as we will
need to represent each rational exactly with two integers. The simplest prin- cipal ring, Z
2
, however, simplifies computation greatly. Here, the coefficients are either 0 or 1, so there is no need for orienting simplices or maintaining
coefficients. A k-chain is simply a list of simplices, those with coefficient 1. Each simplex is its own inverse, reducing the group operation to the symmetric
difference , where the sum of two k-chains c
, d is c + d = c ∪ d − c ∩ d. Consequently, Z
2
provides us with a best system for computing homology of torsion-free spaces.
In fact, nearly all of the spaces in this book are torsion-free. The processes described in Chapter 2 generate subcomplexes of R
3
. R
3
is not compact and creates special cases that need to be handled in algorithms. To avoid these
difficulties, we add a point at infinity and compactify R
3
to get S
3
, the three- dimensional sphere. This construction mirrors that of the two-dimensional
sphere in Definition 4.3. Algorithmically, the one point compactification of R
3
is easy, as we have a simplicial representation of space. Subcomplexes of a triangulation of S
3
do not have torsion.
5
Morse Theory
In the last two chapters, we studied combinatorial methods for describing the topology of a space. One reason for our interest in understanding topology
is topological simplification: removing topological “noise,” using a measure that defines what “noise” is. But as we saw in Section 1.2.3, the geometry
and topology of a space are intricately related, and modifying one may modify the other. We need to understand this relationship in order to develop intelli-
gent methods for topological simplification. Morse theory provides us with a complete analysis of this relationship when the geometry of the space is given
by a function. The theory identifies points at which level-sets of the func- tion undergo topological changes and relates these points via a complex. The
theory is defined, however, on smooth domains, requiring us to take a radical departure from our combinatorial focus. We need these differential concepts
to guide our development of methods for nonsmooth domains. Our exposi- tion of Morse theory, consequently, will not be as thorough and axiomatic as
the accounts in the last two chapters. Rather, we rely on the reader’s familiar- ity with elementary calculus to focus on the concepts we need for analyzing
2-manifolds in R
3
. We begin this chapter by extending some ideas from calculus to manifolds
in Sections 5.1 and 5.2. These ideas enable us to identify the critical points of a manifold in Section 5.3. The critical points become the vertices of a complex.
We define this complex by first decomposing the manifold into regions associ- ated with the critical points in Section 5.4. We then construct the complex in
Section 5.5 and look at a couple of examples.
Spivak and Well’s notes on Milnor’s lectures provide the basis for Morse theory Milnor, 1963. As an introduction to Riemannian manifolds, Morgan
1998 is beautifully accessible. O’Neill 1997 and Boothby 1986 provide good overviews of differential geometry and differential manifolds, respec-
tively. I also use Bruce and Giblin 1992 for inspiration.
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