Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 131–138, 1998

Almost without Holonomy Foliations and Graphs of Groups

Marta Macho-Stadler

In this poster, the holonomy groupoid of an almost without holonomy foliation is described as a graph of abelian groups ([HM] and [Ma2]). As a consequence, and in a similar way to the one described by A. Connes for R-actions [C], we obtain in [HM] and [Ma2] the Baum-Connes conjecture for almost without holonomy foliations.

1 Almost without holonomy foliations

Let (M, F, N), where M^{m} is compact, F a C^{r}-foliation (r ≥ 1) of codimension one, transversally oriented by a vector field Y, tangent to ∂(M) (when ∂(M) ≠ ∅) and N a transverse foliation, orientable, of dimension one, defined by the flow corresponding to Y. (M, F, N) is an \underline{almost without holonomy foliation}, when every non closed leaf L ∈ F, has trivial holonomy. The study of the almost without holonomy foliations can be reduced to the study of the so called models [He1]: (M, F, N) is a \underline{model of almost without holonomy foliation of type 1} \underline{or 2}, if the corresponding condition is verified:

(i) M = L^{m−1} × [0, 1] (L is C^{r}-diffeomorphic to a component of ∂(M)) and N is tangent to [0, 1];

(ii) the foliation F^{∘} induced on M^{∘} by F is a without holonomy foliation and:

(a) F^{∘} is a fibration of M^{∘} over 𝕊^{1}, or

(b) each L ∈F^{∘} is dense in M^{∘}.

132 Marta Macho-Stadler

(M, F, N) is \underline{of finite type}, if F has only a finite number of closed leaves. In the following, (M, F, N) will be an almost without holonomy foliation of finite type. We have the following essential properties ([He1], [He2] and [I]):

(i) a model of type 1 over M = L × [0, 1], is described by an abelian representation of π_{1}(L) in Diff_{+}^{r}([0, 1]);

(ii) if (M, F, N) is a model of type 2 and class C^{r}, r ≥ 2, the leaves of M^{∘} are all diffeormorphic;

(iii) for each L ∈ F, hol(L) is an abelian group;

(iv) F has not exceptional leaves.

Let F = C_{1} ∪ ⋯ ∪ C_{n} be the union of the closed leaves of F (which is closed in M). Then, M −F = U_{1} ∪⋯∪U_{m}, where U_{j} is a connected component of M −F. If F_{j} is (respectively, \overline{F_{j}}) the foliation induced by F over U_{j} (respectively, over \overline{U_{j}}), then F_{j} is a without holonomy foliation, and (\overline{U_{j}}, \overline{F_{j}}) is a model of type 2.

2 (M, F, N) gives a graph of groups Γ(F)

An \underline{oriented graph} is Γ = (Γ^{0}, Γ^{1}, s, r), where:

(i) Γ^{0} is the set of vertices,

(ii) Γ^{1} is the set of edges,

(iii) the maps s, r: Γ^{1} ⟶ Γ^{0} are the incidence maps, where for each a ∈ Γ^{1}, s(a) ∈ Γ^{0} is the source of a and r(a) ∈ Γ^{0} is the end of a (the orientation is included in the definition of r and s). s(a) and r(a) are the extremities of a.

An oriented graph is \underline{finite}, when the sets Γ^{0} and Γ^{1} are finite.

A \underline{finite graph of abelian groups of homeomorphisms of finite type} of ℝ (GHℝ, for brevity), is defined by:

(i) a graph Γ = (Γ^{0}, Γ^{1}, s, r) finite and oriented,

Almost without Holonomy Foliations and Graphs of Groups 133

(ii) for each s ∈ Γ^{0}, G_{s} ⊂ Homeo(ℝ) is an abelian group, of finite type and without fixed points, (iii) for each a ∈ Γ^{1}, H_{a} ⊂ Homeo(ℝ, 0) is an abelian group, of finite type, and

with 0 as unique fixed point,

(iv) for each a ∈ Γ^{1}, there are homomorphisms: S_{a}: H_{a} ⟶ G_{s(a)} and R_{a}: H_{a} ⟶ G_{r(a)}, where R_{a}(f) = log ∘ f ∘ exp and S_{a}(f) = log ∘ f ∘ −exp.

S_{a} and R_{a} are not, in general injective or surjective. Geometrically, the passage of H_{a} to G_{r(a)} consists in to restrict f to (0, ∞) and to reparametrize to obtain an homeomorfism of ℝ, which evidently has not yet fixed points. A \underline{morphism} f: Γ = (Γ^{0}, Γ^{1}, s, r) ⟶ Γ̂ = (Γ̂^{0}, Γ̂^{1}, ŝ, r̂) between two GHℝ, is defined by:

(i) a couple of maps f_{0}: Γ^{0} ⟶ Γ̂^{0} and f_{1}: Γ^{1} ⟶ Γ̂^{1}, which preserve the orientation and verifying ŝ∘ f_{1} = f_{0} ∘ s and r̂∘ f_{1} = f_{0} ∘ r,

(ii) two homomorphisms h_{a}: H_{a} ⟶ Ĥ_{f_{1}(a)} and g_{s}: G_{s} ⟶ Ĝ_{f_{0}(s)}, such that g_{s(a)} ∘ S_{a} = Ŝ_{f_{1}(a)} ∘ h_{a} and g_{r(a)} ∘ R_{a} = R̂_{f_{1}(a)} ∘ h_{a}.

If F = C_{1} ∪ ⋯ ∪ C_{n}, is the union of the closed leaves in F and U = M − F = U_{1} ∪ ⋯ ∪ U_{m}, then for C_{i} (i ∈ {1, ⋯, n}), we observe:

• there is exactly one connected component in U, U_{r(i)}, such that C_{i} is leftadherent to U_{r(i)} (the transverse field is “attracting” in U_{r(i)});

• there is exactly one connected component in U, U_{s(i)}, such that C_{i} is rightadherent to U_{s(i)} (the transverse field is “expanding” in U_{s(i)});

• we can have U_{s(i)} = U_{r(i)}.

The transverse foliation N is defined by a flow φ. If φ: C_{i} × ℝ ⟶ M, F can be lifted on C_{i} × ℝ, in a foliation φ^{∗}(F), such that:

• φ^{∗}(F) is almost without holonomy, but not of finite type,

• the closed leaves of φ^{∗}(F) are isolated and their union K is closed in C_{i} ×ℝ,

• (C_{i} × ℝ) − K = ⋃_{n∈ℕ}W_{n}; if C_{i} (identified with C_{i} × {0}) is left-adherent to

W_{r(i)} and right-adherent to W_{s(i)}, then we have W_{r(i)} ≠ W_{s(i)}.

If W_{i}^{+} = W_{r(i)} ∪ C_{i} and W_{i}^{−} = W_{s(i)} ∪ C_{i}, then W_{i} = W_{i}^{+} ∪ W_{i}^{−} is a neighborhood of C_{i} in C_{i} × ℝ, saturated for φ^{∗}(F). And φ^{∗}(F)|_{W_{i}}, verifies:

• φ^{∗}(F)|_{W_{i}} is an almost without holonomy foliation, • there is exactly one closed leaf C_{i},

• W_{i} fibers over C_{i} with fiber ℝ (after reparametrization of the leaves of φ^{∗}(N)|_{W_{i}} by ℝ),

• φ^{∗}(F)|_{W_{i}} is defined by the suspension of a group of abelian homeomorphisms of ℝ, which represents the holonomy of C_{i}: W_{i} is the \underline{geometric realisation} of the holonomy group of C_{i}.

We have φ: W_{i}^{+} ⟶ V_{i}^{+} = φ(W_{i}^{+}), φ: W_{i}^{−} ⟶ V_{i}^{−} = φ(W_{i}^{−}), and we consider φ(W_{i}) = V_{i}, which is a neighborhood of C_{i} in M. We have two possibilities: V_{i} = V_{i}^{+} = V_{i}^{−} (when U_{s(i)} = U_{r(i)}) or C_{i} = V_{i}^{+} ∩ V_{i}^{−}.

For j ∈ {1, ⋯, m}, F|_{U_{j}} = F_{j} is a without holonomy foliation. If we fix a base point x_{j} ∈ U_{j}, there is a unique transversal N_{j} ∈ N which contains this point. For a convenient election of x_{j}, N_{j} is homeomorphic to ℝ, and we consider a parametrisation p_{j}: ℝ ⟶ N_{j}. F_{j} is defined by a group of homeomorphisms without fixed points, of finite type (thus abelian and archimedian) of N_{j} (thus of ℝ, by reciprocal image): it is the global holonomy group G_{j} of F_{j}. In the same manner, for i ∈ {1, ⋯, n}, the induced foliation F|_{V_{i}}, is defined by an abelian group of homeomorphism of a transversal N_{i}^{∗} (thus, of ℝ, under reparametrisation), of finite type, with 0 as fixed point: it is the holonomy group H_{i} of the leaf C_{i}. And we have an homomorphism R_{i}: H_{i} ⟶ G_{r(i)} , defined by R_{i} = λ_{i} ∘ ρ_{i} ∘ η_{i}, where:

• η_{i}: H_{i} ⟶ H_{i}^{+}, H_{i}^{+} = {f|_{[0,∞)} : f ∈ H_{i}} is the right holonomy group of C_{i} and η_{i} is the restriction to [0, ∞) (η_{i} is not, in general, injective. But, all the η_{i} are injective, by example, when we have a C^{ω}-foliation),

• ρ_{i}: H_{i}^{+} ⟶ H_{r(i)}, where H_{r(i)} is the holonomy of the foliation on W_{r(i)}, which is the reciprocal image by φ of the foliation over U_{r(i)}: we identify H_{r(i)} with the holonomy group of F|_{V_{r(i)}}. ρ_{i} is the restriction to (0, ∞), composed with the exponential map,

• λ_{i}: H_{r(i)} ⟶ G_{r(i)} is the natural inclusion.

We define S_{i}: H_{i} ⟶ G_{s(i)} in a similar manner.

Proposition 1. We have a GHℝ, the graph of (M, F), Γ(F), defined by:

(i) the finite and oriented graph (Γ^{1}, Γ^{0}, s, r), Γ^{1} = {V_{1}, ⋯ V_{n}}, Γ^{0} = {U_{1}, ⋯, U_{m}}, s(V_{i}) ⊂ U_{s(i)} and r(V_{i}) ⊂ U_{r(i)}, where r(V_{i}) is an open set in U_{r(i)}, represented by the inclusion by φ of W_{r(i)} in U_{r(i)} (similarly for s(V_{i})), (ii) for j ∈ {1, ⋯, m}, G_{U_{j}} ⊂ Homeo(ℝ) is the global holonomy group G_{j} of

(U_{j}, F_{j}), which is abelian, of finite type and without fixed points,

(iii) for i ∈ {1, ⋯, n}, H_{V_{i}} ⊂ Homeo(ℝ, 0) is the holonomy group H_{i} of the closed leaf C_{i}, which is abelian, of finite type, and with 0 as unique fixed point,

(iv) for i ∈ {1, ⋯, n}, R_{V_{i}}: H_{V_{i}} ⟶ G_{r(V_{i})} is defined by R_{V_{i}} = R_{i}. r(V_{i}) is open in U_{r(i)} and G_{r(V_{i})} is a subgroup of G_{U_{r(i)}} (similarly for S_{V_{i}}).

Theorem 2. T = (⨆_{j=1}^{m}N_{j})∪(⨆_{i=1}^{n}N_{i}^{∗}) is a total transversal to F, naturally embedded

in M. Γ(F) is the transverse groupoid of F, relatively to T.

Proposition 3. (M, F, N) is in fact without holonomy iff Γ(F) is reduced to a unique vertex s, whose group G_{s} is the holonomy group of the foliation.

Theorem 4. With the obvious notations, (M_{1}, F_{1}, N_{1}) is Morita-equivalent to (M_{2}, F_{2}, N_{2}), iff Γ(F_{1}) and Γ(F_{2}) are isomorphic.

3 A GHℝ gives an almost without holonomy foliation

(M, F) is a classifying foliation, if the holonomy covering of each leaf is contractible, that is, the fibers of the holonomy groupoid are contractible. In this case, the holonomy and homotopy groupoids of the foliation, are isomorphic.

Lemma 5. (M, F, N) is classifying iff all the non closed leaves are contractible and the closed leaves are torus with injective holonomy.

Let Γ be a GHℝ, and consider the subgroups of H_{a}:

H_{a}^{+} = {f|_{[0,∞)} : f ∈ H_{a}} and H_{a}^{−} = {f|_{(−∞,0]} : f ∈ H_{a}}.

If r_{a}: H_{a} ⟶ H_{a}^{+} is the restriction map (r_{a}(f) = f|_{[0,∞)}) and j_{a}: H_{a}^{+} ⟶ G_{r(a)} is defined by j_{a}(h) = log ∘ h ∘ exp, we have R_{a} = j_{a} ∘ r_{a}. We make a similar decomposition for S_{a}: H_{a} ⟶ G_{s(a)}, through H_{a}^{−}. In this manner, we associate to each edge a, two semi-edges a^{+} and a^{−}. We realize geometrically every objet in the graph:

(V) Vertices We fix a vertex s ∈ Γ^{0}. Since G_{s} ⊂ Homeo(ℝ) is abelian, of finite type and without fixed points, the Hölder theorem states that there is n_{s} ∈ ℕ such that G_{s} ≃ ℤ^{n_{s}}. We suspend G_{s} (with fibre ℝ) and we obtain a fibre bundle ℝ → E_{s} → 𝕋^{n_{s}}, with discrete structural group ℤ^{n_{s}}, where E_{s} (diffeomorphic as manifold to 𝕋^{n_{s}} ×ℝ), is provided of a foliation F_{s}, by n_{s}-planes, which is tranverse to the fibers of the fibration.

(SE) Semi-edges If a ∈ Γ^{1}, H_{a}^{+} ⊂ Homeo([0, ∞), 0) is an abelian group, of finite type and having 0 as unique fixed point. Thus, there is p_{a}^{+} ∈ ℕ, such that H_{a}^{+} ≃ ℤ^{p_{a}^{+}}. We suspend this group, and we obtain a fibre bundle [0, ∞) → E_{a}^{+} → 𝕋^{p_{a}^{+}}. E_{a}^{+} (diffeomorphic as manifold to 𝕋^{p_{a}^{+}} × [0, ∞)) is provided of a transverse foliation, F_{a}^{+}, by p_{a}^{+}-planes. We make the analogous construction for H_{a}^{−}.

(VSE) Vertices and open semi-edges Let a ∈ Γ^{1}. The injective homomorphism j_{a}: H_{a}^{+} ⟶ G_{r(a)}, induce a morphism of foliated bundles ĵ_{a}: E_{a}^{+} ⟶ E_{r(a)},

which is an isomorphism between the foliated bundles ĵ_{a}: E^{∘}_{a}^{+} ⟶ j_{a}(E^{∘}_{a}^{+}⊂ E_{r(a)}. On the disjoint sum, E_{a}^{+} ⊔ E_{r(a)}, we define the equivalence relation: x ∼ ĵ_{a}(x) for

x ∈E^{∘}_{a}^{+}: we identify in this manner two open sets in the bundles E_{a}^{+} and E_{r(a)}. The quotient space E_{a}^{+} ⊔ E_{r(a)}/ ∼, is a space E (not yet a fibre bundle), provided of a foliation F_{E} induced by the quotient map: we have glued the foliated bundles E_{a}^{+} and E_{r(a)} together. The same construction is valid for E_{a}^{−} and E_{s(a)}.

(ESE) Edges and semi-edges If a ∈ Γ^{1}, the goal is to glue the foliated bundles E_{a}^{+} and E_{a}^{−}. But H_{a} ⊂ Homeo(ℝ), is an abelian group, of finite type, with 0 as unique fixed point; thus there is q_{a} ∈ ℕ, such that H_{a} ≃ ℤ^{q_{a}}. r_{a}: H_{a} ⟶ H_{a}^{+}, can be extended in an homomorphism r_{a}: 𝕋^{q_{a}} ⟶ 𝕋^{p_{a}^{+}}, which define (since r_{a} is a restriction map) the trivial fibre bundle 𝕋^{q_{a}−p_{a}^{+}} → 𝕋^{q_{a}} →^{r_{a}} 𝕋^{p_{a}^{+}}. If we consider the section ϵ_{a}: 𝕋^{p_{a}^{+}} ⟶ 𝕋^{q_{a}} of this fibre bundle, the compact leaf of the frontier of the fibre bundle E_{a}^{+} has been embedded in 𝕋^{q_{a}}. Finally, we glue 𝕋^{q_{a}} and E_{a}^{+} together, by identifying over the disjoint sum E_{a}^{+}⊔𝕋^{q_{a}} the points (x, 0) and ϵ_{a}(x), for x ∈ 𝕋^{p_{a}^{+}} (that is, we have glued a and a^{+} together, on the graph). We make the same construction for E_{a}^{−}. In this manner we have glued a^{−} and a^{+} together on the graph.

Geometric realization One “model” in the language of graph theory, corresponds to a fixed vertex and all the edges that have it as extremity. Thus, we obtain a “geometric model” by a finite number of identifications of type (VSE). In the language of graph theory, we glue two models together, when they are common vertices: we make this by identifications of type (ESE), by retrieving an edge from the corresponding semi-edges. We obtain in this manner the \underline{geometric realization} of the graph, that is, a “foliated” CW-complex (M_{0}, F_{0}), which is also classifying, by construction. F_{0} is a regular foliation of codimension one over a locally compact space. The dimension of the leaves depends of the different “models”. We can obtain a constant dimension by uniformizing the dimension step by step in the construction of the geometric realization:

• a “model” corresponds to a vertex s of the graph. We consider the finite family {a_{1}, ⋯, a_{k}} of edges having s as extermity. Thus, n_{s} ≥ p_{a_{i}}^{±} for each i ∈ {1, ⋯, k}. For the foliated bundles (E_{a_{i}}^{±}, F_{a_{i}}^{±}), we complete the dimension multiplying the leaves by factors ℝ^{n_{s}−pa_{i}^{±}};

• when we glue different “models” together, there are two dimensions to uniformize. We obtain this, by multiplying the model of less dimension by a convenient factor ℝ^{l} (we embed otherwise the torus 𝕋^{p_{a}^{±}} in 𝕋^{q_{a}}: instead of embedding the torus, we embed a convenient nhood of 𝕋^{p_{a}^{±}} in 𝕋^{q_{a}}).

Then, by induction, we prove that (M_{0}, F_{0}) is Morita-equivalent to (M′_{0}, F′_{0}), where M′_{0} is a non compact manifold. Thus:

Theorem 6. If Γ is a GHℝ, its geometric realization (M_{0}, F_{0}) verifies:

(i) F_{0} is an almost without holonomy and classifying foliation;

(ii) the graph of the foliation (M_{0}, F_{0}), Γ(F_{0}), is isomorphic to Γ.

The Morita-equivalence class of an almost without holonomy foliation (M, F, N) is an invariant of the graph of groups of the foliation:

Theorem 7. The geometric realization of Γ(F) is a foliated space Moritaequivalent to (M, F, N).

The Baum-Connes conjecture for an almost without holonomy foliation (M, F) is now easier to prove: we change (M, F) by its classifying foliation (M_{0}, F_{0}) (Theorems 6 and 7). For a classifying foliation, the statement of the Baum-Connes conjecture is: “The Thom-Connes homomorphism μ: K^{∗}(M) ⟶ K_{∗}(C^{∗}((G)) is a group isomorphism” (G is the holonomy groupoid of F), and using a generalization of the techniques in [Ma1], we prove in [HM] and [Ma2] this result.

References

[C] A. Connes, An analogue of the Thom isomorphism for crossed products of a C*-algebra by an action of ℝ, Advances in Maths. 39, 31-55, 1981.

[He1] G. Hector, Sur les feuilletages presque sans holonomie, C.R. Acad. Sc. Paris 274, 1703-1706, 1972.

[He2] G. Hector, Croissance des feuilletages presque sans holonomie, Springer Lecture Notes in Maths 652, 141-182, 1978.

[HM] Hector et M. Macho Stadler, L’isomorphisme de Thom pour les graphes de groupes abéliens, preprint.

[I] H. Imanishi, Structure of codimension one foliations which are almost without holonomy, J. Math. Kyoto Univ. 16, n^{0}1, 93-99, 1976.

[Ma1] M. Macho Stadler, La conjecture de Baum-Connes pour un feuilletage sans holonomie de codimension un sur une variété fermée, Publicacions Matemàtiques 33, n^{0}3, 445-457, 1989.

[Ma2] M. Macho Stadler, Isomorphisme de Thom pour les feuilletages presque sans holonomie, Thèse de Doctorat, Lyon 1996.

Marta Macho-Stadler

Facultad de Ciencias. Departamento de Matemáticas Universidad del País Vasco-Euskal Herriko Unibertsitatea Apartado 644. 48080 Bilbao Spain

   Email: mtpmastm@lg.ehu.es