ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY
(THE ARNOLD CONJECTURE, CRITICAL POINTS)
Yuli B. Rudyak
Abstract. We prove the Arnold conjecture for closed symplectic manifolds *
*with
ss2(M) = 0 and catM = dimM. Furthermore, we prove an analog of the Luster*
*nik-
Schnirelmann theorem for functions with "generalized hyperbolicity" prope*
*rty.
Introduction
Here we show that the technique developed in [R98] can be applied to the Arn*
*old
conjecture and to estimation of the number of critical points. For convenience*
* of
the reader, this paper is written independently of [R98].
Given a smooth (= C1 ) manifold M and a smooth function f : M ! R, we
denote by critf the number of critical points of f and set CritM = min {critf}
where f runs over all smooth functions M ! R.
The Arnold conjecture [Ar89, Appendix 9] is a well-known problem in Hamil-
tonian dynamics. We recall the formulation. Let (M; !) be a closed symplectic
manifold, and let OE : M ! M be a Hamiltonian symplectomorphism (see [HZ94],
[MS95] for the definition). Furthermore, let FixOE denote the number of fixed p*
*oints
of OE. Finally, let
Arn (M; !) := minOEFixOE
where OE runs over all Hamiltonian symplectomorphisms M ! M. The Arnold
conjecture claims that Arn (M; !) Crit(M).
It is well known and easy to see that Arn (M; !) CritM. Thus, in fact, the
Arnold conjecture claims the equality Arn (M; !) = CritM.
Let catX denote the Lusternik-Schnirelmann category of a topological space X
(normalized, i.e., catX = 0 for X contractible).
Given a symplectic manifold (M2n ; !), we define the homomorphisms
I! : ss2(M) ! Q; I! (x) =
Ic : ss2(M) ! Z; Ic(x) =
where h : ss2(M) ! H2(M) is the Hurewicz homomorphism, c = c1(o M) is the first
Chern class of M and <-; -> is the Kronecker pairing.
______________
1991 Mathematics Subject Classification. Primary 58F05, secondary 55M30, 55N*
*20,.
The author was partially supported by Deutsche Forschungsgemeinschaft, the R*
*esearch Group
"Topologie und nichtkommutative Geometrie"
1
2 YULI B. RUDYAK
Theorem A (see 3.6). Let (Mn ; !); n = dim M be a closed connected symplec-
tic manifold such that I! = 0 = Ic (e.g., ss2(M) = 0) and cat M = n. Then
Arn (M; !) CritM, i.e., the Arnold conjecture holds for M.
It is well known that Crit M 1 + clM for every closed manifold M, where
cl denotes the cup-length, i.e., the length of the longest non-trivial cup-prod*
*uct in
He*(M). So, one has the following weaker version of the Arnold conjecture:
Arn(M; !) 1 + cl(M);
and most known results deal with this weak conjecture, see [CZ83], [S85], [H88],
[F89-1], [F89-2], [LO96]. (Certainly, there are lucky cases when CritM = 1 + cl*
*M,
e.g. M = T 2n, cf. [CZ83].) For example, Floer [F89-1], [F89-2] proved that
Arn (M) 1 + clM provided I! = 0 = Ic, cf. also Hofer [H88]. So, my contribution
is the elimination of the clearance between Crit M and 1 + clM. (It is easy to
see that there are manifolds M as in Theorem A with CritM > 1 + clM, see 3.7
below.)
Let p : P X ! X be the path fibration over a path connected compact metric
space X, and let pk : Pk(X) ! X be the k-fold join over X of p. It is well-known
that pk has a section iff catX < k. Consider the Puppe sequence
Pk(X) -pk!X -jk!Ck(X) := C(pk)
and set
r(X) := sup{m|jm is stably essential}:
It is easy to see that r(X) catX and, moreover, r(X) = catX iff X possesses
a detecting element (as defined in [R98]). In particular, r(M) = catM for every
closed orientable manifold M with catM = dim M, Theorem 2.4.
Actually, in Theorem A we prove that Arn (M; !) 1 + r(M). Then we use a
result of Takens [T68] which implies that CritM = 1 + catM provided catM =
dim M.
After submission of the paper the author and John Oprea proved that catM =
dim M for every closed symplectic manifold (M; !) with I! = 0, see [RO97]. So,
the condition catM = n in Theorem A can be omitted.
Passing to critical points, we prove the following theorem.
Theorem B (see 4.5). Let M be a closed orientable manifold, and let g : M x
Rp+q ! R be a C2-function with the following properties:
(1) There exist disks D+ Rp and D- Rq centered in origin such that
int(M x D+ x D- ) contains all critical points of g;
(2) rg(x) points inward on M x @D+ x intD- and outward on M x intD+ x
@D- .
Then critg 1 + r(M).
In particular, if M is aspherical then critg 1 + catM.y
Notice that functions g as in Theorem B are related to the Conley index theo*
*ry,
see [C76]. I remark that Cornea [Co98] have also estimated the number of criric*
*al
points of functions as in Theorem B.
______________
yRecall that a connected topological space is called aspherical if ssi(X) = *
*0 for every i > 1.
ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 3
We reserve the term "map" for continuous functions of topological spaces, and
we call a map inessential if it is homotopic to a constant map. The disjoint un*
*ion
of spaces X and Y is denoted by X t Y . Furthermore, X+ denotes the disjoint
union of X and a point, and X+ is usually considered as a pointed space where *
*the
base point is the added point.
We follow Switzer [Sw75] in the definition of CW -complexes. A CW -space is
defined to be a space which is homeomorphic to a CW -complex.
Given a pointed CW -complex X, we denote by 1 X the spectrum E = {En}
where En = Sn X for every n 0 and En = pt for n < 0; here Sn X is the n-fold
reduced suspension over X. Clearly, 1 is a functor from pointed CW -complexes
to spectra.
Given any (bad) space X, the cohomology group Hn (X; ss) is always defined to
be the group [X; K(ss; n)] where [-; -] denotes the set of homotopy classes of *
*maps
and the Eilenberg-Mac Lane space K(ss; n) is assumed to be a CW -space.
"Smooth" always means "C1 ".
"Fibration" always means a Hurewicz fibration.
"Connected" always means path connected.
The sign " '" denotes homotopy of maps (morphisms) or homotopy equivalence
of spaces (spectra).
x1. Preliminaries on the Lusternik-Schnirelmann category
1.1. Definition. (a) Given a subspace A of a topological space X, we define
catX A to be the minimal number k such that A U1 [ . .[.Uk+1 where each Ui
is open and contractible in X. We also define catX A = -1 if A = ;.
(b) Given a map f : X ! Y , we define catf to be the minimal number k such
that X = U1 [ . .[.Uk+1 where each Ui is open in X and f|Ui is inessential for
every i.
(c) We define the Lusternik-Schnirelmann category catX of a space X by setti*
*ng
cat X := catX X = cat1X .
Clearly, catf min {cat X; catY }.
The basic information concerning the Lusternik-Schnirelmann category can be
found in [Fox41], [J78], [Sv66].
Let X be a connected space. Take a point x0 2 X, set
fi
P X = P (X; x0) = {! 2 XI fi!(0) = x0}
and consider the fibration p : P X ! X; p(!) = !(1) with the fiber X.
Given a natural number k, we use the short notation
(1.2) pk : Pk(X) ! X:
for the map
pX_*X_._._.*X-pXz______": P_X_*X_._.-.*XzP_X____"----!X
k times k times
where *X denotes the fiberwise join over X, see e.g. [J78]. In particular, P1(*
*X) =
P X.
4 YULI B. RUDYAK
1.3. Proposition. For every connected compact metric space X and every natural
number k the following hold:
(i) pk : Pk(X) ! X is a fibration;
(ii) catPk(X) < k;
(iii) The homotopy fiber of the fibration pk : Pk(X) ! X is the k-fold join
(X)*k;
(iv) If catX = k and X is (q - 1)-connected then pk : Pk(X) ! X is a (kq - 2*
*)-
equivalence;
(v) If X has the homotopy type of a CW -space then Pk(X) does.
Proof. (i) This holds since a fiberwise join of fibrations is a fibration, see *
*[CP86].
(ii) It is easy to see that cat(E1 *X E2) catE1 + catE2 + 1 for every two m*
*aps
f1 : E2 ! X and f2 : E2 ! X. Now the result follows since catP1(X) = 0.
(iii) This holds since the homotopy fiber of p1 is X.
(iv) Recall that A * B is (a + b + 2)-connected if A is a-connected and B is
b-connected. Now, X is (q - 2)-connected, and so the fiber (X)*k of pk is
(kq - 2)-connected.
(v) It is a well-known result of Milnor [M59] that X has the homotopy type
of a CW -space. Hence, the space (X)*k has it. Finally, the total space of any
fibration has the homotopy type of a CW -space provided both the base and the
fiber do, see e.g. [FP90, 5.4.2].
1.4. Theorem ([Sv66, Theorems 3 and 190]). Let f : X ! Y be a map of connected
compact metric spaces. Then catf < k iff there is a map g : X ! Pk(Y ) such that
pkg = f.
x2. An invariant r(X)
Consider the Puppe sequence
Pm (X) -pm-!X -jm-!Cm (X) := C(pm )
where pm : Pm (X) ! X is the fibration (1.2) and C(pm ) is the cone of pm .
2.1. Definition. Given a connected space X, we set
r(X) := sup{m|jm is stably essential}:
(Recall that a map A ! B is called stably essential if it is not stably homotop*
*ic to
a constant map.)
2.2. Proposition. (i) r(X) catX for every connected compact metric space X.
(ii) Let X be a connected CW -space, let E be a ring spectrum, and let ui 2
Ee*(X)); i = 1; : :;:n be elements such that u1 . .u.n6= 0. Then r(X) n. In ot*
*her
words, r(X) clE(X) for every ring spectrum E.
It makes sense to remark that r(X) = catX iff X possesses a detecting elemen*
*t,
as defined in [R96].
Proof. (i) This follows from 1.4.
ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 5
(ii) Because of 1.3(v), without loss of generality we can and shall assume t*
*hat
Cn(X) is a CW -space. We set u = u1 . .u.n 2 Eed(X). Because of the cup-
length estimation of the Lusternik-Schnirelmann category, and by 1.3(ii), we ha*
*ve
p*n(u) = 0. Hence, there is a homotopy commutative diagram
1 jn
1 X ----! 1 Cn(X)
? ?
u?y ?y
dE ________ dE:
Now, if r(X) < n then 1 jn is inessential, and so u = 0. This is a contradic-
tion.
2.3. Lemma. Let f : X ! Y be a map of compact metric spaces with Y connected,
and let jm : Y ! Cm (Y ) be as in 2:1. If the map jm f is essential then catf *
* m.
In particular, if r(Y ) = r and the map
f# : [Y; Cr(Y )] ! [X; Cr(Y )]
is injective then catf r.
Proof. Consider the diagram
X
?
f?y
Pm (Y ) --pm--! Y --jm--! Cm (Y ):
If catf < m then, by 1.4, there is a map g : X ! Pm (Y ) with f = pm g, and hen*
*ce
jm f is inessential. This is a contradiction.
2.4. Theorem. Let Mn be a closed oriented connected n-dimensional PL manifold
such that catM = dim M 4. Then r(M) = catM.
Proof. Let MSP L*(-) denote the oriented PL bordism theory. By the definition
of r(X), it suffices to prove that (jn)* : MSP L*(M) ! MSP L*(Cn(M)) is a non-
zero homomorphism. Hence, it suffices to prove that (pn)* : MSP Ln(Pn(M)) !
MSP Ln(M) is not an epimorphism. Clearly, this will be proved if we prove that
[1M ] 2 MSP Ln(M) does not belong to Im (pn)*.
Suppose the contrary. Then there is a map F : W ! M with the following
properties:
(1) W is a compact (n+1)-dimensional oriented PL manifold with @W = M tV ;
(2) F |M = 1M , F |V : V ! M lifts to Pn(M) with respect to the map pn :
Pn(M) ! M.
Without loss of generality we can assume that W is connected.
Suppose for a moment that ss1(W; M) is a one-point set (i.e., the pair (W; M*
*) is
simply connected). Then (W; M) has the handle presentation without handles of
indices 1, see [St68, 8.3.3, Theorem A]. By duality, the pair (W; V ) has the *
*handle
presentation without handles of indices n. In other words, W ' V [ e1 [ . .[.es
6 YULI B. RUDYAK
where e1; : :;:es are cells attached step by step and such that dim ei n - 1 f*
*or
every i = 1; : :;:s. However, the fibration pn : Pn(M) ! M is n - 2 connected.
Thus, F : W ! M can be lifted to Pn(M). In particular, pn has a section. But
this contradicts 1.4.
So, it remains to prove that, for every membrane (W; F ), we can always find*
* a
membrane (U; G) with ss1(U; G) = * and G|@U = F |@W . Here @U = @W = M t V
and G : U ! M. We start with an arbitrary connected membrane (W; F ). Consider
a PL embedding i : S1 ! intW . Then the normal bundle of this embedding is
trivial. Indeed, w1() = 0 because W is orientable.
Since M is a retract of W , there is a commutative diagram
0 - ---! ss1(M) ----! ss1(W )----! ss1(W; M) - ---! 0
flfl ?
fl ?yF*
ss1(M) ________ss1(M)
where the top line is the homotopy exact sequence of the pair (W; M). Clearly, *
*if
F* is monic then ss1(W; M) = *.
Let ss1(W ) be generated by elements a1; : :;:ak. We set gi := F*(ai)a-1i2 s*
*s1(W )
where we regard ss1(M) as the subgroup of ss1(W ). Then Ker F* is the smallest
normal subgroup of ss1(W ) contained g1; : :;:gk. Now we realize g1; : :;:gk by
PL embeddings S1 ! intW and perform the surgeries of (W; F ) with respect to
these embeddings, see [W70]. The result of the surgery establishes us a desired
membrane.
2.5. Corollary. Let M be as in 2:4, let X be a compact metric space, and let
f : X ! M be a map such that f* : Hn (M; ssn(Cn(M))) ! Hn (X; ssn(Cn(M))) is
a monomorphism. Then catf catM.
Notice that, in fact, catf = catM since catf catM for general reasons.
Proof. We set ss = ssn(Cn(M)). It is easy to see that Cn(M) is simply connected.
Hence, by 1.3(iv) and the Hurewicz theorem, Cn(M) is (n - 1)-connected. Thus,
[M; Cn(M)] = Hn (M; ss). Let : Cn(M) ! K(ss; n) denote the fundamental class.
Then f* can be decomposed as
# *
f* : Hn (M; ss) = [M; Cn(M)] -f-![X; Cn(M)] -! [X; K(ss; n)] = Hn (X; ss):
Since f* is a monomorphism, we conclude that f# is. Thus, by 2.3 and 2.4,
cat f r(M) = catM:
x3. The invariant r(M) and the Arnold conjecture
Recall (see the introduction) that the Arnold conjecture claims that Arn (M;*
* !)
Crit M for every closed symplectic manifold (M; !).
3.1. Recollection. A flow on a topological space X is a family = {'t}; t 2 R
where each 't : X ! X is a self-homeomorphism and 's't = 's+t for every s; t 2 R
(notice that this implies '0 = 1X ).
ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 7
A flow is called continuous if the function X x R ! X; (x; t) 7! 't(x) is co*
*ntin-
uous.
A point x 2 X is called a rest point of if 't(x) = x for every t 2 R. We de*
*note
by Rest the number of rest points of .
A continuous flow = {'t} is called gradient-like if there exists a continuo*
*us
(Lyapunov) function F : X ! R with the following property: for every x 2 X we
have F ('t(x)) < F ('s(x)) whenever t > s and x is not a rest point of .
3.2. Definition (cf. [H88], [MS95]). Let X be a topological space. We define
an index function on X to be any function : 2X ! N [ {0} with the following
properties:
(1) (monotonicity) If A B X then (A) (B);
(2) (continuity) For every A X there exists an open neighbourhood U of A
such that (A) = (U);
(3) (subadditivity) (A [ B) (A) + (B);
(4) (invariance) If {'t}; t 2 R is a continuous flow on X then ('t(A)) = (A)
for every A X and t 2 R;
(5) (normalization) (;) = 0. Furthermore, if A 6= ; is a finite set which *
*is
contained in a connected component of X then (A) = 1.
3.3. Theorem. Let be a gradient-like flow on a compact metric space X. Then
Rest (X)
for every index function on X.
Proof. The proof follows the ideas of Lusternik-Schnirelmann. For X connected
see [H88], [MS95, p.346 ff]. Furthermore, if X = tXi with Xi connected then
X X
Rest = Rest (|Xi) (Xi) (X):
3.4. Corollary. Let be a gradient-like flow on a compact metric space X, let Y
be a Hausdorff space which admits a covering {Uff} such that each Uffis open and
contractible in Y , and let f : X ! Y be an arbitrary map. Then
Rest 1 + catf:
Proof. Given a subspace A of X, we define (A) to be the minimal number m such
that A U1 [ . .[.Um where each Ui is open in X and f|Ui is inessential. It is*
* easy
to see that is an index function on X (normalization follows from the properti*
*es of
Y ). But (X) = 1 + catf, and so, by 3.3, we conclude that Rest 1 + catf.
3.5. Theorem. Let (M; !) be a closed connected symplectic manifold with I! =
0 = Ic, and let OE : M ! M be a Hamiltonian symplectomorphism. Then there
exists a map f : X ! M with the following properties:
(i) X is a compact metric space;
(ii) X possesses a gradient-like flow such that Rest FixOE;
(iii) The homomorphism f* : Hn (M; G) ! Hn (X; G) is a monomorphism for
every coefficient group G.
Proof. This can be proved following [F89-2, Theorem 7]. (Note that the formulat*
*ion
of this theorem contains a misprint: there is typed z*[P ] = 0, while it must b*
*e typed
8 YULI B. RUDYAK
z*[P ] 6= 0. Furthermore, the reference [CE] in the proof must be replaced by [*
*F7].)
In fact, Floer denoted by z : S ! P what we denote by f : X ! M, and he
showed that the homomorphism z* : Z = Hn (P ) ! Hn (S ) is monic. He did it for
Z-coefficients, but the proof for arbitrary G is similar.
Also, cf. [H88] and [HZ94, Ch. 6].
In fact, Floer considered Alexander-Spanier cohomology, but for compact met-
ric spaces it coincides with H*(-). In greater detail, you can find in [Sp66] *
*an
isomorphism between Alexander-Spanier and Cech cohomology and in [Hu61] an
isomorphism between Cech cohomology and H*(-).
Recall that every smooth manifold turns out to be a PL manifold in a canonic*
*al
way, see e.g. [Mu66].
3.6. Theorem. Let (M; !) be a closed connected symplectic manifold with I! =
0 = Ic and such that catM = dim M. Then Arn (M; !) CritM.
Proof. The case dim M = 2 is well known, see [F89-1], [H88], so we assume that
dim M 4. Consider any Hamiltonian symplectomorphism OE : M ! M and the
corresponding data and f : X ! M as in 3.5. Then, by 3.5, Fix OE Rest ,
and hence, by 3.4 and 2.5, Fix OE 1 + catM, and thus Arn (M; !) 1 + catM.
Furthermore, by a theorem of Takens [T68], CritM 1 + dim M. Now,
1 + catM CritM 1 + dim M = 1 + catM;
and thus Arn (M; !) CritM.
3.7. Example (cat M > clM). Let M be a four-dimensional aspherical symplectic
manifold described in [MS95, Example 3.8]. It is easy to see that H1(M) = Z3.
Furthermore, H*(M) is torsion free, and so a2 = 0 for every a 2 H1(X). Hence,
clM = 3. However, catM = 4 because catV = dim V for every closed aspherical
manifold V , see [EG57]. Moreover, for every closed symplectic manifold N we ha*
*ve
cl(M x N) < cat(M x N) because cat(M x N) = dim N + 4 accoring to [RO97].
x4. The invariant r(M) and critical points
Let X be a CW -space and let A; B be two CW -subspaces of X. Then for every
spectrum E we have the cap-product
\ : Ei(X; A [ B) j(X; A) ! Ei-j(X; B);
see [Ad74], [Sw75]. Here *(-) denotes stable cohomotopy, i.e., *(-) is the
cohomology theory represented by the sphere spectrum S.
In particular, if D = Dk is the k-dimensional disk then for every CW -pair (*
*X; A)
we have the cup-product
\ : Ei(X x D; X x @D [ A x D) k(X x D; X x @D) ! Ei-k(X x D; A x D):
Let a 2 k(D; @D) = Z be a generator. We set t = p*a 2 k(X x D; X x @D)
where p : (X x D; X x @D) ! (D; @D) is the projection.
ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 9
4.1. Lemma. For every CW -pair (X; A) the homomorphism
\t : Ei(X x D; X x @D [ A x D) ! Ei-k(X x D; A x D)
is an isomorphism.
In fact, it is a relative Thom-Dold isomorphism.
Proof. If A = ; then \t is the standard Thom-Dold isomorphism for the trivial
Dk-bundle (or the suspension isomorphism, if you want), see e.g. [Sw75]. In oth*
*er
words, for A = ; the homomorphism in question has the form \t : Ei(T ff) !
Ei-k(X xD) where T ff is the Thom space of the trivial Dk-bundle ff. Furthermor*
*e,
the homomorphism in question has the form
Ei(T ff; T (ff|A)) ! Ei-k(X x D; A x D):
Considering the commutative diagram
. .!. Ei(T (ff|A)) ----! Ei(T ff) ----! Ei(T ff; T (ff|A))! . . .
?? ? ?
y\(t|A) ?y\t ?y\t
. .!.Ei-k(A x D) ----! Ei-k(X x D) ----! Ei-k(X x D; A x D) ! . . .
with the exact rows, and using the Five Lemma, we conclude that the homomor-
phism in question is an isomorphism.
4.2. Definition ([CZ83], [MO93]). Given a connected closed smooth manifold M,
we define G Hp;q(M) to be the set of all C2-functions g : M x Rp+q ! R with the
following properties:
(1) There exist disks D+ Rp and D- Rq centered in origin such that
int(M x D+ x D- ) contains all critical points of g;
(2) rg(x) points inward on M x @D+ x intD- and outward on M x intD+ x
@D- .
4.3. Definition ([CZ83], [MO93]). Given g 2 G Hp;q(M), consider the gradient
flow _x= rg(x). Let xoR denote the solution of the flow through x. We choose D+
and D- as in 4.2, set B := M xD+ xD- and define Sg = Sg;B := {x 2 B|xoR B}.
4.4. Theorem (cf [MO93, 4.1]). For every function g 2 G Hp;q(M), there is a
subpolyhedron K of intB such that Sg K and critg 1 + catB K.
Proof. We set S = Sg. Because of of 4.2, S is a compact subset of intB. Fur-
thermore, S is an invariant set of the gradient flow _x= rg(x), and S contains *
*all
critical points of g. Given A S, we define (A) = 1 + catB A. Clearly, is an
index function on S. Thus, by 3.3, (S) critg. Now, let V1; : :V:(S)be a coveri*
*ng
of S such that every Vi is open and contractible in B. Choose any simplicial tr*
*ian-
gulation of B. Then, by the Lebesgue Lemma, there exists a simplicial subdivisi*
*on
of B with the following property: every simplex e with e \ S 6= ; is contained *
*in
some Vi. Now, we set K to be the union of all simplices e with e \ S 6= ;. Clea*
*rly,
1 + catB K (S), and thus critg 1 + catB K. Finally, we can find K intB
because of the collar theorem.
Let r(M) be the invariant defined in 2.1.
10 YULI B. RUDYAK
4.5. Theorem. For every function g 2 G Hp;q(M), the number of critical points
of g is at least 1 + r(M). In particular, critg 1 + catM if M is aspherical.
Proof. Here we follow McCord-Oprea [MO93]. However, unlike them, here we use
certain extraordinary (co)homology instead of classical (co)homology.
Let r := r(M). We choose K as in 4.4 and prove that catB K r. Consider the
Puppe sequence
Pr(M) -pr!M -jr!Cr(M):
Let e : M+ ! Cr(M) be a map such that e|M = jr and e maps the added point to
the base point of Cr(M). Let h : Cr(M) ! C be a pointed homotopy equivalence
such that C is a CW -complex. We set E = 1 C and let ur 2 E0(M) be the stable
homotopy class of the map he : M+ ! C. Then ur 6= 0 since jr is stably essenti*
*al.
We define
f : K B = M x Rp+q -projection-----!M:
4.6. Lemma. If f* ur 6= 0 then catB K r.
Proof. Since (pr; jr) is a Puppe sequence, p*rur = 0. Hence, the map f can't *
*be
lifted to Pr(M), and therefore the inclusion K B can't be lifted to Pr(B). So,
catB K r. The lemma is proved.
We continue the proof of the theorem. Let j : K B be the inclusion. By 4.6,
it suffices to prove that j* : E*(B) ! E*(K) is a monomorphism. Notice that if Y
is a CW -subspace of RN then there is a duality isomorphism
E0(Y ) ~=E-N (RN ; RN \ Y ) := E-N (RN [ C(RN \ Y ))
see e.g. [DP84]. So, it suffices to prove that the dual homomorphism
D(j*) : E*(RN ; RN \ B) ! E*(RN ; RN \ K)
is monic for a certain (good) embedding B ! RN .
We have the following commutative diagram:
E*(RN ; RN \ B) ________E*(RN ; RN \ B)
? ?
h?y~= D(j*)?y
E*(RN ; RN \ intB) - ---! E*(RN ; RN \ K)
x x
e??~= e0??~=
E*(B; @B) - -a*--! E*(B; B \ K)
where all the homomorphisms except D(j*) are induced by the inclusions. Here h *
*is
an isomorphism since the inclusion intB ! B is a homotopy equivalence (the space
B\ {collar} is a deformation retract of intB). Furthermore, e is an isomorphism
since (B; @B) and (RN ; RN \ intB) are cofibered pairs, while e0 is an isomorph*
*ism
by Lemma 3.4 from [DP84]. So, D(j*) is monic if a* is. Since B \ K B \ S, it
suffices to prove that E*(B; @B) ! E*(B; B \ S) is a monomorphism.
ON ANALYTICAL APPLICATIONS OF STABLE HOMOTOPY 11
Let B+ = M x @D+fxiD- , and let B- = M x D+ x @D-f.i Furthermore,
let A+ := {x 2 B fixoR- 2 B} and let A- := {x 2 B fixoR+ 2 B}. Then
B+ \A- = ; = B- \A+ , and so there are the inclusions i+ : (B; B+ ) ! (B; B \A-*
* )
and i- : (B; B- ) ! (B; B \ A+ ). It turns out to be that both i+ and i- are
homotopy equivalences, [CZ83, Lemma 3].
Let t 2 m (B; B- ) be the class as in 4.1, and let t0 := ((i- )*)-1 (t). Si*
*nce
S = A+ \ A- , we have the commutative diagram
Ei(B; @B) ----! Ei(B; B \ S)
? ?
~=?y\t ?y\t0
~=
Ei-q(B; B+ ) ----! Ei-q(B; B \ A- )
where the left map is an isomorphism by 4.1 and the bottom map is the isomorphi*
*sm
(i+ )*. (Generally, (B; B \ S) is not a CW -pair, but nevertheless in our case*
* the
map \t0 is defined, see [DP84, 3.5].) Thus, the top homomorphism is injective.
Finally, if M is aspherical then catM = dim M, [EG57], and so r(M) = catM
by 2.4.
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Mathematisches Institut, Universit"at Heidelberg, Im Neuenheimer Feld 288, D-
69120 Heidelberg 1, Germany
E-mail address: july@mathi.uni-heidelberg.de