DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 345–352

GRAPHS, EHRESMANN CONNECTIONS AND VANISHING CYCLES

ROBERT A. WOLAK

In recent years the graph of a foliation, an object which has been known for a long time, cf. [10], has known new interest. In fact there are two groupoids associated with a foliation, the homotopy groupoid and the holonomy groupoid, sometimes called the graph. It serves as a basis for the construction of the C^{∗}- algebra associated to the foliation. Moreover, the homotopy groupoid of the characteristic foliation of a Poisson manifold is used in the symplectic integration of this Poisson manifold, cf. [22, 12, 15, 2]. For a general foliation we do not know whether its groupoids are Hausdorff manifolds, cf. [23, 21]. Recently P. Dazord and G. Hector proved that the homotopy groupoid is Hausdorff iff the foliation has no vanishing cycles, cf. [12]. In this note study the existence of vanishing cycles for some classes of foliations, including totally geodesic foliations and these which admit an Ehresmann connection.

1. Preliminaries

In this section, for the convenience of the reader, we will recall some basic definitions and results.

1.1. The graph of a foliation

The definition and the basic properties of the graph of a foliation can be found in [23]. Let us recall the definition.

The graph GR(F) of the foliation F is the space of equivalence classes of triples (y, α, x) where x and y are points of the same leaf L of F and α is a path in L linking x to y. Two triples (y, α, x) and (y′, α′, x′) are equivalent iff x = x′, y = y′ and the holonomy of the curve α^{−1} ∗ α′ is trivial. In the same paper the author proved that the graph of a foliation is a manifold of dimension n + p but in general non–Hausdorff. Moreover, if the elements of the holonomy pseudogroup are determined by their jets the graph is a Hausdorff topological space.

1991 Mathematics Subject Classification. 57R30, 53C05, 53C15.

Key words and phrases. Foliation, Ehresmann connection, vanishing cycle, homotopy groupoid, holonomy groupoid.

Supported by the KBN grant /93.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The correspondences

p_{1} : ⟨y, α, x⟩ ⟼ x,

the source projection, and

p_{2} : ⟨y, α, x⟩ ⟼ y,

the target projection, define two submersions

p_{1} : GR(F) ⟶ M and p_{2} : GR(F) ⟶ M.

First Winkelnkemper and then Wolak, cf. [23, 25, 28], studied the local triviality of the submersions p_{1} and p_{2}. We know that under natural assumptions they are local trivial fibre bundles for Riemannian, transversely affine, totally geodesic and some other classes of foliations. Using the method developed in [25, 28] one can prove that these submersions are locally trivial fibre bundles for foliations admitting an Ehresmann connection.

The homotopy groupoid of a foliation is defined in a similar way, we identify leaf curves which are homotopic relative to their ends in the corresponding leaf. Likewise the homotopy groupoid is a manifold, but not necessarily Hausdorff. There is a natural submersion from the homotopy groupoid onto the holonomy groupoid of a given foliation. In [13] one can find an example of a foliation for which the holonomy groupoid is Hausdorff but its homotopy groupoid is not, cf. also [17]. The non-Hausdorffness of the homotopy groupoid is an important obstruction in the symplectic integration of Poisson manifolds, cf. [13, 2], as well as in the quantization, cf. [1].

1.2. Ehresmann connections

The concept of an Ehresmann connection appears for the first time, under a different name, in [10]. However, recent studies of totally geodesic foliations have brought out its importance, cf. [6, 9], in geometry.

The notion of a foliation admitting an Ehresmann connection is a natural generalization of the concept of a totally geodesic foliation, cf. [7], namely:

Let (M, F) be a foliated manifold. A supplementary subbundle Q to T F is called an Ehresmann connection if for any point x ∈ M and any pair of curves α: [0, a] → M and β: [0, b] → M with the same starting point, such that the curve α is a leaf curve, and β is tangent to Q, there exists a smooth mapping σ: [0, a] × [0, b] → M such that

1) the curves σ_{s}: [0, a] → M, σ_{s} = σ|[0, a] × {s}, s ∈ [0, b], are leaf curves;

2) σ_{0} = α;

3) the curves σ^{t}: [0, b] → M, σ^{t} = σ|{t} × [0, b], t ∈ [0, a],

are tangent to Q;

4) σ^{0} = β.

If rectangles as above exist only for a certain class of curves tangent to the subbundle Q we say that Q is a reduced Ehresmann connection. This class must be sufficiently rich to contain curves starting in each direction of Q. For applications it could be usefull to assume an equivalent definition allowing the path β to begin at any point of the curve α, i.e. β(0) = α(t_{0}) for some t_{0} ∈ [0, a]. Then we would have the condition 4’) σ^{t_{0}} = β.

We know that the orthogonal bundle to a totally geodesic foliation on a complete Riemannian manifold is an Ehresmann connection for this foliation, cf. [6].

Very many results concerning Riemannian foliations were proved using the properties of geodesics under the assumption that the Riemannian manifold is complete. However, in most cases it is sufficient to know that the orthogonal subbundle is an Ehresmann connection and to demonstrate that it is an Ehresmann connection it is not necessary to assume that the Riemannian metric is complete. Let us give an example.

Example Let M = R^{3} − {(0, 0)} × R and p: M → R × R be the projection (x, y, z) ↦ (x, y). Then p(M) = R^{2} − {(0, 0)}. With the standard metric g_{0} on M induced from the euclidean space R^{3}, the Riemannian manifold (M, g_{0}) is not complete; however, the orthogonal bundle to the foliation defined by the connected components of the fibres of the submersion p is an Ehresmann connection for this foliation. To put it succinctly, we can define and work with an Ehresmann connection when certain geodesics are defined uniformly along the leaves.

The existence of the smooth mapping (rectangle) σ allows us to push any Qcurve β along the curve α and any leaf curve α along the curve β. Namely, cf. [6],

β_{♯}: α ↦ β_{♯}α = σ_{b}

and

α_{♯}: β ↦ α_{♯}β = σ^{a}

Let K(L, x) be the subgroup of π_{1}(L, x) of the homotopy classes of loops τ in L for which τ_{♯}γ = γ for any curve γ tangent to Q and starting at the point x. In [6], the authors notice that K(L, x) is a normal subgroup of π_{1}(L, x). The quotient group π_{1}(L, x)/K(L, x) we denote by H_{g}(L, x), cf. [7, 9].

At the end of this section let us formulate some basic properties of an Ehresmann connection.

Lemma 1. For any rectangle σ defined by a pair of curves (α, β), the curve σ^{a} depends only on the homotopy class of the curve α and the curve σ_{b} on the homotopy class of β.

The above lemma is a consequence of the following one:

Lemma 2. Let γ: [0, c]×[0, a] → M be a smooth (continuous) family of leaf curves parametrized by the interval [0, c]. Let β: [0, c] × [0, b] → M be a smooth, resp. continuous, family of curves tangent to the subbundle Q such that γ|[0, c] × {0} = β|[0, c] × {0}. Then there exists a smooth, resp. continuous, mapping σ: [0, c] × [0, a] × [0, b] → M such that for any t ∈ [0, c] the mapping σ|{t} × [0, a] × [0, b] is the rectangle for the pair (α^{t}, β^{t}).

1.3. Some examples

For some classes of foliations one can build the rectangle for some pairs of curves, eg. for a transversely affine foliation it is possible for any leaf curve and the other curve being a transverse geodesic of the transversely projectable flat connection, cf. [27, 26], provided that the foliation is transversely geodesically complete. The same is true for any foliation admitting a tranversely complete tranversely projectable connection, cf. [26]. More generally it is also true for foliations admitting foliated systems of ordinary differential equations (FSODE) which are transversely complete, cf. [24, 26]. Moreover, a transverse curve instead of being a geodesic (solution of an FSODE) can be a piecewise geodesic (resp. piecewise solution of an FSODE). A similar argument is valid for foliations parallel with respect to some connection. It is the case of a lagrangian foliation, cf. [18, 15, 16, 29].

2. Vanishing cycles

A vanishing cycle for a foliation F it is a mapping c: S^{1} × [0, 1] → M such that for any t ∈ [0, 1] c_{t} = c|S^{1} × {t} is a loop on a leaf of F and c_{0} is not homotopic to the constant loop in the leaf but for t > 0 the loops c_{t} are, cf. [14].

In [12] it is demonstrated that the non-existence of vanishing cycles is equivalent to the Hausdorffness of the homotopy groupoid of the foliation, see also [2]. Using the same method one can prove the following theorem. Before stating the theorem we have to introduce the notion of a holonomy vanishing cycle.

Definition 1. A holonomy vanishing cycle for a foliation F it is a mapping c: S^{1}× [0, 1] → M such that for any t ∈ [0, 1] c_{t} = c|S^{1} × {t} is a loop on a leaf of F and c_{0} has non-trivial holonomy but for t > 0 the loops c_{t} carry no holonomy.

Theorem 1. Let (M, F) be a foliated manifold. Then the graph of the foliation F is Hausdorff iff F has no holonomy vanishing cycles.

In particular we have the following:

Corollary 1. Let (M, F) be a foliated manifold. If F has no holonomy, then the graph of the foliation F is a Hausdorff manifold.

For foliations admitting a (reduced) Ehresmann connection we have an interesting characterization of the existence of (holonomy) vanishing cycles.

We are going to prove that for the foliations we are interested in the existence of a vanishing cycle implies the existence of some very special vanishing cycles.

Lemma 3. If a foliation F admits a vanishing cycle, then it admits a vanishing cycle c: S^{1} × [0, 1] → M for which the curves c^{t} are tangent to the subbundle Q.

Proof Let us first notice that there exists a vanishing cycle, denoted by the same letter c, based on the same loop c_{0} for which the curve c^{0} is tangent ot the subbundle Q. Then we apply Lemma 2 to the mapping c with the second mapping given by β: [0, 1] × [0, 1] → M β(t, s) = σ^{0}(s + t(1 − s)). From the continuity assured by Lemma 2 we get that c_{0}^{♯}(c^{0}) = c^{0}. This fact ensures that the rectangle defined by the pair (c_{0}, c^{0}) defines a mapping σ from S^{1} × [0, 1] into M. We know that the loops σ_{t} and c_{t} are in the same leaf for any t ∈ [0, 1] and from the continuity we obtain that for t small enough these loops are so near that they are free homotopic. Therefore the loops σ_{t}, t > 0 are nullhomotpic and the mapping σ|S^{1} × [0, t_{0}] → M is a vanishing cycle for a suitable small t_{0} > 0.□

Theorem 2. A foliation admitting an Ehresmann connection has no vanishing cycles.

Proof Let assume that there exists a vanishing cycle and take a one whose existence has been ensured by Lemma 3: σ: S^{1} × [0, 1] → M. For any t ∈ (0, 1] σ_{t}: S^{1} → M is a leaf loop homotopic to the constant loop in the leaf L_{t} passing through the point σ(0, t). Let H: [0, 1] × [0, 1] → L_{1} be the homotopy of σ_{1} to the constant loop. Let ω_{t}: [0, 1] × [0, 1] → M be the rectangle defined by the pair (H_{t}, (σ^{0})^{−1}). For some t ∈ [0, 1] ω_{t}^{0} ≠ ω_{t}^{1}, as otherwise the loop σ_{0} would be homotopic to the constant loop in the leaf L_{0}. Contradiction. □

Corollary 2. The homotopy groupoid of a foliation admitting an Ehresmann connection is a Hausdorff manifold.

The existence of holonomy vanishing cycles can be characterized in the language of holonomy groups of leaves.

Theorem 3. Let F be a foliation admitting a (reduced) Ehresmann connection. Then there exists a holonomy vanishing cycle iff the natural mapping ξ: H_{g}(L) → H_{tr}(L) is not injective for some leaf L of the foliation F.

Proof →) Let σ: S^{1} × [0, 1] → M be a holonomy vanishing cycle. The loop σ_{0} carries nontrivial holonomy. Then there exits a holonomy vanishing cycle based on the same loop σ_{0} for which the curve σ^{0} is tangent to the transverse subbundle Q. Let us denote this new holonomy vanishing cycle by the same letter σ. If in the rectangle c defined by the pair (σ_{1}, (σ^{0})^{−1}) c^{0} is different from c^{1}, the loop σ_{1} and the constant loop in the leaf L_{1} define different elements in the group H_{g}(L_{1}) but both have trivial holonomy. The same is true for any loop σ_{t} and the curves σ^{0}|[0, t] and σ^{0}|[t, 1]. If not, then for any t > 0 σ_{t}^{♯}(σ^{0}) = σ^{0}, and by continuity σ_{0}^{♯}(σ^{0}) = σ^{0}. The loop σ_{0} carries holonomy h, so let y be a point of a transverse submanifold at the point σ(0, 0) for which h(y) ≠ y. Let t be sufficiently small so that there exists a curve β tangent to Q starting at σ(0, t) and ending at the plaque containing y. As the holonomy along σ_{0} maps y to h(y) β(1) must be different from σ_{t}^{♯}(β)(1), so the loop σ_{t} carries no holonomy but defines a nontrivial element in the group H_{g}(L_{t}).

←) Let σ_{0} : [0, 1] → L be a loop in the leaf L which defines a nontrivial element in H_{g}(L) but which carries no holonomy. Thus there exists a (distinguished) transverse curve β such that in the rectangle σ defined by the pair (σ_{0}, β) the transverse curve σ^{0} is different from σ^{1}. However as the holonomy of σ_{0} is trivial there exists ϵ_{0} > 0 such that σ^{0}|[0, ϵ_{0}] = σ^{1}|[0, ϵ_{0}] and it is not the case for any ϵ_{1} > ϵ_{0}, i.e. σ^{0}|[ϵ_{0}, t] ≠ σ^{1}|[ϵ_{0}, t] for any t ∈ (ϵ_{0}, ϵ_{1}]. Therefore the holonomy along the loop σ_{ϵ_{0}} is not trivial. The space of t ∈ [0, ϵ_{0}] for which the loop σ_{t} carries no holonomy is an open subset of [0, ϵ]. Therefore there exists an ϵ > 0 such that for any t ∈ [0, ϵ) the loop σ_{t} has no holonomy but the holonomy of σ_{ϵ} is nontrivial. Thus σ̄(t, s) = σ(t, ϵ − t) for (t, s) ∈ S^{1} × [0, ϵ] is a holonomy vanishing cycle. □

Corollary 3. Let (M, F) be a foliated manifold. If F admits a (reduced) Ehresmann connection, then the holonomy groupoid (graph) of F is Hausdorff iff for any leaf L of the foliation F the mapping ξ: H_{g}(L) → H_{tr}(L) is injective.

Having proved the characterization of the existence of (holonomy) vanishing cycles it is time to check in which cases these conditions are satisfied.

In some papers we studied, cf. [24, 25, 26], foliations admitting foliated systems of ordinary differential equations (FSODE). This class of foliations includes Riemannian, cf. [20], transversely affine, cf. [11, 27, 26], ∇ − G–, cf. [19, 20, 26], transversely projective, transversely conformal, cf. [4], foliations as well as foliations admitting a transversely projectable Cartan connection, cf. [3]. One of the main features of such foliations is that there exists a locally trivial fibre bundle E_{0} → M over M with a foliation F_{E} whose leaves are covering spaces of the corresponding leaves of F, [24, 26]. This bundle is the bundle of transverse initial conditions for the FSODE. In the cases we are interested in the leaves of the foliation F_{E} are holonomy coverings of leaves of the foliation F. If transverse solutions exists globally, the foliation admits a reduced Ehresmann connection, cf. [24, 26]. Namely, let α be any leaf curve and β a transverse solution tangent to the chosen transverse subbundle Q starting at the point x = β(0) = α(0) . Let v ∈ (E_{0})_{x} be the initial condition of the solution β. Let L_{v} be the leaf of the foliation F_{E} passing through the point v. As the leaf L_{v} is a (holonomy) covering space of the leaf L_{x} passing through the point x the leaf curve α can be lifted to a leaf curve ᾱ in the leaf L_{v} starting at the point v. The curve ᾱ provides us with initial conditions along the curve α. The rectangle is defined as follows: σ(t, s) = exp(s; ᾱ(t)) where s ↦ exp(s; η) is the solution of the FSODE with the initial condition η. Let α_{1} and α_{2} be two leaf curves from the point x to the point y and β a solution starting at the point x with the initial condition v. If our curves carry the same holonomy, their lifts ᾱ_{i}, i = 1, 2, respectively, to the point v of the bundle of initial conditions have the same end point, i.e. ᾱ_{1}(1) = ᾱ_{2}(1). Let σ_{1} and σ_{2} be the rectangles associated to the pairs (α_{1}, β) and (α_{2}, β), respectively, then (σ_{1})^{1} = (σ_{2})^{1} as both curve are the solution of the FSODE with the initial condition ᾱ_{1}(1) = ᾱ_{2}(1). Therefore we have proved the following proposition.

Proposition 1. Let (M, F) be a foliated manifold admitting a transversely complete, transversely transitive FSODE with the uniqueness solution property. Then the foliation admits neither holonomy vanishing cycles nor vanishing cycles.

As a corollary of Proposition 1 and Theorem 1 we get the following theorem, cf. [25].

Theorem 4. Let (M, F) be a foliated manifold admitting a transversely complete, transversely transitive FSODE with the uniqueness solution property. Then the homotopy groupoid and the holonomy groupoid (the graph) of the foliation F are Hausdorff manifolds.

As particular cases of this theorem we obtain the fact that under suitable assumptions the homotopy and holonomy groupoids of Riemannian, transversely affine and so on, foliations are Hausdorff manifolds. Therefore we have the following:

Theorem 5. Let F be a transversely complete Riemannian (transversely affine, ∇ − G) foliation, then the foliation admits neither holonomy vanishing cycles nor vanishing cycles.

For codimension 1 foliatons we can draw the following conclusion:

Corollary 4. Let F be a codimension one transversely complete transversely affine (∇ − G) foliation of a compact 3-manifold M, then the foliation does not have any Reeb component.

References

1. F. Alcalde Cuesta, Préquantification de certaines variétés de Poisson, in Geometric Study of Foliations, Tokyo 1993, ed. T. Mizutani and al., World Scientific, Singapore, 1994, 123–143.

2. F. Alcalde Cuesta, G. Hector, Intégration symplectiques des variétés de Poisson sans cycle évanouissant, Israel J. Math. (to appear)

3. R. A. Blumenthal, Cartan connections in foliated bundles, Michigan Math. J. 31(1984), 55–63.

4. R. A. Blumenthal, Stability theorems for conformal foliations, Proc. AMS 91,3 (1984), 485– 491.

5. R. A. Blumenthal, J. J. Hebda, De Rham decomposition theorem for foliated manifolds, Ann. Inst. Fourier 33 (1983), 183–198.

6. R. A. Blumenthal, J. J. Hebda, Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliations, Quart. J. Math. Oxford 35 (1984), 383–392.

7. R. A. Blumenthal, J. J. Hebda, Ehresmann connections for foliations, Indiana Un. Math. J. 33(4) (1984), 597–611.

8. G. Cairns, A general description of totally geodesic foliations, Tohoku Math. J. 38(1986), 37–55.

9. G. Cairns, Feuilletages géodésibles, Thèse, USTL Montpellier 1987.

10. Ch. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloq. Topologie Bruxelles, 1950, 29–55.

11. W. Goldman, M. W. Hirsch, G. Levitt, Invariant measures for affine foliations, Proc. AMS 86 (1982), 511–518.

12. P. Dazord, G. Hector, Integration symplectique des variétés de Poisson totalement asphériques, in Symplectic Geometry, Groupoids and Intregrable Systems, Séminaire Sud Rhodanien à Berkeley (1989), Springer Math. SRIP 20 (1991), Springer-Verlag, 38–72.

13. G. Hector, Une nouvelle obstruction à l’intégrabilité des variétés de Poisson régulières, Hokkaido Math. J. 21 (1991), 159–185.

14. G. Hector, U. Hirsch, Introduction to the Geometry of Foliations, Part A and B, Vieweg, Braunschweig 1981, 1983.

15. H. Hess, Connections on symplectic manifolds and geometric quantization, Springer LN in Math. 836, 153–165.

16. T. Inaba, The tangentially affine structure of lagrangian foliations and the tangentially projective structure of legendrian foliations, preprint 17. C. Lasso de la Vega, Groupoide fondamental et groupoid d’holonomie de certains feuilletages

réguliers, Publ. Mat. UAB 33,3 (1989), 431–443.

18. P. Libermann, Problèmes d’équivalence et géométrie symplectiques, Astérisque 107–108 (1983), 43–68.

19. P. Molino, Propriétés cohomologiques et propriétés topologiques des feuilletages à connexion transverse projetable, Topology 12(1973), 317–325.

20. P. Molino, Riemannian Foliations, Progress in Math. vol 73, Birkhäser, 1988.

21. J. Phillips, The holonomy imperative and the homotopy groupoid of a foliated manifold, Rocky Mountain J. Math. 17 (1987), 151–165.

22. A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. AMS 16 (1987), 101–103.

23. H. Winkelnkemper, The graph of a foliation, Ann. Glob. Anal. Geom. 1 (1983), 51–75.

24. R. Wolak, Foliations admitting transverse systems of differential equations, Comp. Math. 67 (1988), 89–101.

25. R. Wolak, Le graphe d’un feuilletage admettant un système d’équations différentielles, Math. Z. 201 (1989), 177–182.

26. R. A. Wolak, Geometric Structures on Foliated Manifolds, Universidad de Santiago de Compostela, 1989.

27. R. Wolak, Transversely affine foliations compared with affine manifolds, Quarterly J. Math. Oxford 41 (1990), 369–384.

28. R. Wolak, The graph of a totally geodesic foliation, Ann. Pol. Math. 60.3 (1995), 241–247.

29. R. Wolak, Ehresmann connections for lagrangian foliations, to be published in J. Geom. and Physics

Instytut Matematyki, Uniwersytet Jagielloński Wl. Reymonta 4, 30–059 Kraków, POLAND