CATEGORY WEIGHT: NEW IDEAS CONCERNING
LUSTERNIK-SCHNIRELMANN CATEGORY
Yuli B. Rudyak
Introduction
The concept of category weight was introduced by Fadell-Husseini [FH] and
developed by Rudyak and Strom. Here we give a survey, some further development
and applications of category weight.
The Lusternik-Schnirelmann category of a topological space X, catX, is defined
as the minimal number k such that X admits a numerable covering {A1; : :;:Ak+1}
where each Ai is contractible in X. Lusternik and Schnirelmann [LS] introduced
the invariant catX for manifolds. They proved that, for every connected smooth
(=C1 ) closed manifold M,
1 + catM CritM := min{critf|f 2 C1 (M; R)}
where critf is the number of critical points of a smooth real-valued function f*
* on
M. Afterwards Fox [Fox] suggested considering catX as an invariant of a space
X. The basic information concerning the Lusternik-Schnirelmann category can be
found in [Fox], [Sv], [J].
This (homotopy) invariant is quite far from other invariants like homotopy and
homology groups, so it is difficult to compute catX. For example, in [G], Ganea
asked whether cat(X x Sn) = catX + 1, n > 0, X connected. The affirmative
claim is usually referred to as the Ganea conjecture. Recently it was disproved*
* by
Iwase [I], but it is still unclear whether it is true for manifolds. Here is a*
*nother
rather naive question. Let f : M ! N be a map of degree 1 of closed manifolds.
Is it true that catM catN?
One of the favorite and famous ways to estimate the Lusternik-Schnirelmann ca*
*t-
egory is a so-called cup-length estimation (Froloff-Elsholz [FE], Eilenberg). N*
*amely,
if u1. .u.n6= 0 for some ui2 eH*(X) then catX n. The idea of the proof is quite
simple: if X = A1[ . .[.An where each Aiis contractible in X then ui|Ai= 0, and
so u1. .u.n|(A1 [ . .A.n) = 0. However, the cup-length estimation is not perfec*
*t.
For example, if L = L2n+1pis the lens space with p an odd prime, then catL = 2n*
*+1
(Krasnosel'ski, 1955), while the cup-length estimation gives us catL n + 1 onl*
*y.
Fadell-Husseini refined the cup-length estimation by suggesting that elements ui
be equipped with "weights". Speaking informally, we say that the category weight
of u (cwgt u) is k if
u|(Ai1[ . .[.Aik) = 0
_____________
1991 Mathematics Subject Classification. Primary 55M30, secondary 55N20, 55S*
*30, 58F05.
Typeset by AM S-TEX
1
2 YULI B. RUDYAK
forPAi as above. Clearly, catX cwgtu if u 6= 0. Furthermore, cwgt(u1. .u.n)
cwgt ui, and, since cwgtui 1, we conclude that
X
catX cwgt(u1. .u.n) cwgt ui n
provided u1. .u.n6= 0. Certainly, this improves the cup-length estimation. For
example, this establishes a short calculation of catL2n+1p, see [FH]. So, it se*
*ems
reasonable to find (indecomposable) elements of high category weight and, more
generally, to be able to compute category weight. Unfortunately, category weight
is not a homotopy invariant, i.e., cwgt h*u is not necessary equal to cwgt u fo*
*r a
homotopy equivalence h. This makes category weight difficult for calculations.
Because of this, it makes sense to introduce a homotopy invariant version of
category weight as the author did in a talk in the AMS Summer Research Institut*
*e,
Seattle, July 1996 (for the publication see [R2]). This invariant is called st*
*rict
category weight. A similar concept was also introduced by Strom [S2]. There are
many ways to define swgtu; u 2 H*(X). One of them is:
swgt u = min{cwgt h*u} where h runs over all homotopy equivalences Y ! X.
Clearly, cwgtu swgtu, so we can use swgt to estimate cwgt. Furthermore, swgt
has better multiplicative properties than cwgt, see x3. It turns out that swgt*
* is
quite manageable: for example, it is possible both to find many elements of high
swgt(see x4) and to apply swgt to certain geometric problems (see xx6,7). Also,
notice that strict category weight yields a decreasing filtration {Fn(X)} of H**
*(X),
Fn(X) := {u 2 H*(X)| swgtu n}.
Here is another description of swgt: we have X ' BX, the classifying space
for the loop space of X. Let {BnX} be the Milnor filtration of BX, and let
in : BnX BX ' X. (Notice that in is homotopy equivalent to a map (1.5)
described in x1.) Then
swgtu = sup{k|i*k(u) = 0}:
In other words, the above filtration {Fn(X)} is just the filtration
Ker {i*n: H*(X) ! H*(BnX)}:
This also shows how to apply the Eilenberg-Moore spectral sequence to the study
of the Lusternik-Schnirelmann category, cf. [To].
It is clear that category weight can be defined in more general situations. F*
*or
example, we can consider an arbitrary (extraordinary) cohomology theory instead
of H*(-). In this paper we consider even a little bit more general functors th*
*en
cohomology.
Throughout this paper, we reserve the term "map" for continuous functions and
the term "inessential map" for homotopy trivial maps (i.e., for maps which are
homotopic to constant maps).
We use the sign ' for homotopy of maps, and we write f 'o g when there is a
pointed homotopy between pointed maps f and g.
The reduced cone of a pointed map f is denoted by Cf.
"Connected" always means "path connected".
CATEGORY WEIGHT 3
Given a pointed topological space X, the (reduced) cohomology group Hk(X; ss)
is defined as [X; K(ss; n)] (pointed homotopy classes) where the Eilenberg-Mac
Lane space K(ss; n) is assumed to be ( homotopy equivalent to) a CW -space. Two
exceptions to this agreement (in 3.10 and 4.2) are mentioned explicitly.
Given x 2 Hk(X), the notation x|ssk(X)= 0 means that = 0 for every
a 2 Im{h : ssk(X) ! Hk(X)} where h is the Hurewicz homomorphism.
The paper is organized as follows. In x1 we consider the Lusternik-Schnirelma*
*nn
category. In x2 we give a definition and background properties of (strict) cate*
*gory
weight. In x3 we demonstrate multiplicative properties of (strict) category wei*
*ght.
In x4 we explain how to find elements of high category weight. In x5 we show how
to use swgt for control of the Lusternik-Schnirelmann category and in x6 we app*
*ly
this technique to manifolds. In particular, we prove special cases of the Ganea
conjecture and the degree conjecture. In x7 we apply our technique to the famous
Arnold conjecture about symplectic fixed points. In fact, we idicate how to pro*
*ve
it for closed symplectic manifolds (M; !) with !|ss2(M)= 0 = c1|ss2(M).
1. Lusternik-Schnirelmann category
1.1. Definition. (a) ([Fet], [BG]) Given a map ' : A ! X, we say that cat' k
if there is a numerable covering U1; : :;:Uk+1 of A such that '|Ui is inessenti*
*al for
every i. Then cat' = k if k is minimal with this property. Also, we set cat' = *
*-1
if A = ;.
(b) If i : A ! X is an inclusion then we set catX A := inf{cati} where i runs
over all the inclusions i : U ! X of neighborhoods U of A.
(c) ([LS]) We define the Lusternik-Schnirelmann category catX of a space X by
setting catX := cat1X = catXX.
1.2. Lemma. Let f : X ! Y be a map with Y connected, and let x0 2 X be an
arbitrary point. If catf = k then there is a numerable covering {B1; : :;:Bk+1}
such that f|Bi is inessential and x0 2 B for every i.
Proof. Let {A1; : :;:Ak+1} be a numerable covering of X with f|Aiinessential, a*
*nd
let {'1; : :;:'k+1} be a partition of unity dominated by {A1; : :;:Ak+1}. We set
Ci:= '-1i[1=(2k + 2); 1]
and Bi:= Ci[ {x0}. It is easy to see that {Bi} is the desired covering.
1.3. Proposition ([BG]). (i) For every diagram A -'!Y -f!X we have catf'
min{cat'; catf}. In particular, catf min{catX; catY }.
(ii) If ' ' : A ! X then cat' = cat .
(iii) If h : Y ! X is a homotopy equivalence then cat' = cath' for every
' : A ! X.
Notice that, in view of 1.3(iii), catX is a homotopy invariant of X. Also, it*
* is
easy to see that catX dim X for every connected CW -space X.
1.4. Proposition ([B], [Fox]). For every two connected CW -spaces X; Y we have
max {catX; catY } cat(X x Y ) catX + catY:
4 YULI B. RUDYAK
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
(1.5) pk : Pk(X) ! X
for the map
pX_*X_._.*.XpX_-z______": P_X_*X_._.*.XP-Xz_______"----!X
k times k times
where *X denotes the fiberwise join over X, see e.g. [J]. In particular, P1(X) *
*= P X.
1.6. Theorem ([Sv, Theorems 3 and 190]). Let ' : A ! X be a map with X
connected. Then cat' < k iff there is a map : A ! Pk(X) such that pk = '.
For future references we fix the following simple information, the proofs can*
* be
found e.g. in [R2].
1.7. Proposition. (i) Pk(X) is a fibration over X;
(ii) If X has the homotopy type of a CW -space then P - k(X) does;
(iii) catPk(X) k - 1.
1.8. Proposition ([Sv, Th. 21]). The map p2 : P2(X) ! X (and so the map
w2 : W2(X) ! X) is homotopy equivalent over X to the map " : SX ! X.
The concept of Lusternik-Schnirelmann category can be generalized as follows.
Let T be the category of pointed spaces and pointed maps. Let E be the category
whose objects are pairs (set, subset) and whose morphisms (U; V ) ! (U0; V 0) a*
*re
functions f : U ! U0 with f(V ) V 0. Given a contravariant functor F : T ! E
with F (X) = (U; V ) for some X 2 T , the notation u 2 F (X) means that u 2 U.
We say that u 2 F (X) = (U; V ) is trivial if u 2 V . Given u 2 F (Y ) and f : *
*X ! Y ,
we write f*u for F (f)(u). Furthermore, given A X, we denote i*u by u|A.
1.9. Definition. Let F : T ! E be a contravariant functor satisfying the
following properties:
(1)(weak homotopy property) If f 'o g : X ! Y then f*u is trivial if g*u is.
(2)(triviality property) F (pt) has the form (U; U), i.e., every u 2 F (pt) *
*is
trivial.
Let X be a connected pointed space. Given u 2 F (X), we define the Lusternik-
Schnirelmann category of u, catu, to be the minimal k with the following proper*
*ty:
there is a numerable covering {A1; : :;:Ak+1} where each Ai is a pointed subspa*
*ce
of X and u|Ai is trivial for every i = 1; : :;:k.
1.10. Examples. (a) Given a pointed topological space Y , set F (X) = TY (X) :=
(U; V ) where U is the set of all pointed maps X ! Y and V is the subset of all
inessential (under non-pointed homotopy) maps. (The action of F on morphisms is
given by the composition.) Then, the definition above leads to the invariant ca*
*tf
defined in 1.1.
CATEGORY WEIGHT 5
(b) Every pointed set (U; u0) can be considered as the pair (U; {u0}) of sets.
Thus, every functor from T to the category of pointed sets can be regarded as a
functor T ! E .
(c) Every abelian group can be regarded as a pointed set (with the base point
0). So, every functor from T to the category of abelian groups can be treated a*
*s a
functor T ! E .
(d) (Fary [F]) Because of (c), a reduced cohomology theory (not necessarily
additive) E*(-) on T yields the functor X 7! F (X) := (Ei(X); {0}). Then, given
u 2 Ei(X), we have the invariant catu, the Lusternik-Schnirelmann category of
the class u.
(e) Given X 2 T , set F (X) = (U; V ) where U is the set of fiberwise homotopy
equivalence classes of fibrations over X and V consists of fibrations which adm*
*it a
section. The action of F on morphism is given by passing to induced fibrations.*
* If
u 2 F (X) is the equivalence class of a fibration , then catu is the genus of ,*
* [Sv];
another name is the sectional category, secat, [J].
Notice that the example (e) generalizes example (a) since catf = secatf* where
= {p1 : P Y ! Y } for f : X ! Y .
1.11. Proposition. Let X be a connected pointed space.
(i) catf*u min{catf; catu} for every f : X ! Y in T and every u 2 F (Y ).
(ii) If f 'o g : X ! Y then catf*u = catg*u for every u 2 F (Y ).
(iii) If h : X ! Y is a pointed homotopy equivalence then cath*u = catu for
every u 2 F (Y ).
2. Category weight
2.1. Definition. Let F : T ! E be a functor as in 1.9, and let u 2 F (X). We do
not require X to be connected.
(a) We define the category weight of u (denoted by cwgtu) by setting
fi
cwgtu = sup{k fiu|A is trivial whenevercatXA < k}
where A runs over all pointed closed subsets of X.
(b) We define the strict category weight of u (denoted by swgtu) by setting
fi
swgtu = sup{k fi'*u is trivial for every map ' : A ! X in T withcat' < k}:
Notice that cwgtu = 1 = swgtu for every trivial element u.
In particular, the (strict) category weight of a map (cf. 1.10(a)) and a coh*
*o-
mology class (cf.1.10(d)) is defined. Category weight was defined by Fadell and
Husseini [FH] (for F = H*(-)). Strict category weight was defined by Rudyak for
F = E*(-) as in 1.10(d) and Strom for F as in 1.10(a), cf. [R2], [S2]; Strom ca*
*lls
it essential category weight. I must also note that Strom prefers to say that *
*the
(strict) category weight of a trivial element is not defined.
2.2. Theorem (cf. [R2]). (i) swgt u cwgt u, and cwgt u catX provided u
is not a trivial element. Furthermore, for every map f : Y ! X in T we have
catf swgtu provided f*u is non-trivial. Finally, swgtu 1 for every u.
(ii) For every inclusion i : A ! X in T we have cwgti*u cwgtu.
(iii) For every map f : Y ! X in T we have swgtf*u swgtu.
6 YULI B. RUDYAK
(iv) If f 'o g : X ! Y then cwgtf*u = cwgtg*u and swgtf*u = swgtg*u.
(v) swgtu = swgth*u for every pointed homotopy equivalence h.
Notice that properties (iii) and (v) of 2.2 are not valid for cwgteven if f =*
* H*(-),
see [R2], [S2]. In other words, category weight is not a homotopy invariant, a*
*nd
this is a main motivation for introducing strict category weight. The following
proposition gives us another description of strict category weight.
2.3. Proposition. Let X be a connected pointed space. Then for every u 2 F (X)
we have: (i) swgtu = sup{k|p*k(u) is trivial};
(ii) If swgtu = k then cwgtp*k+1u = k; (iii) swgtu = minf2F {cwgt f*u} where
F is the class of all maps f : Y ! X in T .
Proof. (i) See [R2, 1.8(v)].
(ii) Since swgtu = k, we conclude that, by (i), p*k+1u is non-trivial. Now
k = swgtu swgtp*k+1u cwgtp*k+1u catPk+1X k:
(iii) Clearly, swgt u swgtf*u cwgt f*u. Now, if swgtu = k then, by (ii)
swgtu = cwgtp*k+1u.
2.4. Theorem. Let X be a connected pointed space.
(i) For every f : X ! Y in T and every u 2 F (Y ) we have
catf (swgt u) cat(f*u):
In particular, for every diagram X -f!Y -g!Z we have catf (swgt g) cat(gf).
(ii) For every u 2 F (X) we have
catXA (cwgt u) cat(u|A):
In particular, for every map f : X ! Y we have catX A (cwgt f) cat(f|A).
Formula (ii) was found by Strom [S2], formula (i) is an obvious analog of (ii*
*).
This theorem improves properties 1.11(i) and 2.2(i). Notice that the proof does*
* not
use properties (1) and (2) of 1.9.
Proof. We prove only (i). Let catf = n, swgt u = k > 0. We must prove that
cat(f*u) p := [n=k]. Let {A1; : :;:An+1} be a numerable covering of X such
that f|Ai is inessential for every i. Without loss of generality we can assume *
*Y to
be connected, and so, by 1.2, we can assume that each Ai contains the base poin*
*t.
We set
B1 := A1 [ . .[.Ak;B2 = Ak+1 [ . .[.A2k; : :;:
Bp = A(p-1)k+1[ . .[.Apk;Bp+1 = Apk+1 [ . .[.An+1:
Since cat(f|Bj) < k, we conclude that, by 2.2(i), (f|Bj)*u is trivial. So, (f*u*
*)|Bj
is trivial for every j = 1; : :;:p + 1. Thus, catf*u p.
2.5. Corollary ([S2]). For every f : X ! Y in T with X connected and every
u 2 F (Y ) we have
cwgt f*u (cwgt f) swgtu; swgt f*u (swgt f) swgtu:
2.6. Proposition ([R2], [S2]). Let X be a CW -space, and let f : X ! Y , X 2 T
be an essential map such that ssi(Y ) = 0 for i > m. Then cwgt f m. In
particular, if E is spectrum with ssi(E) = 0 for i > m then cwgtu q +m for eve*
*ry
u 2 Eq(X); u 6= 0.
CATEGORY WEIGHT 7
3. Multiplicative properties
Let F be a functor as in 1.9. Given a Puppe sequence X -f!Y -! Cf in T , we
conclude, by 1.9(1), that the image of the composition
F (Cf) -! F (Y ) -! F (X)
consists of trivial elements.
3.1. Definition. A functor F as 1.9 is called half-exact if, for every pointed
polyhedral pair (X; A), the sequence
* i*
F (X [ CA) -j!F (X) -! F (A)
is "exact", i.e., i*u is trivial iff u = j*v for some v 2 F (X [ CA). Here i : *
*A ! X
and j : X ! X [ CA are the inclusions. Given (U1; V1); : :;:(Un; Vn) 2 E , we s*
*et
(U1; V1) x . .x.(Un; Vn) := (U1 x . .U.n; V1 x . .V.n):
Let E; F be two half-exact functors, and let G be an arbitrary functor as in 1.*
*9.
Suppose that, for every two polyhedra X; Y (not necessarily finite), there is a
natural transformation (where x is as described in (3.2))
(3.3) m : E(X) x F (Y ) ! G(X ^ Y ):
This transformation yields a transformation
o : E(X) x F (Y ) -m!G(X ^ Y ) -! G(X x Y );
where shrinks the wedge X _ Y . We set u v := o(u; v) for u 2 E(X); v 2 F (Y *
*):
If X = Y and : X ! X x X is the diagonal, we set uov := *(u v) 2 G(X).
3.4. Theorem (cf. [FH], [R2]). (i) For every pointed connected polyhedron X and
every u 2 E(X); v 2 F (X) we have
cwgt(uov) cwgtu + cwgtv:
(ii) For every pointed connected CW -space X and every u 2 E(X); v 2 F (X)
we have
swgt(uov) swgtu + swgtv:
Proof. (i) First, assume that cwgt u = k < 1; cwgtv = l < 1. Take a pointed
k+l[
closed subspace A of X with catXA < k + l. Then, clearly, A Vi where each
i=1
Viis open and contractible in X. Since A is closed, there is a subdivision of X*
* with
the following property: every simplex e with e \ A 6= ; is contained in some Vi*
*, cf.
[W, Theorem 35]. We let Ai; i = 1; : :;:rn be the union of all simplexes contai*
*ned
in Vi. Then A A1 [ . .[.Ak+l where each subpolyhedron Ai is contractible in
X. We set K := A1 [ . .[.Ak [ {*}; L := Ak+1 [ . .[.Ak+l[ {*} and let
iK : K X; iL : L X; jK : X X [CK; jL : X X [CL; h : X X [C(K [L)
8 YULI B. RUDYAK
be the inclusions. Consider the commutative diagram
*m
E(X [ CK) x F (X [ CL) ----! G(X [ C(K [ L))
? ?
j*Kxj*L?y ?yh*
*o
E(X) x F (X) ----! G(X);
where, for the top row, we have used the canonical homotopy equivalence
(X [ CX0) ^ (Y [ CY 0) ' X x Y [ C((X x Y 0) [ (X0x Y ))
which holds for any polyhedral pairs (X; X0) and (Y; Y 0). Since both i*Ku and *
*i*Lv
are trivial, u = j*Ku0; v = j*Lv0 for some u02 E(X [ CK); v02 F (X [ CL). Clear*
*ly,
uov = h(u0ov0). Thus, (uov)|A is trivial since A K [ L.
If, say, cwgt u = 1 then we must prove that uov|A is trivial if catX A < 1.
Asserting as above, we conclude that there is a pointed subpolyhedron K X
with A K and catX K < 1. Then u|K is trivial, and hence (uv)|K is (take
L = * in the above diagram).
(ii) Let swgt(uv) = k. Then, by 2.3(ii), cwgt p*k+1(uv) = k. Recall that every
CW -space is homotopy equivalent to a polyhedron, and so, by 1.7(ii), there is a
homotopy equivalence " : Y ! Pk+1(X) such that Y is a polyhedron. We put
f = pk+1" : Y ! X. Then f*(uv) 6= 0, and
k = swgt(uv) cwgtf*(uv) catY k:
Now
swgt(uov)= cwgtf*(uov) cwgtf*u + cwgtf*v swgtf*u + swgtf*v
swgtu + swgtv:
3.5. Theorem ([R2, 1.14]). swgt(u v) swgtu + swgtv.
Notice that the corresponding inequality for cwgt is wrong, cf. 2.2(iii,v).
Certainly, we can consider not only two functors E; F but any finite number of
functors equipped with a natural transformation like (3.3).
3.6. Examples. (a) Given n arbitrary spectra E(1); : :;:E(n) and n pointed
CW -spaces X1; : :;:Xn, we have the homomorphism
: E(1)*(X1) . . .E(n)*(Xn) ! (E(1) ^ . .^.E(n))*(X1 x . .x.Xn):
see [Sw]. Furthermore, consider the correspondence (not a homomorphism!)
r : E(1)*(X1) x . .x.E(n)*(Xn)! E(1)*(X1) . . .E(n)*(Xn);
r(u1; : :;:un)= u1 . . .un:
So, we have a natural transformation
o := r : E(1)*(X1) x . .x.E(n)*(Xn) ! (E(1) ^ . .^.E(n))*(X1 x . .x.Xn):
CATEGORY WEIGHT 9
Now, by 3.5 X
swgt(u1 . . .un) swgtui:
for every ui 2 E(i)*(Xi). Furthermore, given ui 2 E(i)*(X), we have the element
u1o. .o.un 2 (E(1) ^ . .^.E(n))*(X), and
X X
cwgt(u1o. .o.un) cwgt ui; swgt(u1o. .o.un) swgtui
(in the first unequality X is assumed to be a polyhedron).
(b) Given a ring spectrum E with a multiplication u : E ^ E ! E, for every n
we have the iterated multiplication
un : E_^_._.^.E-z____"-!E
n times
which yields a function
o : E*(X1)x. .x.E*(Xn) -! (E ^. .^.E)*(X1x. .x.Xn) -(un)*--!E*(X1x. .x.Xn)
where the first arrow is o from (a). In this case the element u1o. .o.un 2 E*(X*
*) is
usually denoted by u1. .u.n, and we have
X
(3.7) catX cwgt(u1. .u.n) cwgt ui n
if u1. .u.n6= 0. Certainly, this refines the cup-length estimation catX n.
3.8. Example ([RO]). A closed connected symplectic manifold is a pair (M2n; !)
where M is a connected closed smooth 2n-dimensional manifold and ! is a closed
non-degenerate 2-form. Notice that in this case !n is a volume form for M. In
particular, ! yields a non-trivial de Rham cohomology class which we denote also
by !. Since !n 6= 0, we conclude that catM n. Rudyak-Oprea [RO] proved that
swgt! = 2 provided !|ss2(M)= 0. Thus, if (M2n; !) is a symplectic manifold with
!|ss2(M)= 0 then, by (3.7),
catM swgt!n n swgt! = 2n;
and hence catM = 2n (because catM dim M).
3.9. Examples. (a) ([S2]) Given two pointed spaces A; B, set E(-) = TA (-),
F (-) = TB (-) and G(-) = TA^B (-) (see 1.10(a)). We define m : E(X)xF (Y ) !
G(X ^ Y ) by setting m(f; g) = f ^ g, and 3.4 is applicable to this case.
(b) According to 1.10(e), the category weight cwgt of a fibration is defined.
It turns out to be that, for every fibrations ; j over the same base X,
cwgt( *X j) cwgt + cwgtj:
This can be deduced from 3.4, but we indicate a direct proof. Let A; K; L be as*
* in
the proof of 3.4, and let s1 (resp. s2) be a section of over K (resp of j over*
* L).
Take a function ' : A ! [0; 1] such that '-1(0; 1] 2 K and '-1[0; 1) 2 L. We set
s(x) := 2 {the total space of * j}; x 2 A:
Then s is a desired section over A. Now, we can formulate (and, probably, explo*
*it)
the obvious analog of (3.7).
The following result shows how Massey products help to estimate catX. Let
H*(-; R) denote singular cohomology with coefficients in a commutative ring R.
Given a matrix V over H*(X; R), we set cwgtV = min{cwgt v} where v runs over
all entries of V .
10 YULI B. RUDYAK
3.10. Theorem ([R2]). Given X 2 T , let V1; : :;:Vn be matrices over H*(X; R).
Suppose that the matric Massey product is defined. If 0 =2
then
catX mini{cwgt V2i} + mini{cwgt V2i+1}:
My feeling is that this result is somehow related to (3.7), but I can't say h*
*ow
explicitly.
4. Elements of high category weight
The results of the previous section show that it makes sense to search (inde-
composable) elements of high category weight. The first example of this kind was
found by Fadell-Husseini [FH]. Namely, they proved that, for every odd prime p
and connected X,
(4.1) cwgt fiP nu 2 ifu 2 H2n+1(X; Z=p):
(Actually, they proved that swgtfiP nu 2.)
4.2. Theorem. Let X 2 T , and let V1; : :;:Vn be matrices over H*(X; R)
(singular cohomology with coefficients in a commutative ring). Suppose that the
matric Massey product is defined. Then, for every V 2
and every entry u of V , we have cwgtu 2.
This theorem was explicitly formulated in [R2]. Actually, it follows from 1.*
*6,
1.8, and the result of Gugenheim-May [GM] that Ker{"* : H*(X) ! H*(SX)}
contains all the matric Massey products. Note that 4.2 implies (4.1) since fiP *
*nu 2
__ (p times) for every u 2 H2n+1(X; Z=p), Kraines [K].
Let E; F be two spectra, and let : E ! F be a (stable) cohomology operation.
Without loss of generality we can assume that E = {En} and F = {Fn} are -
spectra, i.e., that there are weak homotopy equivalences oen : En ! En+1, etc.
Then yields a family n : En ! Fn of maps such that the diagram
En --n--! Fn
? ?
oeEn?y ?yoeFn
En+1 -n+1---!Fn+1
commutes up to homotopy. Following Strom [S2], set
d = d() = inf{k|k is an essential map}
4.3. Theorem ([S2]). swgt n = 1 for n > d, and swgtd 2.
Notice that d() = 2n + 1 for = fiP n. This gives us another proof of (4.1).
Indeed, if u 2 H2n+1(X; Z=p) then
swgtfiP n(u) = swgt2n+1(u) swgt(2n+1) swgtu 2 swgtu 2:
CATEGORY WEIGHT 11
4.4. Theorem ([S3]). Let G be a discrete group, and let E be a spectrum such
that ssi(E) = 0 for i < m. Then swgtu k + m for every u 2 Ek(BG).
For k = 2 this result goes back to Fadell-Husseini [FH]. Notice that, by 2.6,
swgtu = k for every u 2 Hk(BG); u 6= 0. As an application of 4.4, we note the
following corollary.
4.5. Corollary. Let M be a smooth manifold such that the structure group of its
tangent bundle reduces to a discrete group G. If wI(M) 6= 0 then catM |I|.PHere
I = {i1; : :;:ik}, wI is the Stiefel-Whitney class wi1. .w.ikand |I| = ik.
Proof. It suffices to prove that swgtwI |I|. But this follows from 4.3 and 2.2*
*(iii),
since wI is induced from BG.
The last example of elements of high category weight is as follows.
4.6. Theorem. Let X be a simply connected rational space, and let (; d) be the
minimal Sullivan model for X. If a cohomology class x 2 H*(X; Q) = H*(; d)
has the form x = [a1. .a.k] for some ai2 ; dimai 0, then swgtx k.
Proof. Notice that the aiare not assumed to be cocycles and, hence, we can't ap*
*ply
(3.6). This follows from the result of Felix-Halperin [FeH, x3] that p*k: H*(X;*
* Q) !
H*(Pk(X); Q) annihilates all elements of the form [a1. .a.k].
5. Detecting elements
5.1. Definition. Let F be as 1.9. An element u 2 F *(X) is called a detecting
element for X if swgtu = catX.
We formulate the above definition for the general situation, but really we wi*
*ll
apply it to the case when F is a cohomology theory, as in 1.9(d).
5.2. Theorem ([R2]). Let X; Y be two connected p[ointed CW -spaces and let E; G
be two spectra. Suppose that there are detecting elements u 2 E*(X); v 2 G*(Y ).
If 0 6= u v 2 (E ^ G)*(X x Y ) then cat(X x Y ) = catX + catY , and u v is a
detecting element for X x Y .
By 2.2(i), 3.5 and 1.4,
cat(X x Y ) swgt(u v) swgtu + swgtv = catX + catY cat(X x Y ):
5.3. Corollary. If a pointed CW -space X possesses a detecting element then
cat(X x Sn) = catX + 1 for every n 0. In other words, the Ganea conjecture
holds for X.
Proof. We take a detecting element u 2 E*(X) and apply 5.2 to the case Y =
Sn; Gn(Y ) = n(Sn), and v is given by the identity map Sn ! Sn. One can prove
that u v 6= 0 (see e.g. [R2]), and the result follows.
For example, Strom [S1] proved that a (q - 1)-connected CW -space X possesses
a detecting element if catX = [dim X=q], and so such an X satisfies the Ganea
conjecture.
12 YULI B. RUDYAK
5.4. Theorem ([R2]). Let R be a ring spectrum, and let E be an arbitrary R-
module spectrum. Let Mn ; Nn be two closed connected HZ-orientable PL manifolds,
and let f : N ! M be a map of degree 1 and such that N is R-orientable. If
M possesses a detecting element u 2 E*(M) then catf = catM. In particular,
catN catM.
Proof. It is easy to see that f* : E*(M) ! E*(N) is monic. So, f*u 6= 0, and
hence, by 2.2(i), catf swgtu = catM. Hence, by 1.3, catf = catM.
The results above show that it is useful and important to know whether a space
possesses detecting elements. Consider the Puppe sequence
Pm (X) -pm-!X -jm-!Cm (X) := C(pm )
where pm : Pm (X) ! X is the fibration (1.5) and C(pm ) is the cone of pm .
5.5. Theorem ([R2]). Let X be a pointed connected CW -space with catX = k <
1. If jk is stably essential (i.e., the stable homotopy class of jk is non-zero*
*) then
X possesses a detecting element.
In fact, the stable homotopy class of jm can be treated as a universal (among
cohomology functors) detecting element. Also, we note that, for every X with
catX = k, the fibration pk : Pk(X) ! X is a detecting element (see 1.10(e)).
However, unfortunately, it is difficult to apply analogs of 5.2-5.4 to this cas*
*e.
6. Manifolds
Given a PL manifold M, we denote by M the stable normal bundle of M.
6.1. Theorem ([R2]). Let Mn ; n = dim M 4, be a closed (q - 1)-connected PL
manifold, q 1. Suppose that there is a natural number m such that M |M(m) is
trivial and
n min{2q catM - 4; m + q catM - 1}:
Then M possesses a detecting element.
Putting m = n + 1 and m = 1, we get the following corollary.
6.2. Corollary. (i) Let M be a closed (q - 1)-connected stably parallelizable *
*PL
manifold, q 1. Suppose that 4 dim M 2(q catM - 2). Then M possesses a
detecting element.
(ii) Let q 1, and let Mn ; n = dim M 4 be a closed orientable (q - 1)-
connected PL manifold such that q catM = n. Then M possesses a detecting
element. Moreover, there exists a detecting element u 2 Hn (M; ssn(Cn(M))).
6.3. Corollary. Let M be as in 6:1. Then cat(M x Sm1 x . .x.Smn ) = catM + n
for any natural numbers m1; : :;:mn.
Other results about the Ganea conjecture for manifolds can be found in [R1], [R*
*2],
[S1]. Based on 6.2 and 5.4, we get the following theorem.
6.4. Theorem. (i) Let Mn be as in 6.2(i), and let f : Nn ! Mn be a map of
degree 1 where N is a stably parallelizable PL manifold. Then catf = catM. In
particular, catN catM.
(ii) Let f : N ! M be a map of degree 1 of closed HZ-orientable PL manifolds.
If catM = dim M then catf = catM = catN.
CATEGORY WEIGHT 13
6.5. Corollary. Let M be an oriented PL manifold with catM = dim M, and let
f : X ! M be a map of an arbitrary topological space such that
f* : Hn (M; ssn(Cn(M))) ! Hn (M; ssn(Cn(M)))
is a monomorphism. Then catf = catM.
Proof. By 6.2, there is a detecting element u 2 Hn (M; ssn(Cn(M))), and f*u 6= *
*0.
Thus,
cat f swgtu = catM:
But, by 1.3(i), catf catM.
7. Applications to the Arnold conjecture
In [A, Appendix 9] Arnold proposed a beautiful conjecture concerning the rela-
tion between the number of fixed points of certain self-diffeomorphisms of a cl*
*osed
symplectic manifold (M; !) and the minimum number of critical points of any
smooth (= C1 ) function on M.
Let (M2n; !) be a closed symplectic manifold. A symplectomorphism OE : M !
M (i.e., a diffeomorphism with OE*! = !) is called Hamiltonian (or exact) if it
belongs to the flow of a time-dependent Hamiltonian vector field on M. See [HZ]*
* or
[MS] for details. We define Arn(M; !) to be the minimum number of fixed points
for any Hamiltonian symplectomorphism of M. The Arnold conjecture claims that
the following inequality holds for every closed symplectic manifold (M; !):
Arn(M; !) CritM:
The conjecture, usually (but not universally) weakened by replacing CritM by
the cup-length of M, has been proved under various hypotheses for various class*
*es
of manifolds ([CZ], [H], [Fl1], [Fl2]). Here the following theorem (formulated*
* ex-
plicitely in [R2] and based on Floer's approach) plays the crucial role.
7.1. Theorem. Let (M; !) be a closed connected symplectic manifold such that
!|ss2(M)= 0 = c1(M)|ss2(M). Then there exists a map f : X ! M with the following
properties:
(i) X is a compact metric space;
(ii) 1 + catf Arn(M; !);
(iii) The homomorphism f* : Hn (M; G) ! Hn (X; G) is a monomorphism for
every coefficient group G.
The following theorem is proved in [R3] and [RO].
7.2. Theorem. Let (M2n; !) be a closed connected symplectic manifold with
!|ss2(M)= 0 = c1|ss2(M):
Then Arn(M; !) CritM; i.e., the Arnold conjecture holds for M.
Proof. First, note that, by 3.8, catM = 2n = dim M. Hence,
1 + catM CritM 1 + dimM = 1 + catM
14 YULI B. RUDYAK
(the last inequality is a theorem of Takens [T]). So, 1 + catM = CritM. Thus, in
view of 7.1, it remains to prove that catf catM where f is the map from 7.1.
We give two proofs of this inequality.
First proof. This follows from 6.5.
Second proof. Rudyak-Oprea [RO] proved that swgt! = 2. Thus, since f*!n 6= 0,
catf swgt!n 2n = catM:
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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
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