Submersions, fibrations & bundles Ga"el Meigniez Mathematics Subject Classification : 55R05, 55R10 to appear in Transactions of the A.M.S. Abstract _ When does a submersion have the homotopy lifting property ? When is * *it a locally trivial fibre bundle ? We establish characterizations in terms of consistency in the topology* * of the neighbouring fibres. In differential topology, one meets nonproper submersive maps, and hope that* * they will be fibrations (resp. fibre bundles) under hypotheses of consistency between the homotopy type* * (resp. topology) of the neighbouring fibres. The aim of this paper is to give suitable characterization* *s. I _ Submersions & Fibrations. This first part belongs to the most elementary homotopy theory. Our purpose * *is to establish the following homotopy lifting characterization, and a few corollaries. theorem A _ A surjective map is a fibration if and only if it verifies those t* *hree conditions: it is a homotopic submersion, all vanishing cycles of all dimensions are trivial, and* * all emerging cycles of all dimensions are trivial. Let us first specify definitions, conventions and notations. I.1. _ Definitions. Throughout this paper, "space" means Hausdorff topologic* *al space, äm p" means continuous map, öp lytope" means finite simplicial complex. For every p 0 , d* *enote Bpthe compact p-ball and Sp= @Bp+1the p-sphere. Fix a basepoint * 2 Sp . Let E, B be two spaces and ß : E ! B a map. As usually, by a homotopy for the map f : X ! Y , we mean a map F : X x [0, * *1]! Y such that F(x, 0) = f(x) for every x 2 X ; and we call (E, ß), or ß, a fibration, or equi* *valently we say that it has the homotopy lifting property, if for every map f : X ! E whose source X is a polyt* *ope, every homotopy for ß O f lifts to a homotopy for f. More generally, call ß a r-fibration if this i* *s true for every polytope X of dimension at most r . A 0-fibration is also said to have the path lifting prope* *rty. Here is another generalization of fibrations. Two homotopies F, F0: X x [0,!1]Y are said to have the same germ if they coincide in a neighborhood of the subspa* *ce X x {0} . definition 1 _ Call ß a homotopic submersion, or equivalently say that it has * *the germ-of-homotopy lifting property, if for every map f : X ! E whose source X is a polytope, ever* *y germ-of-homotopy for ß O f lifts to a germ-of-homotopy for f. Every fibration is a homotopic submersion (obvious) as well as every smooth * *submersion in the sense of Differential geometry, and more generally every topological submersion (after l* *emma 6 below.) 1 A "fibre" is the inverse image in E of a point b 2 B, denoted Eb= ß-1(b). A * *map X ! E is vertical if it sends X into a single fibre. A map f : X x Y ! E is fibred if for each point* * y in Y , the map fy : X ! E : x 7! f(x, y) is vertical. definition 2 _ A vanishing p-cycle is a fibred map f : Spx [0,!1]E such that, for each t > 0, the map ftis null-homotopic in its fibre. Call f tri* *vial if f0 is also null-homotopic in its fibre. Observe that f has to be continuous, but that we don't demand the homotopies* * the vertical maps gt: Bp+1! E such that @gt= ft, to depend continuously on t . Dually, definition 3 _ An emerging p-cycle is a fibred map f : Spx]0, 1] ! E such that f(*, t) has a limit for t ! 0 (recall that * denotes the basepoint in* * Sp .) Call it trivial if there exists ffl > 0 and a fibred map f0: Spx [0, ffl[! E such that for each 0 < t < ffl, one has f0(*, t) = f(*, t), and such that the m* *aps ftand f0tare homotopic to each other in their common fibre, relatively to the basepoint f(*, t). Here also, we don't ask any continuity, relatively to t , of the homotopy li* *nking ftto f0t. Thus a nontrivial vanishing (resp. emerging) cycle is a lack of injectivity * *(resp. surjectivity) in the homotopy groups of fibres, when moving from a given fibre to immediately neighb* *ouring ones over some path in the base. Write V (X) for the space of vertical maps from X into E , with the compact-* *open topology. Thus there is a canonical map : ßX : V (X) ! B Obviously, every p-dimensionnal vanishing (resp. emerging) cycle for the map ß * *can as well be considered as a 0-dimensionnal vanishing (resp. emerging) cycle for the map ßSp. They are * *simultaneously trivial. To end with precisions, the pullback of ß by some map f : B0 ! B is, as usua* *lly, defined as f*ß : f*E ! B0 , where f*E is the set of couples (e, b0) 2 E x B0 such that ß(e* *) = f(b0) , and where (f*ß)(e, b0) = b0. It is immediate that if ß is a fibration, or a r-fibration, or a homotopic s* *ubmersion, or has only trivial p-vanishing, or emerging, cycles, then every pullback enjoys the same property. Example 4 _ The three conditions in theorem A are independent. Indeed, let E * * [0,21]be defined by the condition xy = 0 (resp. x 6= 1=2 or y 6= 1=2) (resp. y = 0 or (y = 1 and x * *6= 0)). Let B = [0, 1]and ß be simply (x, y) 7! x . Then the first (resp. second) (resp. third) condition is n* *ot verified, but the two other ones are. I.2 _ Proof of theorem A. I.2.1 : three general remarks about homotopic submersions. An immediate and well-known fact about fibrations is that they automatically* * also have the relative homotopy lifting property. The same argument proves that : lemma 5 _ Every homotopic submersion verifies the relative germ-of-homotopy lif* *ting property : for every polytope X , every subpolytope X0 X , every map f : X ! E, every germ-of-homot* *opy F for ß O f and every germ-of-homotopy ~F0for f|X0 , if ß O ~F0= F|X0 then ~F0extends to a germ* *-of-homotopy ~Ffor f such that ß O ~F= F . 2 lemma 6 _ Assume that ß is open and is a local homotopic submersion, i.e. that* * each x 2 E has a neighborhood U(x) E such that the restriction ß|U(x)is a homotopic submersion* * of U(x) onto ß(U(x)) . Then ß is a (global) homotopic submersion. Proof. Consider a polytope X , a map f : X ! E and a germ-of-homotopy F : X * *x [0, ffl[! B for ß O f . Proceed by induction on p = dimX . The case p = 0 is trivial. Subdividi* *ng X we can assume that for each p-simplex oe , its image f(oe) is contained in an open subset U(oe) * *E such that ß|U(oe)verifies the germ-of-homotopy lifting property. By induction, we have a lifting ~F0for F res* *tricted to the (p-1)-skeletton of X . Since after the previous lemma ß|U(oe)has the relative germ-of-homotopy * *lifting property, ~F0extends to the interior of oe . o A second immediate corollary of lemma 5, is the following tool for handling * *sections. lemma 7 _ Let ß : E ! B be a surjective homotopic submersion whose base B is a * *polytope. Let Y B be a subpolytope. Denote (Y ) the space of (continuous) sections of ß over Y .* * Then : (i: local section extension) Each s 2 (Y ) extends over some neighborhood o* *f Y ; in particular each point of B has a neighborhood over which there exists a section; (ii: sections pasting) Let s 2 (B) and let s02 (Y ) be homotopic to s|Y . * *Then there exists s002 (B) which coincides with s over the complement of an arbitrary small neighborhood o* *f Y , and with s0on Y . I.2.2 : Haefliger's lemma. The following fact, due to Haefliger [4] , is central in the demonstration o* *f theorem A. We give a proof since it seems simpler than the original one. lemma 8 _ If a surjective homotopic submersion ß : E ! B has a polytope base B * *and weakly contractible fibres Eb , then it has a section. Proof. By induction on q = dimB . Obvious if q = 0 . Assume that the lemma i* *s true for q -1 . Since a surjective homotopic submersion has local sections, subdividing B if necessary,* * every q-cell oe is the domain of a section soe. On the other hand, the induction hypothesis gives a section s* * over the (q - 1)-skeletton. Over the boundary of each q-cell oe , the two sections s|@oeand soe|@oeare homo* *topic. Indeed, consider E0 the space of vertical paths c : [0, 1]! Eb such that b 2 @oe , c(0) = soe(b) an* *d c(1) = s(b) . Obviously the map c 7! b is a homotopic submersion of E0onto @oe with weakly contractible fib* *res, so by the induction hypothesis it admits a section, i.e. a homotopy between s|@oeand soe|@oe. Thus * *the section pasting property of the homotopic submersion ß gives an extension of s over oe . o I.2.3 : About the sections of a homotopic submersion over the interval. In this subparagraph, consider a surjective homotopic submersion whose base * *is the interval : ß : E ! B = [0, 1] Fix a basepoint x0 2 E0 . To be short, call a section pointed if s(0) = x0 ; wr* *ite (Y ) (resp. 0(Y ) ) the space of (pointed) sections over Y I ; and call s 2 (Y ), s02 (Y 0) compati* *ble at point b 2 Y \ Y 0if s(b) and s0(b) belong to the same arcwise connected component of Eb. lemma 9 _ Assume that all 0-vanishing cycles are trivial. If s, s02 ([0, 1]) a* *re compatible at some point, then they are compatible at every point. Proof. Recall that if K [0, 1]is a compact subset, we call t 2 K a first-s* *pecies point of K if t is an extremity of a nonempty open interval disjoint from K . The set U [0, 1]of points where s and s0are compatible is open since ß is * *a homotopic submersion, and its complement has no first-species point since 0-vanishing cycles are triv* *ial. Thus U = [0, 1]. o lemma 10 _ Assume that all 0-vanishing cycles and 0-emerging cycles are trivial* *. Then 0([0, 1]) is not empty. Proof. Consider T the set of t's such that there exists a pointed section ov* *er [0, t] . It is open since ß is a homotopic submersion. Thus by contradiction it would be of the form T = [0* *, T[ with T 2]0, 1] . Let sn 2 ([0, T - 1=n]) be pointed. By the previous lemma, since sn(0) = sn+1(0) ,* * the sections sn and sn+1 are also compatible at point T - 1=n . Thus the sections pasting property gives* * a pointed s 2 ([0, T[) . Let s02 (]T - ffl, T]) be any section over a left neighborhood of T . Then s and s* *0together form a 0-emerging cycle at point T . Since it must be trivial, there is a s002 ([T - ffl0, T]) c* *ompatible with s at t - ffl0. The sections pasting property gives a pointed section over [0, T] , contradiction. o 3 lemma 11 _ any surjective homotopic submersion all of whose vanishing 0-cycles * *and emerging 0-cycles are trivial, is a 0-fibration. Proof : immediate by lemma 10 and pullbacks. o lemma 12 _ Assume that all vanishing cycles and emerging cycles of all dimensio* *ns are trivial. Then 0([0, 1]) is weakly contractible. Proof. For every polytope X, recall V (X) , the space of vertical maps X ! E* * . The canonical map ßX : V (X) ! [0, 1]is a homotopic submersion, as well as the restriction map : æp: V (Bp+1) ! V (Sp) Since we assumed that all vanishing (resp. emerging) cycles of ß of dimension p* * + 1 are trivial, it follows that all 0-dimensionnal vanishing (resp. emerging) cycles of æp are also trivia* *l. Indeed, let f : S0x [0,!1]V (Bp+1) (resp. f : S0x]0, 1] ! V (Bp+1)) be a van* *ishing (resp. emerging) 0-cycle for æp . Gluing, for each t 2 [0, 1](resp. each t 2]0, 1]) , f(-1, t) a* *nd f(+1, t) along their common boundary, one gets a vanishing (resp. emerging) (q + 1)-cycle for ß , the trivi* *ality of which implies that f is also trivial. Thus after lemma 11, æp is a 0-fibration as a map V (Bp+1) ! V 0(Sp) , where* * V 0(Sp) is the image of æp , i.e. the space of vertical p-cycles null-homotopic in their fibre. Every given map g : Sp ! 0([0, 1]) , may as well be considered as a section* * s : [0, 1]! V (Sp) (set s(t)(x) = g(x)(t) ). The set of t's such that s(t) 2 V 0(Sp) is open since ß is* * a homotopic submersion, and has no first-species boundary point since p-vanishing cycles are trivial. Thus * *s is a path in V 0(Sp) , and thus lifts to V (Bp+1) , i.e. g extends to Bp+1. o I.2.4 : End of the proof of theorem A . Given a map ß : E ! B verifying the three conditions of the theorem, a map f* * : X ! E and a homotopy F for ß O f , consider E0the space of of pairs (x, c) where x 2 X and * *where c : [0,!1]E verifies c(0) = f(x) and ß(c(t)) = F(x, t) for every t 2 [0, 1]. Then : ß0: E0! X : (x, c) 7! x is a homotopic submersion, as follows from lemma 5. It is surjective by lemma 1* *0, and its fibres are weakly contractible by lemma 12. After lemma 8, this map ß0admits a section; in other * *words F lifts. o I.3 _ Corollaries. The two first ones immediately follow from theorem A. Corollary 13 _ Let ß : E ! B be a surjective homotopic submersion. Under eithe* *r of the following assumptions, ß is a fibration : 1. Each fibre is weakly contractible; 2. The inclusion of each fibre into E is a weak homotopy equivalence. Surprisingly, point 1, essentially identic to Haefliger's lemma, seems among* * our various corollaries, to be the only one to have been previously known. Corollary 14 _ Let ß : E ! B be a surjective homotopic submersion. The followi* *ng properties are equivalent : 1. The map ß is a fibration : 2. For each b in B, the map ß induces a weak homotopy equivalence of pairs (* *E, Eb) ! (B, b) ; 3. (dim-1 base principle for fibrations) For each path c : [0,!1]B, the pull* *back c*E is a fibration over [0, 1]. The dim-1 base principle allows us, whenever useful, to restrict our attenti* *on to the case B = [0, 1]. Corollary 15 _ In a commutative diagram of maps : E-h!E0 ß &. ß0 B 4 assume that ß and ß0 are surjective homotopic submersions. For each b 2 B , co* *nsider Eb = ß-1(b) , E0b= ß0-1(b) , and the restriction hb= h|Eb: Eb! E0b. (i) (Fibrewise fibration principle) If ß , ß0and every hbare fibrations and * *if h is a surjective homotopic submersion, then h is a fibration; (ii) If ß is a fibration and if every hb is a weak homotopy equivalence, the* *n ß0is a fibration; (iii) If ß0is a fibration and if every hb is a weak homotopy equivalence, th* *en ß is a fibration. Proof. (i) We can assume, to simplify, that B is contractible (for example u* *sing the dim-1 base principle). Fix x02 E0, write b = ß0(x0), Ex0= h-1(x0) , and consider the inclusions j : Eb* *! E, j0: E0b! E0. We have a commutative diagram of pairs : (Eb, Ex0)j*-!(E, Ex0) (hb)*# # h* 0 (E0b, x0)j*-!(E0, x0) Since ß, ß0 and hb are fibrations, j*, j0*and (hb)* are weak homotopy equivalen* *ces, and thus so is h* . Corollary 14 concludes. (ii) Using the dim-1 base principle, we can assume, to fix ideas, that B = [* *0, 1]. Then we have a section-lifting lemma : for every subinterval I [0, 1]and every section s0: I* * ! E0 , there is a section s : I ! E compatible with s0, i.e. such that for every t 2 I, the points h(s(t)* *) and s0(t) lie in the same arcwise connected component of E0t. Proof of this lemma : since every hbis a 0-homotopy equivalence, there exist* *s a set-theoric (nonneces- sarily continuous) section oe : I ! E compatible with s0. Since ß is a homotopi* *c submersion, each t 2 I has a neighborhood U(t) in I with a (continuous) section st: U(t) ! E such that st(* *t) = oe(t) . Since ß0is a homotopic submersion, stis compatible with s0over some smaller neighborhood of * *t . Thus I can be divided into subintervals Ii= [ti, ti+1] , domains of sections si: Ii! E compatible wit* *h s0. In particular, since hti is a 0-homotopy equivalence, si-1(ti) and si(ti) lie in the same arcwise connec* *ted component of Eti. Thus the section pasting tool (lemma 7) allows us to build a section s : I ! E compa* *tible with s0. The lemma is proved. Since 0-vanishing and 0-emerging cycles of ß are trivial, and since every hb* *is a 0-homotopy equivalence, the section-lifting lemma gives straightforwardly that 0-vanishing and 0-emergi* *ng cycles of ß0are trivial. Changing E for V (Sp) , the space of vertical maps Sp ! E , and changing E0f* *or V 0(Sp) , the space of vertical maps Sp! E0, we get that p-vanishing and p-emerging cycles of ß0are* * trivial, for every p 0 . Theorem A concludes. (iii) Much like (ii), but simpler. o Corollary 17 _ Let ß : E ! B be a surjective homotopic submersion. The map ß i* *s a fibration if and only if the canonical map from each fibre into the homotopy-theoric fibre of ß * *is a weak homotopy equivalence. Follows immediately from the preceding one, (iii). We can also deduce two characterizations for product maps. Of course, we cal* *l ß : E ! B a fibration over a given subset B0 B if the restriction ß|ß-1(B0): ß-1(B0) ! B0 is a fibration. Corollary 18 _ Assume that ß = (ß1, ß2) : E ! B = B1x B2 is a surjective homot* *opic submersion. The three following properties are equivalent : (i) The map ß is a fibration; (ii) the map ß is, for each b12 B1 , a fibration over {b1} x B2 ; and ß1 is * *a fibration; (iii) the map ß is, for each b1 2 B1 , a fibration over {b1} x B2 ; and, for* * each b2 2 B2 , a fibration over B1x {b2} . Proof. The equivalence of (i) and (ii) is a special case of the fibrewise fi* *bration principle (corollary 15.) It is obvious that (i) implies (iii). 5 Let us assume that (iii) is true. To prove (i),we can, using the dim-1 base* * principle, assume that B2 = [0, 1]. For every b1 2 B1 , since ß is a fibration over {b1} x B2 which is* * contractible, ß-1(b1, 0) is a deformation retract of ß-1({b1} x B2) = ß1-1(b1) . On the other hand, since ß is a fibration over ß2-1(0) , equivalently ß1 re* *stricted to ß2-1(0) is a fibration. It follows from corollary 15 (ii) that ß1 is also a fibration. Thus (ii) is * *verified. Thus (i) is also. o The next and last corollary, two refined forms of the dim-1 base principle, * *will be a crucial tool in part II : Corollary 19 _ Let ß : E ! B be a surjective homotopic submersion. a) Assume that B = Rq . Then ß is a fibration if and only if it is a fibrati* *on over every straight line in the base parallel to one of the axis; b) Assume that B is a C1 (resp. real analytic) (resp. PL) manifold. Then ß i* *s a fibration if and only if its pullback by every C1 (resp. real analytic) (resp. PL) path in B is a fib* *ration over [0, 1]. Proof. a): immediate by the preceding corollary and an induction on q ; b): * *immediate by a). o Example 20 _ in which ß = (ß1, ß2) is not a fibration (its fibres don't have t* *he same homotopy type) although it is a surjective submersion and ß1 , ß2 are both fibrations (their f* *ibres are contractible) : E = { (x, y, z) 2 R3 = z 6= 0 orx > y } and ß : E ! B = R2: (x, y, z) 7! (x, y) . II _ Submersions & Bundles. In this second part we turn to submersions between manifolds, establish a ne* *cessary and sufficient condition for such a map to be a (locally trivial fibre) bundle (theorem B belo* *w), and apply it to several typical situations. The main tool is the analytic understanding of fibrations w* *e got in xI . It is applied to spaces of embeddings of compact domains into the fibres. The first question that raises is probably : for such a submersion, is it th* *e same to be a fibration or to be a bundle ? It was considered by Ferry [3], in a framework slightly dif* *ferent from ours; he built counterexamples. On the other hand, Haefliger's lemma allows much simple ones, * *e.g. : Example 21 _ Let W R3be the Whitehead manifold _ thus an open subset which is* * contractible, but has some ß1 at infinity, and in particular is not homeomorphic to R3 . Let E * *R4 be the set of quadruples (x, y, z, t) such that (x, y, z) 2 W or t 6= 0 . Let ß : E ! B = R : (x, y, z, t) 7! t Then p is a smooth submersion (since E is open in R4) and a fibration (since al* *l fibres are contractible, see xI .) But it is certainly not a locally trivial fibre bundle, since one of the * *fibres is not homeomorphic to the neighbouring ones. II.1. _ Definitions. To fix ideas, work in the smooth (C1 real) differentiab* *ility class. Let Ep+q, Bq be manifolds _ this means paracompact, not necessarily compact, real differenti* *al manifolds. For simplicity, assume that E, B are without boundary. Let ß : E ! B be a surjective smooth sub* *mersion _ that is, its differential at every point of E is onto. Then each "fibre" Eb= ß-1(b) is a p-m* *anifold. Call a submanifold of E a product if it is the image of a fibred (see xI) sm* *ooth embedding X x Y ! E . Recall ß is a (locally trivial, fibre) bundle if every b 2 B has a neighborhood* * Y such that ß-1(Y ) is a product. A popular sufficient condition for ß to be a bundle is that ß be a riemannia* *n submersion [5], [10]. Recall that a riemannian metric on E is called bundle-like if there exists a riemannia* *n metric on the base B such that, for each b 2 B and each x 2 Eb , the differential Dxß establishes a linea* *r isometry from the normal vector space xEb onto the tangent vector space TbB . It is easy, using a parti* *tion of the unity, to make a (maybe noncomplete) bundle-like metric. The 'orthogonal geodesic lemma' asserts* * that if any geodesic line is once normal to the fibre it crosses, then it is forever. It follows easily t* *hat if X Eb is an open subset (to fix ideas) such that the exponential exp(vx) is defined for all x 2 X and a* *ll vx 2 xEb with norm less than a uniform positive constant, then the set of all these exp(vx)'s is a prod* *uct. In particular we get the fundamental : 6 Lemma 22 _ Every compact subset of every fibre has a product neighborhood. One calls ß riemannian if E admits a complete bundle-like metric, and, takin* *g X = Ebabove, one sees that every riemannian submersion is a bundle. There are many generalizations. Our viewpoint is different: we look for purely differential-topologic condit* *ions in terms of the topology of the fibres. In this direction, very few seems to be known, namely : 1. If ß is proper, then it is a bundle (Ehresmann). This follows at once fro* *m lemma 22. 2. If all the leaves are compact and have the same number of connected compo* *nents, then ß is a bundle. Also obvious by lemma 22. 3. If each fibre is diffeomorphic to R , then ß is a bundle. Proof: one may * *assume that B is orientable. Using a partition of the unity, make a nonsingular vector field tangent to the * *fibres, make it integrable, and integrate it. 4. Much less elementary is Palmeira's lemma [9]: if each fibre is diffeomorp* *hic to Rp , p a nonnegative integer, and if the base B has dimension q = 1 , then ß is a bundle. A fibred embedding of X xY into E can be considered as a section, over some * *subset of B , with values in the space V E(X, E) of vertical embeddings of X into E . Thus an approach co* *uld be to start from a large compact domain X in a fibre and from a ball Y B ; to use our knowledge of the* * topology of the fibres to compute as much as possible the homotopy type of the space of embeddings of X i* *nto each fibre, and to use xI to get such sections. Of course there would remain the problem to engulf arb* *itrary large compact subsets of ß-1(Y ) . For instance, in example 21, the canonical map V E(B3, E) ! B is a* * fibration (see the proof of corollary 31 below) and nevertheless the vertical embeddings of B3 exhaust all * *fibres but E0 . This leads us to the following notions. By a vertical domain we mean a p-dimensionnal compact submanifold of a fibre* *, X Eb, with a smooth boundary. Endow V E(X, E) , the set of vertical embeddings of X into E , with t* *he topology of smooth uniform convergence. Let also V E0(X, E) be the connected component of V E(X, E* *) containing the original inclusion X ! Eb. Definitions 23 _ Let a VD = VDb b2B be a family of vertical domains. Call it : a) exhaustive if every compact subset of every fibre is contained in some X * *2 VD ; b) isotopy invariant if for every X 2 VD and every OE 2 V E0(X, E) , we have* * OE(X) 2 VD ; c) r-fibred if, for every two domains X, X02 VDbsuch that X Int(X0) , the * *restriction map æX,X0: V E0(X0, E) ! V E0(X, E) is a r-fibration (i.e. has the homotopy extension property for polytopes of dim* *ension at most r , see xI.) II.2. _ Characterizations and criteria for bundles. theorem B _ A surjective smooth submersion ß : E ! Bq is a (locally trivial fib* *re) bundle if and only if it admits an exhaustive, isotopy invariant, (q - 1)-fibred family of vertical d* *omains. Before the proof, let us precise definitions and establish preliminary lemma* *s. Lemma 24 _ For every pair of domains X Int(X0) , the restriction map æX,X0is * *a topological submersion. This follows at once from lemma 22. For every vertical domain X 2 VDband every subset Y B homeomorphic to a co* *ntractible polytope (in fact it will allways be homeomorphic to Bk for some 0 k q), write (X, * *Y ) for the space of continuous sections fl : Y ! V E0(X, E) . Observe in particular that for fl 2 (X, Y ) , t* *he variable point fl(x)(y) admits continuous partial derivates of all order with respect to x , bur not necessari* *ly with respect to y . Since we 7 shall perform non strictly differentiable operations on the base, it is more co* *nvenient to consider all those sections than merely smooth ones. Write : Imfl = { fl(y)(x) = x 2 X, y 2 Y } Call fl a parametrization of the VD-box Imfl . Recall that X Eb. Call fl pointed (at point b) if b 2 Y and if fl(b) = idX. For every compact subset C E , write C < Imfl if C Imfl and if C \ Ey * *Int(Im fl \ Ey) for every y 2 Y . Given two sections fl 2 (X, Y ) , fl02 (X0, Y 0) , say of course that fl0e* *xtends fl if X X0, Y Y 0, and if fl(y)(x) = fl0(y)(x) for every x 2 X , y 2 Y . lemma 25 _ If fl is pointed and if Imfl < Imfl0, then there exists a pointed r* *eparametrization of Imfl0 extending fl . This is an exercise using the classical [1] : Proposition 26 _ Let X, X0be compact manifolds with X Int(X0) . Let Diff0(X0)* * denote the group of isotopies of X0 and E0(X, X0) denote the canonical connected component of th* *e space of embeddings of X ! X0. Then the restriction map Diff0(X0) ! E0(X, X0) is a fibration (and even* * a principal bundle.) Here is the main tool to engulf large compact subsets. lemma 27 _ Let Ki E ( i = 1,2,3) be three VD-box with base Yi= ß(Ki) . Assume * *that Y1\ Y2= Y3 , that Ki\ ß-1(Y3) < K3 for i = 1, 2 , and that VD is (dimY3)-fibred. Then there * *is a VD-box K4 such that Ki< K4 for i = 1, 2 . Proof. Fix a basepoint b 2 Y3 . For i = 1, 2, 3, let Xi= Ki\ Eb and let fli2* * (Xi, Yi) be a pointed parametrization of Ki. According to lemma 25, we can assume that fl3 extends fl* *1|Y3. Since VD is (dimY3)-fibred, there exists a pointed fl012 (X3, Y1) , extendi* *ng fl1 , and which coincides with fl3 over Y3 . According to lemma 25, there is a reparametrization fl001of * *fl01such that fl001|Y3extends fl2|Y3. In the same way, since VD is (dimY3)-fibred, there exists a fl022 (X3, Y2) * *extending fl2 and which coincides with fl001over Y3 . Since fl001and fl02coincide over Y3, they define together an element of (X3* *, Y1[ Y2), whose image K4 is obviously > K1 and > K2 . o Proof of theorem B. Ö nly if" is trivial: just take for VD the set of all ve* *rtical domains. Thus reciprocally we assume that ß : E ! B is a smooth surjective submersion which admits an exha* *ustive, isotopy invariant, (q - 1)-fibred set VD of vertical domains, and let us prove that it is a fibrat* *ion. For every b 2 B, let Y be a neighborhood of b diffeomorphic to Bq . We first claim : Every compact subset C ß-1(Y ) is < some VD-box. The proof is by induction on k , the smallest integer such that there exists* * Y khomeomorphic to Bk and verifying ß(C) Y k Y . For k = 0 the claim follows from the exhaustiveness of VD . Assume That the claim is proved for k-1 . Identify Y kwith [0,k1]. For each * *y 2 Y , by exhaustiveness of VD there is a vertical domain belonging to VD and containing C \ Ey in its i* *nterior. By lemma 24 this domain is the intersection of Ey with a VD-box Ky whose base ß(Ky) is a neighbo* *rhood of y . Since C is compact, y admits a smaller neighborhood Wy such that C \ ß-1(Wy) < Ky . Thus d* *ividing Y k= [0,k1] into small enough cubes of equal size 1=N , say Q1, . .,.QNk, each C \ ß-1(Qi) * *is < some VD-box, say Ki, of base ß(Ki) = Qi. Consider Ui = Q1 [ . .[.Qi the union of the i first small cubes (ranged of c* *ourse in the natural lexicographic order). Assume by induction on i that C \ß-1(Ui) is < some VD-box* * K0iof base ß(K0i) = Ui. The intersection Ui\ Qi+1is obviously homeomorphic to Bk-1, thus, by the induct* *ion (on k) hypothesis, (Ki[ K0i) \ ß-1(Ui\ Qi+1) is < a VD-box. Since moreover Uiand Qi+1are contracti* *ble and since VD is (k - 1) - fibred , lemma 27 assures the existence of a VD-box K0i+1> C \ ß-1(Ui* *+1) . For i = Nk - 1, we are done : the claim is proved. 8 Fix (Cn) an increasing sequence of compact sets whose union is ß-1(Y ) . The* * previous claim gives a sequence of pointed sections ,n 2 (Xn, Y ) , with Xn 2 VDb, such that : Cn [ Im,n-1< Im,n Changing if necessary ,n to another section close to it, we can assume that the* * embedding (x, y) 7! ,n(y)(x) is smooth. After lemma 25, a convenient reparametrization of ,n gives ,n as an * *extension of ,n-1. Consider , the inductive limit of the ,n's . Its image is ß-1(Y ) since it contains ever* *y Cn . In other words , is a smooth trivialization of E over Y . o Of course, in practice it may be hard to verify that a map such as æX,X0is (* *q - 1)-fibred, since, after part I, this is something like comparing the (q - 1)-homotopy type of the embed* *ding spaces of domains into the different fibres. So our next tasks will be to change this condition to mor* *e handy ones. Corollary 28 (dim-1 base principle for bundles) _ a) A surjective smooth submersion ß : E ! B = Rq is a bundle iff it is a bun* *dle over each straight line in the base parallel to one of the axis. b) A surjective smooth submersion ß : E ! B is a bundle iff for every smooth* * path fl : [0, 1]! B , the pullback fl*ß : fl*E ! [0, 1]is a bundle. c) Theorem B is still true if we change the hypothesis 'VD is (q - 1)-fibred* *' to the weaker one 'VD is 0-fibred'. Proof. To prove a), we make an induction on q . Assume That ß : E ! Rq is a * *surjective submersion and a bundle over each straight line L parallel to an axis . Let VD be the set * *of all vertical domains. After theorem B, it is enough to prove that æ : V E0(X0, E) ! V E0(X, E) is a fibrati* *on for every two vertical domains X, X02 VD such that the first one is contained in the interior of the s* *econd one. By assumption, for every L, our submersion ß is a bundle over L , and in par* *ticular the map V E0(X, ß-1(L)) ! L is a fibration. By corollary 19, ßX : V E0(X, E) ! Rq is a fibration. Write ßX = (ß1X, . .,.ßqX) . Thus the last factor, ßqX: V E0(X, E) ! R is also a fibration. In the same way, ßqX0is also a fibration. But by the induction hypothesis, æ is a fibration over (ßqX0)-1(t) for each * *t 2 R . The fibrewise fibration principle (corollary 15,i) concludes that æ is actually a fibration. Affirmation b) then follows immediately from a); and affirmation c) from the* *orem B and from b). o Also, since our theorem A gave a satisfying analysis of fibrations, one can * *assume that ß is a submersion and a fibration, say a üs bmersion-fibration,ä nd ask for sufficient condition* *s which make it a bundle. Definition 29 _ Call VD engulfing when for every three domains X, X0, X002 VD* * bsuch that X Int(X0) \ Int(X00) , if, in the ambiant space Eb, the domain X0can be pushe* *d into X00by a homotopy relative to X , then it can also by an isotopy relative to X . Observe that we only ask the homotopy, and the isotopy, to be the identity o* *n X , rather than on the whole of X0\ X00. Corollary 30 _ Let ß : E ! Bq be a (surjective, smooth) submersion-fibration. I* *f it admits an exhaustive, isotopy invariant, engulfing set of vertical domains, then it is a bundle. Proof. After the preceding corollary, we can assume that B = [0, 1]. Given* * X , X0 2 VD 0such that X Int(X0) X0 and , 2 (X, [0,)1]pointed, we have to build a pointed ,0* * 2 (X0, [0,)1]such that æX,X0(,0) = , . Consider the set T of t's such that there exists a pointe* *d ,0t2 (X0, [0, t]) such that æX,X0(,0) = ,|[0,t]. Obviously T is an interval, open after lemma 24., an* *d containing 0. Assume, by contradiction, that T = [0, T[ . Since ß is a fibration, there is a section* * fl of the restriction map V 0(X0, E) ! V E0(X, E) , where V 0(X0, E) is the space of vertical maps X0! E * *homotopic to idX0. Since 9 VD is exhaustive, there is a X002 VDT containing fl(T)(X0) in its interior. By * *lemma 24, for t < T close enough to T there is a ,002 (X00, [t, T]) such that ,00(T) = idX00. In the fibre Et, we have three domains ,(t)(X) , ,0(t)(X0), ,00(t)(X00) ; an* *d since ß is a fibration, the second one can be pushed into the third one by a homotopy relative to the first* * one. Since VD is engulfing, it can also by an isotopy relative to the first one. In other words ,0(t) is is* *otopic in Et, relatively to ,(t)(X) , to some embedding OE : X0! ,00(t)(X00) . The section : s 7! ,00(s) O ,00(t)-1O OE belonging to (X0, [t, T]) , is homotopic to ,0at point t, thus after lemmas 7 * *and 24, there exists a pointed section over [0, T] , contradiction. o II.3. _ Applications. We show how the criteria established in the preceding * *paragraph apply in several typical situations. Corollary 31 _ A surjective smooth submersion with each fibre diffeomorphic to * *Rp is a bundle. Proof. The family of all vertical domains diffeomorphic to the compact p-bal* *l is exhaustive since each fibre is diffeomorphic to Rp . This family is obviously isotopy invariant. Recall Alexander's trick : let Mp be any smooth manifold without boundary, l* *et E(Bp, M) be the space of embeddings of the compact p-ball into M , and let F(M) the space of frames o* *f TM . Then the map : J10: E(Bp, M) ! F(M) that to each embedding associates its 1-jet at the origin, is a homotopy equiva* *lence. It follows that the restriction map æX,X0has contractible fibres. By lemma 8* * it is a fibration. Thus theorem B applies. o Corollary 32 _ Let ß : Ep+q! Bq be a (surjective, smooth) submersion-fibration. a) If p = 2 then ß is a bundle; b) If p 5 , if each fibre Eb is topologically finite, i.e. diffeomorphic * *to the interior of a compact p-manifold Mb with smooth boundary @Mb , and if every connected component of ev* *ery @Mb is simply connected, then ß is a bundle; c) (Stabilization)The map E x R3p+1! B : (x, x0) 7! ß(x) is a bundle. Point c) answers a question of [3]. Proof. a) We can assume that the fibres are connected. Then the family of al* *l connected vertical domains is obviously exhaustive and isotopy invariant. The engulfing property is verifi* *ed by connected domains in surfaces _ this is an exercise, using for example the results and methods of [2* *]. Thus corollary 30 applies. b) The set of all vertical domains X Eb such that X is a deformation retra* *ct of Eb is obviously exhaustive and isotopy invariant. After Van Kampen's and Grushko-Neumann's theo* *rem, each component of W = Eb\Int(X) is simply connected. After Poincar'e duality and the h-cobordism * *theorem, W ~=@Xx[0, 1( . The relative engulfing property follows immediately, and corollary 30 applies. c) Given a domain X and a polytope K Int(X) , say that X shrinks to K if, * *for every neighborhood U(K) , the whole domain X can be pushed into U(K) by an istopy of embeddings of* * X into X , relative to a neighborhood of K . For every b 2 B , define VDb the set of (4p + 1)-dimensionnal vertical domai* *ns X E0b= Ebx R3p+1 with the two following properties : 1. X shrinks to a (p - 1)-dimensionnal polytope; 2. Every (p - 1)-dimensionnal subpolytope in Int(X) is contained in a (2p - * *1)-dimensionnal polytope to which X shrinks. Obviously VD is isotopy invariant. To prove that it is also engulfing, let X* *, X0, X002 VDb be as in definition 29. Let K, K0be (p - 1)-dimensionnal polytopes on which X , X0respec* *tively shrink (property 1.) Since dimK + dimK0< dimE0b, by a general position argument, we can chose K,* * K0disjoint. Since X02 VD , by property 2 there is a (2p - 1)-dimensionnal complex L Int(X0) con* *taining K [ K0and to which X0shrinks. 10 By hypothesis, there is a homotopy that pushes X0 into X00and is the identit* *y on X . Again by a general position argument, since 2(dimL + 1) < dimE0b, after a small perturba* *tion if necessary, this homotopy induces, in restriction to L , an isotopy of embeddings of L into E0b,* * and remains the identity on a neighborhood of K . Since X0shrinks to L , we get an isotopy of embeddings of X0into E0b, that p* *ushes it into X00. Moreover it is the identity on a neighborhood of K . Since X shrinks to K , we can chose this isotopy to be identity on the whole* * of X , and the engulfing condition is established. Finally, we prove that VD is exhaustive. It is enough to prove that for ever* *y domain Dp Eb , the product Dpx B3p+1is contained in a domain X 2 VD . Let X be a regular neighborh* *ood of Dpx B3p+1. In particular X shrinks to Dpx B3p+1. If Ebis compact, then, since ß is assumed a fibration, all fibres are compac* *t with the same ß0 , thus, as previously mentionned, ß is already a bundle. Thus we can assume that Eb is not compact, thus @Dp is not empty, and in par* *ticular Dp shrinks to some (p - 1)-subpolytope K . It follows obviously that X also shrinks to K . Finally, let K Int(X) be any (p-1)-polytope. We have to find a larger, (2p* *-1)-dimensionnal polytope L to which X shrinks. After a first isotopy if necessary, we can assume that K * *is contained in Dpx B3p+1. Decompose E0b= Ebx R3p+1as Ebx Rp-1x R2p+2, and consider the canonical projecti* *on of K into the factor EbxRp-1. Since 2 dimK < 2p-1 , after a small perturbation this projectio* *n become an embedding of K into Dpx Bp-1. Thus we have an isotopy of embeddings pushing K into Dpx Bp* *-1. Thus we can assume that K Dpx Bp-1. Then L = Dpx Bp-1fits. In conclusion VD is exhaustive. Thus corollary 30 applies. o Question Assume That ß : E ! B is a submersion-fibration and that a group ac* *ts freely, cocompactly and properly discontinuously on E and permutes the fibres. Does it follow that * *ß is a bundle ? As a last application, in corollary 31, Rp can be replaced by any manifold b* *ounded by the sphere and of large enough dimension : Corollary 33 _ Let M be a compact manifold of dimension p 5 and with smooth b* *oundary homeomorphic to Sp-1. Then any (surjective, smooth) submersion all of whose fib* *res are diffeomorphic to Int(M) , is a bundle. Observe that here we don't assume any more the submersion to be a fibration. Proof. We first claim that for every submanifold X Int(M) abstractly diffe* *omorphic with M, the pair (M, X) is contractible. Indeed, after Van Kampen's and Grushko-Neumann's theorems, W is simply conne* *cted. Also, the Mayer-Vietoris sequence for M = X [ W gives, for each 2 k p - 2 , that Hk(X* *; Z) Hk(W; Z) is isomorphic to Hk(M; Z) . Since Hk(X; Z) is (abstractly) isomorphic to Hk(M; * *Z), it follows from the elementary theory of finitely generated abelian groups that W is (p-2)-connecte* *d. Thus the pair (W, @X) is (p-2)-connected. On the other hand, by Poincar'e duality, Hk(W, @X; Z) is isomo* *rphic to Hn-k(W, @M; Z), thus null for k n - 1 since W is 1-connected and @M is 2-connected. To sum up, (W, @X) has no homology. By excision principle, (M, X) doesn't ei* *ther. The claim is proved. Fix b 2 B . Chose X Eb a core. By lemma 24, there is a neighborhood Y of b* * in B and a section fl 2 (X, B) . By the claim, for every y 2 Y , the vertical domain fl(y)(X) = I* *mfl [ Ey is a deformation retract of Ey . After corollary 15, ß is a fibration over Y . Corollary 32, b) * *concludes. o It follows that Palmeira's conjugation theorem for open manifolds foliated b* *y Rpextends to more general kind of leaves; the (long) proof is exactly as in [9] : Corollary 34 _ Let Mp be as in the preceding corollary. Let (V, F) and (V 0, F* *0) be foliated, simply connected, (p + 1)-dimensionnal, open manifolds whose leaves are all diffeomorp* *hic to M . Assume that the (in general non-Hausdorff) leaf spaces V=F , V 0=F0 are diffeomorphic. Then thi* *s diffeomorphism can be realized by a smooth conjugation between the foliations. II.4. _ Ends of deformation-equivalent manifolds. What do two different fibr* *es Eb, Eb0of a fibration- submersion ß : E ! B must have in common, assuming of course that B is connecte* *d ? Call two such open manifolds, say, deformation-equivalent. Compare [3]. 11 Of course they have the same homotopy type, but not necessarily the same pro* *per homotopy type (example 21.) In fact, every contractible p-manifold U is deformation-equivale* *nt to Rp : proceed as in example 21, but change Rp to U , and W to a small open p-ball in U . Nevertheless, in dimension 2, deformation-equivalent manifolds are necessari* *ly diffeomorphic (corollary 32.) The following proposition answers a question of [3], where some particular c* *ases were obtained. Proposition 35 _ Two deformation-equivalent manifolds necessarily : a) are dif* *feomorphic if one of them is compact; b) have the same tangential homotopy type; c) have the same or* *ientability property w.r.t. every ring R ; d) have isomorphic algebras of compactly supported cohomology wi* *th coefficients in every ring R that makes them orientable; e) have homeomorphic ends spaces. Proof. a) If Eb is compact, then all fibres, having the same dimension and h* *omotopy type as Eb , are compact; moreover they have the same ß0; and we have seen as a corollary of lem* *ma 22 that ß is necessarily a bundle. b) Because the tangent vector spaces TEband TEb0are two pullbacks of a singl* *e p-dimensionnal vector space over E : the kernel of dß . c) Follows from b). d) and e) One can assume that B = Bq and that b , b0 2 Int(B) . Thus E has * *a boundary @E = ß-1(@B) . We shall prove that Eb has the same cohomology algebra and the s* *ame ends space as E ; of course the same will be true for Eb0. We have a commutative diagram : * H*c(E; R) i-! H*c(Eb; R) # # Hp+q-*(E, @E; R)j*-!Hp+q-*(E, E \ Eb; R) Vertical arrows are Poincar'e and Alexander duality and they are one-to-one; i** *is the morphism of restriction to the properly embedded submanifold Eb ; and j* is the inclusion morphism, one* *-to-one because @B is a deformation retract of B \ {b} and because ß is a fibration. Thus i* is one-to-* *one; and in the same time an algebra morphism. This proves d). For every locally compact space S, its space of ends can be defined as the s* *pectrum of an algebra, namely the algebra B(S) of germs, at the neighborhood of infinity, of locally c* *onstant functions with value in R = Z=2Z . On the other hand, we have a commutative diagram whose lines are * *exact : 0 -! R -! B(E) -! H1c(E; R)-! H1(E; R) =# f # i*# g # 0 -! R -! B(Eb) -! H1c(Eb; R)-! H1(Eb; R) where f , i* and g are restriction morphisms. Since ß is a fibration, g is one-* *to-one. By d), i* is one-to-one. By the five lemma, f is also one-to-one. Thus the algebras B(E) , B(Eb) are iso* *morphic, thus they have homeomorphic spectra. o II.5. _ More examples and questions. We end with a few (pleasant) monsters. Example 36. _ Let again W be the Whitehead manifold, let V W be an open 3-bal* *l, and let U V be an open subset diffeomorphic to W . Play the same game as in example 21, but wi* *th W instead of R3 and with U instead of W . Then again ß is a fibration of R4 onto R , but this time * *all fibres are diffeomorphic to W . If it were a bundle, then there would be a 1-parameter family of embeddi* *ngs it: W ! W such that i0= idWand i1(W) = U . But this would imply that every compact subset of W coul* *d be engulfed by the 3-ball V , and W would be diffeomorphic to R3 , a contradiction. Thus ß is not * *a bundle. Example 37. _ Let V R4 an open subset homeomorphic but not diffeomorphic to* * R4 (Casson- Friedman, see for example [6].) Play the same game as in example 21, but with R* *4 instead of R3 and with V instead of W . The total space E is diffeomorphic to R5 since it is 5-dimensi* *onnal, contractible and simply connected at infinity [12]. This (real-analytic !) submersion-fibration * *of R5 onto R is not a bundle in the C1 (or even C1) category, since one of the fibres is not diffeomorphic t* *o the other ones. On the other hand, all our work extends to the Cr categories (r 2 N) . The reader will provi* *de himself a proof of lemma 12 22 in the C0 and the C1 differentiability class. In particular, corollary 31 is* * valid in class C0 ; thus our submersion-fibration is a bundle in the C0 sense. Example 38 _ Let : E0-h!E ß0&. ß B be a commutative diagram of (smooth, surjective) submersions. If ß0and h are fi* *brations, then ß is also _ obvious, since to be a fibration it is enough to have the homotopy lifting pr* *operty for simplicies. Jean Pradines asked if we can change "fibration" to üb ndle". The answer is negative* *, always with the same counterexample: let E, B, ß be as in the example 21, let E0= E x R and let h be* * the first projection. Then h is a bundle. Also ß0is a bundle by corollary 31, since all its fibres ar* *e R3x R or W x R , thus diffeomorphic to R4 . But ß is not. This phenomenon was already observed in [3]. Example 39 _ We can also answer negatively Pradines' question with h a normal, * *infinite cyclic covering. Let E*, B, ß* be as in example 21, let E = E*x S1 , let ß(x, y) = ß*(x) , let E* *0be the universal covering of E and let h : E0! E be the canonical projection. Then h is a normal covering* * and again ß0is a bundle by corollary 31, since W x R is diffeomorphic to R4 . But ß is not, since W x S* *1 does not have the same proper homotopy type as R3x S1. Example 40 _ As Alan Weinstein points out [13], our results also allow to answ* *er negatively Pradines' question with E the quotient of E0by the free, fibrewise, action of a compact g* *roup. Actually, let E, B, ß be as in the example 37 (thus with a fibre being an exotic R4), let E0= E xS3and l* *et h be the first projection. Then h is a bundle. Also, each fibre of ß0is S3x a topological R4 , thus the in* *terior of a 7-dimensionnal compact manifold with (simply connected) boundary, by [11]. By corollary 32 b),* * ß0is a bundle. But ß is not. Question (Weinstein) What about Pradines' question with h a finite covering * *? Or equivalently, if one likes better, a normal one, i.e. the quotient by the free fibrewise action of a* * finite group ? Some of these results have been announced or conjectured in [7] and [8]. [1] Cerf J., Topologie de certains espaces de plongements, Bull. Soc. Mat. Fra* *nce 89 (1961), 227-380. [2] Epstein D.B.A., Curves on 2-manifolds and isotopies, Acta mathematica 115 * *(1966), 83-107. [3] Ferry S., Alexander duality and Hurewicz fibrations, Trans. Amer. Mat. Soc* *. 327, 1 (1991), 201-219. [4] Haefliger A., Groupo"ide d'holonomie et classifiants. in Structures tran* *sverses des feuilletages. Toulouse 1982, Ast'erisque 116 (1984), 70-97. [5] Hermann R., A sufficient condition that a mapping of Riemannian manifolds * *be a fiber bundle. Proc. AMS 11 (1960), 236-242. [6] Kirby R.C., The topology of 4-manifolds, L.N.M. 1374 (1989). [7] Meigniez G., Submersions et fibrations localement triviales. C. R. Acad. S* *ci. Paris, 321, s'erie I (1995), 1363-1365. [8] Meigniez G., Sur le rel`evement des homotopies. C. R. Acad. Sci. Paris, 32* *1, s'erie I (1995), 1497-1500. [9] Palmeira C.F.B., Open manifolds foliated by planes, Ann. of Math. 107 (197* *8), 109-131. [10] Reinhart B.L., Foliated manifolds with bundle-like metrics, Ann. of Math.* * 69, 1 (1959), 119-132. [11] Siebenmann L., Thesis, Orsay. [12] Stallings J., The piecewise linear structure of euclidian space, Camb. Ph* *ilos. 58, 3 (1961), 481-488. [13] Weinstein A., Linearization of regular proper groupoids, preprint, Berkel* *ey (2001.) Laboratoire de Math'ematiques et d'Application des Math'ematiques, Universit* *'e de Bretagne Sud, Campus de Tohannic, Centre de recherche, F-56017 Vannes Cedex, France Gael.Meigniez@univ-ubs.fr 13