POINCARE'S ROLE AS THE FATHER OF
ALGEBRAIC
TOPOLOGY
10-minute talk for "Science and
Art/
Poincare and Duchamp" conference at
Harvard, November
1999
Rhonda invited me to this conference
because of
something that I wrote about
Poincaré in my book, "The =
Nature
and Power of Mathematics." ([D]) This book was
written for a
nontechnical audience, more specifically a course called
"Introduction
to Mathematical Thought" that I taught to liberal arts students
at
Lehigh University. The specific material that Rhonda liked was
about how
2-
dimensional beings on a spherical world could use geometry to tell
that
they
were on a spherical world, and how the same world could be
modeled as a
flat
world with unusual rules of geometry. These are ideas
that
Poincaré presented
in his popular writings such as
"Science and
Hypothesis."([P3])
However, my field of expertise is
algebraic
topology, and since this conference
is half about Poincaré,
and since
Poincaré is generally considered to be the
most important
figure in
the early history of algebraic topology, I decided to
say a few
words about
Poincaré's role in the development of algebraic topology.
Topology, literally the study of
surfaces, is a
form of geometry in which we
don't care about specific measures such
as
length and curvature, but rather we
deal with properties such as the
number
and types of holes, properties which
are not altered by continuous
changes
such as stretching. In algebraic
topology, we often reduce the
questions about these generalized surfaces to
questions in algebra,
and then
study the algebraic questions.
Topology as a subject began to take
shape
between 1850 and 1870 in work of the
German mathematicians Riemann
([R1]),
Listing ([L]), Möbius ([M]), and Klein
([K]) who
succeeded in
showing that every orientable 2-dimensional surface is
equivalent to
the
surface of a doughnut with a certain number of holes. Here
"orientable" is a technical term that means you can have
vectors
sticking out
from the entire surface in a continuous way, and
"equivalent" means that you
can deform one surface into
the other
without any tearing or amalgamating. The
surface of a
doughnut, also
called a torus, is 2-dimensional because if can be
completely
covered by
flexible patches. Think of an innertube. We emphasize
that we
are not thinking of the solid doughnut, only its outer surface.
Here
are pictures of some of these models of 2-dimensional
surfaces.
One advance primarily due to
Riemann ([R2])
was the study of n-dimensional
manifolds, for any positive integer
n.
An n-dimensional manifold is something
that is composed of subsets
each of
which is equivalent to an n-dimensional
ball, just as we have noted
that a
torus is composed of 2-dimensional patches.
We usually deal
with n
dimensions using coördinates, although Poincaré
introduced more
geometric approaches to dimensionality.
In a long paper called
"Analysis
Situs"([P1]), published in 1895, Poincaré
revolutionized
the subject by introducing algebraic quantities, now called the
fundamental
group and homology groups, that can be associated to topological
spaces in
such a way that if two topological spaces differ with respect to any
of
these quantities, then one can say for sure that these spaces are not
equivalent, i.e. that one cannot be deformed to the other. For
example, the
first homology group of the n-holed torus pictured
above is
what is called a
free abelian group of rank 2n, and since, for
different
values of n, these
groups are different, one can assert that the
spaces are
not equivalent. The
first homology group and the
fundamental
group both deal, in slightly
different ways, with the different
sorts of
loops in a topological space.
Essentially, each hole in an
n-holed
torus has two types of loops around it.
Poincaré's
initial
treatment of these ideas was not totally rigorous, but this
paper laid
the groundwork for the next 30 years of work in topology.
Also in this paper and its
supplements,
Poincaré investigated the extent to
which the fundamental
group and
homology groups characterize a space. That is,
if two spaces
have the
same fundamental group and the same homology groups, are
they
necessarily
equivalent topological spaces? After at least one flawed
attempt, he
formulated a conjecture that a 3-dimensional manifold with the same
fundamental group and homology groups as those of a 3-dimensional
sphere
must
be equivalent to a sphere. Two comments are in order
here: The
3-dimensional
sphere is not the sphere that we pictured
earlier. That
was 2-dimensional.
The 3-dimensional sphere is an analogue of
that one
dimension higher. Second,
some care with the precise notion of
equivalence is required. Topological
equivalence here means a
1-1
correspondence between the points of the two
spaces such that points
which
are close together in one space correspond to
points which are close
together in the other.
This problem, known as the
3-dimensional
Poincaré Conjecture, remains unsolved
to this day, probably
the most
famous and important outstanding question in
topology. The
same
question can be raised for manifolds of any dimension, not
just 3,
and
ironically, it has been proved to be true in all dimensions other
than the
dimension, 3, in which it was originally conjectured. It may seem
counterintuitive that the problem is easier for high-dimensional
manifolds
than
for 3-dimensional manifolds; the reason for the difficulty in 3
dimensions is
that there is less room for certain kinds of
modifications to
take place.
In dimensions 5 and above, this
Generalized
Poincaré Conjecture was proved to
be true in 1960 by Stephen
Smale
([S1]), who was then a postdoctoral fellow
associated with research
institutes in Princeton and Rio de Janeiro. A few
years later,
Smale
had to justify to some bureaucrats the way in which their
grant
money was
being spent, and his phrase that his "best known work was done
on the
beaches of Rio de Janeiro" became widely publicized.([S2])
In 1981, the Generalized
Poincaré
Conjecture in dimension 4 was proved to be
true by Michael Freedman
([F]) of
University of California at San Diego. Both
Smale and Freedman
became
extremely famous for their work. Both won the Fields
Medal,
the
mathematical equivalent of the Nobel Prize.
As an offshoot of Freedman's work,
it was proved
by Oxford graduate student
Simon Donaldson in 1982 ([Do]) that there
is a
topological space which is
topologically equivalent to R4 and which
is a
differentiable manifold, meaning
that it has a notion of smoothness,
but in
which the notion of smoothness is
fundamentally different than it is
in the
standard version of R4. This result
was shocking to
mathematicians and
physicists, because in all other dimensions
there is only one
possible
notion of smoothness in Euclidean space. It is
particularly
interesting because space-time in which most physicists work is
4-
dimensional. Donaldson also won the Fields Medal for his
work.
In conclusion, I would say that all
of this work
traces back to Poincaré's
genius.
REFERENCES
D. D.M.Davis,
The Nature and
Power of Mathematics, Princeton (1993).
Do. S.Donaldson, An application of
gauge theory
to four-dimensional topology,
Jour Diff Geom 18 (1983)
279-315.
F. M.Freedman, The topology of
four-dimensional manifolds, Jour Diff Geom 17
(1983)
357-454.
K. F.Klein, Bermerkungen über
den
Fusammenghang der Flächen, Math. Annalen
7
(1874) 549-557.
L. J.B.Listing, Vorstudien zur
Topologie,
(1848).
M. A.F.Möbius, Theorie
der
elementaren Verwandtschaft, Werke 2 (1863) 433-471.
P1. H.Poincaré, Analysis
Situs, J. Ec.
Polytech ser 2, vol 1 (1895) 1-123.
P2. H.Poincaré,
Cinquième
complément à l'analysis situs, Palermo Rend 18 (1904)
45-110.
P3. H.Poincaré, Science et
Hypothesis,
(1902).
R1. G.F.B.Riemann, Grundlagen
für eine
allgemeine Theorie des Functionen einer
veränderlichen
complexen
Grösse, Werke 2nd ed (1851) 3-48.
R2. G.F.B.Riemann, Über die
Hypothesen,
welche der Geometrie zu Grunde liegen,
Werke 2nd ed (1854)
272-287.
S1. S.Smale, Generalized Poincare
Conjecture in
dimensions greater than four,
Annals of Math 74 (1961)
391-406.
S2. S.Smale, The Story of the Higher
Dimensional
Poincare Conjecture (What
actually happened on the beaches of Rio),
Math
Intelligencer 12 (1990) 44-51.