THE H-PRINCIPLE, LECTURE 14: HAEFLIGER S THEOREM CLASSIFYING FOLIATIONS ON OPEN MANIFOLDS
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1 THE H-PRINCIPLE, LECTURE 14: HAELIGER S THEOREM CLASSIYING OLIATIONS ON OPEN MANIOLDS J. RANCIS, NOTES BY M. HOYOIS In this lecture we prove the following theorem: Theorem 0.1 (Haefliger). If M is an open manifold, there is a bijection between (1) codimension q foliations on M up to integral homotopy and (2) homotopy classes of fibrewise surjective vector bundle maps T M NΓ q. 1. Defining the map Recall that Γ q is the topological groupoid of germs of local diffeomorphisms of R q, the topology being that of the étalé space of the discrete sheaf of local diffeomorphisms. The space BΓ q carries a Γ q -structure U with the following universal property: for any space X, f f 1 (U) induces a bijection between homotopy classes of maps [X, BΓ q ] and Γ q -structures on X up to homotopy. To define the map in Haefliger s theorem, we need to promote this bijection at least to an equivalence of groupoids, following the general theory of classifying spaces. To any Γ q -structure on a space X is associated a map f : X BΓ q together with a homotopy H from f 1 (U) to. This data is canonically defined up to homotopy: given (f 0, H 0 ) corresponding to 0, (f 1, H 1 ) corresponding to 1, we can associate to any homotopy from 0 to 1 a homotopy f from f 0 to f 1 and a Γ q -structure K on X [0, 1] 2 with restrictions given by H 0 K H 1 f 1 (U) and the homotopy type of this data depends only on the homotopy type of. There is a map of topological groupoids d: Γ q GL q given by (x, g) dg. or any Γ q -structure with classifying map f, we define its normal bundle N to be the vector bundle classified by Bd f. In other words, if NΓ q is the pullback by Bd of the universal bundle on BGL q, then N = f (NΓ q ). If f 0 and f 1 are part of different classifying data for, then they are related by a canonical homotopy class of homotopy, so there exists a canonical homotopy class of isomorphisms f 0 (NΓ q ) = f 1 (NΓ q ). By definition, the normal bundle is functorial in that N r 1 () = r (N ) for r a continuous map. Let now M be a manifold. A Γ q -structure on M is called smooth if it is represented by a cocycle {U α, φ αβ } in which the maps φ α : U α R q are smooth. If is a smooth Γ q -structure, we can define a bundle map T M N over the covering {U α } as follows: T U α dφ α N U α T R q U α U α φ α R q, Date: Lecture ebruary 14, Last edited on ebruary 19,
2 where the right-hand square is a pullback by definition of N. By construction, this bundle map is natural in M: if r : N M is a smooth map, then the Γ q -structure r 1 () is represented by the cocycle {r 1 (U α ), φ αβ r}, so we have a commutative square T N dr T M N r 1 () N. Our goal is now to classify Γ q -foliations on M, the set of which we denote by ol q (M). These are the smooth Γ q -structures on M whose associated bundle map T M N is fibrewise surjective. Thus to any ol q (M) we can associate an element T M NΓ q M f BΓ q in the space Map surj Vect (T M, NΓ q) of fibrewise surjective bundle maps. We wish to show that this gives a well-defined map (1) ol q (M) π 0 Map surj Vect (T M, NΓ q). Suppose that (f 0, H 0 ) and (f 1, H 1 ) are two choices of classifying data for, defining bundles maps (f 0, 0 ) and (f 1, 1 ), and let (f, K) be associated to the identity homotopy p 1 () of, where p: M [0, 1] M is the projection. Then N K is a bundle on M [0, 1] 2 that restricts to the normal bundles of p 1 () and f 1 (U) on the top and bottom sides. By homotopy invariance of vector bundles this gives an isomorphism N p 1 () = N f 1 (U) = f (NΓ q ), and hence a path T M [0, 1] N [0, 1] = N p 1 () = f (NΓ q ) NΓ q M [0, 1] M [0, 1] f BΓ q in Map surj Vect (T M, NΓ q) between (f 0, 0 ) and (f 1, 1 ). urther, suppose that is an integrable homotopy between 0 and 1, that is, is a Γ q -foliation on M [0, 1] whose restriction to M {t} is a foliation for all t. If f : M [0, 1] BΓ q is a classifying map for, we can choose f t as classifying maps for t = i 1 t (). Let (f, ) be the bundle map associated to f and (f t, t ) that associated to f t. Then T M [0, 1] T (M [0, 1]) M [0, 1] f NΓ q BΓ q is a path from (f 0, 0 ) to (f 1, 1 ) in Map surj Vect (T M, NΓ q), since for each t the restriction of the top map to T M {t} is di t = t which is fibrewise surjective. So the map (1) descends to where is the relation of integrable homotopy. Haef : ol q (M)/ π 0 Map surj Vect (T M, NΓ q) 2
3 2. Proof of the theorem Suppose that M and E are manifolds and that is a Γ q -foliation on E. Then Map (M, E) denotes the space of smooth maps s: M E that are transversal to, i.e., such that the composite T M ds T E NΓ q is fiberwise surjective. This is just the bundle map associated to the smooth Γ q -structure s 1 ( ), so we have a map (2) Map (M, E) ol q (M), s s 1 ( ). What happens if we take homotopy classes? Suppose that s: M [0, 1] E is a smooth homotopy such that s t : M E is transversal to for all t [0, 1]. Then s 1 ( ) is a homotopy between s 1 0 ( ) and s 1 1 ( ), and we claim that it is an integrable homotopy. The restriction of s 1 ( ) to M {t} is s 1 t ( ) which is a foliation by assumption, so it remains to show that s 1 ( ) itself is a foliation, or equivalently that s is transversal to. This follows from the the transversality of all the s t, since the composition T M dit T (M [0, 1]) ds T E NΓ q is fibrewise surjective and each fiber of T (M [0, 1]) is accounted for in this way. Thus, the map (2) descends to π 0 Map (M, E) ol q (M)/. We can summarize the situation by the commutative square (3) π 0 Map (M, E) ol q (M)/ = π 0 Map surj Vect (T M, N ) Haef π 0 Map surj Vect (T M, NΓ q) in which the left-hand map is a bijection when M is open, by the Gromov-Phillips theorem. This square will be used to prove both the surjectivity and the injectivity of Haef, by using appropriate pairs (s, ). These will be given by the following technical lemma. Lemma 2.1. Let be a Γ q -structure on a manifold M. Then there exists a manifold E, a closed embedding s: M E, and a Γ q -foliation on E such that s 1 ( ) =. Proof. Let be represented by the cocycle {U α, φ αβ : U αβ Γ q }. By paracompactness we can assume that the cover {U α } is locally finite. Consider the topological groupoid Γ M q of germs of diffeomorphisms of M R q that are of locally the form (x, v) (x, γ(v)) where γ is a local diffeomorphism of R q, i.e., all germs are of the form id g for some g Γ q. The obvious map of sheaves induces an inclusion i: Γ q Γ M q that sends a germ g to id g. Observe that the diagram φ αβ Γ q i Γ M q U αβ (id,φ α) M R q. is commutative. This means that i φ αβ is a section of Γ M q over the graph of φ α Uαβ. Extending the germs of this section at every point to a neighborhood and then using a partition of unity on U αβ R q, we can extend this section to a global section U αβ R q Γ M q. Such a section corresponds 3
4 to a diffeomorphism Φ αβ defined on U αβ R q, locally of the form (x, v) (x, γ(v)), and whose germ at (x, φ α (x)) is id φ αβ (x) for any x U αβ. or x M, denote by V x M R q the set on which: Φ αα is defined and the identity, whenever x U α ; Φ 1 βα Φ αβ is defined and the identity, whenever x U αβ ; Φ βγ Φ αβ and Φ αγ are defined and equal, whenever x U αβγ. Now because all these identites hold true of the germs at (x, φ α (x)) and our cover is locally finite, V x is a neighborhood of (x, φ α (x)). or each α, therefore, the set x U α V x U α R q is a neighborhood of the graph of φ α : U α R q ; let E α be an open subneighborhood. Let E = E α / α where (α, x, v) (β, y, w) iff x = y in U αβ and Φ αβ (x, v) = (y, w). This is an equivalence relation because of the way we constructed the neighborhoods E α. Clearly there is an induced smooth map π : E M, [α, x, v] x. To make sure that E is actually a manifold, we must still check that it is Hausdorff. So let [α, x, v], [β, y, w] E be distinct. If x y, then the points are separated by π. Otherwise, Φ αβ (x, v) (y, w), so in this case the points are separated by the second component of the inclusion E α U α R q. Define s: M E to be x [α, x, φ α (x)] for x U α. inally, let be defined by the cocycle {E α, φ αβ }, where φ αβ : E αβ Γ q sends [α, x, v] to the germ g such that id g is the germ of Φ αβ at (x, v). It is then straightforward to check that is a Γ q -structure and that s 1 ( ) =. Moreover, φ α : E α R q is just the composition of the open embedding E α U α R q and the projection to the second factor, hence is smooth and a submersion. Proof of surjectivity. Take a bundle map T M NΓ q M f BΓ q in the target, and let = f 1 (U). By the lemma, there exists s: M E and a foliation on E such that s 1 ( ) =. Let f : E BΓ q be a map classifying. Because s is a cofibration, we can choose it so that f = f s. Then the left-hand square in T M N NΓ q M s E f BΓ q is an element of Map surj Vect (T M, N ) mapping to (f, ) by the bottom map in the square (3). This square completes the proof. Proof of injectivity. Let 0 and 1 be Γ q -foliations on M and suppose that there is a path T M [0, 1] NΓ q M [0, 1] 4 f BΓ q
5 between the associated bundles maps (f 0, 0 ) and (f 1, 1 ). We can assume that 0 = f0 1 (U) and 1 = f1 1 (U). Then = f 1 (U) is a homotopy from 0 to 1. By the lemma, there exists s: M [0, 1] E and a foliation on E such that s 1 ( ) =. Since s 1 0 () = 0 and s 1 1 () = 1 are foliations, s 0 and s 1 belong to Map (M, E), and they map to 0 and 1 by the top map of the square (3). Using this square it will suffice to show that the images of s 0 and s 1 in Map surj Vect (T M, N ) are in the same connected component. A path between them is given by the left-hand square in T M [0, 1] N NΓ q M [0, 1] s E f BΓ q, since 0 = ds 0 and 1 = ds 1. This completes the proof of Haefliger s classification. To understand its meaning, let s pretend for a moment that BΓ q is a manifold and that U is a Γ q -foliation on it. Then we have a commutative triangle π 0 Map U (M, BΓ q ) = ol q (M)/ Haef π 0 Map surj Vect (T M, NΓ q) in which the top map is an isomorphism by the theorem of Gromov and Phillips, and Haefliger s theorem is equivalent to the vertical map being an isomorphism. Morally speaking, therefore, Haefliger s theorem is the statement that U is the universal Γ q -foliation. 3. Some consequences The following proposition gives Haefliger s classification a more homotopy-theoretic flavor. Proposition 3.1. If dim M = n, π 0 Map surj Vect (T M, NΓ q) is naturally identified with the set of lifts BΓ q BGL n q Bd id BGL q BGL n q M BGL n τ up to homotopy, i.e., the set of maps M BΓ q BGL n q in the homotopy category of spaces over BGL n. Equivalently, this is π 0 of the space of sections of the fibration τ 1 (BΓ q BGL n q ) on M, where τ classifies the tangent bundle of M. Proof. Such a lift is the same thing as a map M BΓ q BGL n q together with a homotopy between the two maps M BGL n, up to homotopy. This is equivalent to a map f : M BΓ q, a rank n q vector bundle K on M, and an isomorphism T M = K f (NΓ q ), up to homotopy. Any homotopy class of such data has a representative in which the injection K T M is the inclusion of a subbundle. Since a splitting of this inclusion is unique up to homotopy, this data is equivalent to a map f : M BΓ q together with a fibrewise surjective map T M f (NΓ q ), up to homotopy. This is exactly π 0 Map surj Vect (T M, NΓ q). 5
6 Corollary 3.2. If M is contractible, there exists a unique Γ q -foliation on M up to integrable homotopy. Proof. There is a unique lift in Proposition 3.1, since BΓ q BGL n q and BGL n are connected. Corollary 3.3. If M is open and parallelizable, then there exists at least one Γ q -foliation on M. Proof. The map M BGL n factors through a point, so it can be lifted by the previous corollary. Define BΓ q by the homotopy fiber sequence BΓ q BΓ q Bd BGL q. In other words, BΓ q classifies Γ q -structures with trivialized normal bundle. Proposition 3.4. Bd: BΓ q BGL q is q-connected. Proof. Let 0 k q 1. There exists a unique codimension q foliation on the open manifold S k R q k, since it has dimension q. By Proposition 3.1, there exists a unique lift BΓ q S k R q k BGL q up to homotopy. But S k R q k has trivial tangent bundle, so the bottom map is null-homotopic and therefore these lifts identify with homotopy classes of maps S k BΓ q. Corollary 3.5. Let M be open and k-truncated. If q k, then any rank n q subbundle of T M is the tangent bundle of a Γ q -foliation, which is unique up to integrable homotopy if q > k. References [1] Haefliger, André. Lectures on the theorem of Gromov. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp Lecture Notes in Math., Vol. 209, Springer, Berlin, [2] Lawson, Blaine. oliations. Bull. Amer. Math. Soc. 80 (1974),
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