HAVING THE H-SPACE STRUCTURE IS NOT A
GENERIC PROPERTY
JIANZHONG PAN
Abstract. In this note, we answer in negative a question posed
by McGibbon[7] about the generic property of H-space structure.
In fact we verify the conjecture of Roitberg [12]. Incidentally, the
same example also answers in negative the open problem 10 in
McGibbon[7]
1. Introduction
Let X be a connected CW complex, the L-S category of X, cat(X),
of X is the least integer k 0 such that X can be covered by k + 1
open subsets which are contractible in X. Of course the condition that
X is a CW complex is unnecessary for the above definition to have a
meaning . However it is in this context that a rich theory of category
exists. Recent works in rational homotopy theory gave rise to a theory
of rational L-S category. This makes it possible to calculate the rational
L-S category and thus attack the rational Ganea Conjecture[1]. On the
other hand works by N.Iwase, H.Scheerer, and D.Stanley provided the
method to determine the L-S category itself in some case which lead
to the construction of counterexamples to the Ganea Conjecture, see,
____________
Date: June,5,2000.
1991 Mathematics Subject Classification. 55P60,55P45.
Key words and phrases. H-space, Phantom map,Mislin genus.
The author is partially supported by the NSFC project 19701032 and ZD9603
of Chinese Academy of Science .
1
2 JIANZHONG PAN
e.g., [14 ]. Besides the application to the Ganea Conjecture , another
interesting application of these ideas was given by Roitberg[12 ]. To
explain this we have to state one problem posed by McGibbon[7]:
Question 1.1. If X and Y have the same Mislin genus ,i.e. X(p)'
Y(p)for all p where X(p)is the p-localization of X, does it follow that
cat(X) = cat(Y )?
In his paper[12 ], applying some results about category by Iwase[4]
and results about phantom maps , Roitberg was able to answer the
above question negatively. His main result can be stated as follows
Theorem 1.2. Let OE : K(Z, 5) ! S4 be an essential, special, phan-
tom map and X be the mapping cone of OE .Then cat(X) = 2.
Remark 1.3. It is well known that cat(X) = 1 iff X is a co-H-space .
It follows from the above theorem that X is not a co-H-space . On the
other hand it easy to know that S4 _ 2K(Z, 5) has the same Mislin
genus with X and is a co-H-space .
From this point of view , the theorem above answers in negative the
following
Question 1.4. If X and Y have the same Mislin genus and X is co-
H-space , does it follow that Y is also a co-H-space?
Question1.1 has an obvious Eckmann-Hilton dual . However it may
not be a good question at present time since the dual L-S category
is not well developed. A manageable problem is the obvious dual of
Question1.4 which has been posed by McGibbon in [7]. A more precise
conjecture was given by Roitberg in his paper[12 ]. The purpose of this
paper is to establish Roitberg's conjecture and thus also answers in
negative McGibbon's problem.
HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 3
Theorem 1.5. Let _ : K(Z, 2) ! S6 be an essential, special, phan-
tom map and Z be the homotopy fiber of _. Then Z and K(Z, 2)x 2S6
have the same Mislin genus and Z is not an H-space.
Since the dual L-S category is not well developed and Iwase's paper[4]
is unavailable to the author, method from the well developed theory
of H-space will be used in stead. Another feature is the application of
the newly developed Gray index of phantom map [5],[9]. Actually , by
duality , our method also gives an alternative proof of the main results
in [12 ].
Incidentally , the example constructed above combined with Theo-
rem 3.4 in [3] also provides a negative answer to the open problem 10
in [7]: Is X an H-space if each of its Postnikov approximations X(n) is
? Actually we have the following
Theorem 1.6. Let _ : K(Z, 2) ! S6 be an essential, special, phan-
tom map and Z be the homotopy fiber of _. Then Z(n) is an H-space
for each n but Z is not.
Proof. It follows immediately from Theorem 3.4 in [3] that Z(n) and
(K(Z, 2) x 2S6)(n)have the same homotopy type and thus Z(n) is an
H-space for each n.
In this paper all spaces involved are assumed to be 1-connected CW
complexes with finite type.
The author would like to thank Prof. Roitberg for his interest in
this work and for pointing out a fatal error in the earlier version of this
paper . It is to correct that error that we find the application of Gray
index to this work.
4 JIANZHONG PAN
2. Background about H-spaces and Phantom maps
First we will recall some backgrouds about H-spaces , see [17 ] for
details. An H-space is a space X with a map ~ : X x X ! X such
W
that ~|X_X = F where F : X X ! X is the natural folding map.
An H-map between H-spaces is a map of spaces f : X ! Y such that
the following diagram commutes up to homotopy.
fxf
X x X --- ! Y x Y
? ?
~X ?y ~Y?y
f
X --- ! Y
In this case we say that f is a ~X - ~Y H-map. Two elementary but
important results are the followings
Proposition 2.1. Let (X, ~X ) be an H-space. Then, for any space M,
[M, X] is an algebraic loop, i.e., for any f, g 2 [X, Y ] there exists a
unique Df,g2 [M, X] such that
~*(Df,g, g) = f
Proposition 2.2. If f : (X, ~X ) ! (Y, ~Y ) is an H-map , then the
homotopy fiber of f is an H-space.
Thus it is important to know when is a map an H-map or what is
the obstruction for a map to be an H-map.
Definition 2.3. Let (X, ~) and (Y, ~0) be H-spaces and f : X ! Y be
a map of spaces. H-derivation of f is the map
HD(f) 2 [X ^ X, Y ]
which is defined by
HD(f) = Df~,~0(fxf)
where : X x X ! X ^ X is the natural quotient map.
HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 5
Remark 2.4. The definition of H-derivation depends on the H-space
structures on both X and Y .
Remark 2.5. Let (X, ~) and (Y, ~0) be H-spaces and f : X ! Y be
a map of spaces. It is well known that f : (X, ~X ) ! (Y, ~Y ) is an
H-map iff HD(f) = *.
An easy but crucial corollary of this last remark is the following
Corollary 2.6. Let (X, ~) and (X0, ~0) be H-spaces and f : X ! X0
be a map of spaces. Assume further that ßiX0 = 0 for i 2d and X is
(d - 1)-connected . Then f is a ~ - ~0 H-map.
Following is one of the fundamental properties of H-derivation
Proposition 2.7. Let (Xi, ~i) be H-spaces, i = 0, 1, 2 and f : X0 !
X1, g : X1 ! X2 be maps of spaces.
(a)If f : X0 ! X1 is a ~0 - ~1 H-map, then HD(gf) = HD(g)(f ^ f)
(b)If g : X1 ! X2 is a ~1 - ~2 H-map, then HD(gf) = gHD(f)
Another ingredient for the main result is the phantom map . Recall
that a map f from a CW complex X is called an phantom map if
its restriction to the n-th skeleton is inessential for any integer n. Let
P h(X, Y ) denote the set of homotopy classes of phantom maps from X
to Y . The following result which follows from the Sullivan conjecture
provides us many examples of phantom maps.
Theorem 2.8. [8] Let Y = iK and X = jZ such that i, j 0 ,K
is a 1-connected finite CW complex. Then every map from X to Y is
a phantom map if Z is as follows:
o Z is the classifying space of a 1-connected compact Lie group
o Z is an infinite loop space with torsion fundamental group
o Z has only finitely number of nontrivial homotopy groups
6 JIANZHONG PAN
and in this case we have
Y O
P h(X, Y ) = [X, Y ] = [X(0), Y ] = Hn (X, ßn+1(Y ) R)
n>0
Let P h(X, Y ) denote the set of homotopy classes of phantom maps
from X to Y . The p-localization lp induces a natural map
l*p: P h(X, Y ) ! P h(X, Y(p))
It follows that there is a natural map
Y
l : P h(X, Y ) ! P h(X, Y(p))
p
It is well known that l is an epimorphism [15 ] and Ker(l) is nontrivial
iff P h(X, Y ) is nontrivial[7], see also [3],[11 ] . The phantom map in
Ker(l) is called special , following Roitberg[12 ], see also,[6] where it is
called the clone of constant map.
On the other hand , Gray, Le Minh Ha , McGibbon and Strom [2],
[5],[9] introduced the notion of Gray index which is defined as follows:
Definition 2.9. Let f : X ! Y be a phantom map. Then f can be
~f
factorized as the composition X ! X=Xk ! Y for each k . The Gray
index of f , denoted by G(f), is the largest integer k such that the f~
can be chosen to be a phantom map. G(f) = 1 if no such k exists.
Remark 2.10. Let f : X ! Y be a phantom map. Then f can be lifted
to the k-th connected covering for each k and G(f) + 1 is the largest
integer k such that the kth lifting can be chosen to be a phantom map.
A useful fact we need is
Proposition 2.11. Let f : X ! Y be a phantom map. Then
(i)G(f) n if X is n-connected or Y is n + 1-connected.
N N
(ii) G(f) 2 {k|Hn (X, ßn+1(Y ) Q) 6= 0} if Hn (X, ßn+1(Y ) Q) = 0
for n sufficiently large.
HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 7
For the proof of the Proposition above, see [5] and [9].
An immediate corollary of the above Proposition which is crucial to
our purpose is
Corollary 2.12. Let f : K(Z, 2m) ^ K(Z, 2m) ! 1S4m+2 be any
essential map. Then G(f) = 4m where m 1.
Now we are ready to prove the main result.
3. Proof of Theorem1.5
First is a preliminary lemma needed later.
Lemma 3.1. Let X be an H-space which is (e - 1)-connected and
ßi(X) = 0 for i 2e and Y = jK where K is a (d + j - 1)-connected
finite CW-complex with j 1 and d > e 2. Let _ : X ! Y be
any essential map. Then Z(=homotopy fiber of _) is not an H-space
if HD(_) O (i ^ i) is essential where i : Z ! X is the homotopy fiber
of _ .
Proof. If Z is an H-space , then * = _ O i : Z ! Y is an H-map
and HD(*) = 0. Since i is an H-map by Corollary2.6 , it follows by
Proposition2.7 that
* = HD(*) = HD(_ O i) = HD(_) O (i ^ i)
which is in contradiction to the condition.
Remark 3.2. To apply the above lemma it suffices to discuss when _
is not an H-map and when i ^ i induces an injective.
Theorem 3.3. Let X = K(Z, 2m) and Y = 1S4m+2 with m 1.
Then there is no essential H-map from X to Y .
8 JIANZHONG PAN
Proof. By Proposition2.8, the rationalization r : X ! X(0)which is an
H-map induces an isomorphism of groups
r* : [X(0), Y ] ! [X, Y ]
It follows from Proposition2.7 that it suffices to prove that there is no
essential H-map from X(0)to Y .
On the other hand the map h : S4m+2 ! K(Z, 4m + 2) which repre-
sents a generator of H4m+2 (S4m+2 ; Z) ~= Z induces an isomorphism of
groups
( 3h)* : [X(0), Y ] ! [X(0), K(Z, 4m + 1)]
Again the Proposition2.7 implies that it suffices to prove that there is
no essential H-map from X(0)to K(Z, 4m + 1) which is well known to
be equivalent to the injectivity of the following homomorphism
`* : H4m+2 ( X(0); Z) ! H4m+2 ( X(0)^ X(0); Z)
where ` is defined as follows:
Let X * X = X x I x X={(x, 0, y) ~ (x, 0, y0), (x, 1, y) ~ (x0, 1, y)}
be the join . There is a well defined map
k : X * X ! X ^ X
by k[x,t,y]=(x,y,t). It is well known that X * X is homotopy equivalent
to X ^ X. If X is an H-space with multiplication ~ , then ` is the
composite map
X ^ X ' X * X !k (X x X) ! X
where the last map is the map - ß1 + ~ - ß2 and ß1, ß2 are the
projection of X x X to the first and second factors respectively .
HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 9
Consider the following commutative diagram where the horizontal
maps which are isomorphisms come from the universal coefficient The-
orem
H4m+2 ( X(0); Z) --- ! Ext (H4m+1 ( X(0); Z), Z)
? ?
`*?y Ext(`*,Z)?y
H4m+2 ( X(0)^ X(0); Z) --- ! Ext(H4m+1 ( X(0)^ X(0); Z), Z)
It follows that it suffices to prove that the map
`* : H4m+1 ( X(0)^ X(0); Z) ! H4m+1 ( X(0); Z)
or equivalently the map
`* : H4m (X ^ X; Q) ! H4m (X; Q)
is injective . On the other hand, it is well known that the map `* is
dual to the reduced coproduct which is an isomorphism in this case
and thus completes the proof.
The Theorem1.5 is actually a corollary of the following more general
Theorem
Theorem 3.4. Let X = K(Z, 2m) and Y = S4m+2 with m
1. Let _ : X ! Y be any essential, special, phantom map. Then
Z(=homotopy fiber of _) is not an H-space and has the same Mislin
genus with X x Y .
Proof. That Z and X x Y have the same Mislin genus follows from
the condition that _ : X ! Y is a special phantom map.
On the other hand Lemma3.1 and Theorem3.3 apply here. Thus the
Theorem above follows from the following Proposition.
Proposition 3.5. Let X = K(Z, 2m) and Y = S4m+2 with m 1.
Let _ : X ! Y be any essential, special, phantom map which exists by
10 JIANZHONG PAN
Proposition2.8 and the remark after it. Then (i ^ i)* : [X ^ X, Y ] !
[Z ^ Z, Y ] is injective where i : Z ! X is the homotopy fiber of _ .
Proof. Let f : X ^ X ! Y be any essential map . If f O (i ^ i) ~ * we
will prove that this leads to a contradiction which concludes the proof.
Since f is a phantom map , f O (i ^ i) ~ * is also a phantom map.Thus
we have the following commutative diagram up to homotopy.
f
Z ^ Z --i^i-! X ^ X --- ! Y
? ? ?
? ? ?
y y idy
i(0)^i(0) f~
Z(0)^ Z(0) --- - ! X(0)^ X(0) --- ! Y
If f O (i ^ i) ~ * , then f~O (i(0)^ i(0)) is the composite Z(0)^ Z(0)!
Zfi^ Zfi!h Y where Zfiis the homotopy fiber of the rationalization
X ! X(0). On the other hand we claim that
Claim 3.6. Any map h : Zfi^ Zfi! Y factors through a map Zfi^
Zfi! Ffi^ Ffiwhere F = 2S4m+2 .
Assuming this , note that i(0)^ i(0)admits a right inverse, we have
that f is the composite X ^ X ! X(0)^ X(0)! Ffi^ Ffi! Y .
It is easy to know that Ffiis 4m - 1-connected and thus Ffi^ Ffi
is 8m - 2-connected. It follows that f is the composite X ^ X !
X(0)^ X(0)! Y < 8m - 2 >! Y . By Remark2.10, G(f) 8m - 3
which contradicts Corollary2.12.
Remark 3.7. Roitberg has shown us how the use of Gray index can be
avoided.
It remains to prove the Claim3.6 which follows from the following
Lemma 3.8. There is a map g ^ g : Zfi^ Zfi! Ffi^ Ffisuch that
the following map is a weak homotopy equivalence
( g ^ g)* : map*( Ffi^ Ffi, Y ) ! map*( Zfi^ Zfi, Y )
HAVING THE H-SPACE STRUCTURE IS NOT A GENERIC PROPERTY 11
Proof. Roitberg and P. Touhey proved in [13 ] that , if X, Y have the
same Mislin genus, then Xfi' Yfi. So we have a map g which is a
composite
Zfi' (X x F )fiiø!Ffi
where ß : X x F ! F is the projection.
To prove g ^ g induces a homotopy equivalence it suffices to prove
that ßfi: (X x F )fi! (F )fiinduces a homotopy equivalence
( ßfi^ ßfi)* : map*( Ffi^ Ffi, Y ) ! map*( (X x F )fi^ (X x F )fi, Y )
which follows directly from the fact that map*(Xfi, Y ) is weakly con-
tractible and the fact
map*( Xfi, Y ) ' map*( X, ^Y)
which can be found in [16 ] , for a stronger result , see Pan and Woo
[10 ].
_____________________-
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12 JIANZHONG PAN
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_____________________-
Institute of Math.,Academia Sinica ,Beijing 100080, China
E-mail address: pjz@math03.math.ac.cn