POINCARE'S ROLE AS THE FATHER OF
ALGEBRAIC
TOPOLOGY

10-minute talk for "Science and
Art/
Poincare and Duchamp" conference at

Harvard, November 1999

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])

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.

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.

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.

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.

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.

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.

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])

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.

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.

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.

genius.

REFERENCES

D. D.M.Davis, The Nature and Power of Mathematics, Princeton (1993).

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.

Jour Diff Geom 18 (1983) 279-315.

F. M.Freedman, The topology of
four-dimensional manifolds, Jour Diff Geom 17

(1983) 357-454.

(1983) 357-454.

K. F.Klein, Bermerkungen über
den
Fusammenghang der Flächen, Math. Annalen
7

(1874) 549-557.

(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.

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.

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.

Werke 2nd ed (1854) 272-287.

S1. S.Smale, Generalized Poincare
Conjecture in
dimensions greater than four,

Annals of Math 74 (1961) 391-406.

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.

actually happened on the beaches of Rio), Math Intelligencer 12 (1990) 44-51.