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*
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Laboratoire de Math'ematiques et d'Application des Math'ematiques, Universit*
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de Tohannic, Centre de recherche, F-56017 Vannes Cedex, France
Gael.Meigniez@univ-ubs.fr
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