EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2361-2369
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

An Outreach Note on the Poincaré Conjecture for
Non-specialists

Daniele Ettore Otera

Vilnius University, Institute of Data Science and Digital Technologies, Akademijos st. 4,
LT-08663, Vilnius, Lithuania

Abstract. The Poincaré Conjecture, a problem formulated by the French mathematician Henri
Poincaré more than a century ago, has been one of the main challenge of modern mathematics. It
states that any three-dimensional space which is closed on itself and without holes can be deformed
into a sphere of dimension 3.
Even if the conjecture was solved at the beginning of this century, it still remains a mysterious,
appealing and intriguing problem worth to be further studied in detail. The purpose of this short
popularizing note is, on the one hand, to provide a quick overview for non-experts of what we know
today about the Poincaré Conjecture and its related problems in dimension 3, and, on the other
hand, to explain why it has represented a central problem in mathematics.

2020 Mathematics Subject Classifications: 57R60, 57K35
Key Words and Phrases: 3-manifolds, Poincaré conjecture, differential and geometric structures

1. Introduction

On the occasion of the new millennium, and a century after the famous International
Congress of Mathematics held in Paris in 1900 where David Hilbert drew up his famous
list of 23 unsolved mathematical problems at that time, the Clay Mathematics Institute
in Cambridge, Massachusetts, chose a new group of seven difficult problems/conjectures
that were still unsolved in the years 2000, awarding a prize of one million dollars for the
solution of each one of them. The millennium prizes were announced once again in Paris
in the spring of 2000, and among these seven great questions of the new century stand
out the Poincaré Conjecture (which is easy to state and a century old), and the all-famous
Riemann Hypothesis, formulated in 1859, the only conjecture that was already part of
Hilbert’s 23 problems of 1900.

Of all these problems, only one has been solved in the meantime: the Poincaré Conjec-
ture, settled by the Russian mathematician Grigori Perelman in 2003 [3, 4]. The resolution
of this century-old conjecture, along with the fact that he refused the 1-million prize, has
drawn the attention of the general public especially to this problem.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5351

Email address: daniele.otera@mif.vu.lt - daniele.otera@gmail.com (D. E. Otera)

https://www.ejpam.com 2361 © 2024 EJPAM All rights reserved.



D. E. Otera / Eur. J. Pure Appl. Math, 17 (3) (2024), 2361-2369 2362

2. Preliminaries

Since we want to address to an audience of non-experts, we will start from scratch, in
order to be able to state and comment the Poincaré Conjecture and its generalizations.

2.1. Manifolds

We need to start by defining and talking about the central topics of interest for us:
manifolds of dimension n, where, for simplicity, we will consider a manifold as an object
considered in a space of dimension N (greater than n). A manifold of dimension 1 (n = 1)
is a line or a curve (in our standard plane R2 for example), while a manifold of dimension
2 (n = 2) is what we commonly call a surface (in our 3-dimensional space R3). With
more fantasy and abstraction, we can imagine a 3-dimensional manifold M as a subspace
of a space of dimension N ≥ 4 (imagine R4 as Einstein’s space-time), such that, locally,
it looks like our 3-dimensional space (as well as, for instance, a local piece of a surface
resembles a piece of the real plane), and so on for any natural number n.

Just like prime or complex numbers, it turns out that manifolds are also central objects
in the architecture of modern mathematics. They are in fact the basic building blocks of
the branch of mathematics called Topology (literally the study of “places and forms”).
And the Poincaré Conjecture is a cornerstone of the classification of them.

Now, just as a single coordinate is sufficient to identify a point on a curve, two num-
bers (coordinates) are needed to identify a point on a surface. For example, on the earth’s
surface, (which is a two-dimensional sphere S2), we need longitude and latitude. Inci-
dentally, the fact that this parametrization possesses anomalies (e.g. all meridians meet
at the north and south poles, where longitude therefore ceases to be well defined) is a
sign of a basic topological fact: the sphere S2 is topologically different (technically not
“homeomorphic”, see here below) to the torus T 2, which is the surface of a tyre.

Obviously, the sphere S2 differs also from (is not homeomorphic to) the Euclidean
space R2 either, but this is an easier thing to understand, since the intuitive fact that
the sphere is a “closed” surface while the plane R2 is “open” corresponds to a topological
difference that is well encoded (a manifold is said to be closed if it has no boundary and
takes up a finite region of space).

We end this section by giving an idea of what a homeomorphism is. Homeomor-
phisms are equivalences in the category of topological spaces, more precisely bijective
and continuous correspondences in both directions. In particular, homeomorphisms are
those functions which preserve all the topological properties of a given manifold (topologi-
cal space). And it turns out that two manifolds are homeomorphic if one can continuously
(i.e. without cutting or glueing) deform the first manifold into the second one.

The Poincaré conjecture states that the 3-sphere is the only three-dimensional compact
manifold without boundary and without ‘holes’, up to homeomomorphisms, i.e. it is the
only such manifold where any closed path can be contracted to become a point.



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2.2. From the sphere S2 to the ball B3

The reader will certainly be familiar with at least one 3-dimensional manifold, namely
our three-dimensional space, R3, but we can construct many other ones in any dimension.

Indeed, also real world mechanics or physics force us to move from dimension 2 to
dimensions 3, 4 or higher. These higher dimensional manifolds are objects comparable to
surfaces, but where 3, 4 or more (local) coordinates are needed to identify a point. For
example, the motion of three bodies subjected to the force of gravity can be studied as an
18-dimensional manifold, each body being defined by three spatial coordinates and three
velocity coordinates.

If we imagine the two-dimensional sphere as the (exterior) surface (or boundary) of a
three-dimensional ball, then we can conceive spheres in three, four or more dimensions...
And so we realize that the n-dimensional sphere is the boundary of the ball of dimension
n + 1. Also, we may figure out that, in each dimension n, the n-sphere is somehow the
simplest possible closed manifold to study.

To be more precise, the n-sphere Sn is the set of points of the Euclidean Rn+1 space
which are at distance 1 from the origin, and elementary analytic geometry helps us by
providing its explicit equation Sn = {x1, x2, · · · , xn+1 ∈ Rn+1 such that x21 + x22 +
· · ·+ x2n+1 = 1} ⊂ Rn+1; while together with its interior one obtains the n+ 1 ball Bn+1

expressed in Cartesian coordinates as Bn+1 = {x1, x2, · · · , xn+1 ∈ Rn+1 such that x21 +
x22 + · · ·+ x2n+1 ≤ 1} ⊂ Rn+1.

Let us now imagine a sphere S2, like the surface of the earth, deprived of the north
pole, namely S2 − N . With a little more imagination it is not difficult to see that this
punctured sphere is, in some sense, “contractible”. That is: each point p of S2 −N (with
p different from the south pole) belongs to a single meridian (this is not true for both the
poles), and so it can slide continuously along this meridian until it reaches the south pole.

The same thing can be done for a sphere of any dimension n. And even in this case,
the “meridians” of Sn −N meet only at the south pole.

But we can also conceive something more complicated, some kind of spheres Σn where
the continuous flow from Σn−N goes towards the south pole but in a much more compli-
cated way, crossing and intersecting not only in the south pole but also elsewhere. These
“sorts of spheres” are called homotopy spheres (they are objects which are similar to
spheres but different from them, because they are shaped in a different way). In other
words, a homotopy sphere is a closed manifold Σn such that Σn − P can be deformed
continuously (always remaining in Σn − P ) to a point (like normal spheres).

3. Formulation of the Poincaré Conjecture

The Poincaré Conjecture states that the only homotopy sphere Σ3 of dimension 3
is the sphere S3 (up to homeomorphisms) [7]. In other words, this means that one can
always find a continuous flow from Σ3−N (the north pole) towards the south pole without
intersections, crossovers or overlaps, except at the south pole. This is also equivalent to
saying that Σ3−N = R3 (this is clear for S3, by means of the “stereographic projection”).



D. E. Otera / Eur. J. Pure Appl. Math, 17 (3) (2024), 2361-2369 2364

To be more precise, Poincare’s own formulation in 1904 [6], although equivalent to the
one just mentioned, was a little bit different: Poincaré indeed conjectured that the only
closed and “simply connected” 3-dimensional manifold was just (homeomorphic to) S3.
Where a manifold V is said to be simply connected (a new topological notion introduced
by Poincaré himself) if every closed curve (i.e. a loop) in V can be deformed continuously
to a point, remaining within V .

Note that the first statement given above at the beginning of this section has the ad-
vantage of being valid in every dimension. In particular, one can formulate the generalized
Poincaré Conjecture, which says that for every n, a homotopy sphere Σn is (homeomorphic
to) Sn. But the step from n = 3 to any n, appeared some thirty years after Poincaré, and
moreover, no further progress was made in any dimension greater than 2 until the 1950s.

On the other hand, although formulated in a very different way, the case n = 2,
which is much more easy, was already known since the middle of the 19th century. More
precisely, in dimension 2, the corresponding statement is that in any closed surface which
is different from a 2-sphere S2 one can find at least a loop which cannot be continuously
contracted to a point. This result actually follows from a far more detailed and deeper
theorem: the classification of closed and connected two-dimensional manifolds, which was
proved in different forms since the 1860s, and which says that every compact surface is
homeomorphic to a sphere with some number of handles or cross-caps attached.

3.1. TOP versus DIFF

Now, we come back to the sphere S2, seen as the surface of our globe B3. This object
is obviously homeomorphic, i.e. topologically equivalent, to the surface of an ellipsoid (a
rugby ball), but also to the surface of a cube (because we can imagine a play dough sphere,
which we can model, without breaking it, as either an ellipsoid or a cube). However, the
ellipsoid is smooth, like the sphere, while the surface of the cube is not, because there are
edges, corners and points.

Hence, we can say that the equality between the sphere and the ellipsoid is realized by
functions that possess continuous derivatives (and these functions are called diffeomor-
phisms = differentiable homeomorphisms, a more restrictive notion than that of homeo-
morphism). The sphere and the ellipsoid are both homeomorphic and diffeomorphic. On
the contrary, the topological equivalence between the sphere and the cube surface is only
possible through simple continuous functions that do not admit derivatives: these two sur-
faces are homeomorphic, but not diffeomorphic. The sphere is smooth and differentiable,
while the cube is not.

Now, if we only consider manifolds of dimension less than or equal to 3, then this
distinction is an unnecessary pedantry, because in those cases the categories of topological
and differentiable manifolds coincide (i.e. we know how to round off edges). Conversely,
in dimensions greater than 3, it turns out that there are obstructions to rounding objects,
and there exist examples of non-smootable manifolds.

For example, in 1956 the famous mathematician J. Milnor proved that on the sphere of
dimension 7 coexist several (actually 28) differentiable structures that are not diffeomorfic



D. E. Otera / Eur. J. Pure Appl. Math, 17 (3) (2024), 2361-2369 2365

to each other. Therefore there exist “exotic” spheres, i.e. spheres admitting differential
structures that are different from the standard one (see [3]).

Hence, the general idea at the time was that, as the dimension increases, the difficulties
could only increase too. But then, around 1960, S. Smale realized the opposite, and proved
at once the Generalized Poincaré Conjecture in any dimension at least 5: a manifold which
is a homotopy sphere Σn is homeomorphic to a sphere Sn, for every n ≥ 5.

The reasons why the difficulties decrease can be explained very simplistically as follows:
in large dimensions, there is a lot of empty space to manoeuvre around the problems and
to develop the needed geometric constructions to solve them and to establish the equality
to be proved. While in dimensions smaller than 3 (1 or 2), there is not enough space to
create problems. Finally, in dimension 3, namely the starting point of Poincaré, there is
both the possibility of having problems but also very little space to act...

The limit situation is dimension 4. This is a world of its own, very different from the
others (dimension 2, 3 or higher dimensions, see [5]). In this case it actually took a great
tour-de-force to prove the Poincaré Conjecture: in 1982 M. Freedmann proved that Σ4

is topologically equal (homeomorphic) to S4. Actually, thanks to Freedmann’s work, we
know nowadays that precisely in dimension 4 (the dimension of our space-time) there is
a great mystery between the topological and the differential pictures. And the smooth
4-dimensional Poincaré Conjecture is still an open problem, far from being solved.

Summarizing, we know that the generalized Poincaré Conjecture can be true or false in
the different categories (TOP or DIFF), depending on the dimension, thank to the work of
several esteemed mathematicians, such as John Milnor, Steve Smale, Michael Freedman,
and Grigori Perelman (see [8]), all of whom have been awarded the Fields Medal, which
is the most prestigious prize in mathematics, the equivalent of Nobel Prize.

In particular, in the category TOP (i.e. for topological manifolds) the generalized
Poincaré Conjecture is true in all dimensions! Whereas in the DIFF category (i.e. for
differential manifolds) it is true in dimensions 1, 2, 3, 5 and 6, it is still open in dimension
4, whereas in the other cases it is generally false.

Notice that, even if the original Poincaré Conjecture in dimension 3 has been solved
a century after its first formulation, in all that time, various ways to attack the problem
have been developed and tried, as it is related to various areas of mathematics, from group
theory to differential equations, from physics to general relativity. And even though it has
remained unproven for a hundred years, many profound new results have emerged from
the techniques developed to solve it (see [1–4, 8–10]). In few words, all the work devoted
to the conjecture improved the deep understanding of the world of 3-manifolds.

4. Geometric Structures in dimension 3

The Poincaré Conjecture is a purely topological problem. Nevertheless, all efforts by
topologists to prove it have failed, and in fact no topological proof exists to this day.
Therefore, for its resolution, a good idea is to leave the topological framework, and to use
geometric or analytic methods in order to have more tools to attack the problem.



D. E. Otera / Eur. J. Pure Appl. Math, 17 (3) (2024), 2361-2369 2366

One of the most suitable strategies is to equip the manifolds with some geometric
structures. And in fact, it was just this approach that has proved successful in the end.

This path is very much related to another work by Poincaré himself, his famous Uni-
formization Theorem. This result concerns surfaces (hence we are in dimension 2) and,
roughly speaking, tells us that on each surface we can put a geometry (i.e. a way of mea-
suring distances and angles) that, locally, is like one of the three classical geometries with
constant curvature: Euclidean geometry (the plane with zero curvature), non-Euclidean
geometry called Lobachevsky’s hyperbolic geometry (the one where several lines parallel
to a given line pass through a point, called the plane with negative curvature -1), and that
of the round sphere (called elliptic geometry with positive curvature +1).

The obvious question is now: can we say something similar in dimension 3? In the 1970s
the great American mathematician William Thurston [9] conceived a very spectacular
program in order to “geometrize” all closed 3-dimensional manifolds, as Poincaré did in
dimension two, with the difference that in dimension three the possible geometries should
be 8 instead of 3, and among them, the most important being the hyperbolic one.

4.1. Curvature

In order to provide a geometry to a manifold, a natural procedure is to specify, at each
point of the manifold, how the distance between two very close points is expressed. In a
more mathematical way, one must specify, at each point, the “metric tensor”. Introduced
by the German mathematician Bernhard Riemann in the 19th century, this tensor (a
generalization of the notion of vector) is used to determine lengths, angles, areas, volumes,
etc. on the given manifold. For a manifold of dimension n, it is a table of n2 numbers (a
matrix n× n), which is used to calculate the curvature of the manifold.

In dimension 2, the curvature of a surface is an intuitive notion, made rigorous by the
German mathematician Friedrich Gauss around 1830. He defined the curvature R as a
number obtained by an expression R(p) of the curvature of a surface at its point p. And
this number defines and measures the “gap” between the geometry of the surface near
the point p and the Euclidean classical geometry (the geometry of the standard plane).
For instance, the curvature of a two-dimensional sphere is positive, that of a plane (or a
cylinder or a cone) is zero, while that of the surface of a saddle has negative curvature.

If the curvature is independent on the point p of the surface, as in these examples,
we speak of elliptic geometry, Euclidean geometry, and hyperbolic geometry. Notice that,
these three different geometries differ in the shape of their triangles (in particular in the
sum of the inner angles): triangles in the surface of the sphere are fat, those in the plane
are standard, while in the hyperbolic case they are slim.

Around 1850, Riemann generalized the notion of curvature to manifolds of any dimen-
sion n. But when n is greater than or equal to 3, it is no longer just a number, as in the
case of Gauss, but a tensor. Let us suppose that in the neighborhood of a point p of a
manifold V n, we have chosen a local system of coordinates x1, x2, · · · , xn. The Riemann
curvature at a point p of V n is expressed as a table of n4 numbers, each directly dependent
on the (Riemannian) metric defined on the manifold itself, and its derivatives.



D. E. Otera / Eur. J. Pure Appl. Math, 17 (3) (2024), 2361-2369 2367

Note that Riemann’s ideas have long seemed far too abstract. Yet, it is precisely on
these notions that the general relativity, Einstein’s great masterpiece, is based.

4.2. The Geometrization Conjecture

More than a century after Riemann’s innovations, the geometer W. Thurston proposed
a new vast classification project in dimension 3, started in the 1970s, in order to prove the
Poincaré Conjecture and to deeply understand the set of closed 3-manifolds.

Thurston started highlighting, in dimension 3, eight geometries which are particularly
symmetric, three of which are those already defined in the case of surfaces. He therefore
devised the so-called Geometrization Conjecture, according to which any closed man-
ifold of dimension 3 can be broken, in a unique way, into a finite number of pieces, each
of which supports one of the 8 geometries. It is thus a generalization, in dimension 3, of
Poincaré’s uniformization theorem for surfaces mentioned above. (For his work in the field
of topology and geometry in dimension 3, Thurston also received the Fields Medal).

Now, the Geometrization Conjecture is a far more general result than Poincaré Conjec-
ture. In fact, Thurston’s conjecture states, among other things, that among the 8 special
geometries, the only one that a closed and simply connected 3-manifold may carry is that
of constant curvature +1. And it is known that a closed and simply connected 3-manifold
equipped with a metric of constant curvature +1 is topologically equivalent to a sphere.
Thus, one can prove the Poincaré Conjecture also by solving Thurston’s conjecture.

4.3. The Ricci flow

Consider a manifold equipped with a metric. Is it possible to find a process that
modifies its geometry to make it as symmetric as possible?

The idea is to continuously deform the metric at each point p of the manifold so that
the average curvature at point p decreases. This brings us to the work of R. Hamilton in
the 1980s [2]. He introduced an equation (a non-linear partial differential equation) called
the Ricci Flow, which turns out to be very useful [4].

On a 3-manifold, we can define a time-dependent metric, and, at each instant of time,
we can associate to this metric a certain curvature, the so-called Ricci curvature, which
corresponds to a sort of average of Riemann curvatures. The metric and the curvature,
being two tensors of the same type, can be entered into an equation that dictates that the
instantaneous rate of change of the metric corresponds to the opposite of the change of
the Ricci curvature.

Imposing the Ricci flow equation means evolving the metric toward a more regular and
symmetric geometry over time. In dimension 2, Hamilton proved that the Ricci flow for
any metric in a surface evolves, in finite time, toward a metric of constant curvature.

In dimension 3, things are far more difficult, because the flow may “explode”, making
infinite quantities appear. Hamilton’s abstract program, developed and completed by
Perelman, consists just in proving that, as a consequence of these explosions, the manifold
V 3 breaks into pieces on which the Ricci flow may continue to evolve, and that, after a



REFERENCES 2368

finite amount of time and a finite number of explosions, one obtains the starting manifold
decomposed into pieces, each endowed with one of Thurston’s 8 geometries.

Perelman finally managed, with fine methods of non-linear analysis, to control the
explosions of the Ricci flow, and to demonstrate that the whole process of the Ricci flow
extinguishes in a finite time, thus completing Hamilton’s strategy, and proving both the
geometrization of Thurston for 3-manifolds, and the Poincaré Conjecture!

5. Conclusion

Mathematicians call “open problems” those on which they struggle unsuccessfully for
a long time. But an open problem is not just a simple unsolved problem. In fact various
new results are demonstrated by mathematicians every year, and numerous new questions
arise also every year, but (almost) none of them receive such a designation. An open
problem is a problem regarded as exceptional, noble, elusive, but whose comprehension is
fundamental for the development of the research fields that surround it.

The Poincaré conjecture was the prototype of such a problem. It was really a venerable
major question both in classical and modern mathematics. Thanks to it, mathematics has
evolved in different branches: from the birth of algebraic topology to the deep and vast
world of higher dimensional geometry and topology. And, at the end, with the help of fine
and sophisticated analytic tools, to the growth of geometric analysis.

References

[1] D. Gabai. Valentin Poénaru’s Program for the Poincaré Conjecture. In ST Yau,
editor, Geometry Topology and Physics for Raoul Bott, pages 139–169. International
Press, 1994.

[2] R. Hamilton. Three-manifolds with positive Ricci curvature. J. Differ. Geom.,
17(2):255–306, 1982.

[3] J.P. Morgan. Recent progress on the Poincaré Conjecture and the classification of
3-manifolds. Bull. AMS, 42:57–78, 2004.

[4] J.P. Morgan and G. Tian. Ricci Flow and the Poincaré Conjecture. Clay Mathematics
Monographs. Vol. 3. Providence, RI: American Mathematical Society, 2007.

[5] V. Poénaru. The problems of dimension four, and some ramifications. Mathematics,
11(18), 2023.

[6] H. Poincaré. Cinquième complément à l’analysis situs. Rendiconti del Circolo Matem-
atico di Palermo, 18:45–110, 1904.

[7] H. Poincaré. Papers on Topology: Analysis Situs and Its Five Supplements. History
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[8] J. Stillwell. Poincaré and the early history of 3-manifolds. Bull. AMS, 49(4):555–576,
2012.

[9] W.P. Thurston. Three-Dimensional Geometry and Topology. Vol. 1. Princeton Math-
ematical Series. 35. Princeton, NJ: Princeton University Press, 1997.

[10] J.H.C. Whitehead. A certain open manifold whose group is unity. Q. J. Math., Oxf.
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