Essential Category Weight1
Jeffrey A. Strom
University of Wisconsin, Madison
strom@math.wisc.edu
Introduction
Everybody's favorite theorem on LusternikSchnirelmann category is Eilenberg's *
*cup product
theorem: if there is a nonzero Nfold cup product in eh*(X), then cat(X) > N. U*
*nfortunately,
this theorem often fails to detect the full category of a space X; this is the *
*case, for example,
when X is a lens space Lp (with p odd) or X = Sp(n).
In [4], Fadell and Husseini introduced the concept of the category weight o*
*f a cohomology
class, written cwgt(u). The key propertiesPof category weight were that 1 cwg*
*t(u) <
cat(X), and that cwgt(u1. .u.n) cwgt (ui). Thus, Eilenberg's theorem follow*
*s as the
trivial case of the product formula, and much better lower bounds on cat(X) are*
* made
possible. One of the main theorems of [4] shows the effect of cohomology opera*
*tions on
category weight. As a consequence, they show that there are indecomposable clas*
*ses with
cwgt (u) > 1.
In this paper, we generalize category weight to arbitrary maps f : X ! Y ,*
* and expand
the results of [4] and [11]. We also observe in the first two sections that cwg*
*t(f) (or cwgt(u))
is not a satisfactory concept in one important respect: it can be altered dram*
*atically by
composition with a homotopy equivalence. This makes cwgt(f) very difficult to c*
*ompute..
We introduce a new concept, the essential category weight of a map f : X !*
* Y (E(f))
which also satisfies 1 E(f) < cat(X) and a product formula, and which is prese*
*rved by
homotopy equivalences.
This paper is the beginning of a theory of (essential) category weight. Th*
*is theory
unifies and strengthens previous results on LusternikSchnirelmann category and*
* leads to
new theorems. The main results of the general theory are Theorem 3.7 and Theore*
*m 5.1.
Theorem 3.7 is extremely important when considering the effect of cohomology op*
*erations.
Theorem 3.7 If f : X ! Y and g : Y ! Z, then
cwgt(g O f) E(g) . cwgt(f);
and
E(g O f) E(g) . E(f):
There is a corresponding statement for cohomology classes, in which f represent*
*s the coho
mology class u, and g represents the cohomology operation :
Theorem 4.2 If and u are as above, then
cwgt ((u)) E() . cwgt(u)
_____________________________________
1Primary: 55M30, 55P50; Secondary: 55P42
1
and
E((u)) E() . E(u):
In light of this result, we see that the conclusion of Theorem 3.12 of [4] *
*can be changed
from cwgt((u)) 2 to cwgt((u)) 2 . cwgt(u), clearly a substantial improvement.
Theorem 5.1 Let f : X ! K and g : X ! L. Let ^ : K x L ! K ^ L be the quoti*
*ent
map. Then
E(^ O (f x g)) E(f) + E(g);
and
cwgt(^ O (f x g) O d) cwgt(f) + cwgt(g):
Many of the most useful lower bounds on cat(X) are of the product formula t*
*ype. The
basic examples of this kind of result are Eilenberg's cup product theorem, Whit*
*ehead's
theorem that cat(X) is bounded from below by the nilpotence class of [X; G] (Th*
*eorem 2.10
of [16]), and Steenrod's theorem that cat(X) > N if is a normal cohomology ope*
*ration of
N variables and (u1; : :;:uN ) 6= 0 (see [12]).
Theorem 5.1 will be seen to be the common ancestor of all of the product fo*
*rmula type
lower bounds on cat(X). In fact, each of the theorems mentioned above may be de*
*rived from
a corollary of Theorem 5.1 by making the trivial observation that if f 6' *, th*
*en cwgt(f) 1.
If it is known that cwgt(fi) > 1 in any of these situations, then substantially*
* better lower
bounds follow immediately. It is here that Theorems 3.7 and 4.2 are particularl*
*y useful.
Having established the usefulness of essential category weight for unifying*
* and extending
the existing theory, we also show how to obtain new results, by applying essent*
*ial category
weight to Ganea's conjecture. An easy application of Theorem 5.1 yields the fol*
*lowing useful
result.
Proposition 6.1 Let X be a space with cat(X) = N. If there is a map f : X ! Y*
* with
E(f) = N  1, and Snf 6' *, then
cat(X x Sk) = cat(X) + 1
for every 0 < k n.
This leads directly to the following partial solution to Ganea's conjecture, wh*
*ich first ap
peared as Theorem 3.2 of [14].
Theorem 6.3 Suppose X is ndimensional, (p  1)connected and cat(X) = N. If
$ %
n
N = __ + 1
p
and n 6 1 mod p, then
cat(X x Sk) = cat(X) + 1
2
for all k > 0.
The proof amounts to showing that under the conditions imposed, bdN1: X ! X(N*
*1) is
stably nontrivial; it is easy to see that E(dbN1) N if bdN6' *.
I would like to thank Sufian Husseini, Edward Fadell and Doug Lepro, who li*
*stened
to many of these arguments as they were forming. Thanks also to Professor Huss*
*eini for
suggesting that I study category weight.
3
1 Background
In this section we establish some notation and recall some results which will f*
*igure promi
nently in this paper.
1.1 Notation
Spaces are pointed, and have the pointed homotopy type of CW complexes. The se*
*t of
pointed homotopy classes of maps f : X ! Y is denoted [X; Y ]. Basepoints ar*
*e always
denoted by *. The constant map is also denoted *.
The (reduced) mapping cylinder and mapping cone of the map f are denoted Mf*
*, and
Cf respectively. The (reduced) suspension of X is SX, the suspension of f is Sf.
The suspension spectrum of X is denoted (X), the image of a map under the s*
*uspension
spectrum functor is denoted (f).
The diagonal map is denoted dN : X ! XN (or simply d). The map ^ : X x Y *
*! X ^ Y
is the canonical quotient map. The composition ^ O f is denoted bf. The project*
*ions from a
product are denoted pX : X x Y ! X and pY : X x Y ! Y .
The notation H*(X) means ordinary cohomology with arbitrary coefficients. T*
*he dimen
sion of u as a cohomology class is denoted u (that is, u = degree(u)).
The greatest integer less than or equal to x is denoted bxc; the least inte*
*ger greater than
or equal to x is denoted dxe.
1.2 LusternikSchnirelmann Category
In this section, we recall some basic facts concerning LusternikSchnirelmann c*
*ategory. See
[10] for a more complete survey.
Definition Let X be a CW complex, and let f : X ! Y be a map. The category *
*of f,
denoted cat(f), is the least integer N such that
X = X1 [ . .[.XN
where each Xi is a subcomplex of X, and fXi ' *.
Clearly, if f ' g, then cat(f) = cat(f0); if g : Y ! Z, then cat(g O f) c*
*at(f). If g is a
homotopy equivalence, then cat(g O f) = cat(f), for
cat(f) = cat(g1 O (g O f)) cat(g O f):
Definition Let i : A ,! X be the inclusion of a subcomplex. The category of A *
*in X is
catX (A) = cat(i). We write cat(X) = catX(X).
It is equivalent to say catX(A) is the least integer N such that
A = A1 [ . .[.AN ;
where each Ai is a subcomplex of X, and Ai is contractible to a point in X.
4
Write
T NX = {(x1; : :;:xN ) 2 XN  at least onexi= *}:
The following result is due to Berstein and Ganea (see Proposition 1.8 of [2]).
Proposition 1.1 Let f : X ! Y . Then cat(f) N if and only if there is a lif*
*t (up to
homotopy) in the diagram
T NY



f d ?
X ______Y ______YN
The following upper bound on cat(f) follows easily from the cellular approx*
*imation the
orem (see Theorem 6.8 of [15]).
Proposition 1.2jkIf f : X ! Y where Y is (p  1)connected, and X is ndimen*
*sional,
then cat(f) n_p+ 1. In particular, if A X, then
$ %
dim (A)
catX(A) _______ + 1:
p
There is a related concept, which plays an important role in this paper.
Definition Let X be a CW complex. The geometric category of X is the least in*
*teger
N such that X = X1 [ . .[.XN where each Xi is a contractible subcomplex. We w*
*rite
gcat(X) = N.
The usefulness of gcat comes from the following result, due to Berstein and*
* Ganea (see
Propostion 1.7 of [2]).
Theorem 1.3 Let f : X ! Y . Then cat(f) N if and only if f factors as in the*
* diagram
f
X __________Y
@
@R f0
X0
in which gcat(X0) N.
5
1.3 Spectra
Many of our applications use cohomology. We will state our results for cohomolo*
*gy classes
in the greatest generality possible, and state results for ordinary cohomology *
*as corollaries
or examples. Therefore, we must give a brief account of generalized cohomology *
*theories and
spectra. See [1] or [15] for definitions and details.
For any spectrum F , there is a (reduced) cohomology theory defined on spec*
*tra by
F *E = [E; F ]. If X is a CW complex, we define
Fe*(X) = F *((X)) = [(X); F ]
where (X) is the suspension spectrum of X. The Brown Representation Theorem sta*
*tes
that every cohomology theory arises in this way (see Theorem 9.12 of [15]).
Theorem 1.4 Let eh*be a reduced cohomology theory. Then there is a loop spectr*
*um L
and a class 2 h*(L) so that the natural transformation
eL*(E) = [E; L] ffl!eh*(E)
given by f 7! f* is an isomorphism for every spectrum E.
Theorem 1.5 Any spectrum is equivalent to a loop spectrum. If E = (X) and F is*
* a
loop spectrum, then
Fen(X) = [(X); F ]n ~=[X; Fn]:
The inclusion in : (Ln) ,! L gives us classes n = i*n(), and the map
[X; Ln]ffln!Ln(X)
given by f 7! f*(n) is an isomorphism.
This observation can be used to reduce questions about spectra to questions*
* about spaces.
Definition The category of a map u : (X) ! F is the least N such that X = X1[*
*. .[.Xn,
where each Xi is a subcomplex of X, and u(Xi) = 0.
Lemma 1.6 Let u : (X) ! F , and let L be a loop spectrum equivalent to F . Su*
*ppose
u = ffl(f). Then cat(u) = cat(f).
Corollary 1.7 cat(u) N if and only if u factors as in the diagram
(X) _______________Fu
Q Q jj3
Qs j u0
(X0)
where gcat(X0) N.
6
2 Category Weight
In the first part of this section, we give the definition of the category weigh*
*t of a map
f : X ! Y , and prove some basic properties. In the second part, we give a pro*
*cedure for
finding maps f : X ! Y with large category weight.
2.1 Definition and Basic Properties
In this section, we give the definitions of the category weight of a map of spa*
*ces or spectra.
We also give some examples and easy estimates. The category weight of a cohomol*
*ogy class
first appeared in [4], and has been further studied in [11]. The category weigh*
*t of a map is
a simple generalization of their idea.
Definition Let X be a CW complex, and let f : X ! Y with f 6' *. The category*
* weight
of f (cwgt(f)) is the largest integer N such that fA ' * whenever catX(A) N.
Notice that we do not require that Y be a CW complex.
To talk about the category weight of a map, the map must first be nontrivia*
*l. Whenever
the symbol cwgt(f) appears, we are tacitly assuming that f 6' *.
Clearly, it is always true that
1 cwgt(f) < cat(X):
It is easy to see that if f ' g, then cwgt(f) = cwgt(g).
We can also define the category weight of a cohomology class. Since any co*
*homology
theory is represented by a spectrum F , it is equivalent to define the category*
* weight of a
map u : (X) ! F , where F is an arbitrary spectrum. (The difficulty with defin*
*ing cwgt(u)
for a map u : E ! F of arbitrary spectra is that there is no obvious way to de*
*fine the
catE (A) where A is a subspectrum of E.)
Definition Let X be a CW complex and let u : (X) ! F be a (nonzero) map where*
* F is
an arbitrary spectrum. The category weight of u is the largest integer N such t*
*hat u(A) = 0
whenever catX(A) N.
Just as above, when we write cwgt(u), we are assuming u 6= 0. Also,
0 cwgt(u) cat(X)  1;
and cwgt(u) = 0 if and only if u 62 eh*(X). For this reason, we will usually co*
*nsider classes
u 2 eh*(X).
Category weight gives us a new method of computing lower bounds on the cate*
*gory
of a CW complex X. If f : X ! Y and cwgt(f) = N, then cat(X) > N; similarly *
*for
cohomology classes. Of course, this begs the question of how to find such maps *
*and classes.
It is possible to define category weight of elements of F (X), where F : CW*
* ! Set* is
any contravariant homotopy functor from the category of CW complexes to the cat*
*egory
of pointed sets, as follows. If ff 2 F (X), then cwgt(ff) is the largest integ*
*er N such that
F (i)(u) = * whenever i : A ,! X and catX(A) N. It is clear that when F is a c*
*ohomology
theory or the functor F = [; Y ], this definition reduces to the ones given ab*
*ove.
7
Many of our results could be proved for very general functors of this type,*
* but in all of
our examples, the functor F can be expressed F (X) = [X; Y ] for an appropriate*
* space Y .
Therefore, we will state our results in terms of maps.
Example 2.1 Let h* be a cohomology theory with products. For any ui2 eh*(X),
u1. .u.NA = 0
for any A X with catX(A) N, by Eilenberg's theorem. Therefore, cwgt(u1. .u.N)*
* N.
It is possible to express the category weight of a map (X) ! E in terms of*
* category
weight of maps of spaces.
Proposition 2.2 Let F be a spectrum, and suppose L is a loop spectrum equivale*
*nt to
F . Let n 2 eF(nLn) be as in Theorem 1.5. If u = f*n, then cwgt(f) = cwgt(u).
Proof
First suppose cwgt(f) = N, and let iA : A ,! X, with catX(A) N. The diagram
[X; Ln]______eF(nX)ffl~=
 
 
#  *
iA iA
 
 
? ?
[A; Ln]______eF(nA)ffl~=
commutes, so
i*A(u) = i*A(ffl(f)) = ffl(i#A(f)) = ffl(f O iA) = ffl(*) = *
*0:
Thus, uA = 0, and so cwgt(u) cwgt(f).
Now suppose that cwgt(u) = N, and let A X with catX(A) N. Then
0 = i*A(u) = ffl(i#A(f)) = ffl(f O iA):
Since ffl is an isomorphism, it's injective, and so f O iA ' *. *
* 
It is natural to ask about the category weight of a composition; indeed, th*
*is is the
motivation for most of the results in this paper. The following is our first r*
*esult in this
direction.
Proposition 2.3 Let f : X ! Y , and g : Y ! Z, and i : A ,! X. Then
1. cwgt(g O f) cwgt(f);
2. cwgt(f O i) cwgt(f).
The analagous results hold for maps of spectra. If u : (X) ! F and v : F ! E,*
* then
8
10. cwgt(v O u) cwgt(u);
20. cwgt(u O (i)) cwgt(u).
Proof
The proofs are easy. *
* 
We conclude this section with some useful estimates for cwgt(f) and cwgt(u).
In Proposition 2.4, we require that X be a CW complex. It is not enough tha*
*t X simply
be homotopy equivalent to a CW complex, as we will see in Example 2.10 below.
Proposition 2.4 (cf. [11], Proposition 1.11) Let X be a CW complex. Suppose ss*
*r(Y ) = 0
for r > m and that X is (p  1)connected, and let f : X ! Y . Then
$ %
m
cwgt (f) __ :
p
If u 2 F *(X), then $ %
u + m
cwgt(u) _______ :
p
Proof *
* j k
To prove the first assertion, it suffices to find a subcomplex A X with catX(A*
*) m_p+ 1
so that uA 6= 0.
Since ssr(Y ) = 0 for r > m, the inclusion of the m skeleton i : X(m) ,! X *
*induces an
injection
i* : [X; Y ] ! [X(m); Y ]:
Since f 6' *, fX(m) 6' *. This proves the first assertion, since
i j $m %
cat X(m) __ + 1
p
by Proposition 1.3.
To prove the second assertion, let L be a loop spectrum equivalent to F , a*
*nd write
u = n. Then u corresponds to a map
f : X ! Ln;
where ssr(Ln) ~=ssrn(F ) for all r. In particular, ssr(Ln) = 0 for r > m+n. Th*
*e first assertion
now implies the second, using Proposition 2.2. *
* 
It follows from Proposition 2.4 that if ss*(F ) is bounded above, and u 2 F*
*e*(X), then
cwgt (u) < 1. Rudyak has observed that if cwgt(u) = 1, then u must be a phantom*
* class;
the same is true for maps. It is possible to show, for example, that if G is a *
*finite dimensional
topological group and f : BG ! Y is a phantom map, then cwgt(f) = 1. In part*
*icular,
this is true for the phantom maps f : CP1 ! S3 described by Gray in [7].
Lastly, we describe the effect of basic arithmetic operations on the catego*
*ry weight of
cohomology classes.
9
Proposition 2.5 Let u; v 2 eF(*X). Then
cwgt(u + v) min (cwgt(u); cwgt(v));
if cwgt(u) 6= cwgt(v), then strict equality holds. For any a 2 eF(*S0),
cwgt(au) cwgt(u):
Proof
The proof is easy. *
* 
2.2 Existence of Maps with Maximal Category Weight
In this section we will produce maps f : X ! Sn with cwgt(f) = cat(X)  1 for *
*suitable
spaces X. The basic observation is Theorem 2.6.
Theorem 2.6 Suppose that X = Y [g Dn (we do not require that dim(Y ) < n). Let
f : X ! X=Y = Sn
be the quotient map. If g is surjective, then cwgt(f) = cat(X)  1.
Proof
If A X with catX(A) < cat(X), then A 6= X. Since g is surjective, Dn 6 A, and *
*since A is
a subcomplex, it is closed in X, so A misses an interior point of Dn. Therefore*
* fA : A ! Sn
is not surjective, and so fA ' *. *
* 
Corollary 2.7 Let X be as above and suppose u = f*(v) 2 Fen(X). Then cwgt(u) =
cat(X)  1.
Proof
This follows immediately from Theorem 2.6 and Proposition 2.3. *
* 
The next theorem shows that Theorem 2.6 applies to any compact manifold. R*
*ecall
that any compact connected ndimensional manifold can be given a CW decompositi*
*on with
exactly one ncell.
Theorem 2.8 Let M be a compact connected nmanifold, and give it a CW decompos*
*ition
with precisely one ncell. Let
f : M ! M=M(n1)= Sn
be the quotient map. Then f 6' *, and cwgt(f) = cat(M)  1. Also, if M is Rori*
*entable,
and u 2 Hn(M; R), then cwgt(u) = cat(M)  1.
Proof
Since every point of an nmanifold has an ndimensional neighborhood, the attac*
*hing map
for the n cell must be surjective. By Theorem 2.6, cwgt(f) = cat(M)  1 if f 6'*
* *.
10
To see that f 6' *, observe that if M is Rorientable, then
f* : Hn(Sn; R) ! Hn(M; R)
is an isomorphism. Since every compact manifold is Z=2orientable, it follows t*
*hat f 6' *.
If u 2 Hn(M; R), and M is Rorientable, then u = f*v, so cwgt(u) = cat(M) *
* 1 by
Corollary 2.7. *
* 
Example 2.9 Consider RP2n1. By Corollary 2.8, if u 2 H2n1(RP2n1; Z) then cw*
*gt(u) =
2n  1, even though it is not decomposable as a product.
The condition that g be surjective is not usually preserved by homotopy. Th*
*is is our first
indication that changes within homotopy type can change the category weight of *
*a map.
The next example is a dramatic demonstration of this phenomenon.
Example 2.10 Let X be any finite CW complex, and write cat(X) = N. Let Sn Rn+1
as usual, and let q : Sn ! [1; 1] be the projection on the first coordinate.
Since X is a finite CW complex, there is a surjective map r : [1; 1] ! X *
*(by the
HahnMazurkiewicz Theorem, Theorem 3.30 in [9]). We can even assume that both q*
* and r
preserve basepoints. Let f = r O q : Sn ! X. Then f is clearly nullhomotopic, *
*and so there
is a homotopy equivalence
h : X [f Dn+1'! X _ Sn+1:
Let oe generate fH*(Sn+1; G) fH*(X _ Sn+1; G), and let o = h*(oe). Since f is *
*surjective,
Corollary 2.7 tells us that cwgt(o) = N  1. On the other hand, since oeSn+1 *
*6= 0, and
catX_Sn+1(Sn+1) cat(Sn+1) = 2, cwgt(oe) = 1.
Taking n = 1 and cat(X) > 3 in the above example gives us a counterexample *
*to the
assertion which results from letting X in Proposition 2.4 be simply homotopy eq*
*uivalent to
a CW complex.
This procedure can be adapted to find maps f : X ! Sn whose category weigh*
*t detects
the full category of X.
We need the following lemma.
Lemma 2.11 Let X be a finite CW complex. Then every map f : Sn ! X is homotop*
*ic
to a surjective map.
Proof
Let h : [1; 1] ! X be a surjective pointed map (such a map exists by the Hahn*
*Mazurkiewicz
Theorem). Since [1; 1] is contractible, h ' *. Now let f : Sn ! X represent t*
*he homotopy
class ff. Then ff = ff + 0 is represented by the composition
pinch f_h fold
Sn ________Sn _ Sn ______X_ X ________X:
Since h is surjective, so is this representative of the class ff. *
* 
11
Next we recall a result from [3]. Suppose X is a simply connected CW compl*
*ex with
homology groups Hn(X; Z) = Hn, and let A be the space constructed as follows. L*
*et A1 = *.
Given An1, there is a cofibration
jn1
M(Hn; n  1) ! An1! An
where M(Hn; n  1) is the Moore space with
ae
fHq(M(Hn; n  1); Z) = Hn if q = n  1
0 otherwise.
and the map jn1 induces the zero map in homology (in general there are many ch*
*oices for
jn1). Then A = [An. According to Theorem 2.1 of [3], the maps jk can be chosen*
* so that
A ' X.
It follows that if Hq(X) = 0 for q n and Hn1(X) is free (so that we may t*
*ake
M(Hn1; n  2) to be a wedge of spheres), then X is homotopy equivalent to a CW*
* complex
with no cells of dimension n or higher.
Proposition 2.12 If X is homotopy equivalent to a simply connected finite CW c*
*omplex,
then there is a space X0' X and a map f : X0! Sn such that cwgt(f) = cat(X)  *
*1.
Proof
First, replace X by a homotopy equivalent CW complex, and assume that it has be*
*en given
a CW decomposition with no cells of dimension greater than n. We may assume tha*
*t there
is no CW complex homotopy equivalent to X which does not have n dimensional cel*
*ls.
Write Y = X(n1), so X = Y [gDn1[. .[.Dnk. Let A = Y [Dn1and B = Y [Dn2[. .*
*[.Dnk.
We will prove that the map
f : X ! X=B = Sn
is detected by homology in some coefficients. An easy argument using the Mayer*
*Vietoris
exact homology sequence shows that it is sufficient to show that the map
f0 : A ! A=Y = Sn
is detected by homology in some coefficients.
If we give Sn the cellular decompostion Sn = * [ Dn, we see that the map of*
* cellular
chain complexes induced by f0 is
0 ! Cn(A)?= Z @n! Cn1(A)? ! . . .
?y~= ?y
0 ! Cn(Sn) = Z ! 0 ! . . .
Therefore, it suffices to show that Hn(A) 6= 0 for some coefficients.
Suppose that Hn(A) = 0 for any coefficients. Then Hn1(A) must be free. By *
*Theorem
2.1 of [3], X is homotopy equivalent to an (n1) dimensional CW complex, which *
*contradicts
our assumption that there is no CW complex homotopy equivalent to X which does *
*not have
any n dimensional cells.
12
Therefore, f* 6= 0 for some coefficients; it follows that f 6' *. To finish*
* the proof, we use
Lemma 2.11 to replace the attaching map g : Sn1 ! Y for Dn1by a map g0 : S2n*
*1! B
which is surjective, and which is homotopic in B to g. Then
X ' B [g Dn1' B [g0Dn1= X0;
and f0 : X0! Sn is nontrivial, and hence cwgt(g0) = cat(X)  1 by Theorem 2.6.*
* 
I am grateful to Chuck McGibbon for bringing [3] to my attention.
3 Essential Category Weight
We saw in Example 2.10 that composition with a homotopy equivalence can dramati*
*cally
alter the category weight of a map. We constructed a space Y ' X _ Sn in which *
*the the
category weight of the cohomology class in H*(Y ) corresponding to the generato*
*r of H*(Sn)
was larger than it `should' have been. In Example 3.1, we find a space Y ' RP2n*
*1 in which
the category weight of a generator in H2n1(Y ; Z) is smaller than it `should' *
*be.
These examples motivate us to look for a concept similar to category weight*
*, but better
behaved under composition. The result is essential category weight, denoted E(f*
*).
In 3.1, we give the definitions, some examples, and a list of equivalent de*
*finitions. In 3.2,
we examine in detail the behavior of E(f) under composition.
3.1 Definition and Basic Properties
We begin with Example 3.1 which serves as motivation for the concept of essenti*
*al category
weight.
Example 3.1 Let p : S2n1! RP2n1be the standard double cover. Let 2 H2n1(R*
*P2n1; Z)
and oe 2 H*(S2n1; Z) be generators. Then p*() = 2oe. Recall from Example 2.9*
* that
cwgt () = 2n  1.
Now consider Mp, the mapping cylinder of the map p. The diagram
Mp


i 
' q


p ?
S2n1 ______RP2n1
13
commutes. Let 0 = q*(). Then 0i(S2n1)= p*() = 2oe 6= 0. Since catMp(S2n1)
cat(S2n1) = 2, cwgt(0) = 1. This shows that category weight can be altered dra*
*matically
by composition with a homotopy equivalence.
We can do even worse. Define an inclusion j : Mp ,! RP2n1x D2n by
j : [x; t] 7! (p(x); tx):
Let 00= p*RP2n1(). Then 00Mp = 0. By Lemma 2.3, cwgt (00) cwgt (0) = 1, so
cwgt (00) = 1. We have obtained arbitrarily large drops in category weight by *
*taking a
product with a disk.
This phenomenon is actually quite general. If f : X ! Y and g : Z ! X s*
*uch that
cwgt (f O g) N, then the maps f0 : Mg! Y and f00: X x CZ ! Y satisfy cwgt(f0*
*) N
and cwgt(f00) N. If Z can be embedded in some sphere, CZ can be replaced by Dn*
* for
large enough n.
Thus, category weight is not preserved by even very straightforward homotop*
*y equiva
lences. This shortcoming motivates the concept of essential category weight.
Definition Let f : X ! Y with f 6' *. The essential category weight of f is
E(f) = min (cwgt(f O g) g : Z ! X):
In this definition, the spaces Z must be CW complexes; we do not require that X*
* or Y be
CW complexes.
Example 3.2 If bdN6' *, then E(dbN) N. To see this, it suffices to show that *
*bdNO g ' *
whenever cat(g) N (by Proposition 3.5, below). Let g : Z ! X with cat(g) N. *
*Then
we have the following commutative diagram.
T NX




g d ?
Z _______X ______XN
@ 
@ ^
bdN@@R ?
X(N)
Since cat(g) N, there is a lift up to homotopy into T NX by Proposition 1.1. T*
*herefore
dbN O g ' *.
Just as for category weight, there are several equivalent ways to define th*
*e essential
category weight of a cohomology class. The definition we give applies to maps *
*between
arbitrary spectra; this generality will be useful later when we discuss cohomol*
*ogy operations.
14
Definition Let f : E ! F be a map of spectra, and n be an integer. The degree*
* n essential
category weight of f is
E(f; n) = min (cwgt(f O g) g : (Z) ! F is a map of degree n):
A cohomology class u 2 F n(X) corresponds to a map u : (X) ! F of degree n. *
*The
essential category weight of u is
E(u) = min (cwgt(g*(u)) g : Z ! X):
Notice that, according to our convention, these minimums are taken over all g s*
*uch that
f O g 6' * or g*u 6= 0, respectively. As before, the spaces Z must be CW comple*
*xes.
It follows immediately that E(u) E(u; 0). It would follow immediately from*
* the Freyd's
generating hypothesis that E(f; n) = 1 for any map f between finite spectra.
Example 3.3 Let F be a ring spectrum. Let f : Z ! X, and let ui2 F *(X). Then
f*(u1. .u.N) = (f*u1) . .(.f*uN );
and cwgt(f*(u1) . .f.*(uN )) N by Example 2.1. Therefore
E(u1. .u.N) N:
We can express the essential category weight of a map of spectra in terms o*
*f the essential
category weight of a map of spaces.
Proposition 3.4 Let u : E ! F be a map of spectra, and let L; K be loop spec*
*tra
equivalent to E and F respectively. The map u corresponds to a sequence of map*
*s fn :
Ln ! Kn+k. With this notation,
E(u; n) = E(fn):
If u : (X) ! F is of degree n, then u corresponds to a map f : X ! Kn. Then
E(u) = E(f):
Proof
The proof is strictly parallel to the proof of Proposition 2.2. *
* 
Using Proposition 3.4, we see that every result for essential category weig*
*ht of maps will
have an analogue for essential category weight of maps of spectra: simply repla*
*ce the maps
of spectra in question by maps of spaces and apply the result for spaces.
Clearly E(f) cwgt(f) and similarly for cohomology classes. Also
1 E(f) cwgt(f) < cat(X);
just as for cwgt, E(u) = 0 if and only if u 62 eF(*X). In fact, replacing cwgt *
*with E in each
of the results of Section 2.1 yields a true statement for essential category we*
*ight. To prove
these statements, take minimums over the given proofs.
15
The next result is very useful for computing essential category weight.
Proposition 3.5 Let f : X ! Y . The following are equivalent:
1. E(f) N;
2. f O g ' * whenever cat(g) N;
3. f O g ' * whenever g : Z ! X with gcat(Z) N;
4. cwgt(f O g) N whenever g : X0' X.
Let u : E ! F be a map of spectra. The following are equivalent:
10. E(u; n) N;
20. g*u = 0 whenever g : (Z) ! E is a map of degree n and cat(g) N;
30. g*u = 0 whenever g : (Z) ! E is a map of degree n and gcat(Z) N;
40. (assuming E = (X) for some CW complex X) cwgt(g*u) N whenever g : X0' *
*X.
Proof
Let g : Z ! X with cat(g) N. By Theorem 1.3, g factors through a space Z0 wi*
*th
gcat(Z0) N, so that the diagram
Z0
i 
g0

g ? f
Z ______X ______Y
commutes up to homotopy. Since E(f) N, cwgt(g0O f) N. Since cat(Z0) gcat(Z0)*
* =
N, g0O f ' *, and so f O g ' *.
Suppose now that f O g ' * whenever cat(g) N, and let g : Z ! X with gcat*
*(Z) N.
By Theorem 1.4, cat(g) N, so f O g ' * by 2.
Now assume that f O g ' * whenever g : Z ! X with gcat(Z) N. Let f : Z !*
* X, and
let A Z with catZ(A) N. The inclusion map iA : A ,! Z has cat(iA) = catZ(A) *
*N,
and so it factors through a space A0with gcat(A0) N. Since (f O g)A factors t*
*hrough A0,
f O gA ' *. Therefore E(f) N.
We have now shown that 1, 2, and 3 are equivalent. Finally, we prove that 4*
* is equivalent
to 1.
Clearly
E(f) = min (cwgt(f O g) g : Z ! X) min (cwgt(f O g) g : X0'!X):
It remains to prove the reverse inequality.
Let g : Z ! X such that cwgt(f O g) = E(f) = k. Then there is a subspace *
*A Z
such that catZ(A) = k + 1, and f O gA 6' *. Let X0 = Mg, and let p : Mg! X b*
*e the
standard retraction, which is a homotopy equivalence. Then catX0(A) catZ(A) = *
*k + 1,
and f O pA = f O gA 6' *, so cwgt(p O f) k.
16
The proof that 10, 20, 30, and 40 are equivalent is an easy adaptation of t*
*he proof just
given, using Proposition 3.4.
For example, to prove that 10implies 20, we proceed as follows. Let K; L be*
* loop spectra
equivalent to E and F , respectively. Then
Een(X) ~=[X; Ln]
as in Theorem 1.5. If g : (X) ! E corresponds to h : X ! Ln, then cat(g) = ca*
*t(h). Also,
the map u corresponds to a map f : Kn ! Lm , and E(u; n) = E(f).
Therefore, u O g : (X) ! F corresponds to the map f O h. Since E(f) N a*
*nd
cat(h) N, we are done, using our proof that 1 implies 2. *
* 
3.2 Behavior Under Composition
Having defined essential category weight with the hope of getting better compos*
*tion prop
erties, our next task is to describe those composition properties. The main re*
*sult of this
section is Theorem 3.7.
Before we can prove Theorem 3.7, we must establish some numerical relations*
* between
cat(f) and cwgt(f). The basic principle is that in a perfect world, cat(f) . cw*
*gt(f) would be
equal to cat(X).
Proposition 3.6 Let f : X ! Y , and let A X. Then
& '
catX(A)
cat(fA) ________
cwgt (f)
and $ %
catX(A)
cwgt(f) ____________ :
cat(fA)  1
The corresponding results hold when f : (X) ! F is a map of spectra: if u : (X*
*) ! F ,
and A X, then & '
catX(A)
cat(u(A) ) ________
cwgt (u)
and $ %
catX(A)
cwgt(u) ______________ :
cat(u(A) )  1
Proof
Suppose cwgt(f) = p and catX(A) = n. Write A = U1 [ . .[.Un, with each Ui close*
*d and
contractible in X. Let V1 = U1 [ . .[.Up, V2 = Up+1[ . .[.U2p, and so on, up to
Vdn=pe= Up(dn=pe1)+1[ . .[.Un:
Observe that catX(Vi) p for each i, so fVi ' *. Since there are precisel*
*y bn=pc of
them, this proves the first formula.
The second formula follows by arithmetic, and the statements for maps of sp*
*ectra follow
using Propositions 2.2 and 3.4. *
* 
17
Next we relate the category weight and essential category weight of a compo*
*sition of two
maps to the category weight and essential category weight of the individual map*
*s.
Theorem 3.7 Let g : X ! Y , f : Y ! Z. Then
cwgt(f O g) E(f) . cwgt(g)
and
E(f O g) E(f) . E(g):
If u : (X) ! F and v : F ! G, then
cwgt(v O u) E(v; u) . cwgt(u); E(v O u) E(v; g) . E(u):
If u : E ! F , v : F ! G, then
E(v O u; n) E(v; n + u) . E(u; n):
Proof
Write E(f) = p, cwgt(g) = q, and let A X, where catX(A) pq. By Proposition 3.6
& ' & '
catX(A) pq
cat(gA) ________ ___ = p:
cwgt (g) q
Since E(f) = p,
(f O g)A = f O gA ' *
by Proposition 3.5. This proves that cwgt(f O g) pq.
The second formula is a formal consequence of the first. The proofs for map*
*s of spectra
are analagous. *
* 
In the next section, we will use Theorem 3.7 to compute the category weight*
* of some
cohomology classes by showing that they are in the image of cohomology operatio*
*ns whose
representatives 2 F *(F ) have E(; n) > 1.
The hoped for invariance of essential category weight under compostion foll*
*ows immedi
ately from theorem 3.7.
Corollary 3.8 Let f : X ! Y , and g : Z ! X. Then
E(f O g) E(f):
The corresponding results for maps of spectra also holds: if u : E ! F and v :*
* F ! G
E(v O u; n) E(v; n + u);
if u : (X) ! F , and f : F ! G, then E(f O u) E(f). Let f : X ! Y , and let*
* g : Z ! X
be a homotopy equivalence, then E(f) = E(f O g). Similarly, if u : E ! F and g*
* : G'! E,
then E(g*u) = E(u).
18
Proof
The inequalities follow from Theorem 3.7 on observing that if f 6' *, then E(f)*
* 1.
To prove invariance under homotopy equivalences, it suffices to show that E*
*(f) E(fOg).
Let h be a homotopy inverse for g. Then
E(f) = E ((f O g)O h) E(f O g):
The proof for maps of spectra is identical. *
* 
It follows that if g is a homotopy equivalence, then E(g) = 1.
We also obtain the following invariance of essential category weight under *
*the inverses
of maps induced by inclusions.
Proposition 3.9 Let X be a CW complex, and let g : X(m+1)! Y . If ssr(Y ) = *
*0 for
r > m, then the map g extends to a map
f : X ! Y;
and E(f) = E(g). If ssr(F ) = 0 for r > m, and u 2 F *(X(m)), then there is a c*
*lass v 2 F *(X)
such that vX(m) = u, and E(v) = E(u).
Proof
By basic obstruction theory, the map
[X; Y ] ! [X(m+1); Y ]
induced by the inclusion X(m+1),! X is an isomorphism for any space X. This pro*
*ves that
the map g extends to a map f : X ! Y .
To prove E(f) = E(g), let h : Z ! X, where cat(h) E(g). By cellular appro*
*ximation,
we may assume that the map h is cellular. Since the diagram
*
[X; Y ]_________[Z;hY ]
 
 
~ ~
= =
 
? (h )* ?
Z(m+1)
[X(m+1); Y ]______[Z(m+1);Y ]
commutes, it suffices to show that (hZ(m+1))* O g ' *. Since cat(hZ(m+1)) ca*
*t(h) E(g),
this follows immediately from assertion 2 of Proposition 3.5.
The statement for cohomology classes follows from the statement for maps, u*
*sing Propo
sitions 2.2 and 3.4. *
* 
19
4 Essential Category Weight in Cohomology
In this section we will describe the effect of cohomology operations on the (es*
*sential) category
weight of cohomology classes.
Recall that a cohomology operation is a natural transformation : F n(X) !*
* Em (X).
The essential category weight of the cohomology operation is
i j
E() = min cwgt((u))  u 2 eF(*X):
Let ffi denote the connecting homomorphism in the cohomology long exact sequenc*
*e. A stable
cohomology operation is a collection = {n} of cohomology operations n : F n(X)*
* ! Em (X)
such that n O ffi = ffi O n1. If is a stable operation, we define
E(; n) = min (cwgt((u))  u = n)= E(n):
Now we need to recall some basic classifications of cohomology operations. *
*Let E and F
be spectra, and let : L ! F be an equivalence, where L is a loop spectrum. Th*
*en we have
maps n defined by the composition
(Ln) ,! L ! F:
The map
ffl : [X; Ln] ! eF(nX)
given by ffl : f 7! f*n is an isomorphism. It is easy to see that if On = { : e*
*F(nX) ! eE*(X)}
(that is, the cohomology operations with domain F n(X)) then the map On ! eE*(*
*Ln) given
by 7! (n) is an isomorphism. In particular, if K is a loop spectrum equivalen*
*t to E,
then the cohomology operations : F n(X) ! Em (X) are in one to one correspond*
*ence with
Eem (Ln) ~=[Ln; Km ].
Theorem 4.1 With the notation as above, if f : Ln ! Km corresponds to the coh*
*omology
operation , then
E() = E(f):
If is a stable operation corresponding to the map f : F ! E, then
E(; n) = E(f; n):
Proof
This follows from Proposition 3.4. *
* 
From now on, we adopt the convention that if : F n(X) ! Em (X) is a cohom*
*ology
operation defined only in dimension n, then E(; n) = E(), and E(; k) is meaning*
*less for
k 6= n.
A straightforward application of Theorem 3.7 yields the following important*
* observation.
Theorem 4.2 For any class u 2 F n(X), and any cohomology operation : F *(X) *
*! E*(X),
cwgt ((u)) E(; n) . cwgt(u)
20
and
E((u)) E(; n) . E(u):
Proof
Let L be a loop spectrum equivalent to F , and let 2 eE*(Ln) be the class corr*
*esponding to
the operation . Suppose u corresponds to the map u : E(X) ! F of degree n. Th*
*en (u)
corresponds to the composition
(X) u!(Ln)n! E:
The theorem now follows immediately from Theorem 3.7. *
* 
Recall that Fadell and Husseini suggested that we say that an operation is*
* universal in
dimension n for category weight k if cwgt((u)) k for every u 2 ehn(X). In our *
*terminology,
this says E(; n) k. In view of Theorem 4.2, we see that the conclusion to thei*
*r Theorem
3.12 can be improved from cwgt((u)) 2 to cwgt((u)) 2 . cwgt(u).
Next, we show that a given stable cohomology operation will have E(; n) > *
*1 for at
most one integer n; we also give a useful characterization of this dimension n.
Theorem 4.3 (cf. Theorems 3.6 and 3.12 of [4]) Let 2 Ek(F ) be a stable opera*
*tion of
degree k. Then E(; n) > 1 if and only if (Fenr(X)) = 0 for any CW complex X an*
*d any
r > 0.
Proof
Suppose there is some r > 0 such that (Fenr(X)) 6= 0. Then (Fen(SrX)) 6= 0. *
*Since
cat(SrX) = 2, E(; n) = 1. Thus, if E(; n) > 1, then is identically 0 in every *
*dimension
less than n.
Now we assume that (Fenr(X)) = 0 for all r > 0, and show that E(; n) > 1. *
* By
Proposition 3.5, it suffices to show that (Fen(Z)) = 0 for any CW complex Z wit*
*h gcat(Z)
2. Let Z = A [ B with both A and B contractible subcomplexes. Then we have
ffi 
0 __________Fen1(A\ B) ________Fen(Z)~=___________0
 
 
 
 
 
? ?
ffi 
0 _________Een1+k(A\ B) ______Een+k(Z)~=__________0
from the Mayer Vietoris exact sequence. Since (Fen1(A \ B)) = 0, vanishes on *
*eF n(Z). 
We now introduce a notation for the first dimension in which a stable cohom*
*ology class
is nonzero. For a stable cohomology operation 2 F *(F ), let
d() = min (n  (F n(X))6= 0 for someX):
21
Thus, ae
1 if n 6= d()
E(; n) =
> 1 if n = d():
Observe that in the case is an admissible monomial in Steenrod powers, d() is *
*equal to
the excess (e()) of the monomial (see [13]). Thus, we recover Theorems 3.6 and *
*3.12 of [4].
Next we see that for some spectra, the condition d() > 1 is obvious.
Lemma 4.4 Let F be a spectrum with ssr(F ) = 0 for r > m and let be a stable
cohomology operation. Then there is an integer d() such that (F r(X)) = 0 for a*
*ny CW
complex X and any r < d().
Proof
By induction on the skeleta of X, F r(X) = 0 for r < m. *
* 
Example 4.5 We can use this result to compute of the category of the lens spac*
*es. Let
L = S2n1=Zp, where p is an odd prime. The (ordinary) cohomology ring with Zp c*
*oefficients
is
H*(L; Zp) = (x) ZpZp[y]=(yn);
where x = 1 and y = fi(x). Then E(y) 2, since fi is a stable operation, and*
* d(fi) =
e(fi) = 1. The class xyn1 2 H*(L; Zp) is nonzero, and
E(xyn1) 1 + (n  1)2 = 2n  1;
so cat(L) 2n. On the other hand, dim(L) = 2n  1, so cat(L) 2n by Proposition*
* 1.3.
thus cat(L) = 2n.
5 Product Formulae
As we observed in the introduction, many of the most useful lower bounds on cat*
*(X) are of
the product formula type. In this section, we realize each of these results as *
*consequences
of a single statement concerning (essential) category weight. This main theorem*
* is Theorem
5.1.
In 5.1, we prove Theorem 5.1 and some corollaries. In 5.2, we use Theorem *
*5.1 to
prove Theorem 5.6, which is our generalization of Steenrod's theorem on normal *
*cohomology
operations (see [12]). In 5.3, we derive Whitehead's result on the nilpotence o*
*f [X; G] (see
[16]) and Eilenberg's cup product lower bound on cat(X) as corollaries of categ*
*ory weight
formulas which are in turn corollaries of Theorem 5.6.
22
5.1 Product Formulae For Maps
Let f : X ! K and g : X ! L. Then we have the diagram
d ______^
X ________X x X X ^ X
 Q [fxg 
fxg  Q Q f^g
 Q 
? ^ Qs ?
K x L ______K ^ L
Let X be a space and let E and F be spectra. Suppose u : X ! E and v : X ! F *
*. Then
we have an analagous diagram
(d) (^)
(X) _______________(Xx X) ______(X^ X) ' (X) ^ (X)
H H 
HH [uxv 
H H u^v
HH 
HHj ?
E ^ F
The results of this section all follow from Theorem 5.1, which describes the ca*
*tegory weight
and essential category weight of various maps in these diagrams.
Theorem 5.1 Let f : X ! K and g : X ! L. Then
E(f"x g) E(f) + E(g)
and i j
cwgt (f"x g) O d cwgt(f) + cwgt(g):
The analagous results hold for maps of spectra. If u : (X) ! E, and v : (X) !*
* F , then
E(u"x v) E(f) + E(g)
and i j
cwgt (u"x v) O (d) cwgt(u) + cwgt(v):
Proof
Write E(f) = p, E(g) = q and suppose gcat(Z) p + q. Write Z = A [ B with gcat(*
*A) p
and gcat(B) q. Let h : Z ! X x X. Then
(f x g) O h = (f O h) x (g O h) O d:
23
By assumption, f O h ' f0, where f0(A) = *, and g O h ' g0, where g0(B) = *. Th*
*erefore,
(f x g) O h = (f O h) x (g O h) O d ' (f0x g0) O d:
Since ((f0x g0) O d)(Z) K _ L,
(f"x g) O h ' ^ O (f0x g0) O d = *:
The proof of the second assertion is analagous. Let cwgt(f) = p and cwgt(g)*
* = q, and
let A X with catX(A) p + q. By writing A = B [ C with catX(B) p and catX(C) *
* q
and proceeding as above, we can find h0' (f x g) O d such that h0(A) K _ L.
To prove the statement for maps of spectra, let K and L be loop spectra equ*
*ivalent to
E and F respectively. Then the maps u and v correspond to maps (of spaces) u0: *
*X ! Kn
and v0: X ! Lm . Observe that the diagram
(X x X) _________(X ^ Y ) ________(X)^'(Y )
   H H H u^v
(u0xv0) (u0^v0)  H
   H H
? (^) ? ? j HHj
(Ln x Km ) _______(Ln ^ Km ) ______(Ln)^'(Km ) ________E ^ F
commutes, so "u x v= (u"0x v0)*(j). Since E(u; n) = E(u0) and E(v; m) = E(v0),*
* we are
done by the first case and Proposition 3.4.
The proof of the second assertion for maps of spectra is similar. *
* 
Observe that the statement E((f"x g) O d) E(f) + E(g) follows immediately *
*from
Theorem 5.1 and Corollary 3.8.
Corollary 5.2 Let f : X ! K, g : Y ! L. Then
E(f"x g) E(f) + E(g):
The statement remains true for X, Y , K and L spectra.
Proof
Apply Theorem 5.1 to f O pX : X x Y ! K and g O pY : X x Y ! L. Since E(f O p*
*X ) E(f)
and E(g O pY ) E(g),
E(f"x g) = E ((f O pXx)(g O pYO)d) E(f O pX ) + E(g O pY ) E(f) + E(g);
which completes the proof of the first assertion. The proof of the statement fo*
*r spectra is
identical. *
* 
24
The following lemma is very useful when it comes to applying Corollary 5.2.
Lemma 5.3 Let ^ : X x Y ! X ^ Y , and let Z be a space. Then
^* : [X ^ Y; Z] ! [X x Y; Z]
is weakly injective (in the sense that ^*(f) = * if and only if f = *).
Proof
This is Lemma 1.2 of [14]. *
* 
The next Proposition is very useful when trying to apply Corollary 5.2
Proposition 5.4 The map "f x g6' * if and only if f ^ g 6' *.
Proof
The diagram
fxg
X x Y ______Kx L
 
^ ^
 
? f^g ?
X ^ Y ______K ^ L
is commutative. Proposition 5.4 now follows immediately from Lemma 5.3. *
* 
We immediately derive the following result, which was first observed by Fox*
* (see [5]).
Corollary 5.5 Let X and Y be spaces such that X ^ Y 6' *. Then
cat(X x Y ) 3:
Proof
Let ^ : X x Y ! X ^ Y . Since 1X^Y = 1X ^ 1Y 6' *, E(^) 2 by Corollary 5.2. *
* 
It follows from Corollary 5.5 that Ganea's X x Sk conjecture holds for any *
*space with
cat(X) = 2 and SX 6' *.
5.2 Operations of Several Variables
In [12], Steenrod introduced the concept of a normal cohomology operation of se*
*veral vari
ables, and he proved that if is a normal cohomology operation of n variables, *
*then (in our
language)
cwgt((u1; : :;:un)) n:
This suggests the more general formula
X
cwgt((u1; : :;:un)) cwgt(ui)
25
and its corollary X
E ((u1; : :;:un)) E(ui):
We will prove a result of this form. Unfortunately, we will have to use a slig*
*htly stronger
definition of a normal operation than the one Steenrod uses. On the other hand,*
* our definition
has the added generality of applying to any functor of the form F (X) = [X; Y ]*
* and not just
to cohomology theories.
Let
: [X; F1] x . .x.[X; Fn] ! [X; F ]
be a natural transformation. There is a natural equivalence
~=
[X; F1] x . .x.[X; Fn] ! [X; F1 x . .x.Fn]
given by (f1; : :;:fn) 7! (f1 x . .x.fn) O d. This gives us a natural transform*
*ation
0: [X; F1 x . .x.Fn] ! [X; F ]
which is induced by the map
0= 0(1) : F1 x . .x.Fn ! F:
Definition Let : [X; F1] x . .x.[X; Fn] ! [X; F ] be a natural transformatio*
*n. We say
is normal if the map
0: F1 x . .x.Fn ! F
factors as in the diagram.
0
F1 x . .x.Fn _________________________F
HH *
H H 00
^ HHj
F1 ^ . .^.Fn
Notice that, using our definition, if any fi' *, then (f1; : :;:fn) = 0. It*
* is this property
that Steenrod used as the definition of a normal operation.
We now give our generalization of Steenrod's result.
Theorem 5.6 If is normal (in our sense), then
X
cwgt ((f1; : :;:fn)) cwgt(fi):
Similarly, X
E ((f1; : :;:fn)) E(fi):
The analagous statement is true for spectra.
26
Proof
Write f = f1 x . .x.fn. By definition,
(f1; : :;:fn)= 0O (f1 x . .x.fn) O d
= 00O bfO d:
Thus, Theorem 5.6 follows directly from Theorem 5.1. *
* 
5.3 Commutators and Cup Products
We now show how Whitehead's theorem on the nilpotence of [X; G] and the classic*
*al cup
product lower bound on cat(X) follow from useful corollaries of Theorem 5.6.
We begin with the following result, which sharpens Whitehead's theorem (see*
* [16]). Recall
that a space G is grouplike if it satisfies the group axioms up to pointed homo*
*topy. If G is
grouplike, [X; G] has a natural group structure.
Corollary 5.7 Let G be grouplike, let f; g 2 [X; G], and denote the commutator*
* of f and
g by [f; g]. Then
cwgt([f; g]) cwgt(f) + cwgt(g)
and
E([f; g]) E(f) + E(g):
Proof
This operation is induced by the commutator map [; ] : GxG ! G. Since [G; 1] =*
* [1; G] = 1,
[; ] factors through (G x G)=(G _ G) = G ^ G, and hence is normal. *
* 
Corollary 5.8 If cat(X) N, then [X; G] is nilpotent with nilpotence class at *
*most N 1.
Proof
By Corollary 5.7, all Nfold commutators have category weight at least N. Since*
* cat(X)
N, they are trivial. *
* 
The cup length formulae for category weight and essential category weight a*
*lso follow
from Theorem 5.6.
Corollary 5.9 (cf. Theorem 3.5 [4]) Let ui2 Fi*(X), and consider
u1 ^ . .^.un 2 (F1 ^ . .^.Fn)*(X):
Then X
cwgt(u1 ^ . .^.un) cwgt(ui)
and X
E(u1 ^ . .^.un) E(ui):
Proof
Since the cup product is the map
bd u1^...^un
(X) _________(X^ . .^.X) ' (X) ^ . .^.(X) ______F1^ . .^.Fn
27
this is just a restatement of Theorem 5.1. Alternatively, since is a evidentl*
*y a normal
operation, the corollary follows from Theorem 5.6. *
* 
It follows that any Nfold cup product will vanish on spaces with catX(A) *
*N, since
cwgt (ui) 1 for any ui2 eF(*X).
Just as in Corollary 5.2 the corollary to the previous theorem yields a gen*
*eralization for
essential category weight.
Corollary 5.10 Let ui2 Fi*(Xi); then
u1 x . .x.un = (p*1u1) ^ . .^.(p*nun) 2 (F1 ^ . .^.Fn)*(X1 x . .x.Xn)
satisfies X
E(u1 x . .x.un) E(ui):
Proof
This is just a restatement of Corollary 5.2. *
* 
It is not true in general that cwgt(u x v) cwgt(u) + cwgt(v), as the follo*
*wing example
shows.
Example 5.11 Consider RP2n1 as Example 3.1 (also continue the notation of Exa*
*mple
3.1). Let
e = p*RP2n1() 2 H*(RP2n1x S2n; Z):
The inclusion D2n S2n (as the Northern hemisphere, say) induces an inclusion R*
*P2n1x
D2n RP2n1x S2n, and eRP2n1xD2n= 00. By Proposition 2.3, cwgt(e) cwgt(00) =*
* 1,
so cwgt(e) = 1. In fact, we know that the inclusion
S2n1 Mp RP2n1x D2n RP2n1x S2n
induces e 7! 2oe. If o 2 H2n(S2n; Z) generates H2n(S2n) and eo= p*S2n(o), then *
* x o = e. eo,
and so
x oS2n1xS2n= 2oe x o 6= 0:
Since cat(S2n1x S2n) = 3, cwgt( x o) = E( x o) = 2.
6 Application to Ganea's Conjecture
In this section, we show how to apply the results of this chapter to Ganea's X *
*xSk conjecture.
Recall that for any CW complex X,
cat(X) cat(X x Y ) cat(X) + cat(Y )  1:
(See [James] for a proof.) In the case Y = Sk, we have
cat(X) cat(X x Sk) cat(X) + 1:
28
The only known examples where the lower bound holds are cases where X and Y are*
* P and
Q local for complementary sets P and Q of primes. This serves as motivation for*
* Ganea's
conjecture (see Problem 2 in [6]) that
cat(X x Sk) = cat(X) + 1
for all spaces X.
The conjecture is trivial to prove if the category of X is represented by c*
*up length; that
is, if there is a nonzero (cat(X)  1)fold cup product in the cohomology of X.*
* Of course, it
is not always true that the category of X is represented by cup length.
On the other hand, we have seen in Proposition 2.13 that for (many) finite *
*CW complexes
X, the category of X can be represented by category weight (in particular, the *
*category of
a compact manifold is represented by category weight by Theorem 2.8). Unfortuna*
*tely, this
is not enough, as Example 5.11 shows.
The appropriate generalization of cup length is essential category weight.
Proposition 6.1 Let X be a space with cat(X) = N, and suppose there is a map f*
* :
X ! Y such that E(f) = N  1, and that Snf 6' *. Then
cat(X x Sk) = cat(X) + 1
for all 0 < k n. If u is a cohomology class with E(u) = N  1, then
cat(X x Sk) = cat(X) + 1
for all k > 0.
Note Rudyak calls a cohomology class such that E(u) = cat(X)  1 a detecting c*
*lass, and
proves the second part of this proposition (see Theorem 2.5 of [11]).
Proof
Since a cohomology class is represented by a stably nontrivial map, the second *
*part follows
from the first.
Consider the map f"x 1Sk: X x Sk ! SkY . By Theorem Corollary 5.2,
E(f "x 1Sn) E(f) + E(1Sn) = E(f) + 1;
assuming f"x 1Sn6' *. By Proposition 5.4, it suffices to show that f ^1Sn = Skf*
* is nontrivial.
This is the case because Snf 6' *, and n k. *
* 
Proposition 6.1 shows how to fit the main result of [14] into the general f*
*ramework of
category weight. Say X is ndimensional and (p  1)connected. We have seen in *
*Example
3.2 that a good candidate for a map with E(f) = N  1 is
db: X ! X(N1):
Suppose N n+4_2p; then since X(N1) is (Np  1) connected, bdNwill be stably n*
*ontrivial if it
is nontrivial. Therefore, if N n+4_2p, it suffices to show that bdN16' * or, *
*equivalently, that
cat(X) = wcat(X) (recall that wcat(X) is the least integer N such that bdN' *).
29
The following is Theorem 2.2 of [14].
Theorem 6.2 If X is ndimensional and (p  1)connected, (with p > 1) and
$ %
n
cat(X) = __
p
and n 6 1 mod p, then
cat(X) = wcat(X):
Proof
The conditions imposed allow us to use the BlakersMassey Excision Theorem to c*
*onclude
that the sequence
[X; T N1X] ! [X; XN1 ]^*![X; X(N1)]
is exact. By assumption, d 2 [X; XN1 ] does not lift into [X; T N1X], so ^*(d*
*) = bd6' *. 
Now Theorem 3.3 of [14] follows immediately from Proposition 6.1.
Theorem 6.3 If X is ndimensional and (p  1)connected (with p > 1)
$ %
n
cat(X) = __ + 1
p
and n 6 1 mod p, then
cat(X x Sk) = cat(X) + 1
for every k > 0.
Proof
The argument given above does not apply to certain spaces with cat(X) = 2; in t*
*hese cases
the conjecture follows from Corollary 5.5. *
* 
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*sity
of Chicago Press 1974.
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*, Math.
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Mich. J. Math 6 (1959), 313330.
30
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University of WisconsinMadison
Madison, WI 53706
U. S. A.
31