THE CONE LENGTH OF A PRODUCT OF CO-H-SPACES
AND A PROBLEM OF GANEA
Martin Arkowitz and Donald Stanley
Abstract. It is proved that the cone length or strong category of a product*
* of two
co-H-spaces is less than or equal to two. This yields the following positiv*
*e solution
to a problem of Ganea: Let ff 2 ss2p(S3) be an element of order p, p a prim*
*e 3,
and let X(p) = S3 [ffe2p+1. Then X(p) x X(p) is the mapping cone of some map
' : Y ! Z, where Z is a suspension.
x1. Introduction
The (Lusternik-Schnirelmann) category of a topological space X; denoted catX;
is a numerical invariant of homotopy type which has been extensively studied (s*
*ee
[Ja1] and [Ja2] for surveys). It follows easily from the definitions that catX *
*= 0
if and only if X is contractible and catX 1 if and only if X is a co-H-space.
A related numerical invariant is the strong category or cone length of a space *
*X;
denoted cl X [Ga1]. This is defined as follows: cl X is the least integer n 0 *
*such
that there exists n cofibration sequences
Li- ! Xi- ! Xi+1; 0 i < n;
with X0 contractible and Xn having the homotopy type of X: Then cl X 1 if
and only if X has the homotopy type of a suspension. Also cl X 2 if and only
if there exists a cofibration sequence L1 -'!X1 -! C'; where X1 is a suspension
and the mapping cone C' of ' has the homotopy type of X: It can happen that
catX 6= clX [St1], but it has been shown [Ga1] that
catX clX catX + 1:
____________
1991 Mathematics Subject Classification. Primary 55M30, 55P50. Secondary 55P*
*45.
Key words and phrases. Lusternik-Schnirelmann category, strong category, con*
*e length, co-H-
space.
We would like to thank Hans Scheerer and the Freie Universit"at of Berlin fo*
*r their hospitality
during time this work was begun.
Typeset by AM S-TEX
1
2 MARTIN ARKOWITZ AND DONALD STANLEY
In addition, there are inequalities for the category and the cone length of a
cartesian product, namely,
cat(X x Y ) catX + catY and
cl(X x Y ) catX + max {clY; 1}:
The first is due to Bassi [Fo] and the second to Takens [Ta]. Thus if X and Y a*
*re
co-H-spaces, cat(X x Y ) 2 and cl(X x Y ) 3. The following theorem, which is
our main result, shows that the cone length is in fact less than or equal to 2.
Theorem. If A and A0are co-H-spaces of the homotopy type of 1-connected CW
complexes, then cl (A x A0) 2:
The theorem is proved by constructing a map from a mapping cone C' to AxA0,
where ' maps into a suspension, and then showing that this map induces homology
isomorphisms (Corollary 3.3).
>From the theorem we obtain a positive solution to the following problem of
Ganea. Let p be a prime 3 and let ff 2 ss2p(S3) be an element of order p. We
attach a cell to S3 by ff and form the cell complex X(p) = S3 [ e2p+1. It is kn*
*own
that X(p) is a co-H-space but not a suspension [B-H], and so catX(p) = 1 and
clX(p) = 2. Ganea asked [Ga2, Problem 8] if X(p) x X(p) can be the mapping
cone C' of some map ' : Y ! Z, where Z is a suspension, i.e., if cl(X(p) x X(p))
is less than or equal to 2. This is answered affirmatively by the above theore*
*m.
We note that in [F-S] Fernandez-Suarez proved by a different method the followi*
*ng
related result: If Y is the 3-localization of X(3), then cl(Y x Y ) 2.
Ganea does not give any reasons for stating Problem 8, but it seems clear tha*
*t it
is related to Takens's inequality. More precisely, if X is a space such that ca*
*tX = 1,
clX = 2 and cl(X x X) 2, then Takens's inequality is strict for X x X. Until
recently the only known examples where Takens's inequality is strict occurred w*
*hen
both of the spaces had torsion in their homology. Thus an affirmative answer to
Problem 8 shows that there are spaces X without homological torsion such that
cl (X x X) < catX + clX. Hence by the theorem the X(p) are such spaces and by
[F-S] so is the 3-localization of X(3). Of course the theorem provides many more
examples of this phenomenon. Incidentally, there are torsion-free examples where
cl (X x X) and cat X + clX differ by 2 [St2].
For the remainder of this section we present our notation and conventions. All
topological spaces will be based and have the based homotopy type of connected
CW -complexes: All maps and homotopies will preserve base points. The standard
THE CONE LENGTH OF A PRODUCT OF CO-H-SPACES 3
notation of homotopy theory will be used: `'for reduced suspension, `_ 'for the
wedge of spaces and `^ 'for the smashed product of spaces. We write ` 'to
denote natural homeomorphism of spaces or isomorphism of groups and ` 'to
denote same homotopy type of spaces. For spaces R and S; the following natural
homeomorphisms will be frequently used:
(R) ^ S (R ^ S) R ^ (S):
The identity map of a space R is written id:R ! R and the constant map is writt*
*en
0 : R ! S. We denote the natural inclusion by : R _ S ! R x S and the natural
projection by O : R x S ! R ^ S; so that R _ S -! R x S -O! R ^ S is a cofibra*
*tion
sequence. By a commutative diagram of spaces and maps we will mean one that is
commutative up to homotopy. For maps f and g, we write f = g to denote equality
or homotopy of maps.
x2. Preliminaries
Our standing hypothesis in this section is that there are spaces X; X0; A; A0*
*; W
and W 0such that X A _ W and X0 A0_ W 0: We shall use the following
notation: i : A ! X and j : W ! X are the inclusions and p : X ! A and
q : X ! W are the projections. Using X0; A0and W 0in place of X; A and W;
we obtain analogous maps i0; j0; p0and q0:
2.1. Definition. We respectively define maps h0 : A ^ X0 ! A ^ W 0and k0 :
A ^ X0! X _ X0 as the following compositions:
0 p^ id
A ^ X0i^-id!(X) ^ X0 X ^ (X0) id^q-!X ^ (W 0) (X) ^ W 0-! A ^ W 0
and
A ^ X0i^-id!(X) ^ X0 (X ^ X0) -w!X _ X0;
where w is the generalized Whitehead product map [Ar].
2.2. Lemma. There is a commutative diagram of homology groups
H* (A ^ X0) H* A ^ (X0)
#(h0)* #(id^q0)*
H* (A ^ W 0) H* A ^ (W 0) ;
where the horizontal isomorphisms are induced by the natural homeomorphisms.
Consequently, (h0)* is onto and, if e is the composition
0
A ^ A0 id^i-!A ^ (X0) (A ^ X0);
4 MARTIN ARKOWITZ AND DONALD STANLEY
then Kernel(h0)* = Image e*:
0
Proof. The proof consists of showing that the maps (A ^ X0) h-! (A ^ W 0)
0
A ^ (W 0) and (A ^ X0) A ^ (X0) id^q-!A ^ (W 0) are homotopic. This is a
long, but straightforward calculation, and hence omitted.
2.3. Lemma. k0*= 0 : H*(A ^ X0) -! H*(X _ X0):
Proof. This is a consequence of w* = 0: The latter is seen as follows: If O : X*
* x
X0- ! X ^ X0 is the projection, then w O : (X x X0) -! X _ X0 is defined
as a commutator using the suspension structure of (X x X0) (see [Ar] for detail*
*s).
Therefore (w O)* = 0: Since (O)* is onto, we obtain w* = 0:
We note that A is a co-H-space since it is a retract of X: Thus A ^ X0 is a
co-H-space. Therefore one can add two maps defined on A ^ X0:
2.4. Definition. We define h; k : A ^ X0 -! (X _ X0) _ (A ^ W 0) as the
respective compositions
0 i2
A ^ X0-h! A ^ W 0 -! (X _ X0) _ (A ^ W 0) and
0 i1
A ^ X0-k! X _ X0 -! (X _ X0) _ (A ^ W 0);
where i1 and i2 are the inclusions. Then define g : A^X0- ! (X _X0)_(A^W 0)
by g = h + k:
2.5. Lemma. g* = h* : H*(A ^ X0) ! H* (X _ X0) _ (A ^ W 0) :
Proof. This follows from Lemma 2.3 since g* = h* + k*.
x3. The Main Result
The standing hypothesis of x2 also holds in this section. We consider the com-
THE CONE LENGTH OF A PRODUCT OF CO-H-SPACES 5
mutative diagram
A ^ X0 -g! X _ X0_ (A ^ W 0) -a! Cg -b! (A ^ X0)
#id #r #"r
0
A ^ X0 -k! X _ X0 -! Ck0
#i^id
(X) ^ X0 #id #s
#
(X ^ X0) -w! X _ X0 -! X x X0
#p_p0 #pxp0
A _ A0 -! A x A0 -O!A ^ A0;
where the first three horizontal lines are mapping cone sequences, r is the pro*
*jec-
tion, eris induced by the upper left hand square and s is induced by the lower *
*left
hand square. Note that by [Ar] the mapping cone Cw of the generalized White-
head product map w is homotopically equivalent to X x X0: For notational
convenience set
ss = (p _ p0)r and = (p x p0)ser
so we have a commutative diagram
X _ X0_ (A ^ W 0) --a--! Cg
?? ?
yss ?y
A _ A0 ----! A x A0:
Now let ` be the composition
0 i1
W _ W 0j_j-!X _ X0- ! (X _ X0) _ (A ^ W 0)
and form the mapping cone sequence
W _ W 0a`-!Cg -c!Ca`;
that is, Ca` is the mapping cone of a` with inclusion c : Cg ! Ca`: Then a` =
ss` = (p _ p0)ri1(j _ j0) = 0 since ri1 = id; pj = 0 and p0j0= 0: Thus there ex*
*ists
an f : Ca`! A x A0such that the following diagram commutes
Cg
#c &
Ca` -f! A x A0:
6 MARTIN ARKOWITZ AND DONALD STANLEY
3.1. Proposition. The map f induces a homology isomorphism f* : H*(Ca`) !
H*(A x A0): Consequently, f is a homotopy equivalence.
Proof. The proof proceeds by a sequence of numbered steps.
(1) There exists a map e : (A ^ X0) ! A ^ A0 such that the following diagram
commutes
Cg ---b-! (A ^ X0)
?? ?
y ?y"
A x A0 ---O-! A ^ A0:
By Lemmas 2.2 and 2.5,
Imageb* = Kernel(g)* = Kernel(h)* = Kernel(h0)* = Image e*;
0
where e is A ^ A0id^i-!A ^ X0 (A ^ X0):
(2) We show that e*e* = id: H*(A ^ A0) -! H*(A ^ A0): By [Ru, Theorem 1], the
following square commutes
Ck0 -! (A ^ X0)
#(i^id)
#s ((X) ^ X0)
#
X x X0 -O! X ^ X0:
Since = (p x p0)ser; it follows that e can be taken to be the composition
0 0p^p0 0
(A ^ X0) (i^-id)! (X) ^ X ) X ^ X -! A ^ A :
Therefore ee is
0 (i^ id) p^p0
A ^ A0id^i-!A ^ X0 (A ^ X0) -! (X ^ X0) X ^ X0 -! A ^ A0:
But this latter composition is just (p ^ p0)(i ^ i0) = id; and so e*e* = id:
(3) Now we show that f* is onto. Let z 2 H*(A x A0): Then
O*(z) = e*b*(u) for some u 2 H*(Cg) by (1) and (2)
= O**(u):
THE CONE LENGTH OF A PRODUCT OF CO-H-SPACES 7
Therefore
z - *(u)= *(v) for some v 2 H*(A _ A)
= *ss*(w) for some w 2 H*(X _ X0_ (A ^ W 0))
= *a*(w):
Therefore
z = * u + a*(w) = f*c*(u + a*(w) :
Thus z 2 Image f*; and so f* is onto.
(4) Next we consider the exact homology mapping cone sequence
0 0 a* b* 0
H*(A ^ X0) -g*!H* X _ X _ (A ^ W ) -! H*(Cg) -! H* (A ^ X ) :
Now g* = i2*h0*by Lemma 2.5 and h0*: H*(A ^ X0) -! H*(A ^ W 0) is onto by
Lemma 2.2. Thus we have a short exact sequence
0 -! H*(X _ X0) (ai1)*-!H*(Cg) -b*!Imageb* = Image e* -! 0:
(5) Now we show Kernel * Kernelc*: Suppose x 2 H*(Cg) and *(x) = 0:
Therefore
e*b*(x) = O**(x) = 0:
But b*(x) = e*(y) for some y 2 H*(A ^ A0) by (1). Thus by (2)
0 = e*e*(y) = y
and so b*(x) = 0: By (4), there exists z 2 H*(X _ X0) such that a*i1*(z) = x:
Consequently,
*a*i1*(z) = *(x) = 0:
Therefore, *ss*i1*(z) = 0; and so ss*i1*(z) = 0: >From the definition of ss; we*
* have
(p _ p0)*(z) = 0: Thus z = (j _ j0)*(w) for some w 2 H*(W _ W 0): Hence
x = a*i1*(z) = a*i1*(j _ j0)*(w) = (al)*(w):
By exactness, c*(x) = 0; so x 2 Kernelc*: This shows Kernel* Kernelc*:
(6) Next we see that c* : H*(Cg) -! H*(Ca`) is onto. Consider the mapping cone
sequence of a`
W _ W 0a`-!Cg -c!Ca`- t!(W _ W 0) (a`)-!Cg:
8 MARTIN ARKOWITZ AND DONALD STANLEY
Since (a`)* is a monomorphism by (4), (a`)* is a monomorphism. Therefore
t* = 0 : H*(Ca`) ! H* (W _ W 0) , and so c* is onto.
(7) Finally, we show that f* : H*(Ca`) ! H*(AxA0) is one-one. Suppose f*(x) = 0
for x 2 H*(Ca`): By (6), x = c*(y) for some y 2 H*(Cg): Thus
0 = f*(x) = f*c*(y) = *(y):
Consequently y 2 Kernel* and so y 2 Kernelc* by (5). Hence x = c*(y) = 0:
Therefore f* is one-one.
This proves that f* is an isomorphism. To complete the proof we first note th*
*at
A and A0are 1-connected since they are retracts of X and X0respectively. Thus
f is a homotopy equivalence since Ca`is also 1-connected.
3.2. Theorem. Let A; W and X be spaces such that X A _ W and A0; W 0
and X0 be spaces such that X0 A0_ W 0and set
Y = (A ^ X0) _ W _ W 0_ (W ^ W 0) and
Z = X _ X0_ (A ^ W 0) _ (W ^ W 0):
Define ' : Y ! Z by
fi fi fi fi
'fiA^X0= i0g; 'fiW = i1j; 'fiW0 = i2j0; and 'fi(W^W0) = i4;
where g : A ^ X0 ! X _ X0 _ (A ^ W 0) is given by Definition 2.4 and i0 :
X _ X0_ (A ^ W 0) ! Z; i1 : X ! Z; i2 : X0! Z and i4 : (W ^ W 0) ! Z
are all inclusions. If C' is the mapping cone of '; then A x A0 C':
fi
Proof. Let '0= 'fi(A^X0)_W_W0 : (A^X0)_W _W 0! X _X0_(A^W 0):
Then ' = '0_ id: Therefore C' C'0: But by Proposition 3.1, we have C'0
Ca` A x A0:
The following corollary immediately implies the theorem stated in the introdu*
*c-
tion.
3.3. Corollary. Given 1-connected co-H-spaces A and A0. Then there exists a
space Y , a suspension Z and a map ' : Y ! Z such that A x A0 C', the
mapping cone of '.
Proof. By [Ta], there exist spaces X; X0; W and W 0such that X A _ W and
X0 A0_ W 0. If ' : Y ! Z is as defined in Theorem 3.2, then A x A0 C'.
To complete the proof we show that Z is a suspension:
THE CONE LENGTH OF A PRODUCT OF CO-H-SPACES 9
Z = X _ X0_ (A ^ W 0) _ (W ^ W 0)
0
X _ X0_ (A _ W ) ^ W
X _ X0_ (X ^ W 0)
0 0
X _ X _ (X ^ W ) :
3.4. Remarks. (1) Corollary 3.3 implies that if A and A0are co-H-spaces that are
not suspensions, then Takens's inequality
cl(A x A0) catA + max {clA0; 1}
is strict since the right hand side is 3: In particular, we could take A = A0= *
*X(p),
for p an odd prime.
(2) As observed earlier in this section, the product R x S of two suspensions
has the homotopy type of the mapping cone of a map Y ! R _ S: In fact,
Y = (R ^ S) (R) ^ S and the map is the generalized Whitehead product
map. Rutter [Ru] has shown how to extend this result to the case of the product
of a suspension and a co-H-space. He proves that if A0 is a co-H-space, then
(R) x A0 has the homotopy type of the mapping cone of a map Y ! R _ A0;
where Y = R ^ A0: This raises the question (see [F-S, x1]) of whether the produ*
*ct
A x A0 of two co-H-spaces can be represented as the mapping cone of some map
Y ! A_A0: The theorem in x1 shows that AxA0can be represented as a mapping
cone.
(3) The theorem in Section 1 shows that
cl(A x A0) catA + catA0
in the case when catA = catA0= 1. We ask: Is this formula true for other values
of catA and catA0?
References
[Ar]Arkowitz, M., The generalized Whitehead product, Pac. J. Math. 12 (1962), 7*
*-23.
[B-H]Berstein, I. and Hilton, P., Category and generalized Hopf invariants, Ill*
*. J. Math. 4 (1960),
437-451.
[F-S]Fernandez-Suarez, L., On a problem of Ganea about strong category, Top. an*
*d Applications
(to appear).
[Fo]Fox, R., On the Lusternik-Schnirelmann category, Annals of Math. 42 (1941),*
* 333-370.
10 MARTIN ARKOWITZ AND DONALD STANLEY
[Ga1]Ganea, T., Lusternik-Schnirelmann category and strong category, Ill. J. Ma*
*th. 11 (1967),
417-427.
[Ga2]Ganea, T., Some problems on numerical homotopy invariants, Springer Lectur*
*e Notes in
Math. 249 (1971), 23-30.
[Ja1]James, I., On category in the sense of Lusternik-Schnirelmann, Topology 17*
* (1978), 331-
348.
[Ja2]James, I., Lusternik-Schnirelmann category, Handbook of Algebraic Topology*
*, North Hol-
land, 1996, pp. 1293-1310.
[Ru]Rutter, J., A coclassifying map for the inclusion of the wedge in the produ*
*ct, Math. Zeit.
129 (1972), 173-183.
[St1]Stanley, D., Spaces with Lusternik-Schnirelmann category n and cone length*
* n+1, Topology
(to appear).
[St2]Stanley, D., On the Lusternik-Schnirelmann category of maps (to appear).
[Ta]Takens, F., The Lusternik-Schnirelmann categories of a product space, Comp.*
* Math. 22
(1970), 175-180.
Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA
E-mail address: Martin.Arkowitz@dartmouth.edu
Department of Mathematical Sciences, University of Alberta, Edmonton, Al-
berta, Canada T6G 2G1
E-mail address: stanley@math.ualberta.ca