The topology of two-dimensional manifolds or surfaces was well understood in
the 19th century. In fact there is a simple list of all possible smooth compact
orientable surfaces. Any such surface has a well-deﬁned genus g ≥ 0, which can
be described intuitively as the number of holes; and two such surfaces can be put
into a smooth one-to-one correspondence with each other if and only if they have
the same genus.
The corresponding question in higher dimensions is much more
e was perhaps the ﬁrst to try to make a similar study
of three-dimensional manifolds. The most basic example of such a manifold is
the three-dimensional unit sphere, that is, the locus of all points (x, y, z, w) in
four-dimensional Euclidean space which have distance exactly 1 from the origin:
= 1. He noted that a distinguishing feature of the two-dimensional
sphere is that every simple closed curve in the sphere can be deformed continuously
to a point without leaving the sphere. In 1904, he asked a corresponding question
in dimension 3. In more modern language, it can be phrased as follows:
Question. If a compact three-dimensional manifold M
has the property that every
simple closed curve within the manifold can be deformed continuously to a point,
does it follow that M
is homeomorphic to the sphere S
He commented, with considerable foresight, “Mais cette question nous entraˆıne-
Since then, the hypothesis that every simply connected closed
3-manifold is homeomorphic to the 3-sphere has been known as the Poincar´
many advances in our understanding of the topology of manifolds.
For deﬁnitions and other background material, see, for example,  or , as well as .
See [36, pages 498 and 370]. To Poincar´
e, manifolds were always smooth or polyhedral, so
that his term “homeomorphism” referred to a smooth or piecewise linear homeomorphism.
2. Early Missteps
From the ﬁrst, the apparently simple nature of this statement has led mathe-
maticians to overreach. Four years earlier, in 1900, Poincar´
e himself had been the
ﬁrst to err, stating a false theorem that can be phrased as follows.
False Theorem. Every compact polyhedral manifold with the homology of an n-
dimensional sphere is actually homeomorphic to the n-dimensional sphere.
But his 1904 paper provided a beautiful counterexample to this claim, based
on the concept of fundamental group, which he had introduced earlier (see [36,
pp. 189–192 and 193–288]). This example can be described geometrically as fol-
lows. Consider all possible regular icosahedra inscribed in the two-dimensional
unit sphere. In order to specify one particular icosahedron in this family, we must
provide three parameters. For example, two parameters are needed to specify a
single vertex on the sphere, and then another parameter to specify the direction
to a neighboring vertex. Thus each such icosahedron can be considered as a single
“point” in the three-dimensional manifold M
consisting of all such icosahedra.
This manifold meets Poincar´
e’s preliminary criterion: By the methods of homology
theory, it cannot be distinguished from the three-dimensional sphere. However, he
could prove that it is not a sphere by constructing a simple closed curve that cannot
be deformed to a point within M
. The construction is not diﬃcult: Choose some
through angles 0 ≤ θ ≤ 2π/5. This deﬁnes a simple closed curve in M
Figure 2. The Whitehead link
The next important false theorem was by Henry Whitehead in 1934 . As
part of a purported proof of the Poincar´
e Conjecture, he claimed the sharper state-
ment that every open three-dimensional manifold that is contractible (that can be
continuously deformed to a point) is homeomorphic to Euclidean space. Following
e’s footsteps, he then substantially increased our understanding of the
topology of manifolds by discovering a counterexample to his own theorem. His
counterexample can be brieﬂy described as follows. Start with two disjoint solid
in the 3-sphere that are embedded as shown in Figure 2, so that
each one individually is unknotted, but so that the two are linked together with
linking number zero. Since T
is unknotted, its complement T
can be deﬁned as the coset space SO(3)/I
is the group of all rotations of Euclidean 3-space and where I
is the subgroup consisting of the 60
of all homotopy classes of loops from a point to itself within M
, is a perfect group of order 120.
is another unknotted solid torus that contains T
. Choose a homeomorphism h of
the 3-sphere that maps T
onto this larger solid torus T
. Then we can inductively
construct solid tori
⊂ · · ·
by setting T
). The union M
of this increasing sequence is
the required Whitehead counterexample, a contractible manifold that is not home-
omorphic to Euclidean space. To see that π
can be shrunk to a point (after perhaps crossing through itself) within the
larger solid torus T
. But every closed loop in M
must be contained in some T
and hence can be shrunk to a point within T
. On the other hand, M
not homeomorphic to Euclidean 3-space since, if K ⊂ M
is any compact subset
large enough to contain T
, one can prove that the diﬀerence set M
K is not
Since this time, many false proofs of the Poincar´
e Conjecture have been proposed,
some of them relying on errors that are rather subtle and diﬃcult to detect. For a
delightful presentation of some of the pitfalls of three-dimensional topology, see .
3. Higher Dimensions
The late 1950s and early 1960s saw an avalanche of progress with the discovery
that higher-dimensional manifolds are actually easier to work with than three-
dimensional ones. One reason for this is the following: The fundamental group
plays an important role in all dimensions even when it is trivial, and relations
between generators of the fundamental group correspond to two-dimensional disks,
mapped into the manifold. In dimension 5 or greater, such disks can be put into
general position so that they are disjoint from each other, with no self-intersections,
but in dimension 3 or 4 it may not be possible to avoid intersections, leading to
Stephen Smale announced a proof of the Poincar´
e Conjecture in high dimensions
in 1960 . He was quickly followed by John Stallings, who used a completely
diﬀerent method , and by Andrew Wallace, who had been working along lines
quite similar to those of Smale .
Let me ﬁrst describe the Stallings result, which has a weaker hypothesis and
easier proof, but also a weaker conclusion. He assumed that the dimension is seven
or more, but Christopher Zeeman later extended his argument to dimensions 5 and
is a ﬁnite simplicial complex of dimension
of the sphere S
and is locally piecewise linearly
homeomorphic to the Euclidean space R
, then M
is homeomorphic to S
the complement M
(point) is piecewise linearly homeomorphic to R
The method of proof consists of pushing all of the diﬃculties oﬀ toward a single
In order to check that a manifold M
has the same homotopy type as the sphere S
, we must
) = 0, but also that it has the same homology
as the sphere. The example of the product S
shows that it is not enough to assume that
) = 0 when n > 3.
The Smale proof, and the closely related proof given shortly afterward by Wal-
lace, depended rather on diﬀerentiable methods, building a manifold up inductively,
starting with an n-dimensional ball, by successively adding handles. Here a k-handle
can be added to a manifold M
with boundary by ﬁrst attaching a k-dimensional
sphere into the boundary of M
, and then thickening and smoothing corners so as
and canceling such handles. (Compare the presentation in .)
Figure 3. A three-dimensional ball with a 1-handle attached
Smale Theorem. If M
is a diﬀerentiable homotopy sphere of dimension n ≥ 5,
. In fact, M
is diﬀeomorphic to a manifold
This was also proved by Wallace, at least for n ≥ 6. (It should be noted that
the ﬁve-dimensional case is particularly diﬃcult.)
The much more diﬃcult four-dimensional case had to wait twenty years, for the
work of Michael Freedman . Here the diﬀerentiable methods used by Smale and
Wallace and the piecewise linear methods used by Stallings and Zeeman do not
work at all. Freedman used wildly non-diﬀerentiable methods, not only to prove
the four-dimensional Poincar´
e Conjecture for topological manifolds, but also to give
a complete classiﬁcation of all closed simply connected topological 4-manifolds. The
integral cohomology group H
of such a manifold is free abelian. Freedman needed
= Z is a symmetric
integer mod 2 that vanishes if and only if the product manifold M
× R can be
Freedman Theorem. Two closed simply connected 4-manifolds are homeomor-
phic if and only if they have the same bilinear form β and the same Kirby–Sieben-
mann invariant κ. Any β can be realized by such a manifold. If β(x ⊗ x) is odd
for some x ∈ H
, then either value of κ can be realized also. However, if β(x ⊗ x)
signature of β.
In particular, if M
is a homotopy sphere, then H
= 0 and κ = 0, so M
. It should be noted that the piecewise linear or diﬀeren-
every smooth homotopy 4-sphere is diﬀeomorphic to S
; it is not known which 4-
when this structure is essentially unique. The major results on these questions are
due to Simon Donaldson . As one indication of the complications, Freedman
showed, using Donaldson’s work, that R
admits uncountably many inequivalent
In dimension 3, the discrepancies between topological, piecewise linear, and dif-
ferentiable theories disappear (see , , and ). However, diﬃculties with
the fundamental group become severe.
4. The Thurston Geometrization Conjecture
In the two-dimensional case, each smooth compact surface can be given a beauti-
ful geometrical structure, as a round sphere in the genus zero case, as a ﬂat torus in
the genus 1 case, and as a surface of constant negative curvature when the genus is 2
or more. A far-reaching conjecture by William Thurston in 1983 claims that some-
thing similar is true in dimension 3 . This conjecture asserts that every compact
orientable three-dimensional manifold can be cut up along 2-spheres and tori so as
to decompose into essentially unique pieces, each of which has a simple geometri-
cal structure. There are eight possible three-dimensional geometries in Thurston’s
program. Six of these are now well understood,
and there has been a great deal of
The eighth geometry,
For this geometry, we have the following extension of the Poincar´
Thurston Elliptization Conjecture. Every closed 3-manifold with ﬁnite funda-
mental group has a metric of constant positive curvature and hence is homeomorphic
to a quotient S
/Γ, where Γ ⊂ SO(4) is a ﬁnite group of rotations that acts freely
e Conjecture corresponds to the special case where the group Γ ∼
) is trivial. The possible subgroups Γ ⊂ SO(4) were classiﬁed long ago by
5. Approaches through Differential Geometry
and Differential Equations
In recent years there have been several attacks on the geometrization problem
e Conjecture) based on a study of the geometry of the
inﬁnite dimensional space consisting of all Riemannian metrics on a given smooth
See, for example, , , [38, 39, 40], , , and .
See , , , , and . The pioneering papers by  and  provided the basis
for much of this work.
Added in 2004
By deﬁnition, the length of a path γ on a Riemannian manifold is computed, in
terms of the metric tensor g
, as the integral
From the ﬁrst and second derivatives of this metric tensor, one can compute the
, and the scalar curvature R. (As an example, for the ﬂat
= R = 0, while for a round three-dimensional sphere
studies the total scalar curvature
R dV as a functional on the space of all
the metrics of constant curvature (see ).
A diﬀerent approach, initiated by Richard Hamilton studies the Ricci ﬂow [15,
16, 17], that is, the solutions to the diﬀerential equation
crease in directions of positive curvature. This is essentially a parabolic diﬀerential
equationa and behaves much like the heat equation studied by physicists: If we heat
one end of a cold rod, then the heat will gradually ﬂow throughout the rod until
it attains an even temperature. Similarly, a naive hope for 3-manifolds with ﬁnite
fundamental group might have been that, under the Ricci ﬂow, positive curvature
would tend to spread out until, in the limit (after rescaling to constant size), the
manifold would attain constant curvature. If we start with a 3-manifold of posi-
tive Ricci curvature, Hamilton was able to carry out this program and construct a
metric of constant curvature, thus solving a very special case of the Elliptization
Conjecture. However, in the general case, there are very serious diﬃculties, since
this ﬂow may tend toward singularities.
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