The Poincaré Conjecture Clay Research Conference Resolution of the Poincaré Conjecture Institut Henri Poincaré Paris, France, June 8–9, 2010 - bet 19
|a simpliﬁed version of Novikov’s proof and  for a diﬀerent approach to the
topological Pontryagin classes.)
Novikov conjectured (among other things) that a similar result holds for an
arbitrary closed manifold Y with contractible universal covering. (This would imply,
in particular, that if an oriented manifold Y
is orientably homotopy equivalent to
such a Y , then it is bordant to Y .) Mishchenko (1974) proved this for manifolds
Y admitting metrics of non-positive curvature with a use of an index theorem for
operators on inﬁnite dimensional bundles, thus linking the Novikov conjecture to
(Hyperbolic groups also enter Sullivan’s existence/uniqueness theorem of Lip-
schitz structures on topological manifolds of dimensions
A bi-Lipschitz homeomorphism may look very nasty. Take, for instance, inﬁn-
itely many disjoint round balls B
, ... in
→ 0, take a diﬀeomorphism
f of B
ﬁxing the boundary ∂
) an take the scaled copy of f in each B
resulting homeomorphism, ﬁxed away from these balls, becomes quite complicated
whenever the balls accumulate at some closed subset, e.g. a hypersurface in
one can extend the signature index theorem and some of the Donaldson theory to
this unfriendly bi-Lipschitz, and even to quasi-conformal, environment.)
The Novikov conjecture remains unsolved. It can be reformulated in purely
group theoretic terms, but the most signiﬁcant progress which has been achieved
so far depends on geometry and on the index theory.
In a somewhat similar vein, Atiyah (1974) introduced square integrable (also
) cohomology on non-compact manifolds ˜
X with cocompact discrete group
actions and proved the L
-index theorem. For example, he has shown that
if a compact Riemannian 4k-manifolds has non-zero signature,
then the universal covering ˜
X admits a non-zero square summa-
ble harmonic 2k-form.
-index theorem was extended to measurable foliated spaces (where “measur-
able” means the presence of transversal measures) by Alain Connes, where the two
basic manifolds’ attributes– the smooth structure and the measure—are separated:
the smooth structures in the leaves allow diﬀerential operators while the transversal
measures underly integration and where the two cooperate in the “non-commutative
world” of Alain Connes.
If X is a compact measurably and smoothly n-foliated (i.e. almost all leaves
are smooth n-manifolds) leaf-wise oriented space then one naturally deﬁnes Pon-
tryagin’s numbers which are real numbers in this case.
(Every closed manifold X can be regarded as a measurable foliation with the
“transversal Dirac δ-measure” supported on X. Also complete Riemannian man-
ifolds of ﬁnite volume can be regarded as such foliations, provided the universal
coverings of these have locally bounded geometries .)
There is a natural notion of bordisms between measurable foliated spaces, where
the Pontryagin numbers are obviously, bordism invariant.
Also, the L
-signature, (which is also deﬁned for leaves being
bordism invariant by Poincar´
The corresponding L
= n/4, satisﬁes here the Hirzebruch formula
with the L
-signature (sorry for the mix-up in notation: L
(X) by the Atiyah-Connes L
-index theorem .
It seems not hard to generalize this to measurable foliated spaces where leaves
are topological (or even topological
Questions. Let X be a measurable leaf-wise oriented n-foliated space with
zero Pontryagin numbers, e.g. n
≠ 4k. Is X orientably bordant to zero, provided
every leaf in X has measure zero.
What is the counterpart to the Browder-Novikov theory for measurable folia-
Measurable foliations can be seen as transversal measures on some universal
topological foliation, such as the Hausdorﬀ moduli space X of the isometry classes of
pointed complete Riemannian manifolds L with uniformly locally bounded geome-
tries (or locally bounded covering geometries ), which is tautologically foliated
by these L. Alternatively, one may take the space of pointed triangulated manifolds
with a uniform bound on the numbers of simplices adjacent to the points in L.
The simplest transversal measures on such an X are weak limits of convex
combinations of Dirac’s δ-measures supported on closed leaves, but most (all?)
known interesting examples descent from group actions, e.g. as follows.
Let L be a Riemannian symmetric space (e.g. the complex hyperbolic space
as in section 5), let the isometry group G of L be embedded into a locally
compact group H and let O
⊂ H be a compact subgroup such that the intersection
∩G equals the (isotropy) subgroup O
⊂ G which ﬁxes a point l
∈ L. For example,
H may be the special linear group SL
(R) with O = SO(N) or H may be an adelic
Then the quotient space ˜
= H/O is naturally foliated by the H-translate
copies of L
This foliation becomes truly interesting if we pass from ˜
X to X
/Γ for a
discrete subgroup Γ
⊂ H, where H/Γ has ﬁnite volume. (If we want to make sure
that all leaves of the resulting foliation in X are manifolds, we take Γ without
torsion, but singular orbifold foliations are equally interesting and amenable to the
general index theory.)
The full vector of the Pontryagin numbers of such an X depends, up to rescaling,
only on L but it is unclear if there are “natural (or any) bordisms” between diﬀerent
X with the same L.
Linear operators are diﬃcult to delinearize keeping them topologically inter-
esting. The two exceptions are the Cauchy–Riemann operator and the signature
operator in dimension 4. The former is used by Thurston (starting from late 70s)
in his 3D-geometrization theory and the latter, in the form of the Yang–Mills equa-
tions, begot Donaldson’s 4D-theory (1983) and the Seiberg–Witten theory (1994).
The logic of Donaldson’s approach resembles that of the index theorem. Yet, his
∶ Φ → Ψ is non-linear Fredholm and instead of the index he studies the
bordism-like invariants of (ﬁnite dimensional!) pullbacks D
(ψ) ⊂ Φ of suitably
These invariants for the Yang-Mills and Seiberg-Witten equations unravel an
incredible richness of the smooth 4D-topological structures which remain invisible
from the perspectives of pure topology” and/or of linear analysis.
The non-linear Ricci ﬂow equation of Richard Hamilton, the parabolic relative
of Einstein, does not have any built-in topological intricacy; it is similar to the
plain heat equation associated to the ordinary Laplace operator. Its potential role
is not in exhibiting new structures but, on the contrary, in showing that these do
not exist by ironing out bumps and ripples of Riemannian metrics. This potential
was realized in dimension 3 by Perelman in 2003:
The Ricci ﬂow on Riemannian 3-manifolds, when manually redi-
rected at its singularities, eventually brings every closed Rie-
mannian 3-manifold to a canonical geometric form predicted by
(Possibly, there is a non-linear analysis on foliated spaces, where solutions of, e.g.
parabolic Hamilton-Ricci for 3D and of elliptic Yang-Mills/Seiberg-Witten for 4D,
equations fast, e.g. L
, decay on each leaf and where “decay” for non-linear objects
may refer to a decay of distances between pairs of objects.)
There is hardly anything in common between the proofs of Smale and Perel-
man of the Poincar´
e conjecture. Why the statements look so similar? Is it the
e conjecture” they have proved? Probably, the answer is “no” which
raises another question: what is the high dimensional counterpart of the Hamilton-
To get a perspective let us look at another, seemingly remote, fragment of
mathematics – the theory of algebraic equations, where the numbers 2, 3 and 4 also
play an exceptional role.
If topology followed a contorted path 2
→5...→4→3, algebra was going straight
→2→3→4→5... and it certainly did not stop at this point.
Thus, by comparison, the Smale–Browder–Novikov theorems correspond to
non-solvability of equations of degree
≥ 5 while the present day 3D- and 4D-theories
are brethren of the magniﬁcent formulas solving the equations of degree 3 and 4.
What does, in topology, correspond to the Galois theory, class ﬁeld theory, the
modularity theorem... ?
Is there, in truth, anything in common between this algebra/arthmetic and
It seems so, at least on the surface of things, since the reason for the particu-
larity of the numbers 2, 3, 4 in both cases arises from the same formula:
+ 2 ∶
a 4 element set has exactly 3 partitions into two 2-element subsets and where,
< 4. No number n ≥ 5 admits a similar class of decompositions.
In algebra, the formula 4
+2 implies that the alternating group A(4) admits
an epimorphism onto A
(3), while the higher groups A(n) are simple non-Abelian.
In geometry, this transforms into the splitting of the Lie algebra so
(3) ⊕ so(3). This leads to the splitting of the space of the 2-forms into self-
dual and anti-self-dual ones which underlies the Yang–Mills and Seiberg–Witten
equations in dimension 4.
In dimension 2, the group SO
(2) “unfolds” into the geometry of Riemann
surfaces and then, when extended to homeo
), brings to light the conformal ﬁeld
In dimension 3, Perelman’s proof is grounded in the inﬁnitesimal O
of Riemannian metrics on 3-manifolds (which is broken in Thurston’s theory and
even more so in the high dimensional topology based on surgery) and depends on
the irreducibility of the space of traceless curvature tensors.
It seems, the geometric topology has a long way to go in conquering high
dimensions with all their symmetries.
11. Crystals, Liposomes and Drosophila
Many geometric ideas were nurtured in the cradle of manifolds; we want to
follow these ideas in a larger and yet unexplored world of more general “spaces”.
Several exciting new routes were recently opened to us by the high energy and
statistical physics, e.g. coming from around the string theory and non-commutative
geometry—somebody else may comment on these, not myself. But there are a few
other directions where geometric spaces may be going.
Inﬁnite Cartesian Products and Related Spaces. A crystal is a collection
of identical molecules mol
positioned at certain sites γ which are the
elements of a discrete (crystallographic) group Γ.
If the space of states of each molecule is depicted by some “manifold” M , and
the molecules do not interact, then the space X of states of our “crystal” equals
the Cartesian power M
If there are inter-molecular constrains, X will be a subspace of M
more, X may be a quotient space of such a subspace under some equivalence rela-
tion, where, e.g. two states are regarded equivalent if they are indistinguishable by
a certain class of “measurements”.
We look for mathematical counterparts to the following physical problem.
Which properties of an individual molecule can be determined by a given class
of measurement of the whole crystal?
Abstractly speaking, we start with some category
M of “spaces” M with Carte-
sian (direct) products, e.g. a category of ﬁnite sets, of smooth manifolds or of al-
gebraic manifolds over some ﬁeld. Given a countable group Γ, we enlarge this
category as follows.
Γ-Power Category Γ
. The objects X
are projective limits of ﬁnite
Cartesian powers M
∈ M and ﬁnite subsets Δ ⊂ Γ. Every such X is
naturally acted upon by Γ and the admissible morphisms in our Γ-category are
Γ-equivariant projective limits of morphisms in
Thus each morphism, F
∶ X = M
→ Y = N
is deﬁned by a single morphism
M, say by f ∶ M
→ N = N where Δ ⊂ Γ is a ﬁnite (sub)set. Namely, if we
think of x
∈ X and y ∈ Y as M- and N-valued functions x(γ) and y(γ) on Γ then
the value y
(γ) = F (x)(γ) ∈ N is evaluated as follows:
⊂ Γ to γΔ ⊂ Γ by γ, restrict x(γ) to γΔ and apply
f to this restriction x
∣γΔ ∈ M
In particular, every morphism f
∶ M → N in M tautologically deﬁnes a morphism
, denoted f
has many other morphisms in it.
Which concepts, constructions, properties of morphisms and objects, etc. from
M “survive” in Γ
for a given group Γ? In particular, what happens to topological
invariants which are multiplicative under Cartesian products, such as the Euler
characteristic and the signature?
For instance, let M and N be manifolds. Suppose M admits no topological
embedding into N (e.g. M
= [0, 1] or M = RP
). When does M
admit an injective morphism to N
in the category
(One may meaningfully reiterate these questions for continuous Γ-equivariant
maps between Γ-Cartesian products, since not all continuous Γ-equivariant maps
Conversely, let M
→ N be a map of non-zero degree. When is the corresponding
equivariantly homotopic to a non-surjective map?
Γ-Subvarieties. Add new objects to
deﬁned by equivariant systems of
equations in X
, e.g. as follows.
Let M be an algebraic variety over some ﬁeld
F and Σ ⊂ M × M a subvariety,
say, a generic algebraic hypersurface of bi-degree
(p, q) in CP
Then every directed graph G
= (V, E) on the vertex set V deﬁnes a subvariety,
, say Σ
(G) ⊂ M
which consists of those M -valued functions x
(v), v ∈ V ,
)) ∈ Σ whenever the vertices v
are joined by a directed
∈ E in G. (If Σ ⊂ M × M is symmetric for (m
) ↔ (m
), one does
not need directions in the edges.)
Notice that even if Σ is non-singular, Σ
(G) may be singular. (I doubt, this
ever happens for generic hypersurfaces in
.) On the other hand, if
we have a “suﬃciently ample” family of subvarieties Σ in M
× M (e.g. of (p, q)-
) and, for each e
∈ E, we take a generic representative
(e) ⊂ M × M from this family, then the resulting generic subvariety in
× M, call it Σ
(G) is non-singular and, if F = C, its topology does not depend
on the choices of Σ
We are manly interested in Σ
(G) and Σ
(G) for inﬁnite graphs G with a
coﬁnite action of a group Γ, i.e. where the quotient graph G
/Γ is ﬁnite. In partic-
ular, we want to understand “inﬁnite dimensional (co)homology” of these spaces,
F = C and the “cardinalities” of their points for ﬁnite ﬁelds F (see  for
some results and references). Here are test questions.
Let Σ be a hypersurface of bi-degree
(p, q) in CP
= Z. Let P
denote the Poincar´
e polynomial of Σ
(G/kZ), k = 1, 2, .... and let
(s, t) =
(s) = ∑
Observe that the function P
(s, t) depends only on n, and (p, q).
(s, t) meromorphic in the two complex variables s and t? Does it satisfy
some “nice” functional equation?
F = F
, we ask the same question for the generating function in
two variables counting the
-points of Σ
Γ-Quotients. These are deﬁned with equivalence relations R
⊂ X × X where
R are subobjects in our category.
The transitivity of (an equivalence relation) R, and it is being a ﬁnitary deﬁned
sub-object are hard to satisfy simultaneously. Yet, hyperbolic dynamical systems
provide encouraging examples at least for the category
M of ﬁnite sets.
M is the category of ﬁnite sets then subobjects in M
, deﬁned with subsets
⊂ M × M are called Markov Γ-shifts. These are studied, mainly for Γ = Z, in the
context of symbolic dynamics , .
Γ-Markov quotients Z of Markov shifts are deﬁned with equivalence relations
) ⊂ Y ×Y which are Markov subshifts. (These are called hyperbolic and/or
ﬁnitely presented dynamical systems , .)
= Z, then the counterpart of the above P (s, t), now a function only in t, is,
essentially, what is called the ζ-function of the dynamical system which counts the
number of periodic orbits. It is shown in  with a use of (Sinai-Bowen) Markov
partitions that this function is rational in t for all
Z-Markov quotient systems.
The local topology of Markov quotient (unlike that of shift spaces which are
Cantor sets) may be quite intricate, but some are topological manifolds.
For instance, classical Anosov systems on infra-nilmanifolds V and/or expand-
ing endomorphisms of V are representable as a
Z- Markov quotient via Markov
Another example is where Γ is the fundamental group of a closed n-manifold V
of negative curvature. The ideal boundary Z
(Γ) is a topological (n−1)-sphere
with a Γ-action which admits a Γ-Markov quotient presentation .
Since the topological S
→ V associated to the universal covering,
regarded as the principle Γ bundle, is, obviously, isomorphic to the unit tangent
bundle U T
(V ) → V , the Markov presentation of Z = S
deﬁnes the topological
Pontryagin classes p
of V in terms of Γ.
Using this, one can reduce the homotopy invariance of the Pontryagin classes
of V to the ε-topological invariance.
Recall that an ε-homeomorphism is given by a pair of maps f
, such that the composed maps f
ε-close to the respective identity maps for some metrics in V
and a small ε
depending on these metrics.
Most known proofs, starting from Novikov’s, of invariance of p
morphisms equally apply to ε-homeomorphisms.
This, in turn, implies the homotopy invariance of p
if the homotopy can be
“rescaled” to an ε-homotopy.
For example, if V is a nil-manifold ˜
/Γ, (where ˜
V is a nilpotent Lie group
) with an expanding endomorphism E
∶ V → V (such a V is a
Z-Markov quotient of a shift), then a large negative power ˜
V of the lift
V brings any homotopy close to identity. Then the ε-topological invariance
implies the homotopy invariance for these V . (The case of V
∶ ˜v → 2˜v is used by Kirby in his topological torus trick.)
A similar reasoning yields the homotopy invariance of p
for many (manifolds
with fundamental) groups Γ, e.g. for hyperbolic groups.
Questions. Can one eﬀectively describe the local and global topology of Γ-
Markov quotients Z in combinatorial terms? Can one, for a given (e.g. hyperbolic)
group Γ, “classify” those Γ-Markov quotients Z which are topological manifolds or,
more generally, locally contractible spaces?
For example, can one describe the classical Anosov systems Z in terms of the
combinatorics of their
Z-Markov quotient representations? How restrictive is the
assumption that Z is a topological manifold? How much the topology of the local
dynamics at the periodic points in Z restrict the topology of Z (e.g. we want to
incorporate pseudo-Anosov automorphisms of surfaces into the general picture.)
It seems, as in the case of the hyperbolic groups, (irreducible)
tients becomes more scarce/rigid/symmetric as the topological dimension and/or
the local topological connectivity increases.
Are there interesting Γ-Markov quotients over categories
M besides ﬁnite sets?
For example, can one have such an object over the category of algebraic varieties
Z with non-trivial (e.g. positive dimensional) topology in the spaces of its
Liposomes and Micelles are surfaces of membranes surrounded by water which
are assembled of rod-like (phospholipid) molecules oriented normally to the surface
of the membrane with hydrophilic “heads” facing the exterior and the interior of a
cell while the hydrophobic “tails” are buried inside the membrane.
These surfaces satisfy certain partial diﬀerential equations of rather general
nature (see ). If we heat the water, membranes dissolve: their constituent
molecules become (almost) randomly distributed in the water; yet, if we cool the
solution, the surfaces and the equations they satisfy re-emerge.
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