Foliations and 3-Manifolds*
David Gabai
California Institute of Technology, Pasadena, CA 91125, USA
The goal of this talk is two fold. First in Sections 1 and 2 I survey some
key results relating foliations and the topology of 3-manifolds. In particular in
Section 2 I will follow the evolution of one enormously important idea "foliated
height functions" as it has evolved over the last century. Second in Section 3 I
discuss a recent development, the essential lamination.
1. 1-Dimensional Foliations
Since x(M) = 0 for all closed 3-manifolds M, it follows that M has a smooth
vector field, hence a smooth codimension-2 foliation. By requiring a bit of
structure on the foliation one imparts great structure to the 3-manifold. I briefly
discuss one such structure.
A Seifert fibred space is a compact 3-manifold M which is almost a S 1 bundle
over a compact surface, i.e. there exists a projection n : M -> N such that for
each x G N there exists a D2 neighborhood of x such that n~l(D2) = D2 x S1
and n((r, 0i), (1,02)) = (r,p#i + qQ-f) where p ^ 0 and p, q are relatively prime and
depend on x and 0 G R mod 2n.
Remark. Seifert fibred spaces were classified up to fibre preserving homeomorphism in 1928 by Seifert [S]. Their topological classification was obtained by
Orlik, Vogt, and Zieschang [OVZ] in 1967. Seifert fibred spaces have one of six
geometric structures as discussed in Scott's survey article [Sci].
In 1972 Epstein [E] showed that M has a C1 foliation by circles if and only
if M is a Seifert fibred space.
Seifert fibred spaces play a central role in 3-dimensional topology. In fact
Thurston's geometrization conjecture [Th3] for 3-manifolds asserts that a closed
oriented 3-manifold which has no torus [JS], [Jo] or sphere decomposition [M],
[K] is either a Seifert fibred space or a hyperbolic 3-manifold.
Very recently (fall 1990) Casson and the author [Gl] have independently announced proofs of the conjecture that S1 convergence groups [GM] are Fuchsian
groups. This result implies the Seifert fibred space conjecture, i.e. if M is a closed
* Partially Supported by NSF grant DMS-8902343 and a Sloan foundation research
fellowship.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
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David Gabai
orientable irreducible 3-manifold, then M is a Seifert fibred space if and only if
7Ti (M) contains an infinite cyclic normal subgroup. [Me] using the work of [Sc2]
and others reduced the Seifert fibred space conjecture to the convergence group
conjecture.
Regarding the qualitative types of codimension-2 foliations on 3-manifolds I
cite the following two remarkable results.
Theorem (Schweitzer [Sw] 1976). S3 has a C1 foliation by lines.
Theorem (Vogt [V] 1989). R 3 has a C1 foliation by circles.
2. 2-Dimensional Foliations
I will try to give a chronological account of codimension-1 foliations in 3manifolds. For simplicity, unless otherwise stated all manifolds will be closed and
orientable.
In 1863 Möbius proved the following result (see [St]).
Theorem. If S is a smooth closed surface in R3, then S is diffeomorphic to a surface
of genus g.
Idea of Proof. Find enough parallel planes to chop S up into discs, annuii, and
spheres with 3-holes. Manipulate the pieces to get a standard form.
D
This is a foliations proof of a topological theorem. Möbius uses a foliation
of R 3 by parallel planes to decompose S, then rebuilds S in a recognizable way.
This "idea" will be generalized many times over for increasingly sophisticated
applications. Here is one such generalization.
Theorem (Alexander 1923 [A]). A PL embedded S2 in R 3 bounds a 3-Ball. (The
PL Shoenflies theorem).
Idea of Proof. Consider a foliation of R 3 by parallel planes. A generic such
foliation induces a foliation on S2 which has a finite number of critical points
at isolated levels. Analyzing the foliation and the embedding one observes that
saddles can be cancelled with centers after an isotopy of S2, thus, after isotopy,
one obtains an S2 with exactly one maximum and one minimum. This S2 evidently
bounds a 3-cell, hence so does the original one.
D
The study of foliations did not begin as a recognized subject in its own
right until the late 1940s with the work of Ehresmann and Reeb. (Perhaps as an
offshoot of the study of fibre bundles which was a hot new topic in the 1940s.)
The story goes that Ehresmann suggested the following thesis problem to his
student Reeb. Show that there exists no codimension-1 foliation on S3.
Ehresmann-1940s
- Introduced the idea of holonomy [Eh]. Advisor to Reeb and influenced Haefliger's thesis work (whose advisor was DeRham).
Foliations and 3-Manifolds
611
Reeb-1948 [R]
- Reeb foliation of S3.
- Reeb Stability theorem - If F is a codimension-1 foliation on M and L is a
compact leaf with trivial holonomy group, then L has a neighborhood in M
homeomorphic l o L x / and F | L x J is the product foliation.
- If F is transversely orientable and F contains a S2 leaf, then M = S2 X S1.
(He more generally showed that the set of compact leaves with finite m is
closed.)
Remark. Later Lickorisch [L], Zieschang [N], and Wood [W] showed that all
closed 3-manifolds have codimension-1 foliations.
Haefliger 1959 [H]
- Showed that if y is a closed curve transverse to F, an analytic foliation of M,
then 0 ^ y G n\ (M). In particular this implied that M has infinite fundamental
group.
- Showed that the union of compact leaves of a codimension-1 foliation is
closed in M.
Novikov 1962 [N]. If M ^ S2 x S1, F is a C 2 foliation and has no Reeb
components then
-
7E 2 (M)=0.
-
n\(leaves) inject into ni(M).
y transverse to F implies that 0 ^ y 6
m(M).
Remark. It was known to experts that C 2 was not crucial, being used only to
push curves normally off of surfaces. Thus a transverse line field provides the
needed structure in the C° case. See also [So] for a C° proof.
Sacksteder 1962 [Sa]. If F is a C 2 foliation on M then
- F has no holonomy implies that F is defined by a closed 1-form. In particular
the lifted foliation on the universal cover M is the product foliation R 2 x R.
- exceptional minimal sets have résiliant leaves, i.e. leaves with holonomy elements which are contractions.
Remark. A C° "version" of Sacksteder's first theorem can be found in [I]. Tischler
[Ti] showed that a foliation defined by a closed 1-form can be perturbed slightly
to obtain a fibration over S1.
Stallings 1962 [St]
- If M is irreducible, dM ^ 0 and the inclusion map of T -> M induces an
isomorphism on m where T is a component of boundary M, then M = T XL
Rosenberg 1968 [Ro]
- If F is a foliation without Reeb components, then M is irreducible.
- (with Sondow). If M is C 2 foliated by planes, then M is the 3-torus.
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David Gabai
Remark. The Rosenberg-Sondow theorem uses C2 in an essential way for it
invokes Sacksteder's theorem. Although unstated there (see Sect. 3) a C° proof
follows by [I].
The main result of Novikov's paper is that a Reeb component is the obstruction to isotoping a disc rei boundary into a leaf, where the induced foliation of
the D2 is the standard foliation by circles with one singular point at the origin.
(Consider a meridianal disc transverse to a Reeb foliated D2 x S1.)
Rosenberg's key observation in the proof of his first theorem is that Novikov's
result implies that there is no obstruction to carrying out Alexander's argument
to show that a smoothly imbedded S2 in a foliated manifold without Reeb
components bounds a B3.
A generalization of the Möbius, Alexander, Rosenberg line is the following
result of
Rousserie and Thurston 1972 [Rou, Thl]
- If T is a compact embedded m injective surface in M and F is a foliation on
M which has no Reeb components, then T can be isotoped to be transverse
to F except at a finite number of saddle and circle tangencies.
Remark. A circle tangency looks exactly like the rim of a volcano. Rousserie
further shows that the obstruction to isotoping away the circle tangencies in the
case of T a torus is the existence of a cylindrical component, i.e. an annulus
bundle over S1 whose boundary components are leaves and whose interior leaves
are annuii whose ends spiral "in the same direction" about the ends. More
generally it is well known to experts that the obstruction to eliminating the circle
tangencies is the existence of generalized Reeb components, i.e. of bundles over
S1 with fibre a compact surface S with boundary. The boundary tori are leaves
o
and the interior leaves are homeomorphic to S, and nearly tangent to S except
near the ends which spiral in the same direction about the boundary tori.
Thurston 1976 [Th2]
- Defined the Thurston norm on H2(M). \\z\\ = {min-^S') \ [S] = z e H2(M)
where S; = S — S2 components}.
- compact leaves of foliations without Reeb components are Thurston norm
minimizing, i.e. compact leaves are topologically minimal in their homology
class.
Remark. Corresponding definitions and results were given for manifolds with
boundary. Thurston's Pseudo-Anosov theory [FLP] is also of central importance.
Palmeira 1978 [P]
- If F is a foliation on M without Reeb components, then the universal
covering of M is R 3 . Furthermore the induced foliation on R 3 is topologically
equivalent to a product of a foliation on R 2 and R.
Sullivan 1979 [Su]
- If F is C2 and taut, (i.e. there exists a closed curve transverse to F which hits
all the leaves), then there exists a Riemannian metric on M such that all the
leaves are minimal, i.e. locally area minimizing.
Foliations and 3-Manifolds
613
Remark. Sullivan's result was inspired by a letter from Herman Gluck. This result
was generalized by Harvey and Lawson [HL] who showed that F was in fact
calibrated, i.e. there exists a metric so that compact portions of leaves are minimal
area in their homology class. A corresponding topological statement can be made
regarding the Thurston norm.
Gabai [G2, G3]
- 1981 - If T is a Thurston norm minimizing surface in an oriented irreducible
manifold M, (dM possibly nonempty) then there exists a finite depth taut
transversely oriented foliation F on M such that F is a leaf.
- 1 9 8 5 - I f T i s a minimal genus surface for a knot in S3, then there exists a
o
finite depth taut transversely oriented foliation F of S3 — N(k) such that T is
a leaf of F and F \ dN(k) is a foliation by circles.
Remark. The difference between the 1985 and 1981 theorems is that the foliation
F | dN(k) given by the former would be a suspension of a homeomorphism of
the circle with some fixed points. See [G3] for other constructions.
Applications
- If z G Ü2(M,dM), then g(z) = 2t(z) where g denotes the gromov norm
and t denotes the Thurston norm. Viewed another way we obtain a positive
solution to a conjecture of Thurston [Th2] that the norm on homology based
on singular or immersed surfaces is equal to the "classical" norm based on
embedded surfaces. In particular the immersed genus of a knot equals the
embedded genus of a knot, which is the higher genus analogue to Dehn's
lemma. [Gi]
- Positive solutions to Property R and Poenaru conjectures. More generally
we show that a homology S2 x S1 manifold JV obtained by surgery on
a knot k in S3 is irreducible or S2 x S1. Furthermore the extension of a
minimal genus surface for k becomes a Thurston norm minimizing surface
representing the generator of JV. One can view Property R as asserting that
{S3 knot theory} n {S2 x S{ knot theory} = 0. [G3]
- Knots in S 2 x S 1 or torus bundles over S1, which are not contained in 3-cells,
are determined by their complements. [G4]
- If T is a Thurston norm minimizing surface in M disjoint from a boundary
component P, then T remains norm minimizing (and hence m injective) in
all but at most one of the manifolds obtained by Dehn filling on P. [G4]
- Sela showed by example that the bad surgeries corresponding to distinct norm
minimizing surfaces may be distinct. [Se]
- Computing the genus of an oriented link in S3. [G5], [G6]
- Property P for satellite knots. [G7]
- Superadditivity of knot genus under band connnect sum. [G8], [Sch]
- The classification of knots in solid tori such that non trivial surgery yields
a solid torus. In particular there exists a unique knot in D2 x S1 such that
exactly two surgeries yield D2 x S1. [G7] reduced the problem to 1-bridge
braids in solid tori and Berge [B] solved the problem for 1-bridge case. See
also [G9].
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David Gabai
Key elements of the proofs of the theorems.
- Sutured manifold theory. Just as a Haken 3-manifold can be decomposed
to a 3-cell via splitting along incompressible surfaces, the content of sutured
manifold theory is that a 3-manifold M with nontrivial Ü2(M,dM) can be
decomposed, with control, along taut sufaces to obtain a 3-cell. Dual to this
decomposition is a finite depth foliation such that the cores of the finite depth
leaves correspond to the splitting taut surfaces.
- Finite depth foliations "are like" compact surfaces, i.e. technically finite depth
foliations can be treated more like incompressible surfaces rather than "thick"
foliations.
- Height function arguments. A crucial moment in [G3] was the defining the
notion of thin presentation of a knot in S3. A thin presentation is essentially
a bridge presentation where the local maxima are required to be as low as
possible and the minima as high as possible, thus from the point of view
of the horizontal planes in S3 the knot is as thin as possible. With respect
to a thin presentation a lamination can be put into a normal form. In
particular there will be level planes where the induced lamination will have
no inessential curves. These arguments should be viewed as sophisticated
versions of Möbius, Alexander, Rosenberg, . . . .
- Combinatorial arguments.
I close this section with the following results, whose proofs push variants of
the foliated height function arguments to high technical levels.
Gabai 1984 [G10]
- (Simple Loop Conjecture) / : S —> T is a map of closed surfaces and 0 ^ ker
/ # : ni(S) -> ni(T), then there exists a simple loop in ker /#.
Gabai Kazez 1985 [GK1]
- (Homotopy Classification of Maps of Surfaces) If / , g : S —> T are maps of
positive degree of closed oriented surfaces, then there exists a homeomorphism
h : S —> S such that / is homotopic to g oh if and only if f#(ni(S)) = g#(ni(S))
and deg / = deg g.
Remark. The idea behind these results is to view / (or g) as a branched immersion
in T x I, then put the double curve locus into normal form. See [GK 2] for the
version for maps of closed surfaces.
Gordon Luecke 1988 [GL]
- Knots in S3 are determined by their complements.
Remark. The proof involved two steps.
i) If there exists two knots with distinct complements, then each knot complement has a foliation by planar surfaces coming from the foliation of S3 by
level S2's. When these foliations are viewed in a single knot complement one
finds two planar surfaces which intersect each other in an essential way. This
step was done independently by myself and [GL] using thin presentations.
ii) There do not exist such planar surfaces in S3. This step required a deep
combinatorial topology argument.
Foliations and 3-Manifolds
615
3. Essential Laminations
Example. If / : T —• T is a pseudo anosov homeomorphism of a closed surface
T and A is the /-invariant (stable measured) foliation, then X, the suspension of A
in M, the T bundle over S1 defined by / , is a singular foliation on M. One can get
singular foliations in different manifolds by doing Dehn surgery to the singular
circles, Fried was aware of this construction in the late 70's. Independently (at
the same time) Ghys and Thurston knew how to create nonsingular foliations
when the singular locus consisted of 2w prongs.
Remark.
i) ([Gil] 1981) X is an example of a singular foliation with a finite number of
singular circles. X restricted to cross sections of neighborhoods of the circles
look like index n/2 singularities of line fields where n = 1 or n < — 1. The
author observed that Novikov's theorem extends to such singular foliations
provided n < 0. (There exists a singular foliation of S3 with a single singular
circle of type 1/2 and no compact leaves. Consider the foliation transverse
to the fibration of the (-2,3,7) Pretzel knot.)
ii) ([Gil] 1986) Good singular foliations can be constructed transverse to finite
depth foliations. In some sense most manifolds contain such foliations.
iii) The notion of essential lamination [GO] developed with Oertel generalizes
the above notion of "good" singular foliation. Recently Hatcher and Oertel
have shown that essential laminations in non Haken manifolds can be viewed
as singular foliations (of a possibly more general type).
Definition. A codimension-1 lamination X of M 3 is a decomposition of a closed
subset of M by surfaces (called leaves) such that M is covered by charts of the
form R 2 x R and leaves of X pass through in R 2 x pts. X is an essential lamination
[GO] of M if
i) M — X is irreducible.
ii) No leaf of X is a torus bounding a solid torus.
iii) If V is the closure (in the path metric) of a complementary region of X in
M, then dV is incompressible and end incompressible.
Theorem [GO] 1987. Let M be a closed oriented 3-manifold with an essential
lamination X, then
i) M is irreducible.
ii) m (leaves) inject into ni(M).
iii) m (transverse efficient loops) inject into TüI(M), transverse efficient arcs cannot
be homotoped rei boundary into a leaf.
iv) the universal cover of M is R 3 .
Remark. Essential laminations generalize the notions of foliations without Reeb
components and incompressible surfaces. Like finite depth foliations essential
laminations can be manipulated as compact surfaces. While in some sense most
3-manifolds do not contain incompressible surfaces, most 3-manifolds do contain
essential laminations.
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David Gabai
Example. Essential laminations can be obtained by "blowing air" into the singular
loci of the above good singular laminations. See [GO, Ha, O, Na, D] for other
constructions.
Conjecture. If M is a closed aspherical 3-manifold, then M is finitely covered by a
manifold with an essential lamination.
Conjecture. If M is closed oriented and aspherical, then M has an essential lamination or is Seifert fibred.
Remark. The second optimistic conjecture implies the first. The first conjecture is
implied by Waldhausen's conjecture that every aspherical 3-manifold is finitely
covered by a Haken manifold. There is also Thurston's conjecture that every
hyperbolic manifold is finitely covered by a surface bundle over S 1 . The following
result combined with [M2, W2] (see [JN]) shows that there exist examples of
aspherical Seifert fibred spaces with out essential laminations.
Theorem (Brittenham [B], Claus [C] 1990). If X is an essential lamination of a
Seifert Fibred Space, then there exists a sublamination which is either horizontal or
vertical (after isotopy).
Remark. Vertical means that it is a union of Seifert fibres and horizontal means
that it is transverse to the Seifert fibration.
The case for X a compact surface was due to [Wa]. Thurston [Thl] obtained
the result when X was a C 2 foliation in a S 1 bundle. [EHN] generalized that
result to C 2 foliations in Haken Seifert fibred spaces.
Theorem ([GK3] 1990). If M has an essential lamination and O ^ y G ni(M), then
o
D x Sl is the covering space of M with ni generated by y.
Theorem (Gabai-Kazez 1990). If M has a pseudoanosov flow and f,g : M -> M
are homotopic homeomorphisms, then f is isotopie to g.
Remark. It suffices to consider the case that / = id. The hypotheses imply the
existence of two transverse (stable and unstable) essential laminations. The proof
proceeds by first showing that / is isotopie to / ' which fixes a simple closed curve
common to both laminations.
Homotopy implies isotopy for irreducible 3-manifolds is an old conjecture.
Waldhausen [Wa] established the case for M Haken, a number of authors
established the case for M a spherical space form or more generally a Seifert
fibred space, see [Sc3] and [BO]. See also [HS].
Theorem. IfX is an essential lamination of the closed 3-manifold M by planes, then
M is the 3-torus.
Proof. The closure of each complementary region of X, with the path metric, is
o
D x J, therefore X extends to a foliation also called X of M by planes. It follows
Foliations and 3-Manifolds
617
from Theorem 3.1 [I], the lift X of X to the universal covering of M has space of
leaves R. Since each leaf of X is a plane, 711 (M) acts freely and order preserving
on R, thus by a theorem of Holder n\(M) is archemedian and hence free abelian.
This last argument is essentially Proposition 4.1 of [I]. Now argue as in [Ro] to
conclude that M = S1 x S1 x S1.
P
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