• Aim• Date and Place• Programs• Titles and Abstracts• Exercises• Contact• Access• Link“Seminar on Geometry, Dynamics and Foliations”(Sept. 21, 23 and 25, 2013 at Chuo University)“Geometry and Foliations 2013”(Sept. 9 - 14, 2013 at University of Tokyo, Komaba)• Aim
The objective of the BGamma School is first of all to review the principal results in the homotopy
theory of foliations such as h-principles as well as to present new progresses. Also we search for the
future developments. Some introductory lectures and exercise classes are planned.
• Date and Place
September 17th (Tues.) – September 19th (Thu.), 2013
Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551,
Japan
Building No.5 ground floor• Program
17 (Tue)
10:00-11:00
11:20-12:20
12:20-14:00
13:00-14:00
14:00-15:00
15:20-16:20
16:40-17:40
18:00-19:30

Vogt-1
Meigniez-1
MitsumatsuEC

18 (Wed)Vogt-2
Asuke-1
Lunch
EC
Meigniez-2
Eynard-Bontemps
Vogel/Bowden
EC

EC : Exercise Class with tutors
WP : Wine Party (18:00-21:00)

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19 (Thu.)
Vogt-3
Asuke-2
Lunch
EC
Meigniez-3
Schweitzer
MoritaWP• Title and Abstract
Elmar Vogt (Freie Universität Berlin):
1. Why BΓ and what is it?
2. Simplicial constructions: other models for BΓ
3. De Rham theory for BΓ[abstract]Gaël Meigniez (Université de Bretagne-Sud):
Thurston’s h-principle[abstract]

Yoshi Mitsumatsu (Chuo University):Thurston’s h-principle in (4, 2)-dimension: Applications and proof[abstract]

Taro Asuke (University of Tokyo):Construction of secondary characteristic classes for foliations using the Chern-Weil theory[abstract]
Hélène Eynard-Bontemps (Université Pierre et Marie Curie):
Connected components of the space of smooth codimension one foliations on a closed 3-manifold[abstract]
Thomas Vogel (Max Planck Inst. Bonn) / Jonathan Bowden (Augsburg University):
Contact structures and foliations on 3-manifolds[abstract]

Paul A. Schweitzer, S.J. (PUC-Rio) :Gelfand-Fuks cohomology and the cohomology of BΓq[abstract]

Shigeyuki Morita (University of Tokyo, Emeritus):
A few problems on the classifying spaces of foliations• Exercise class (click here )2[abstract]• Contact
e-mail : bgamma13 AT math.chuo-u.ac.jp
tel : +81-3-3817-1745 : Department of Mathematics, Chuo University
fax : +81-3-3817-1746 : Department of Mathematics, Chuo University• Access
About a 15-minute walk form JR Soubu Line Suidoubashi Station
About a 5-minute walk form Tokyo Metro Marunouchi Line or Nanboku Line Kourakuen Station
About a 7-minute walk form Tokyo Metro Mita Line or O-edo Line Kasuga Station
Click here for a map (in Japanese)

Click here for a map (in English)• Scientific Committee
Yoshi Mitsumatsu (Chuo University)Gaël Meigniez (Université de Bretagne-Sud)
Elmar Vogt (Freie Universität Berlin)• Organizing Committee
Yoshi Mitsumatsu (Chuo University)
Shigeaki Miyoshi (Chuo University)
Noboru Ogawa (Chuo University)3

Abstracts
Lecture 1. Why BΓ and what is it?
Lecture 2. Simplicial constructions: other models for BΓ
Lecture 3. De Rham theory for BΓ
Elmar Vogt
1. Why BΓ and what is it?
We introduce Haefliger structures and indicate the rôle they play in answering certain classification
problems for foliations. Finally after having given a second description of Haefliger structures we describe
a model for the classifying space BΓq of codimension q Haefliger structures together with the universal
codimension q Haefliger structure on BΓq .

2. Simplicial Constructions: other Models for BΓ
Using diﬀerent realizations of simplicial spaces we describe models for classifying spaces of topologicalgroupoids or, more generally, of topological categories. We investigate some manipulations with thesespaces which lead to an outline of a proof of the theorem of Graeme Segal (Classifying spaces related tofoliations , Topology 17, 1978, 367-382) that BΓq is weakly homotopy equivalent to the classifying spaceof the discrete monoid of smooth embeddings Rq → Rq . If time permits we indicate how this approach
might be used to give a proof of the Thurston-Mather theorem relating the discrete group of compactly
supported diﬀeomorphisms of Rq with BΓq . Gaël Meigniez will give his recent proof of this theorem in thesecond lecture of this morning.

3. De Rham Theory for BΓ
The groupoid BΓq of germs of local diﬀeomorphisms of Rq is a smooth but non-Hausdorﬀ q-manifold.
Thus the nerve of the corresponding category is a simplicial manifold, and thus using the de Rham complex
on each of the manifolds of the simplicial manifold we obtain a double complex. The cohomology of theassociated total complex should have some relationship with the real singular cohomology of the classifying
space. If all manifolds are Hausdorﬀ it actually computes the real singular cohomology. If not there is
at least some map from the de Rham type cohomology to the singular cohomology, and the secondary
characteristic classes are actually in this image. But going a little further using open covers of our nonHausdorﬀ manifolds by open Hausdorrf subsets leads if the covers are properly arranged to be compatible
with the simplicial boundary maps to a calculation of the singular cohomology of the classifying space.
Apart from the very last step all the above is contained in Bott, Shulman, Stasheﬀ: On the de Rham
Theory of Certain Classifying spaces, Advances of Math 20, 1976, 43-56. The final result is a special caseof a theorem of Crainic and Moerdijk from 1998 but uses a simpler language.

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Thurston’s h-principle
Gaël Meigniez

These talks aim to report Thurston’s proof (to this day, the only one) of his h-principle for foliations ofcodimension two and more. Then, depending on time, I’ll explain the new proof of the Mather-Thurston
homology equivalence by the same techniques.

Contact structures and foliations on 3-manifolds
Thomas Vogel / Jonathan Bowden

Eliashberg and Thurston have shown that all foliations (except foliations by spheres on S 2 × S 1 ) can beapproximated by contact structures provided that the foliation is suﬃciently smooth and the underlying
manifold is closed. We will explain the main ideas of this theorem and discuss the uniqueness of the contact
structure approximating a foliation. This will then be applied to deduce topological properties of the spaceof taut foliations. (This talk is somewhat independent from the lectures about Haefliger’s classifying spaces
since we will discuss an equivalence relation on the space of taut foliations on a closed 3-manifold which is
diﬀerent from the equivalence relation relevant for BΓ.)A few problems on the classifying spaces of foliations
Shigeyuki Morita

This is a continuation of my talk at the conference “Geometry and Foliations, 2013” given in thepreceding week. Here we expand some part of it and focus on the diﬃcult problem of determining homotopy
types of various universal spaces which appear in the theory of foliations. The most important one is
Haefliger’s classifying space for Γ1 -structures in the smooth category.

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Applications and a special proof of Thurston’s h-principle forcodimension q > 1 in the case of 2-dim foliations on 4-manifolds
Yoshihiko Mitsumatsu

This talk has the following purposes:• By presenting some applications of Thurston’s h-principle for foliations of codimension q > 1, we
would like to draw attentions and to provide motivations for the homotopy theory of foliations.
• By explaining a special proof of the principle for the case of 2-dimensional foliations on 4-manifolds,
which is due to Haefliger and Thurston, the talk serves as an introduction to especially Gaël Meigniez’s
minicourse and also presents an interesting application of the simplicity of some group of diﬀeomorphisms.
Section 1.
First we recall the statement of Thurston’s h-principle and specialize it to the case of 2-dimensional
foliations. It turns out to be that if we have a plane field on a manifold, then it is homotopically deformed
as plane fields into an integrable one, ı.e., a 2-dim foliation. This specialization stands on the (q + 1)connectedness of BΓq which is due to Haefliger (up to q-connectedness) and Mather-Thurston. Some partsof this connectivity might be presented as exercises which are common to the minicourses by Elmar Vogt
and by Gaël Meigniez.
Section 2.
Then as an application in (4,2)-dim setting, we show that the existence of a foliation with prescribedcompact leaf of a closed transversal is determined by cohomological informations.
Section 3.
As a further application of this, we can formulate and assure the existence of turbulization and othermodifications. Also T 2 -knot in R4 is always possible to realize as a compact leaf of a foliation on R4 which
is trivial at infinity. For some special T 2 -knot such as the trivial one or spun-knots we can give an explicit
construction of foliations. The method originated in George Reeb’s construction.
Section 4.
Almost the same construction is used in the special proof of the h-principle in (4,2)-dimensional case,which is the final topic in this talk. The simplicity and the simply-connectedness of Diﬀ c (R2 ) play importantroles. From three major steps of Thurston’s proof for the general case, namely, jiggling lemma, civilization,
and, inflation, we do not need the final one. For the first two, we should refer to Gaël Meigniez’s minicourse.

REFERENCES[1] André Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183–194.
[2] Y. Mitsumatsu and E. Vogt, Foliations and compact eaves on 4-manifolds I. Realizationand self-intersection of compact leaves, in Groups of Diﬀeomorphisms, 415–442. Advanced
Studies in Pure Mathematics 52, Math. Soc. Japan, 2008.
[3] W. P. Thurston, Foliations and groups of diﬀeomorphisms, Bull. Amer. Math. Soc., 80(1974), 304–307.
[4] W. P. Thurston, The theory of foliations of codimension greater than 1, Comment. Math.
Helv., 49 (1974), 214–231.

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Construction of secondary characteristic classes for foliations
using the Chern-Weil theory
Taro Asuke

T.B.A

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Connected components of the space of smooth codimension one
foliations on a closed 3-manifold
Hélène Eynard-Bontemps
We are interested in the topology of the space Fol(M ) of C ∞ transversely oriented codimension one
foliations on a given closed oriented 3-manifold M . We make no diﬀerence between such a foliation andits tangent plane field, that is between foliations and integrable plane fields, and thus think of Fol(M ) asa subset of the space P(M ) of smooth plane fields on M , endowed with the usual C ∞ topology.
Most plane fields are not integrable: non integrable plane fields actually form a dense open subset ofP(M ). It is known since the late 60’s, however, that any closed 3-manifold admits a smooth codimension
one foliation. Moreover, according to works of J. Wood and W.P. Thurston, any smooth plane field canι∗
be deformed into a smooth foliation. In other words, the map π0 Fol(M ) →π0 P(M ) induced by theι
inclusion Fol(M ) ,→ P(M ) is surjective. It is then tempting to ask whether this inclusion is actually aweak homotopy equivalence – or equivalently whether foliations satisfy Gromov’s h-principle –, especiallyas such a result was obtained by Eliashberg in the closely related field of (overtwisted) contact structures.
In this talk, we will present a first step in this direction:
Theorem The inclusion of Fol(M ) in P(M ) induces a bijection between the sets of connected components
of both spaces.

8Gelfand-Fuks cohomology and the cohomology of BΓq
Paul A. Schweitzer, S. J.

1. Definition of Gelfand-Fuks cohomology. The Gelfand-Fuks cohomology of a smooth manifold M n is thecohomology of the Lie algebra X (M ) of smooth vector fields on M with coeﬃcients in R [2]. If Ak (M )denotes the vector space of continuous alternating k-forms on X (M ), then the diﬀerential dk : Ak (M ) →Ak+1 (M ) for α ∈ Ak (M ) is defined as usual by setting∑(−1)i+j α([Xi , Xj ], X0 , . . . , X̂i . . . , X̂j , . . . , Xk )dα(X0 , X1 , . . . , Xk ) =0≤i≤j≤kwhere X1 , . . . , Xk ∈ X (M ). Then ddefined to be

k+1◦ dk = 0 and the Gelfand-Fuks cohomology of M in degree k isk
HGF(M ) = ker(dk )/im(dk−1 ).
∗Gelfand and Fuks showed that for a compact manifold M the cohomology ring HGF(M ) is finite-dimensional[2].
∗
For M = Rn , the Gelfand-Fuks cohomology HGF(Rn ) turns out to be the real cohomology of a finite
CW-complex Yn . If p : E → B = BUn is the universal principal Un bundle and Xn is the 2n-skeleton of thebase space B (which is well-defined since B is a CW-complex with only even-dimensional Schubert cells),∗
then Yn = p−1 (Xn ). At first it seemed that HGF(M ) might be an essentially new invariant of smooth∗manifolds, but that turned out to be false. In fact, Bott’s conjecture aﬃrming that HGF(M ) is isomorphic
to the real cohomology of the space of sections of the fiber bundle with fiber Yn associated to the tangent
bundle T M was proven by several authors. (Note that On ⊂ Un acts on Yn , so the bundle in question iswell-defined.)∑q∗
2. Calculation of HGF(Rq ). Using the Lie derivative with respect to the radial vector field R = i=1 ∂/∂xi
on Rq , one shows that the inclusion Aqpt Rq ⊂ Aq Rq of the alternating q-forms with support at the origin
induces an isomorphism on homology. Now Aqpt Rq is generated by the Dirac operators δ i , which are defined∑i, etc. This permits explicit calculation ofδ i ( i fi (x)∂/∂xi = fi (0), and their partial derivatives δji , δjk∗q
HGF (R ) (see Bott’s notes [3] pp. 125-151).
Let E(u1 , u3 , . . . , u2k+1 ) be the exterior algebra on the elements of odd index up to 2k + 1, the largest
odd integer not greater than q. Let R[c1 , . . . , cq ] be the polynomial algebra generated by the elements ciwith deg(ci ) = 2i and dui = ci , and denote the truncated polynomial algebra R[c1 , . . . , cq ]/I2q , where I2q
b 1 , . . . , cq ]. Then defineis the ideal of elements of degree greater than 2q, by R[cb 1 , . . . , cq ].
W On = E(u1 , u3 , . . . , u2k+1 ) ⊗ R[ckTheorem HGF(Rq ) ≈ H ∗ (W Oq ).
kCorollary HGF(Rq ) has nonzero elements in degrees k = 2q + 1 and k = q 2 + 2q, but vanishes fork < 2k + 1 or k > q 2 + 2q.
2q+1
The nonzero class [u1 cq1 ] ∈ HGF(Rq ) corresponds to the Godbillon-Vey class.
∗
3. Characteristic classes of foliations. The cohomology classes in HGF(Rq ) act as characteristic classesn∗of a codimension q foliation F on a manifold M , since there is a homomorphism φF : HGF(Rq ) →∗H (M ; R). (See Bott-Haefliger [1].) Since the homomorphisms φF are functorial, they induce a universal
homomorphism∗
φ : HGF(Rq ) → H ∗ (BΓq ; R)

where BΓq is the Haefliger classifying space for codimension q foliations. Bott and others conjectured
that the homomorphism φ should be injective, so that the characteristic classes given by the Gelfand-Fuks
9cohomology would be independent. Partial results by Heitsch, Kamber and Tondeur, and Hurder, among
others, support this conjecture, but the complete proof still seems to be very distant.
The cohomology of BΓq is not countably generated, as shown by Thurston’s examples giving continuous
variation of the Godbillon-Vey class for foliations on S 3 and other characteristic classes which vary continuously. Perhaps there is a relationship with coeﬃcient automorphisms of R as a Q-vector space which also
show that H ∗ (BΓq ; R) is immense.
To define φF for a foliation F on M , consider the Lie algebra a(Γ) of “formal Γ vector fields” as theinverse limit of the k-jets at 0 of smooth vector fields:a(Γ) = lim ak (Γ).
←−
Here Γ is the pseudogroup of diﬀeomorphisms of open sets in Rq . The cohomology of the continuousalternating forms on a(Γ) coincides with that of A∗pt Rq . If the foliation F is defined locally by submersions
fU : U → Rq , the k-jets of fU at points of U can be used to define φF . (See Bott-Haefliger [1] p. 1041.)
There are homomorphisms analogous to φ for the classifying spaces of foliations that are transversely
holomorphic or transversely symplectic or have trivialized normal bundles. Partial results on independenceof the corresponding characteristic classes have been obtained by Asuke in the first case and by various
others in the second case.

REFERENCES[1] R. Bott and A. Haefliger, On the characteristic classes of Γ-foliations, Bull. Amer. Math.
Soc. 78 (1972), 1039-1044.
[2] I.M. Gelfand and D.B. Fuks, Cohomologies of the Lie algebra of tangent vector fields of asmooth manifold, Functional Anal. Appl. 3 (1969), 194-210.
[3] M. Mostow and J. Perchik, Notes on Gel’fand Fuks cohomology and characteristic classes, inProceedings of the 11th Annual Holiday Symposium at New Mexico State Univ., Dec. 27-31,
1973, Dept. of Mathematical Sciences, New Mexico State Univ., (1975), 1-220.

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Exercise sets 1 and 2
For the lectures by Elmar Vogt, Gael Meigniez, and Yoshi Mitsumatsu(provisional version)BΓq is the homotopy fibre of the classifying map BΓq → BOq of the normal bundle and homotopicallyclassifies Γq -structures with trivialized normal bundles.
It is known that BΓq is (q + 1)-connected, i.e., πi (BΓq ) = 0 for 0 ≤ i ≤ q + 1. The q-connectedness isdue to Haefliger. Up to q − 1-connectedness it is shown using point foliations (i.e., codimension-q foliations
on q-manifolds) and standing on Haefliger’s theorem and the theory of Gromov-Phillips.
Exercise 1-1(easy) Show π0 (BΓ1 ) = 0 in the following sense. Show that a parallelized S 0 × R1 (there
are 4 such structures) is a parallelized submanifold of R1 with the standard parallelization.
Exercise 1-2 Show π1 (BΓ2 ) = 1 in the following sense. Show that with any parallelization S 1 × R1
(there are Z-many such structures) is immersed into R2 with the standard parallelization.
Now Haefliger’s theorem implies any Γ2 -structure on S 1 × R1 is contractible and thus also π1 (BΓ2 ) = 1.
Exercise 1-3 By using 1-2, show the following result due to Claude Roger.
Any R2 -bundle over S 2 can be the normal bundle to some Γ2 -structure on S 2 .
This is nothing but the surjectivity of the homomorphism π2 (BΓ2 ) → π2 (BO2 ) so that from the homotopy
long exact sequence formally it follows immediately from 1-2.
Exercise 2-1 Show π1 (BΓ1 ) = 1 in the following way.
First confirm that any Γ1 -structure on S 1 is realized as a smooth foliation on S 1 ×R1 which is transverseto {∗} × R1 .
Next change the zero-section homotopically which draws a very wild sine-like curve and make shorter
the R1 -component so that the foliation becomes so simple.
Then embed the resultant foliation into a trivial foliation on S 1 × R1 and extend it over D2 as Γ1 structure.
Exercise 2-2 Toward π2 (BΓ1 ) = 0.
Assume such a singular foliation (Γ1 -structure) on S 2 is given that it has two centers and away from thesecenters it can be regarded as an foliated interval bundle over S 1 .
Show that this foliation is homologically trivial. Being combined with 2-1, this implies that the given(Γ1 -structure) on S 2 extends to D3 .
The key ingredients are the perfectness and the contractibility of Diﬀ c R.

REFERENCES[1] André Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183–194.
[2] Anthony Phillips, Smooth maps transverse to a foliation, Bull. Amer. Math. Soc., 76(1970), 792–797.
[3] Mikhael Gromov, Partial diﬀerential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin,
1986.

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