Two Special Cases of Ganea's Conjecture1
Jeffrey A. Strom
University of WisconsinMadison
strom@math.wisc.edu
Abstract Ganea conjectured that for any finite CW complex and any k > 0, cat(X*
* xSk) =
cat(X) + 1. In this paper we prove two special cases of this conjecture. The ma*
*in result is
the following. Let X be a (p  1)jconnectedkn dimensional CW complex (not nece*
*ssarily
finite). We show that if cat(X) = n_p+ 1 and n 6 1 mod p (which implies p > *
*1), then
cat(X x Sk) = cat(X) + 1. This is proved by showing that wcat(X x Sk) = wcat(X)*
* + 1 in a
much larger range, and then showing that under the conditions imposed, cat(X) =*
* wcat(X).
The second special case is an extension of Singhof's earlier result for manifol*
*ds.
Introduction
For any CW complex X, and any k > 0
cat(X) cat(X x Sk) cat(X) + 1
(see [5], Proposition 2.3). Ganea conjectured that cat(X x Sk) = cat(X) + 1 for*
* any finite
complex X and any k > 0 (see [2], Problem 2).
Singhof proved the conjecture for X a compact differentiable or PL manifold*
* and k
lying in a bounded (possibly empty) interval depending on the dimension, connec*
*tivity and
category of X (see [8], Corollary 6.7 or [6], Theorems 1 and 2). More recently,*
* Hess proved
that the conjecture holds for simply connected spaces when cat is replaced with*
* cat0, the
rational version of cat, and k > 1 (see [4], Theorem 3). It follows that the co*
*njecture holds
for simply connected rational spaces and k > 1.
In this paper, we will prove the following.
j*
* k
Theorem 3.2 Let X be a (p1) connected n dimensional CW complex. If cat(X) = *
*n_p+1
(i.e. cat(X) is as large as possible) and n 6 1 mod p (which forces p > 1), t*
*hen
cat(X x Sk) = cat(X) + 1
for all k > 0.
This follows easily from two subsidiary results, which are interesting on their*
* own.
j k
Theorem 2.2 If cat(X) = n_p+ 1 and n 6 1 mod p, then
cat(X) = wcat(X):
_____________________________________
1Primary: 55M30, 55P50; Secondary: 55P42
1
The proof is a simple application of the BlakersMassey Excision theorem. In [*
*5], James
stated that cat(X) and wcat(X) agree in a stable range, but did not give any de*
*tails. We
assume that this is the result he had in mind.
Theorem 3.1 If wcat(X) n+2_2p+ 1, then
wcat(X x Sk) wcat(X) + 1
for all k > 0.
With one exception, our result is the first general result on this problem *
*which allows
X to be a CW complex without any additional structure (as opposed to a manifold*
* or a
rational space). Also, our result holds for arbitrarily large values of k, in c*
*ontrast to Singhof's
theorem.
The exception is the following. In [5] it is proved that if X is (p  1) c*
*onnected with
cat(X) = N and dim (X) = (N  1)p, then the (N  1)stpower of the fundamental c*
*lass
2 Hp(X; ssp(X)) is nonzero. It follows that the conjecture holds in this case.
The proof of Theorem 3.1 can be interpreted in terms of product length in a*
*n appropriate
cohomology theory. Let E and Sk denote the suspension spectra of X and Sk, resp*
*ectively,
and let ff 2 eE*(X x Sk) and fi 2 eS*k(X x Sk) be the stable classes of p1 : X *
*x Sk ! X and
p2 : X x Sk ! Sk respectively. Then
z___N1_"_____
ff ^ . .^.ff^fi 6= 0 2 (E(N1)^ Sk)*(X x Sk):
The generalized cohomology version of the classical lower bound on (weak) categ*
*ory given
by cup length now finishes the proof.
It is interesting to observe that our proof comes from an analysis of the r*
*elationship
between category and weak category, while Singhof's proof arises from an examin*
*ation of
the relationship between category and geometric (or strong) category.
In section 4, we will show how the following result for manifolds follows d*
*irectly from
Singhof's theorem.
Corollary 4.2 Let X be a compact differentiable or PL manifold which is (p1) *
*connected
and n dimensional, with n 4 and cat(X) 4. If
n + 4p + 3
cat(X) __________;
2p
then
cat(M x Sk) = cat(M) + 1
for every k > 1. If p = 1, then we may take k = 1 as well.
Thus, applying Theorem 3.3 to a manifold improves on Singhof's result if X is s*
*imply
connected and k = 1, or if n 4p  1. For example, if X = Sp(2), Singhof's theo*
*rem does
not apply, but Theorem 3.3 does apply. It follows that cat(Sp (2) x Sk) = 5 for*
* all k > 0.
I have recently received a preprint of a paper by Yu Rudyak [7], in which h*
*e gives a result
similar to Corollary 4.2, with a similar proof.
2
I'd like to thank Professor Edward Fadell, Professor Sufian Husseini, and D*
*oug Lepro
for their help in preparing this paper. I am also grateful to Professor Alejand*
*ro Adem for
pointing out that the proof Theorem 3.2 does not require the use of generalized*
* cohomology
theories.
1 Preliminaries
In this section, we will describe our notation and recall some basic facts conc*
*erning Lusternik
Schnirelman category.
1.1 Notation
Throughout this paper, the space X is a (not necessarily finite) pointed CW com*
*plex. We
denote the set of pointed homotopy classes of maps f : X ! Y by [X; Y ]. We de*
*note the
smash product of 2 spaces (or spectra) by A ^ B. The Nfold smash product of A *
*with itself
is
z_____N_"_______
A(N)= A ^ A ^ . .^.A:
We will never refer explicitly to the skeleta of X, so this will not cause any *
*confusion. The
canonical projection
X1 x X2 x . .x.XN ! X1 ^ X2 ^ . .^.XN
is denoted by ^. The composition ^ O f will be abbreviated ^f. The suspension o*
*f the space
X or the map f will be denoted SX or Sf, respectively. The loopspace of the spa*
*ce X is
denoted X (we suppress the base point).
1.2 LusternikSchnirelman Category
Next, we recall some basic facts about LusternikSchnirelman category. For a mo*
*re complete
survey, see [5].
Definition Let X be a CW complex with base point *. Write
T NX = {(x1; : :;:xn)  xi= * for somei} XN :
The category of X (denoted cat(X)) is the least integer N such that there is a *
*lift up to
3
homotopy in the diagram
T NX




d ?
X ______XN
There is a related notion, the weak category of X (see [3]); wcat(X) is the lea*
*st N so that
d^: X ! X(N) is nullhomotopic. Since X(N) = XN =T NX, cat(X) wcat(X).
It follows immediately from the cellularjapproximationktheorem that if X is*
* n dimensional
and (p  1) connected, then cat(X) n_p+ 1 (see [5], Proposition 5.1).
Note 1.1 The conditions that appear in our theorems are of the form cat(X) n+*
*2_p. An
easy numerical argument shows that for an integer N
$ %
n_+_2_ n
N __ + 1
p p
j k
if and only if N = n_p+ 1 and n 6 1 mod p.
1.3 Homotopy Classes of Maps
Recall that a map OE : S ! T of pointed sets is weakly injective if OE(s) = * *
*if and only if
s = *.
Lemma 1.2 Let ^ : X x Y ! X ^ Y . Then
^* : [X ^ Y; Z] ! [X x Y; Z]
is weakly injective for any space Z.
Proof
The sequence
* @* (Si)*
[X x Y; Z]^ [X ^ Y; Z] [S(X _ Y ); Z] [S(X x Y ); Z]
is an exact sequence of pointed sets (see [10], Theorem 2.41). Therefore, it su*
*ffices to show
that @* = 0, or equivalently (Si)* is surjective.
4
Since the suspension functor S and the loop space functor are adjoint (see*
* [10], Corol
lary 2.8), we have the commutative diagram
~=
[S(X x Y ); Z]_____[X x Y; Z]
 
(Si)* i*
 
? ~= ?
[S(X _ Y ); Z]_____[X _ Y; Z]
Thus it suffices to show that i* is surjective.
Recall that X is an H space with multiplication given by juxtaposition of l*
*oops (see
[10], Example 2.15). Given f _ g : X _ Y ! X, define h : X x Y ! Z by the for*
*mula
h(x; y) = f(x) . g(y). Then
i*(h) = h O i = hX_Y ' f _ g:
This proves the lemma. *
* 
2 Category and Weak Category
In this section, we will give a conditon under which cat(X) = wcat(X). The key *
*ingredient
is Lemma 2.1, which is equivalent to the BlakersMassey Excision Theorem.
j q
Lemma 2.1 Let A ! X ! B be a cofibration, and let F ! Y ! B result from co*
*nverting
j to a fibration. Assume A; X, and B are simply connected. Then there is a comm*
*utative
diagram j
A ______X ______B
  
i ' g 1
  
? ? q ?
F ______Y ______B
If A is (a  1) connected, and B is (b  1) connected, then i is an (a + b  2)*
* equivalence.
5
Proof
The commutativity of the diagram yields an infinite commutative ladder
. . .______ssr+1(X;A)______ssr(A)_______ssr(X)______ssr(X; A)________.*
*. .
   
g* i* ~= g*
   
? ? ? ?
. . .______ssr+1(Y;F )_____ssr(F)_______ssr(Y )______ssr(Y;F )_______.*
*. .
By the 5 lemma it suffices to show that g* is an (a + b  1) equivalence.
We have the diagram
g
(X; A) ______(Y;F )
 
 
 
j  q
 
 
? ?
(B; *)______(B;1*)
in which q is a homotopy isomorphism. Since all spaces involved are simply conn*
*ected, j is
an (a + b  1) equivalence by the BlakersMassey Excision Theorem (Theorem I of*
* [1]). It
follows that g is an (a + b  1) equivalence. *
* 
Now we can prove the main theorem of this section.
Theoremj2.2k Let X be an n dimensional (p  1) connected CW complex. If cat(X) =
n ___
p + 1 and n 6 1 mod p (which forces p > 1), then
cat(X) = wcat(X):
Proof
Write N = cat(X). By Note 1.1, the condition on cat(X) is equivalent to N n+2_*
*p.
Let F ! E ! X(N1)result from converting the map ^ : XN1 ! X(N1)to a f*
*ibration.
6
Then we have the diagram
g
T N1X ______F
 
 
h j
 
 
d ? i ?
X ______XN1 ______E'
@ 
@ 
^@d ^ 
@@R ?
X(N1)
Since T N1X is (p  1) connected and X(N1) is (N  1)p  1 connected, g is an
(N  1)p + p  2 = Np  2
equivalence by Lemma 2.1. It follows that g* : [X; T N1X] ! [X; F ] is a surj*
*ection because
!
n + 2
n = _____ p  2 Np  2:
p
Suppose wcat(X) < cat(X). Then ^d' *, and so there is a lift (up to homotop*
*y) of
i O d into F . Let 2 (g*)1(). Then
i O h O ' j O g O = j O ' i O d:
Since i is a homotopy equivalence, it follows that h O ' d; that is, cat(X) < *
*N. This is a
contradiction. *
* 
3 The Main Theorem
In this section and the next, we assume that X is a (p  1) connected n dimensi*
*onal CW
complex.
Theorem 3.1 If wcat(X) n+2_2p+ 1 then
wcat(X x Sk) wcat(X) + 1
for all k > 0.
7
Proof
Write wcat(X) = N; then the map ^d: X ! X(N1)is nontrivial. Since X(N1)is (N*
* 1)p1
connected, and ! !
n + 2
n = 2 _____+ 1  1 p  2 2(N  1)p  2
2p
this map is stably nontrivial by the Freudenthal Suspension Theorem.
We need to show the map ^d: X x Sk ! (X x Sk)(N)is nontrivial. Consider th*
*e commu
tative diagram
d ______^
X x Sk ________(X x Sk)N (X x Sk)(N)
QQ  
Q Q p1x...xp1xp2 p ^...^p ^p
dx1 QQ   1 1 2
QQs ? ?
^
XN1 x Sk ______X(N1)^Sk:
It is certainly enough to show that the composite
X x Skdx1!XN1 x Sk^! X(N1)^ Sk
is nontrivial. From the commutative diagram
dx1
X x Sk ______XN1 x Sk
 
 
^  ^
 
? Sk^d ?
X ^ Sk ______X(N1)^Sk
we see that it is enough to show that the composite
Skf (N1) k
X x Sk^! X ^ Sk ! X ^ S
represents a nontrivial class in [X x Sk; X(N1)^ Sk]. Since ^dis stably nontri*
*vial,
Skd^6= * 2 [X ^ Sk; X(N1)^ Sk]:
By Lemma 1.2, ^* is injective. This proves the theorem. *
* 
8
Remark. It is easy to view this proof in terms of product length using appropr*
*iate coho
mology theories. We refer the reader to [10], Chapters 8, 9 and 13 for informa*
*tion about
generalized cohomology.
Let E and Sk denote the suspension spectra of X and Sk, respectively. Let *
*p1 : X x
Sk ! X and p2 : XxSk ! Sk be the projections, and let ff 2 eE*(XxSk) and fi 2*
* eS*k(XxSk)
be the stable classes of the maps p1 and p2, respectively.
The product
z__N1_"_____
ff ^ . .^.ff^fi 2 (E(N1)^ Sk)*(X x Sk)
is the stable homotopy class of the map
((p1 ^ . .^.pi) ^ p2)O ^d: X x Sk ! X(N1)^ Sk:
We saw in the proof of Theorem 3.1 that this map is nontrivial.
In fact it is stably nontrivial. We have seen that ^dis stably nontrivial; *
*let ffi 2 (E(N1)^
Sk)*(X ^ Sk) denote the corresponding cohomology class. Then ff ^ . .^.ff ^ fi *
*= ^*ffi which
is nonzero by the cohomology analogue of Lemma 1.2. The existence of an Nfold *
*product
in the cohomology of X x Sk immediately implies wcat(X x Sk) N + 1.
This argument is easily extended to products of spheres. Let oeibe the clas*
*s of the identity
map in (Seki)*(Ski), and write
fii= p*i+1(oei) 2 (E(N1)^ Sk1^ . .^.Skm)*(X x Sk1x . .x.Skm):
It now follows just as above that ff ^ . .^.ff ^ fi1 ^ . .^.fim 6= 0 and so
wcat(X x Sk1x . .x.Skm) wcat(X) + m:
j*
* k
Theorem 3.2 Let X be a n dimensional (p1) connected CW complex. If cat(X) = *
*n_p+1
and n 6 1 mod p (forcing p > 1), then
cat(X x Sk) = cat(X) + 1
for all k > 0.
Proof
If n < 2p, then cat(X) 2, and the result is trivial. If cat(X) = 1, then X x S*
*k ' Sk, so
cat(X x Sk) = 2. If X is nontrivial, there will be a nonzero class u 2 fH*(X),*
* and hence
a nontrivial cup product in H*(X x Sk). Therefore, we assume n 2p; it follows*
* that
n+2______n+2_
p > 2p + 1.
By Theorem 2.2, cat(X) = wcat(X). By Note 1.1, cat(X) n+2_p, and so
n + 2 n + 2
wcat(X) = cat(X) _____ _____ + 1;
p 2p
so Theorem 3.1 applies. Therefore,
cat(X) + 1 cat(X x Sk) wcat(X x Sk) wcat(X) + 1 = cat(X) + 1;
and the theorem is proved. *
* 
9
Using the remarks following Theorem 3.1, we can extend this argument to show
cat(X x Sk1x . .x.Skm) = cat(X) + m
when X satisfies the hypotheses of Theorem 3.2.
Example 3.3 Let X = Sp(2). Then n = dim(Sp (2)) = 10, and Sp(2) is 2 connected*
*, so
p = 3. By Example 4.4 of [9], cat(Sp (2)) = 4. Thus
10
cat(Sp (2)) = 4 = ___ + 1 = 4;
3
so Theorem 3.2 applies. It follows that
cat(Sp (2) x Sk1x . .x.Skm) = 4 + m
for all ki> 0.
4 Extending Singhof 's Theorem
We continue to assume that X is n dimensional and (p  1) connected.
The results of [8] and [6] which concern Ganea's conjecture are explicitly *
*stated only for
the case p = 1. The arguments given apply to larger p, and significantly better*
* results follow
immediately. From this point of view, the most complete statement of Singhof's *
*theorem is
as follows (cf. [8], Theorems 6.1 and 6.2, and Corollary 6.7 or [6], Theorems 1*
* and 2).
Theorem (Singhof) If X is an n dimensional (p  1) connected compact PL or dif*
*feren
tiable manifold with n 4, cat(X) = N 4, and
( n+4k+3_
if k p
N n+2kk+3p+3_
2p if k p
then cat(X x Sk) = N + 1.
This section is devoted to extending the values of k for which Singhof's th*
*eorem applies.
For the rest of this section, we assume that X is an n dimensional (p1) connec*
*ted compact
PL or differentiable manifold, that n 4, and that N = cat(X) 4.
We will show that if Singhof's theorem applies to the manifold X and Sp, th*
*en
cat(X x Sk) = cat(X) + 1
for all k > 1. We first show that cat(X x (Sp)M ) = cat(X) + M. For each k > p_*
*2, Singhof's
theorem will apply to X x (Sp)M and Sk if M is large enough. It is then a simp*
*le matter to
conclude that cat(X x Sk) = cat(X) + 1.
The case k p_2is more difficult. For this, we choose M large enough that
p
_e
cat X x (Sp)M x Sd 2 = cat(X) + M + 1
10
p_e
and such that Singhof's Theorem applies to X x(Sp)M xSd2 . Then the previous ca*
*se shows
that we may take any k > p_2. An easy induction completes the argument.
In [7], Rudyak uses the Singhof's theorem as stated in [8] and an argument *
*similar to
ours to prove the case p = 1 of our Corollary 4.2 (see [6], Theorem 3.7).
Lemma 4.1 Let X be as above, and suppose
n + 4p + 3
N __________:
2p
Let k > p_2. Write X0 = X x (Sp)M x Sk, n0 = dim (X0), N0 = cat(X0), and deno*
*te the
connectivity of X0 by (p0 1) (of course, these may vary with M). If M is large*
* enough then
N0 = cat(X0) = N + M + 1
and
n0+ 4p0+ 3
N0 ___________:
2p0
Proof
By Singhof's theorem applied to X and Sp,
cat(X x Sp) = cat(X) + 1:
Furthermore, since
!
(n + p) + 4p + 3 n + 4p + 3 1
cat(X x Sp) = N + 1 _______________= __________ + __;
2p 2p 2
Singhof's theorem also applies to X x Sp and Sp. Inductively, we obtain
i j
cat X x (Sp)M = N + M:
We have to consider 2 cases. First assume k p. To apply Singhof's theore*
*m to
X x (Sp)M and Sk, we require
i j (n + Mp) + k + 3p + 3
N + M = cat X x (Sp)M _____________________
2p
or
n + k + 3p + 3 M
N ______________ ___:
2p 2
Clearly, if M is large enough, this inequality will hold. The case p_2< k p is*
* similar. Now
we require
i j (n + Mp) + 4k + 3
N + M = cat X x (Sp)M _________________
2k
or
n + 4k + 3 p
N __________ M 1  ___ :
2k 2k
11
Since k > p_2, this inequality will hold for M large enough. This proves the fi*
*rst assertion.
To prove the second assertion in the case k p, we require
(n + Mp + k) + 4p + 3
N + M + 1 _____________________
2p
or
n + k + 4p + 3 M
N + 1 ______________ ___:
2p 2
In the case p_2< k p, we require
n + Mp + k + 4k + 3
N + M + 1 ____________________
2k
or
n + 5k + 3 p
N + 1 __________ M 1  ___ :
2k 2k
In either case, the condition holds for M large enough. *
* 
Corollary 4.2 If X is as above, and N n+4p+3_2p, then
cat(X x Sk) = cat(X) + 1
for all k > 1. If p = 1, we may allow k = 1.
Proof
We begin by defining a decreasing sequence of positive integers by p1 = p, and
( pj+1
____ if pj is odd
pj+1 = pj2_ :
2+ 1 if pj is even
It is easy to see that if p > 1 this sequence decreases to 2, where it stabiliz*
*es and if p = 1, it
is constantly 1. Notice that pj+1 > pj_2.
We now define a sequence of spaces Xj. They will be nj dimensional, (pj 1)*
* connected,
and cat(Xj) = Nj. The key properties of the Xj are the following:
1. Xj+1 = Xjx (Spj)Mj x Spj+1;
2. Nj+1 = Nj+ Mj+ 1;
3. Nj nj+4pj+3_2pj.
To construct the Xj, first set X1 = X. Given Xj, define Xj+1 by property 1; by *
*Lemma 4.1
we can find an Mj large enough that 2 and 3 are satisfied.
First we consider the case p > 1. Let j be large enough that pj = 2, and co*
*nsider Xj.
By definition, 0 1
j1Y
Xj = X x @(Sp)M1 x (Spi)Mi+1A x S2:
i=2
Q p M +1
Write S = (Sp)M1 x (S i) i and s = cat(S). By property 2, we see Nj = N + s *
*+ 1.
12
Let k 2. By Lemma 4.1, there is an M large enough that
i j
cat Xjx (S2)M x Sk = Nj+ M + 1 = N + s + M + 2:
On the other hand,
i j
cat (X x Sk) x S x (S2)M+1 cat(X x Sk) + s + M + 1:
Together, these inequalities imply that cat(X x Sk) N + 1. It follows that cat*
*(X x Sk) =
cat(X) + 1.
The case p = 1 is easier: we apply Lemma 4.1 directly to X and argue as in *
*the previous
paragraph. *
* 
By increasing the Mj if necessary, we can use the same argument to show tha*
*t under the
conditions of Corollary 4.2,
cat(X x Sk1x . .x.Skm) = cat(X) + m:
If p > 1, we need to require ki 2, but if p = 1, we allow ki= 1 also. We omit t*
*he details.
Example 4.3 Let X = Sp(2), as in Example 3.3. Since
10 + 4 . 3 + 3 25
cat(Sp (2)) = 4 < ____________ = ___;
2 . 3 6
neither Singhof's theorem or Corollary 4.2 apply to Sp(2).
5 References
[1] Blakers and W. Massey: The homotopy groups of a triad, II. Ann. Math. (*
*1952)
192200?.
[2] T. Ganea: Some problems on numerical homotopy invariants. Lecture Notes in*
* Math
ematics 249 (1971) 2330.
[3] W. J. Gilbert: Some examples for weak category and conilpotency, Ill. J. *
*Math. 12
(1968), 421432.
[4] K. P. Hess: A proof of Ganea's conjecture for rational spaces. Topology,*
* 30 (1991),
205214.
[5] I. M. James: On category in the sense of Lusternik and Schnirelmann. Topo*
*logy, 17
(1978), 331348.
[6] L. Montejano: A quick proof of Singhof's cat(MxS1) = cat(M)+1 theorem. Man*
*uscripta
Math., 42 (1983), 4952.
[7] Y. Rudyak: Some remarks on category weight, Heidelberg, Heft 135.
13
[8] W. Singhof: Minimal coverings of manifolds with balls. Manuscripta Math., *
*29 (1979),
385415.
[9] P. A. Schweitzer: Secondary cohomology operations induced by the diagonal *
*mapping.
Topology. 3 (1965), 337355.
[10] R. Switzer: Algebraic Topology: Homotopy and Homology. SpringerVerlag (1*
*975).
[11] G. W. Whitehead: Elements of Homotopy Theory. SpringerVerlag (1978).
University of WisconsinMadison
Madison WI 53706
U.S.A.
strom@math.wisc.edu
14