Algebraic Topology - Computer Graphics and Geometric Modeling - page 389

since the rank (B i (K)) terms cancel each other in the sum. This proves the theorem
because b q (K) = rank(H q (K)).

It follows from Theorem 7.2.3.10 that this combinatorially defined number c(K)
is a topological invariant associated to the underlying space ΩKΩ. In fact, it is
more than that and actually depends only on the homotopy type of ΩKΩ because that
is the case for the Betti numbers.
Theorem 7.2.3.10 suggests the following definition of a well-known invariant of a
polyhedron.

Definition. If X is a polyhedron, the Euler-Poincaré characteristic of X , c( X ), is

defined by

dim X

Â

q

() = () ()

c X 1 b X .

q

q = 0

The typical way to compute the Euler-Poincaré characteristic of a polyhedron X
is of course to use a simplicial complex K that triangulates X and use the numbers
n q (K). This is also how it is often defined. Our definition has the advantage that the
property of it being an intrinsic invariant of a polyhedron that is independent of any
triangulation is built into the definition.

7.2.4 Cell Complexes

The homology theory we developed was based on simplices, but as we have men-
tioned before, we could have used other spaces as our basic building blocks, such as
n-dimensional cubes, for example. The main advantage of simplices is a theoretical
one. They simplify some formulas and constructions. A big practical disadvantage of
simplices, however, is the fact that the simplicial complexes that triangulate spaces
typically contain a great many simplices. Even a simplicial complex that triangulates
a simple space such as the basic n-dimensional simplex already has an exponential
number of simplices (as a function of n). Any algorithm for computing homology
groups based on simplices would quickly be overwhelmed by their number for all but
relatively low-dimensional spaces. Fortunately, one can define homology groups based
on more efficient decompositions of spaces.

Definition. An open k-cell is any space c that is homeomorphic to R k . The integer k is
called its dimension is denoted by dim c . An open cell is an open c-cell for some k. A cell
decomposition of a space is a collection of disjoint open cells whose union is the space.

Note that the dimension of an open cell is well defined by Theorem 7.2.3.5.
A straightforward generalization of simplicial complexes is to look for cell decom-
positions where we allow curved cells rather than just linear cells like the simplices.
Actually, we shall go a step further.

Definition. A map f : ( X , A ) Æ ( Y , B ) is called a relative homeomorphism if f : X Æ Y
is a continuous map that maps X - A homeomorphically onto Y - B .