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
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)
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.