ON CATEGORY WEIGHT
AND ITS APPLICATIONS
Yuli B. Rudyak*
Abstract. We develop and apply the concept of category weight which was
introduced by Fadell and Husseini. For example, we prove that category *
*weight
of every Massey product , ui 2 eH*(X) is at least 2 provid*
*ed X is
connected.
Furthermore, we remark that elements of maximal category weight ena*
*ble us
to control the Lusternik-Schnirelmann category of a space. For example,*
* we prove
that if f : N ! M is a map of degree 1 of closed stable parallelizable *
*manifolds
and dimM 2 catM - 4 then catN catM.
We also prove that if M is a closed manifold with dimM 2 catM - 3 *
*then
cat(M x Sm1 x . .x.Smn ) = catM + n, i.e., the Ganea conjecture holds f*
*or M.
INTRODUCTION
This is a revised version of my preprint [19].
Let catX denote the Lusternik-Schnirelmann category of a space X (we consid*
*er
the normalized category, i.e., catX = 0 if X is contractible). It is well known*
* (Froloff-
Elsholz [9]) that if u1 . .u.n6= 0 for some ui 2 H*(X); dim ui > 0 then catX n.
Fadell-Husseini [5] developed this observation as follows. They defined the cat*
*egory
weight (denoted by cwgt) of an element u 2 H*(X); dim u > 0 to be the maximal k
such that u|A = 0 whenever A A1[ . .[.Ak, where each Ai is open and contractib*
*le
in X. It is easy to see that, for any u1; : :u:k2 H>0 (X), we have
X
catX cwgt(u1 . .u.k) cwgt ui
provided 0 6= u1 . .u.k2 H*(X). Since cwgt ui 1, this yields the Froloff-Elsh*
*olz
estimation catX k. However, if cwgt ui > 1 for some (indecomposable) ui, then *
*we
obtain a better estimation of catX than just the inequality catX k. For exampl*
*e,
Fadell-Husseini [5] proved that, for every odd prime p and every admissible seq*
*uence
I, cwgt(StIu) 2 if u 2 He(I)(X; Z=p) where e(I) is excess of I. This enables t*
*hem
to compute the Lusternik-Schnirelmann category for some new spaces.
Massey products yield other examples of elements of higher category weight,*
* see
Theorem I below.
_____________
The author was partially supported by Deutsche Forschungsgemeinschaft, the *
*Research Group
"Topologie und nichtkommutative Geometrie"
1
2 YULI B. RUDYAK
This approach can be generalized for an arbitrary spectrum (cohomology theo*
*ry)
E. Namely, for every u 2 E*(X) we can define cwgt u similarly to the above (but
we must let X be a CW -space and consider only CW -subspaces A of X). Moreover,
given another spectrum F and an element v 2 F *(X), we have the element uov 2
(E ^ F )*(X), and
cwgt(uov) cwgt u + cwgt v:
So, in principle, one can expect that in this way we can find suitable estimati*
*ons of
the Lusternik-Schnirelmann category.
Furthermore, category weight with values in arbitrary cohomology theories e*
*nables
us to control the Lusternik-Schnirelmann category of spaces as follows. We say *
*that
u 2 Ek(X) is a detecting element for (a connected CW -space) X if cwgt u = catX*
* and
cwgt f*u cwgt u for every map f : Y ! X. The following result demonstrates the
usefulness of detecting elements. Given u 2 E*(X); v 2 F *(Y ), consider the pa*
*iring
o : E*(X) F *(Y ) ! (E ^ F )*(X x Y ) and set u v := o(u v).
Theorem A (see 2.2). Let X; Y be two connected CW -spaces. Suppose that the*
*re
are detecting elements u 2 E*(X); v 2 F *(Y ). If 0 6= u v 2 (E ^ F )*(X x Y )*
* then
cat(X x Y ) = catX + catY . Furthermore, in this case u v is a detecting eleme*
*nt
for X x Y .
In particular, we have the following corollary.
Corollary B (see 2.3). If a CW -space X possesses a detecting element then
cat(X x Sm1 x . .x.Smn ) = catX + n for any natural numbers m1; : :;:mn.
Recall that the long-standing Ganea conjecture [10] claims that cat(X x Sm *
*) =
catX + 1 for every connected finite CW -space X and every m > 0.
So, it is useful to know conditions which imply the existence of detecting *
*elements.
Let p : P X ! X be the path fibration over X, and let pk : Pk(X) ! X be the k-f*
*old
join over X of p. It is well-known that pk has a section iff catX < k.
Theorem C (see 2.4). Let X be a connected CW -space with catX = k < 1.
Consider the Puppe sequence
Pk(X) -pk!X -jk!C(pk):
If jk is stably essential (i.e., the stable homotopy class of jk is non-zero) t*
*hen X
possesses a detecting element.
In particular, the hypotheses of theorem C hold if jk is essential, X is (q*
* - 1)-
connected, and dim X 2kq - 2.
In fact, clearly, the stable homotopy class of the map jk is the universal *
*detecting
element for X.
Because of Theorem C, it is useful to introduce the invariant
r(X) = sup{m|jm is stably essential}:
It is easy to see r(X) catX and, moreover, r(X) = catX iff X possesses a detec*
*ting
element. Furthermore, for every ring spectrum E we have clE(X) r(X) where
clE(X) is the cup-length of X with respect to E, i.e., the length of the longes*
*t non-
trivial product in eE*(X).
Using surgery theory, we deduce from theorem C the following results.
ON CATEGORY WEIGHT AND ITS APPLICATIONS 3
Theorem D (see 3.3). (i) Let M be a closed (q -1)-connected stably parallel*
*izable
PL manifold, q 1. Suppose that 4 dim M 2(q catM - 2). Then M possesses a
detecting element.
(ii) If M is an arbitrary closed orientable (q - 1)-connected PL manifold. *
*q 1.
If q catM = dim M 4 then M possesses a detecting element.
Corollary E (see 3.4). Let M be as in theorem D. Then cat(M x Sm1 x . .x.
Smn ) = catM + n for any natural numbers m1; : :;:mn.
Theorem F (see 3.6). (i) Let Mn be as in theorem D, part (i), and let f : N*
*n !
Mn be a map of degree 1 where N is a stably parallelizable PL manifold. Then
catN catM.
(ii) Let f : N ! M be a map of degree 1 of closed HZ-orientable PL manifold*
*s.
If catM = dim M 4 then catM = catN.
We note that for q = 1 corollary E can be strengthened.
Theorem G (see 3.8). Let M be a closed connected PL manifold. Suppose that
dim M 2 catM - 3. Then cat(M x Sm1 x : :x:Smn ) = catM + n for any natural
numbers m1; : :;:mn.
Singhof [21] proved that cat(M x S1) = catM + 1 for M in theorem G, and our
proof uses this result. The case n = 1 is published in [20], see also [24].
We also consider relationships between category weight and Massey products.*
* It
is more or less known that every Massey product is trivial (i.e., it contains t*
*he zero)
on each space X with catX 1. See [11] and [27] about related things. The conce*
*pt
of category weight establishes an adequate context for this matter. Namely, rou*
*ghly
speaking, the category weight of any Massey product is 2.
Given a matrix M = (mij) over H*(X), let cwgt M = min{cwgt (mij)}. Here we
prove the following theorems.
Theorem H (see 4.6). Let X be a CW -space, and let V1; : :;:Vn be matrices
over H*(X). Suppose that the matric Massey product is defined. *
*If
0 =2 then
catX mini{cwgt V2i} + mini{cwgt V2i+1}:
Theorem I (see 4.9). Let X be a connected CW -space, and let V1; : :;:Vn be
matrices over eH*(X). Suppose that the matric Massey product is d*
*efined.
Then cwgt V 2 for every V 2 . In particular, if there exists V*
* 2
, V 6= 0, then catX > 1.
Because of a result of Kraines [13], theorem I implies the above mentioned *
*result
of Fadell-Husseini: cwgt (StIu) 2 if dim u = e(I), see 5.9.
As we shall see below, category weight is not a homotopy invariant, i.e., t*
*he
equality cwgt h*u = cwgt u fails for arbitrary u 2 E*(X) and arbitrary homotopy
equivalence h : Y ! X. However, there is a homotopy invariant version of catego*
*ry
weight. Namely, we define the strict category weight (denoted by swgt) of an el*
*ement
u 2 E*(X) to be the maximal k such that '*u = 0 for every ' : A ! X with the
following property: A = A1 [ . .[.Ak where each Ai is an open subspace of A and
4 YULI B. RUDYAK
'|Ai : Ai ! X is inessential. In other words, to do this modification, we cons*
*ider
arbitrary maps A ! X instead of inclusions A X. Clearly, strict category weigh*
*t is
a homotopy invariant, i.e., swgt h*u = swgt u for every homotopy equivalence h.
The paper is organized as follows. In section 1 we expose preliminaries on *
*category
weights (both cwgt and swgt). In section 2 we discuss detecting elements and t*
*heir
applications. In section 3 we apply our theory to manifolds. In section 4 we co*
*nsider
relationships between category weights and Massey products.
We follow Switzer [26] in the definition of CW -complexes. A CW -space is d*
*efined
to be a space which is homeomorphic to a CW -complex. A CW -inclusion is an
inclusion i : A X such that A is a CW -subcomplex with respect to a certain
CW -decomposition of a CW -space X.
We work entirely in the category of compactly generated weak Hausdorff spac*
*es.
So, we take the direct products and functional spaces in this category, see [17*
*]. In
particular, the direct product of two CW -spaces is a CW -space.
The disjoint union 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.
Given a pointed CW -space 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. See [1] or [26] for more details. Clearly, 1 is a functor*
* from
pointed CW -spaces to spectra.
"Fibration" always means a Hurewicz fibration.
"Connected" always means path connected.
The sign " '" denotes homotopy of maps (morphisms) or homotopy equivalence
of spaces (spectra).
Given an element u 2 E*(X) and an inclusion i : A ! X, we denote i*u by u|A.
We agree that, for every u 2 E*(X), u|A = 0 if A = ;.
I am grateful to Dieter Puppe for useful discussions. In particular, he exp*
*lained
me that category weight is not a homotopy invariant.
1. PRELIMINARIES ON CATEGORY WEIGHT
1.1. Definition. (a) (cf. 2], [6]) Given a map ' : A ! X, we say that cat' *
* k
if A can be covered by open sets A1; : :;:Ak+1 such that '|Ai is inessential fo*
*r 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 := cati.
(c) (cf. [15]) We define the Lusternik-Schnirelmann category catX of a spac*
*e X
by setting catX := cat1X .
The basic information concerning the Lusternik-Schnirelmann category can be
found in [7], [12], [25].
1.2. Proposition ([2]). (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 ' = cat h' for every
' : A ! X.
ON CATEGORY WEIGHT AND ITS APPLICATIONS 5
1.3. Lemma ([7]). For every two connected CW -spaces X; Y we have
max {catX; catY } cat(X x Y ) catX + catY:
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
pk : Pk(X) ! X (1.*
*4)
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. [12]. In particular, P1(X*
*) = P X.
1.5. Proposition. For every connected 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 X is (q - 1)-connected then pk : Pk(X) ! X is a (kq - 2)-equivalenc*
*e;
(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, *
*[3].
(ii) It is easy to see that cat(E1 *X E2) catE1 + catE2 + 1 for every two *
*maps
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) The space (X)*k has the homotopy type of a CW -space since X does,
Milnor [18]. Furthermore, the total space of any fibration has the homotopy typ*
*e of a
CW -space provided both the base and the fiber do, see e.g. [8, 5.4.2].
1.6. Theorem ([25, Theorems 3 and 190]). Let ' : A ! X be a map of CW -
spaces with X connected. Then cat' < k iff there is a map : A ! Pk(X) such th*
*at
pk = '.
1.7. Definition. Let E be an arbitrary spectrum (cohomology theory). Let X
be a CW -space, and let u 2 Eq(X) be an arbitrary element.
(a) (cf. [5]) We define the category weight of u (denoted by cwgt u) by set*
*ting
fi
cwgt u = sup{k fiu|A = 0 for every CW -subspace A of X with catX A < k}:
(b) We define the strict category weight of u (denoted by swgt u) by setting
fi
swgtu = sup{k fi'*u = 0 for every maps ' : A ! X with cat' < k}
where A is assumed to be a CW -space.
6 YULI B. RUDYAK
The concept of strict category weight was introduced by the author in his t*
*alk at
the AMS Summer Recearch Institute, Seattle, July 1996.
We remark that cwgt u = 1 = swgt u if u = 0. Also, swgt u = 0 if '*u 6= 0 f*
*or
some inessential map ' : A ! X.
Notice that catX X(n) cat X(n) n for every CW -complex X. Hence, if
cwgt u = 1 then u|X(n) = 0 for every n, i.e., u is a phantom, u 2 lim-1{E*(X(n)*
*)}.
1.8. Theorem. (When we write wgt it means that the corresponding claim is
valid for both cwgt and swgt :) Let u 2 E*(X) be as in 1:7. Then the following *
*hold:
(i) swgt u cwgt u, and cwgt u catX provided u 6= 0. Furthermore, for every
map f : Y ! X of CW -spaces we have catf swgt u provided f*u 6= 0. Finally, if
X is connected then swgt u 1 whenever u 2 eE*(X) E*(X).
(ii) For every CW -inclusion i : A ! X we have cwgt i*u cwgt u.
(iii) For every map f : Y ! X of CW -spaces we have swgt f*u swgt u.
(iv) swgt u = swgt h*u for every homotopy equivalence h.
(v) If X is connected then swgt u = sup{k|p*k(u) = 0}, where pk is a fibrat*
*ion
(1:4).
(vi) If X = tXffthen wgt u = minff{wgt u|Xff}.
(vii) If ae : E ! F is a morphism of spectra and u 2 E*(X) then wgt ae*u w*
*gt u.
(viii) For any elements ui 2 E*(X); i = 1; : :;:n, we have wgt (u1 + . .+.u*
*n)
mini{wgt ui}.
(ix) Let E be a spectrum such that ssi(E) = 0 for i > m. If u 2 Eq(X); u 6*
*= 0,
then wgt u q + m. In particular, if 0 6= u 2 Hq(X) then wgt u q.
Proof. (i) This is obvious.
(ii) This follows because catA B catX B for every CW -inclusions B A X.
(iii) This follows from 1.2(i).
(iv) This follows from 1.2(iii). fi
(v) By 1.6, swgt u k if p*ku = 0. So, swgt u sup{k fip*ku = 0}. Conversel*
*y, let
swgt u = l 1. Then, by (iii) andf1.5(ii),ip*lu = 0 if l < 1, and p*ku = 0 for *
*every
k 2 N if l = 1. Thus, l sup{k fip*ku = 0}.
(vi, vii, viii) This is obvious.
(ix) By 1.8(i), it suffices to prove the claim for cwgt. Let X(n) be the n-*
*skeleton
of X, and let in : X(n-1) -! X(n) be the inclusion. The Puppe sequence X(n-1) -*
*in!
X(n) ! _Sn yields the exact sequence
*
. .-.!Eeq(_Sn ) -! Eq(X(n)) -in!Eq(X(n-1)) -! eEq+1(_Sn ) -! . .:.
Since ssi(E) = 0 for i > m, we conclude that Eeq(Sn ) = 0 for n > q + m. So, t*
*he
homomorphism
i*n: Eq(X(n)) ! Eq(X(n-1))
is an isomorphism if n > q+m+1 and a monomorphism if n = q+m+1. So, for every q
the inverse sequence {Eq(X(r))} stabilizes as r ! 1, and hence lim-1r{Eq(X(r))}*
* = 0,
see e.g. [26]. Considering the Milnor exact sequence (see loc.cit.)
0 ! lim-1r{Eq(X(r))} ! Eq(X) -ae!lim-{Eq(X(r))} ! 0;
r
ON CATEGORY WEIGHT AND ITS APPLICATIONS 7
we conclude that the canonical epimorphism ae : Eq(X) ! lim-{Eq(X(r))} is an is*
*o-
morphism. But, by the above, lim-{Eq(X(r))} = Eq(X(n)) if n > q + m. Hence, the
restriction homomorphism Eq(X) ! Eq(X(n)) is an isomorphism for n > q + m.
Now, suppose that cwgt u > q + m. Then u|X(q+m) = 0. But, by the above, the
restriction homomorphism Eq(X) ! Eq(X(q+m) ) is monic, and hence u = 0.
1.9. Corollary. Given a CW -space X and an arbitrary spectrum E, for every
u 2 E*(X) we have
swgt u = minf2F{cwgt f*u} = minh2H{cwgt h*u}
where F is the class of all maps f : Y ! X and H is the class of all homotopy
equivalences h : Z ! X of CW -spaces.
Proof. Because of 1.8(v), we can assume that X is connected. Firstly, we pr*
*ove
that min f2F{cwgt f*u} = minh2H {cwgt h*u}. Clearly,
minf2F{cwgt f*u} minh2H{cwgt h*u}:
Conversely, choose any cellular map g : Y ! X such that min f2F {cwgt f*u} =
cwgt g*u. We represent g as a composition Y -i!M -h!X where i is a CW -inclusion
and h is a homotopy equivalence. Now, because of 1.8(ii),
cwgt g*u = cwgt i*h*u cwgt h*u;
and hence
minh2Hcwgth*u = minf2F{cwgt f*u}:
Now, let k := minf2F {cwgt f*u}. By 1.8, swgt u swgt f*u cwgt f*u for eve*
*ry
f, and so swgt u k. Conversely, by the above, there is a homotopy equivalence
h : Z ! X such that cwgt h*u = k := minf2F {cwgt f*u}. Then
cwgt p*kh*u cwgt h*u = k;
and so, by 1.5(ii), p*kh*u = 0. Hence, by 1.8(v), swgt h*u k, and thus, by 1.8*
*(iv),
swgt u k.
1.10. Example (D. Puppe). Let p be an odd prime. Consider the lens space
ffi
L = S2n+1 (Z=p)
where Z=p acts freely on S2n+1 . It is well known that catL = 2n+1, Krasnoselsk*
*i [14].
Let b 2 H2n+1 (L; Z=2) = Z=2 be the generator. We prove that cwgt b = 2n + 1 wh*
*ile
swgt b = 1. Indeed, since b|A = 0 for every A L, we conclude that cwgt b =
catL = 2n + 1. Now, consider the universal covering map f : S2n+1 ! L. This is a
map of degree p, and so f*b 6= 0. On the other hand, swgt f*b catS2n+1 = 1, and
so swgt b = 1 since swgt b swgt f*b.
1.11. Notation. Given n spectra E(1); : :;:E(n) and n CW -spaces X1; : :;:X*
*n,
we have the homomorphism
o : E(1)*(X1) . . .E(n)*(Xn) ! (E(1) ^ . .^.E(n))*(X1 x . .x.Xn):
See [1] or [26]. Given ui 2 E(i)*(Xi); i = 1; : :;:n, we denote o*(u1 . . .un*
*) by
u1 . . .un.
8 YULI B. RUDYAK
Furthermore, if Xi = X for every i = 1; : :;:n then we set
u1o. .o.un := *(u1 . . .un) 2 (E(1) ^ . .^.E(n))*(X)
where : X ! Xn is the diagonal.
Finally, let E be a ring spectrum with the multiplication : E ^ E ! E. We
define the morphism
n : E_^_._.^.E-z____"-!E
n times
by induction, by setting 1 = and n+1 = O(n ^ 1E ). Then for every CW -space
X we have the homomorphism
(n)* : (E ^ . .^.E)*(X) -! E*(X):
Now, given ui 2 E*(X); i = 1; : :;:n, we set
u1 . .u.n:= (n)*(u1o. .o.un) 2 E*(X):
1.12. Theorem. Let X be a polyhedron. When we write wgt it means that the
corresponding claim is valid for both cwgt and swgt.
(i) For any elements ui 2 E(i)*(X); i = 1; : :;:n, we have wgt (u1o. .o.un)
Xn
wgt ui.
i=1
(ii) Let E be a ring spectrum. Then for any ui 2 E*(X); i = 1; : :;:n, we *
*have
Xn
wgt (u1 . .u.n) wgt(u1o. .o.un). In particular, wgt(u1 . .u.n) wgt ui.
i=1
Proof. (i) (cf. [5]) We start with the case of cwgt. First, assume that cwg*
*t ui < 1
for every i. We set ck := cwgt uk and
r1 := 1; rk := 1 + c1 + . .+.ck-1
P
for 1 < k n. Take a CW -subspace A X with catX A < cwgtui = rn. Then,
rn[
clearly, A Vi where each Vi is open and contractible in X. Since A is clos*
*ed, there
i=1
is a subdivision of X with the following property: every simplex e with e \ A 6*
*= ; is
contained in some Vi, cf. [29, Theorem 35]. Let Ai; i = 1; : :;:rn be the union*
* of all
simplices contained in Vi. Then
rn[
A Ai
i=1
and each polyhedron Ai is contractible in X. For every i = 1; : :;:n we set
ae; if c = 0;
Bi = i
Ari[ Ari+1[ . .[.Ari+ci-1 if ci > 0:
Now, we set B = B1 [ . .[.Bn, and it suffices to prove that u|B = 0. Clearly,
catX Bi < ci = cwgt ui. So, ui|Bi = 0. Let ji : (X; ;) ! (X; Bi) and j : (X; *
*;) !
ON CATEGORY WEIGHT AND ITS APPLICATIONS 9
(X; B) be the inclusions. Consider the commutative diagram
*o0
E(1)*(X; B1) . . .E(n)*(X; Bn) ----! (E(1) ^ . .^.E(n))*(X; B)
? ?
j*1...j*n?y ?yj*
*o00
E(1)*(X) . . .E(n)*(X) ----! (E(1) ^ . .^.E(n))*(X) :
Since ui|Bi = 0, we conclude that ui = j*vi for some vi 2 E(i)*(X; Bi). Clearl*
*y,
u1o. .o.un = j*(v1o. .o.vn), i.e., u 2 Im j*. Thus, u1o. .o.un|B = 0.
Now assume that, say, cwgt u1 = 1. We must prove that u1o. .o.un|A = 0
whenever catX A < 1. But this holds since u1|A = 0.
Now we prove the inequality for swgt. By 1.9, there is a homotopy equivalen*
*ce h :
Y ! X such that swgt(u1o. .u.n) = cwgt h*(u1o. .o.un). Without loss of generali*
*ty
we can assume that Y is a polyhedron. Now
X
swgt (u1o. .o.un)= cwgt h*(u1o. .o.un) cwgt h*ui
X X
swgt h*ui = swgt ui:
(ii) This follows from 1.8(vii).
1.13. Lemma. Let u1 . . .un 2 (E(1) ^ . .^.E(n))*(X1 x . .x.Xn) be as in
1:11. Let ssi : X1 x . .x.Xn ! Xi; i = 1; : :n:, be the projection. Then
u1 . . .un = (ss*1u1)o. .o.(ss*nun)
where ss*i: (E(1) ^ . .^.E(n))*(Xi) ! (E(1) ^ . .^.E(n))*(X1 x . .x.Xn).
Proof. We consider the case n = 2 only: the general case can be considered
similarly. So, let ss1 : X x Y ! X and ss2 : X x Y ! Y be the projections. We m*
*ust
prove that u v = (ss*1u)o(ss*2v).
Let : X x Y ! X x Y x X x Y be the diagonal. Consider the diagram
*ss*
E*(X) F *(Y ) -ss1--2-! E*(X x Y ) F *(X x Y )
? ?
o0?y ?yo00
*
(E ^ F )*(X x Y )-(ss1xss2)-----!(E ^ F )*(X x Y x X x Y )
flfl ?
fl ?y*
(E ^ F )*(X x Y ) ________ (E ^ F )*(X x Y ):
The top square commutes because of naturality of o, the bottom square commutes
since (ss1 x ss2)O = 1XxY . Now,
u v = o0(u v) = *(ss1 x ss2)*o0(u v) = *o00(ss*1u ss*1v) = *(ss*1u ss*1*
*v)
= ss*1uoss*2v:
10 YULI B. RUDYAK
1.14. Corollary. Let u1 . . .un 2 (E(1) ^ . .^.E(n))*(X1 x . .x.Xn) be
as in 1:11. Then n
X
swgt (u1 . . .un) swgt ui:
i=1
Proof. Firstly, let X be a polyhedron. Then, by 1.12(i), 1.13 and 1.8(i),
X X
swgt(u1 . . .un) = swgt((ss*1u)o. .o.(ss*nun)) swgt ss*iui swgt ui:
Now the general case follows from 1.8(iv), since every CW -space is homotopy eq*
*uiv-
alent to a polyhedron (e.g., to the geometric realization of its simplicial set*
*).
1.15. Remark. Obviously, category weight can be defined in more general
situations. Namely, given a full category T of the category of topological sp*
*aces
and maps, a class P of pairs (X; A) with X; A 2 T , and a functor F : T ! A =
{abelian groups and homomorphisms }, one can define cwgt u and swgt u for any *
*u 2
F *(X). (Moreover, it suffices even to consider functors F whose target is the *
*category
of pointed sets.) Furthermore, if we assume that F is homotopy invariant, then *
*certain
analogs of 1.5 and 1.8 can be proved for suitable T and P (recall that 1.6 hol*
*ds for
any Hausdorff paracompact spaces X; Y ). Finally, given a pairing E(X) F (Y )*
* !
G(X x Y ) with E; F; G : T ! A, one can try to prove multiplicative properties*
* of
category weight, like 1.12-1.14. However, here we should suppose that E and F a*
*re
exact functors, and we must be careful with the choice of T : one must be able*
* to
construct a pairing
E(X [ CA) F (Y [ CB) ! G((X x Y ) [ C(X x B [ A x Y )):
For most of applications it suffices to require that E(X) = [X; 2Y ] (pointed h*
*omotopy
classes), etc. In this way one can prove certain versions of 2.2 and 2.3 below *
*for more
general spaces. But we do not need such generality and so do not dwell these th*
*ings.
2. DETECTING ELEMENTS
2.1. Definition. Let X and E be as 1.7. An element u 2 E*(X) is called a
detecting element for X if swgt u = catX.
2.2. Theorem. Let X1; : :;:Xn be connected CW -spaces, and let E(1); : :;:E*
*(n)
be arbitrary spectra. Suppose that there are detecting elements ui 2 E(i)*(Xi)*
*; i =
1; : :;:n. If
0 6= u1 . . .un 2 (E(1) ^ . .^.E(n))*(X1 x . .x.Xn)
Xn
then cat(X1 x . .x.Xn) = catXi. Furthermore, in this case u1 . . .un is a
i=1
detecting element for X1 x . .x.Xn.
Proof. By 1.8(i) and 1.14,
X X
cat(X1 x . .x.Xn) swgt(u1 . . .un) swgt ui = cat Xi:
P
Thus, by 1.3, cat(X1x . .x.Xn) = cat Xi. Furthermore, u1 . . .un is a detecti*
*ng
element for X1 x . .x.Xn since swgt(u1 . . .un) = cat(X1 x . .x.Xn).
ON CATEGORY WEIGHT AND ITS APPLICATIONS 11
2.3. Corollary. Suppose that a connected CW -space X possesses a detecting
element. Then cat(X x Sm1 x . .x.Smn ) = cat X + n for any natural numbers
m1; : :;:mn.
Proof. Firstly, we consider the case n = 1, i.e., we prove that cat(X x Sm*
* ) =
catX + 1. Let u 2 Ek(X) be a detecting element for X. We apply 2.2 to Y = Sm
and F = S, the sphere spectrum. Recall that E ^ S ' E. Let v 2 F m(Y ) =
m (Sm ) = Z = em (Sm ) be a generator. By 1.8(i), 1 swgt v catSm = 1, i.e.,
swgt v = 1 = catSm , i.e., v is a detecting element for Sm . So, by 2.2, it su*
*ffices to
prove that u v 6= 0 2 Ek+m (X x Sm ). (This seems to be a well-known general f*
*act,
but I can't give a reference and therefore I give a proof.)
Let " : (Sm )+ ! Sm be a map such that "|Sm = 1Sm . Recall that the doma*
*in
(Sm )+ of " is a pointed space. We choose a base point in the range Sm of " so*
* that "
turns into a pointed map. Then the morphism
b = 1 " : 1 ((Sm )+ ) ! 1 Sm = m S
of spectra yields the element v. So, u v is represented by a morphism
1 ((X x Sm )+ ) = 1 (X+ ^ (Sm )+ ) ' 1 (X+ ) ^ 1 ((Sm )+ )
-a^b-!kE ^ m S = k+m E;
where a : 1 X+ ! kE represents u. Now, let e : Sm ! (Sm )+ be the inclusion.
We assume that the domain Sm is a pointed space. Of course, e is not a pointed*
* map,
but the map
S2(1 ^ e) : S2(X+ ^ Sm ) -! S2(X+ ^ (Sm )+ )
is homotopic to a pointed map, and this pointed map is unique up to pointed ho-
motopy. We choose any such a pointed map and denote it by f. Then we obtain a
morphism
j := -2 1 f : 1 (X+ ^ Sm ) ! 1 (X+ ^ (Sm )+ ):
Consider the following homotopy commutative diagram where j0 and j00are induced
by j:
1 ((X xfSmi)+f)i ' 1 (X+ ) ^ 1f((Smi)+f)i a^b--!kE ^ mfSi=fk+miEfifi
fifififi fifififi fifififi fifififi
1 (X+ ^ (Smx)+ ) ' 1 (X+ ) ^ 1x((Sm )+ ) a^b--!kE ^ mfSi=fk+miEfifi
j0?? ??j00 fifififi fifififi
1 (X+ ^ Sm ) = 1 (X+ ) ^ 1 (Sm ) a^1--!kE ^ m S = k+m E:
Since the bottom line gives us the element u 2 Ek(X); u 6= 0, we conclude that *
*the
top line is an essential morphism, and so u v 6= 0. Thus, cat(X x Sm ) = catX *
*+ 1.
Moreover, by 2.2, u v is a detecting element for X x Sm .
Let vi 2 mi (Smi ) = Z be a generator. We prove by induction the following:
cat(X x Sm1 x . .x.Smn ) = catX + n, and u v1 . . .vn is a detecting element
for X x Sm1 x . .x.Smn . This is done for n = 1. Suppose that our claim is vali*
*d for
some n. Then, similarly to above, we conclude that
cat(X x Sm1 x . .x.Smn x Smn+1 ) = cat(X x Sm1 x . .x.Smn ) + 1 = catX + n + 1:
12 YULI B. RUDYAK
Furthermore, uv1. .v.nvn+1 = (uv1. .v.n)vn+1 is a detecting element
by 2.2. The induction is confirmed.
The results above shows that it is useful and important to know whether a s*
*pace
possesses detecting elements. Consider the Puppe sequence
Pm (X) -pm-!X -jm-!Cm (X) := C(pm )
where pm : Pm (X) ! X is the fibration (1.4) and C(pm ) is the cone of pm .
2.4. Theorem. Let X be a 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 pos*
*sesses a
detecting element.
Proof. By 1.5(v), there is a pointed homotopy equivalence h : Ck(X) ! C whe*
*re
C is a pointed CW -complex. Let e : X+ ! C be a map such that e|X = hjk and e
maps the added point to the base point of C. The morphism
1 e
1 X+ - --! 1 C
of spectra gives us an element uk 2 (1 C)*(X), and uk 6= 0 since jk is stable e*
*ssential.
Clearly, p*kuk = 0, and hence, by 1.8(v), swgt uk k. So, by 1.8(i), swgt uk = *
*k since
k = catX.
We give another interpretation of the above. We set
r(X) := sup {swgt u|u 2 E*(X); u = 0} (2.*
*5)
(E;u)
2.6. Proposition. Let X be an arbitrary CW -space.
(i) r(X) catX, and r(X) = catX iff X possesses a detecting element.
(ii) r(X x Sn ) r(X) + 1 for every n.
(iii) Let u1o. .o.un 2 (E(1) ^ . .^.E(n))*(X) be as in 1:11. Suppose that *
*X is
connected and that ui 2 eE(i)*(X) E(i)*(X). Then r(X) n if u1o. .o.un 6= 0. In
particular, r(X) clE(X) for every ring spectrum E.
(iv) If X is connected then
r(X) = sup{m|p*mu = 0 for some u 2 E*(X); u 6= 0}
= sup{m|jm is stably essential}:
Proof. (i) This is obvious.
(ii) The case r(X) = 1 is trivial. So, we assume that r(X) = r < 1. Then th*
*ere
is u 2 E*(X) with swgt u = r. The proof of 2.3 shows that u v 6= 0 where v 2
n(Sn ) = Z is a generator. Now, by 1.8(i) and 1.14, r(X x Sn ) swgt(u v) r +*
* 1.
(iii) This follows from 1.8(i) and 1.12(i).
(iv) The first equality follows from 1.8(v). Furthermore, as in the proof *
*of 2.4,
each map jm yields an element um 2 (1 C)0(X) where C is homotopy equivalent to
Cm (X). Then p*mum = 0, and every element u 2 Ek(X) with p*mu = 0 can be induced
from um (via a morphism kE ! 1 C). Hence, the second equality holds.
2.7. Remark-Example. If q > 1 then every (q - 1)-connected CW -space
X with dim X = qk; k = cat X possesses a detecting element. Namely, it is the
ON CATEGORY WEIGHT AND ITS APPLICATIONS 13
characteristic class u 2 Hn (X; ssn-1 ((X)*k)) of the fibration pk. See also [1*
*2] where
James considered the k-th power of the fundamental class 2 Hq(ssq(X)).
2.8. Remark. Similarly to 2.5, one can introduce the invariant
ae(X) = sup{m|jm is essential}
Clearly, r(X) ae(X) catX. Furthermore, 1 + ae(X) wcat X where wcat denotes
the so-called weak category (non-normalized), see e.g. [12] for the definition.
2.9. Remark. There is a natural question whether every space possesses a
detecting element. The answer is: no, but... . Firstly, why no. Clearly, i*
*f X is a
homology disk then eE*(X) = 0 for every E, and so there is no detecting element*
*s in
E*(X). But, we can consider cohomology (ordinary) of X with local coefficients.*
* In
this case we can define, and really find, detecting elements for X. Moreover, t*
*here are
examples of simply connected spaces X without detecting elements. However, we c*
*an
consider extraordinary analog of cohomology with local coefficients, i.e., coho*
*mology
theories over X, and try to find detecting elements there.
3. APPLICATIONS TO MANIFOLDS
Given a PL manifold M, we denote by M the stable normal bundle of M.
3.1. Theorem. 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.
Proof. We denote catM by k. Let Pk(M) -pk! M jk-!Ck(M) be the Puppe
sequence as in 2.4. By 2.4, it suffices to prove that jk : M ! Ck(M) is stable
essential. To do this, it suffices, in turn, to find a homology theory E*(-) su*
*ch that
(pk)* : E*(Pk(M)) ! E*(M) is not an epimorphism. Let ' : BP L|m ! BP L be the
m-connective covering of BP L. We set E*(-) to be the (BP L|m; ')-bordism theor*
*y,
see [23, Ch. 4, Example 17]. Since M |M(m) is trivial, M admits a (BP L|m; ')-
structure (not unique). We fix a (BP L|m; ')-structure on M and regard the iden*
*tity
map M ! M as an element 2 En(M). We prove that =2Im(pk)*.
Suppose not. Then there is a map F : W ! M with the following properties:
1. W is a compact (n + 1)-dimensional (BP L|m; ')-manifold with @W = M t V ;
2. the map F |V : V ! M can be lifted to Pk(M) with respect to pk;
3. F |M = 1M .
Without loss of generality we can assume that W is connected.
Let [-] denote the entire function. We set a = min {m; [(n - 1)=2]} and lea*
*ve it
to the reader to prove that
n - a kq - 1: (3.*
*2)
Suppose for a moment that ssi(W; M) = 0 for i a. Clearly, a n - 3 = n + 1 - 4,
and so (W; M) has a handle presentation without handles of indices a, see [22,*
* 8.3.3,
Theorem A]. By duality, the pair (W; V ) has a handle presentation without hand*
*les
of indices n - a + 1. In other words, W ' V [ e1 [ . .[.es where e1; : :;:es a*
*re cells
attached step by step and such that dim ei n-a for every i = 1; : :;:s. Howeve*
*r, the
14 YULI B. RUDYAK
fibration pk : Pk(M) ! M is (kq - 2) connected. Thus, in view of (3.2), F : W !*
* M
can be lifted to Pn(M). In particular, pn has a section. But this contradicts 1*
*.6.
So, it remains to prove that, for every membrane (W; F ), we can always fin*
*d a
membrane (U; G) with ssi(U; G) = 0 for i a and G|@U = F |@W . Here @U = @W =
M t V and G : U ! M. We start with an arbitrary connected membrane (W; F ).
Since M is a retract of W , there is a commutative diagram
0 ----! ssi(M) ----! ssi(W )----! ssi(W; M) ----! 0
flfl ?
fl ?yF*
ssi(M) ________ssi(M)
where the top line is the homotopy exact sequence of the pair (W; M). Clearly, *
*if F*
is monic then ssi(W; M) = 0 (one-point set for i = 1).
First, consider i = 1. Let ss1(W ) be generated by elements a1; : :;:ak. We*
* regard
ss1(M) as the subgroup of ss1(W ) and set gi := F*(ai)a-1i2 ss1(W ). Then Ker F*
** is
the smallest normal subgroup of ss1(W ) containing g1; : :;:gk. (This is valid *
*because
Ker F* = {F*(x)x-1 |x 2 ss1(W )}.) We realize g1; : :;:gk by PL embeddings ji :*
* S1 !
intM; i = 1; : :;:k. Then the normal bundle of each embedding ji is trivial si*
*nce
W |W (1)is trivial. Hence,_we_are_able to perform the surgeries of (W; F ) with*
* respect
to these_embeddings._Let (W ; F) be the result of this sequence of surgeries. *
*Clearly,
ss1(W ; F) is the one-point set. Moreover, one can perform the above surgeries*
* so that
__W|W (m) is trivial, see [28].
Now, we proceed by induction and kill the kernel of F* : ssi(W ) ! ssi(M); *
*i a.
This yields the desired membrane (U; G).
3.3. Corollary. (i) Let M be a closed (q - 1)-connected stably paralleliza*
*ble
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)-con*
*nected
PL manifold such that q catM = n. Then M possesses a detecting element. Moreove*
*r,
there exists a detecting element u 2 Hn (M; ssn(Cn(M))).
Proof. (i) Put m = n + 1 in 3.1.
(ii) Put m = 1 in 3.1. Then, by 3.1 and 2.4, we have a detecting element in
(1 C)0(M), where C is a CW -space which is homotopy equivalent to Cn(M). It is
easy to see that Cn(M) is simply connected, and hence, by 1.5(iv) and the Hurew*
*icz
Theorem, Cn(M) is (n - 1)-connected. Thus, (1 C)0(M) = Hn (M; ssn(Cn(M))).
Note that the case q > 1 is already considered in 2.7.
3.4. Corollary. Let M be as in 3:1. Then cat(M xSm1 x. .x.Smn ) = catM +n
for any natural numbers m1; : :;:mn.
Proof. This is a direct consequence of 2.3.
3.5. Lemma. 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 l*
*et
f : N ! M be a map of degree 1. If N is R-orientable then f* : E*(M) ! E*(N)
is a monomorphism.
ON CATEGORY WEIGHT AND ITS APPLICATIONS 15
Proof. It is easy to see that f*[N] is an R-orientation of [M] if [N] is a*
*n R-
orientation of N. Hence, the transfer f!: E*(N) ! E*(M) is defined, see e.g [4]*
*, and
we have f!f*(x) = (deg f)x for every x 2 E*(X). Thus, f* is monic.
3.6. Theorem. (i) Let Mn be as in 3.3(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 manifold*
*s.
If catM = dim M then catf = catM = catN.
Proof. (i) By 1.2(i), it suffices to prove that catf catM. By 3.3(i), M h*
*as a
detecting element, say, u 2 E*(M) with some E. We apply 3.5 to the case R = S,
the sphere spectrum. It is possible because every spectrum is a module over S. *
*Recall
that every stably parallelizable manifold is R-orientable for every ring spectr*
*um R,
see e.g. [26, 14.40]. In particular, N is S-orientable, and so, by 3.5, f*u 6= *
*0. Now,
by 1.8(i)
catf swgt u = catM:
(ii) This is similar to (i). We just use the existence of the detecting el*
*ement
u 2 Hn (M; ssn(Cn(M))) and the monomorphicity of
f* : Hn (M; ssn(Cn(M))) ! Hn (N; ssn(Cn(M))):
3.7. Theorem ([21, Corollary 6.7]). (i) Let m > 0, and let M be a closed
connected PL manifold such that
dim M 2 catM - m - 2:
Then cat(M x Sm ) = catM + 1.
(ii) In particular, if M is a closed connected PL manifold such that
dim M 2 catM - 3
then cat(M x S1) = catM + 1.
3.8. Theorem. Let M be a closed connected PL manifold such that
dim M 2 catM - 3
Then cat(M x Sm1 x . .x.Smn ) = catM + n for any natural numbers m1; : :;:mn.
Proof. Let T rdenote the r-dimensional torus. We recall that catT r= r, and*
* so,
by 1.3, cat(X x T r) r for every connected CW -space X.
Firstly, we prove that cat(M x T r) = catM + r. We prove this by induction.*
* For
r = 1 this is 3.7(ii). Now, suppose that cat(M x T r) = catM + r. Then
dim (M x T r) = dim M + r dim M + 2r 2(catM + r) - 3 = 2 cat(M x T r) - 3:
So, by 3.7(ii),
cat(M x T r+1) = cat(M x T rx S1) = cat(M x T r) + 1 = catM + r + 1:
Now we prove the theorem by induction. Firstly, we prove that cat(M x Sm ) =
catM + 1 for every m > 0. Choose r such that
dim(M x T r) 2(catM + r) - m - 2 = 2 cat(M x T r) - m - 2;
16 YULI B. RUDYAK
(for example, r >> m + dim M). Then, by 3.7(i),
cat(M x T rx Sm ) = cat(M x T r) + 1 = catM + r + 1:
Now, if cat(M x Sm ) 6= catM + 1 then, by 1.3, cat(M x Sm ) catM. But then
cat(M x Sm x T r) cat(M x Sm ) + catT r catM + r:
This is a contradiction.
Now, fix a natural n and suppose by induction that
cat(M x Sm1 x . .x.Smk ) = catM + k
whenever dim M 2 catM - 3 and k n. Given natural numbers m1; : :;:mn+1 ,
choose l such that
dim (M x Sm1 x . .x.Smn x T l) 2 cat(M x Sm1 x . .x.Smn x T l) - 3:
Then cat(M x Sm1 x . .x.Smn x T lx Smn+1 ) = cat(M x Sm1 x . .x.Smn x T l) + 1.
But
cat(M x Sm1 x . .x.Smn x T l) = cat(M x T lx Sm1 x . .x.Smn )
= cat(M x T l) + n = catM + l + n:
The second equality holds since dim(M x T l) 2 cat(M x T l) - 3, the third one*
* holds
since dim M 2 catM - 3. Thus,
cat(M x Sm1 x . .x.Smn x T lx Smn+1 ) = catM + n + l + 1:
Now, asserting just as in the case n = 1, we conclude that
cat(M x Sm1 x . .x.Smn x Smn+1 ) = catM + n + 1:
The induction is confirmed.
4. CATEGORY WEIGHT AND MASSEY PRODUCTS
Throughout the section we fix a commutative ring R. (We restrict the notion*
* of
ring to rings which are associative and unital, i.e., possess a multiplicative *
*identity
element.) Given a space X, let C*(X) denote the singular cochain complex C*(X; *
*R)
unless some other is said explicitly. Similarly, we write H*(X) for H*(X; R), *
*the
singular homology with coefficients in R. We assume that C*(X) is equipped with
the usual associative cup product pairing. This cup product turns H*(X) into a
commutative ring. Given an element a 2 C*(X), we denote by [a] the cohomology
class of a. So, [a] 2 H*(X).
For simplicity, the zero matrix (of any size) over C*(X) or H*(X) will be d*
*enoted
just by 0. Furthermore, we assume that every matrix over C*(X) or H*(X) has
homogeneous entries (but different entries are allowed to be of different dimen*
*sions).
We recall the definition of matric Massey products. Given a 2 C*(X),_we set
a = (-1)|a|a. Given a matrix M = (mij) with entries in C*(X), we set M = (__mi*
*j)
and ffiM = (ffimij) where ffi : C*(X) ! C*(X) is a coboundary operator. Finally*
*, we
set [M] := ([mij]).
Let U = (uij) be an m x n-matrix and V = (vij) be an n x p matrix, both over
H*(X). We say that U and V are compatible if deg uikvkj does not depend on k for
ON CATEGORY WEIGHT AND ITS APPLICATIONS 17
each i and j. A sequence {V1; : :;:Vn} of matrices over H*(X) is called compati*
*ble if
each pair (Vi; Vi+1) is compatible.
4.1. Definition (cf. [16]). Let V1; : :;:Vn; n 2, be a compatible seque*
*nce
of matrices over H*(X). A defining system for V1; : :;:Vn is a family A = {Aij*
*} of
matrices over C*(X); 1 i < j n + 1, with the following properties:
1. [Ai;i+1] = Vi for every i;
j-1X
2. ffiAij= AirArj if i + 1 < j < n + 1.
r=i+1 n
X
Consider the matrix c(A) := A1r Ar;n+1. One can prove that its entries are c*
*ocy-
r=2
cles, and so we have the matrix [c(A)] over H*(X). We define
fi
:= {[c(A)] fiA runs over all defining systems} H*(X):
The family is called the matric Massey n-tuple product of V1; : :*
*;:Vn.
The indeterminacy of the Massey product is the subgroup
fi
Indet := {x - y fix; y 2 }:
of H*(X). So, is contained in a coset with respect to Indet.
Clearly, every matrix of has size m x p, where V1 has size mx*
*? and
Vn has size ? x p.
S
4.2. Proposition ([16, Prop. 2.3]). Indet
where runs over all matric Massey (n - 1)-tuple products.
4.3. Examples._ (a) For every compatible matrices U; V over H*(X) we have
__ = U V .
(b) If V1 is a row-matrix and Vn is a column-matrix then is a*
* subset
in H*(X), i.e., every element of is just an element of H*(X). Mor*
*eover,
there is the important special case when every Vi is 1 x 1-matrix, i.e., it is *
*just an
element of H*(X). In this case we write ui rather than Vi and ra*
*ther
than .
(c) Let p be an odd prime, let fi : Hi(-; Z=p) ! Hi+1(-; Z=p) be the Bokste*
*in
homomorphism, and let P n : Hi(-; Z=p) ! Hi+2n(p-1)(-; Z=p) be the Steenrod
power operation. Then for every u 2 H2n+1 (X; Z=p) we have fiP n(u) 2 ____
(p times), Kraines [13].
4.4. Lemma. Given a CW -space X, let ff 2 H*(X) have category weight k.
Then, for every open subset A of X with catX A < k, there exists a 2 C*(X) such
that [a] = ff and a|A = 0, i.e., a 2 C*(X; A) C*(X).
* i*
Proof. In the exact sequence H*(X; A) -j! H*(X) -! H*(A) of the pair (X; A)
we have i*ff = 0, and so ff = j*fi for some fi 2 H*(X; A). Now, fi = [a] for s*
*ome
a 2 C*(X; A) C*(X).
Given a matrix M = (mij) over H*(X), we set
cwgtM := min{cwgt mij}:
18 YULI B. RUDYAK
4.5. Theorem. Given a CW -space X, let V1; : :;:Vn be matrices over H*(X).
Suppose that the matric Massey product is defined. If 0 =2
then
catX mini{cwgt V2i} + mini{cwgt V2i+1}:
Proof. The case n = 2 follows from 1.8(i) and 1.12(ii), so we assume that n*
* > 2.
Let k = min{cwgt V2i}; l = min{cwgt V2i+1}. Suppose that catX < k +l. Then there
is an open covering {A1; : :;:Ak+l} of X such that every Ai is contractible in *
*X. Set
[k k+l[
A := Ai; B := Ai
i=1 i=k+1
and note that A [ B = X while catX A < k and catX B < l.
Given i, consider a family {Ai;i+1} of matrices over C*(X) such that [Ai;i+*
*1] = Vi.
Since cwgt V2i k, there exists, by 4.4, a matrix A2i;2i+1such that A2i;2i+1|A =*
* 0 for
every i. Similarly, there exists a matrix A2i-1;2isuch that A2i-1;2i|B = 0 for *
*every i.
Routine arguments shows that there is a commutative diagram
C*(X; A) C*(X; B) ----! C*(X; A [ B) ________0
?? ?
y ?y
C*(X) C*(X) ----! C*(X)
where the bottom arrow is the cup product pairing. Therefore, because of we sa*
*id
above, A2i-1;2iA2i;2i+1= 0 = A2i;2i+1A2i+1;2i+2for every i. So, we get a defin*
*ing
system {Aij} for the Massey product if we put Aij= 0 whenever j >*
* i+1.
Now, since n > 2, X
A1jAj;n+1= 0:
and thus 0 2 . This is a contradiction.
4.6. Lemma. Let X be a connected CW -space, and let V1; : :;:Vn be an arbit*
*rary
compatible sequence of matrices over He*(X). If catX 1 then the matric Massey
product is defined, has zero indeterminacy, and = {*
*0}.
Proof. Firstly, we prove that is defined and 0 2 . There
are A; B X such that X = A [ B and both A; B are open and contractible in
X. Since X is connected and each Vi is a matrix over He*(X), we conclude that
Vi|A = 0 = Vi|B for every i. Now, following the proof of 4.5, we choose represe*
*ntatives
Ai;i+1over C*(X) for Vi such that A2i;2i+1|A = 0 = A2i-1;2i|B for every i. Now,*
* again
as in 4.5, we put Aij= 0 whenever j > i + 1. Then A = {Aij} is a defining system
for the Massey product . Furthermore, 0 2 since c(A)*
* = 0.
Now we prove that has zero indeterminacy, i.e., *
* = {0}.
Clearly, this holds for n = 2. Suppose by induction that = {0} f*
*or
every k < n and every compatible U1; : :;:Uk over He*(X). Then h*
*as
zero indeterminacy since, by 4.2, the last one is covered by the Massey products
. The induction is confirmed.
ON CATEGORY WEIGHT AND ITS APPLICATIONS 19
4.7. Corollary. Given a connected CW -space X, let V1; : :;:Vn be matrices
over He*(X). Suppose that the matric Massey product is defined. *
* Let
f : Y ! X be an arbitrary map of CW -spaces with Y connected and catY 1. Then
f*V = 0 for every V 2 .
Proof. f*V 2 f* = {0}, the last equality fo*
*llows
from 4.6.
4.8. Theorem. Let X be a connected CW -space, and let V1; : :;:Vn be matri-
ces over He*(X). Suppose that the matric Massey product is defin*
*ed.
Then swgt V 2 for every V 2 . In particular, if there exists V*
* 2
; V 6= 0, then catX 2.
Proof. Let p2 : P2(X) ! X be the fibration (1.4). By 1.5(v), there is a hom*
*otopy
equivalence h : W ! P2(X) where W is a CW -space. By 1.5(i),
catW = catP2(X) 1:
Consider any V 2 and take any entry u of V . Then, by 4.7, h**
*p*2u =
0, and hence p*2u = 0. Thus, by 1.8(v), swgt u 2.
Fadell and Husseini [5] related category weight with Steenrod operations. *
*Here
we interpret their results in terms of Massey products.
Fix an odd prime p. Let fi : Hi(-; Z=p) ! Hi+1(-; Z=p) be the Bokstein ho-
momorphism, and let P n: Hi(-; Z=p) ! Hi+2n(p-1)(-; Z=p) be the Steenrod power
operation. Recall that a sequence I = {a1; : :;:an} of non-negative integers is*
* called
admissible if ai = 2ni(p - 1) + "i where "i = 0 or 1 and ai pai+1. Set
aefiP ni if " = 1;
Stai = i
P ni if "i = 0;
P
and StI = Sta1. .S.tan. Finally, set |I| = ai and e(I) = 2n1p + 2"1 - |I|.
4.9. Corollary. Let p be an odd prime, and let X be a connected CW -space.
(i) For every u 2 H1(X; Z=p) we have cwgt(fiu) = swgt(fiu) = 2 provided fiu*
* 6= 0.
(ii) If u 2 H2n+1 (X; Z=p) then swgt(fiP nu) 2.
(iii) If dim u = e(I) for some u 2 H*(X; Z=p) then swgt StIu 2.
Proof. Clearly, (i) follows from (ii) and 1.8(ix). We prove (ii). As we rem*
*arked in
4.3(c), Kraines [13] proved that fiP nu 2 ____ (p times). Now the result*
* follows
from 4.8.
We prove (iii). In fact, this is a formal consequence of (ii). Firstly, s*
*uppose
StI = fiP n1StJ. Now, if dim u = e(I) then dim StJu = 2n1 + 1, and so, by (ii),
swgt StIu = swgt fiP n1(StJu) 2. Furthermore, suppose StI = P n1StJu. Now, if
dim u = e(I) then dim StJu = 2n1, and so StIu = (StJu)p. Thus, swgt(StIu) p.
Since swgt cwgt, 4.9 implies Theorems 3.6 and 3.12 from [5].
4.10. Proposition. Let X be a nilpotent space, and let X be the (rational )
Sullivan model for X. Let a; b 2 X be two closed elements (i.e., da = 0 = db), *
*and
let a2 = 0. Then for every x 2 X with dx = ab we have swgt [ax] 2.
Notice that x is not a cocycle, and so we cannot apply, say, 1.12(ii).
20 YULI B. RUDYAK
Proof. Clearly, [ax] 2 <[a]; [a]; [b]>.
4.11. Example. Let be a 2m-dimensional vector bundle over S2m with O() 6=
0; here O denotes the Euler class. Let E be the total space of the unit sphere *
*bundle
of the vector bundle x over S2m x S2m . We prove that catE = 3.
Firstly, E is (2m - 1)-connected, and dim E = 8m - 1. So,
cat E 8m_-_1_2m< 4;
see e.g. [12]. Hence, catE 3.
The Sullivan model for E is a free graded commutative differential algebra *
*gener-
ated by {a; b; x; y; z}, where
dima = 2m = dim b; dim x = dim y = dim z = 4m - 1;
da = 0 = db; dx = a2; dy = b2; dz = ab:
Here {a; b; x; y} is the Sullivan model for the base S2m x S2m , and z gener*
*ates
H4m-1 (S4m-1 ; Q); the equality dz = ab holds since O( x ) 6= 0. Now, [az - xb]*
* 2
<[a]; [a]; [b]>, and so cwgt u swgt[az - xb] 2. Furthermore, for dimensional *
*reasons,
[az -xb] 6= 0. Since E is a simply connected manifold, there is, by Poincare du*
*ality, an
element u 2 H2m (E; Q) such that [az - xb]u 6= 0. Thus, catE cwgt [az - xb]u *
*3.
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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
__