390 12. MODERN GEOMETRIES
homology group. Moreover, he noted, while the order in which the cycles in a chain
were traversed was irrelevant, the fundamental group was not necessarily commu-
tative. He suggested redefining the term simply connected to mean having a trivial
fundamental group. He gave examples to show that the homology groups do not
determine the topological nature of a manifold, exhibiting three three-dimensional
manifolds all having the same homology groups, but different fundamental groups
and therefore not topologically the same (homeomorphic). He then asked a number
of questions about fundamental groups, one of which has become famous. Given
two manifolds of the same number of dimensions having the same fundamental
group, are they homeomorphic? Like Fermat's last theorem, this question has been
attacked by many talented mathematicians, and proofs have been proposed for a
positive answer to the question, but—at least until recently—all such proofs have
been found wanting.^46
4.5. Point-set topology. Topology is sometimes popularly defined as "rubber-
sheet geometry," in the sense that the concepts it introduces are invariant under
moving and stretching, provided that no tearing takes place. In the kinds of com-
binatorial topology just discussed, those concepts usually involve numbers in some
form or other—the number of independent cycles on a manifold, the Euler charac-
teristic, and so forth. But there are also topological concepts not directly related
to number.
Continuity and connectedness. The most important of these is the notion of con-
nectedness or continuity. This word denotes a deep intuitive idea that was the
source of many paradoxes in ancient times, such as the paradoxes of Zeno. As
we shall see in Chapter 15, it is impossible to prove the fundamental theorem of
algebra without this concept.^47 For analysts, it was crucial to know that if a cer-
tain function was negative at one point on a line and positive at another, it must
assume the value zero at some point between the two points. That property eventu-
ally supplanted earlier definitions of continuity, and the property now taken as the
definition of continuity is designed to make this proposition true. The clarification
of the ideas surrounding continuity occurred in the early part of the nineteenth
century and is discussed in more detail in Chapter 17. Once serious analysis of this
concept was undertaken, it became clear that many intuitive assumptions about the
connectedness of curves and surfaces had been made from the beginning of deduc-
tive geometry. These continuity considerations complicated the theory of functions
of a real variable for some decades until adequate explanations of it were found. A
good example of such problems is provided by Dedekind's construction of the real
numbers, discussed in Chapter 8, which he presented as a solution to the problem
of defining what is meant by a continuum.
(^46) As of this writing, evidence begins to accumulate that the Russian mathematician Grigorii
Perlman of the Steklov Institute in St. Petersburg has settled the Poincare conjecture (Associated
Press, January 7, 2004). As a graduate student at Princeton in 1964, when a mathematician came
to town claiming to have proved this elusive result, I discussed it with Norman Steenrod (1910-
1971), one of the twentieth century's greatest topologists. He told me that proving the conjecture,
although difficult, would be a rather uninteresting thing to do, since it would only confirm what
people already thought was true. It would have been much more exciting to disprove it.
(^47) Even the second of the four proofs that Gauss gave, which is generally regarded as a purely
algebraic proof, required the assumption that an equation of odd degree with real coefficients has
a real solution—a fact that relies on connectedness.