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In-Depth Information
CHAPTER 7
Algebraic Topology
7.1
Introduction
The central problem of algebraic topology is to classify spaces up to homeomorphism
by means of computable algebraic invariants. In the last chapter we showed how two
invariants, namely, the Euler characteristic and orientability, gave a complete classi-
fication of surfaces. Unfortunately, these invariants are quite inadequate to classify
higher-dimensional spaces. However, they are simple examples of the much more
general invariants that we shall discuss in this chapter.
The heart of this chapter is its introduction to homology theory. Section 7.2.1
defines the homology groups for simplicial complexes and polyhedra, and Section
7.2.2 shows how continuous maps induce homomorphisms of these groups. Section
7.2.3 describes a few immediate applications. In Section 7.2.4 we indicate how homol-
ogy theory can be extended to cell complexes and how this can greatly simplify some
computations dealing with homology groups. Along the way we define CW complexes,
which are really the spaces of choice in algebraic topology because one can get the
most convenient description of a space with them. Section 7.2.5 defines the incidence
matrices for simplicial complexes. These are a fundamental tool for computing
homology groups with a computer. Section 7.2.6 describes a useful extension of
homology groups where one uses an arbitrary coefficient group, in particular,
Z
2
. After
this overview of homology theory we move on to define cohomology in Section 7.3.
The cohomology groups are a kind of dual to the homology groups. We then come to
the other major classical topic in algebraic topology, namely, homotopy theory. We
start in Sections 7.4.1 and 7.4.2 with a discussion of the fundamental group of a topo-
logical space and covering spaces. These topics have their roots in complex analysis.
Section 7.4.3 sketches the definition of the higher-dimensional homotopy groups and
concludes with some major theorems from homotopy theory. Section 7.5 is devoted
to pseudomanifolds, the degree of a map, manifolds, and Poincaré duality (probably
the single most important algebraic property of manifolds and the property that sets
manifolds apart from other spaces). We wrap up our overview of algebraic topology
in Section 7.6 by telling the reader briefly about important aspects that we did not
have time for and indicate further topics to pursue. Finally, as one last example,
