PHANTOM ELEMENTS AND ITS APPLICATIONS
JIANZHONG PAN AND MOO HA WOO
Abstract. In our previous work[11] , a relation between Tsukiyama
problem about self homotopy equivalence was found by using a
generalization of phantom map. In this note , fundamental result
is established for such a generalization. This is the first time one
can deal with phantom maps to space not satisfying finite type
condition. Application to Forgetful map is also discussed briefly.
1. Introduction
The main aim of this paper is to study phantom map , its gener-
alization and applications. After the discovery of the first example
of phantom map by Adams and Walker[1] , theory of phantom map
receives a lot of attention . The main aim of these previous studies
is however to understand it , e.g., the computation and the proper-
ties of phantom maps. The first application of theory of phantom
maps was given by Harper and Roitberg[5],[12 ] who applied it to com-
pute SNT (X) and Aut(X). Recently applications are also found by
Roitberg[13 ] and Pan[10 ] where several conjectures of McGibbon were
settled. On the other hand , a remarkable connection was established
by Pan and Woo[11 ] between Tsukiyama problem about self homotopy
equivalence and a generalization of phantom map. A byproduct of this
connection is that a special case of Tsukiyama problem is almost equiv-
alent to the famous Halperin conjecture in rational homotopy theory[4].
____________
Date: Aug. 30, 2000.
1991 Mathematics Subject Classification. 55P10,55P60,55P62,55R10.
Key words and phrases. Phantom map,Forgetful map, Halperin conjecture.
The first author is partially supported by the NSFC project 19701032 and
ZD9603 of Chinese Academy of Science and the second author wishes to acknowl-
edge the financial supports of the Korea Research Foundation made in the program
year of (1998) and TGRC 99.
1
2 JIANZHONG PAN AND MOO HA WOO
A well known characterization of map between nilpotent spaces of
finite type to be phantom map is the following
Theorem 1.1. Let X, Y be nilpotent CW complexes of finite type with
Y 1-connected and f : X ! Y be any map. Then the followings are
equivalent
o f is a phantom map
o e O f ' * where e : Y ! ^Y is the profinite completion
o f O ø ' * where ø : Xfi! X is the homotopy fiber of the
rationalization
On the other hand , in our previous paper[11 ], we generalized the
concept of phantom map to that of phantom element and announced
a theorem characterizing an element to be a phantom element which
generalizes Theorem1.1 . In this paper we will generalize further so
that we can deal with space which is not of finite type.
Theorem 1.2. Let X be nilpotent CW complex of finite type , Y be
1-connected such that ßn(Y ) is reduced group for n 2 and g : X ! Y
be any map.Then the followings are equivalent:
o ff 2 ßj(map*(X, Y ); g) is a phantom element
o (e*)# (ff) = 0 where (e*)# : ßj(map*(X, Y ); g) ! ßj(map*(X, ^Y); ^g)
o (ø *)# (ff) = 0 where (ø *)# : ßj(map*(X, Y ); g) ! ßj(map*(Xfi, Y ); gf*
*i)
Note that the assumption that Y is 1-connected is not a real restric-
tion by an observation of Zabrodsky[15 ]. We will give a complete proof
of this theorem in this paper.
As an application we have(Corollary of Proposition 3.3)
Corollary 1.3. Let P be 1-connected finite dimensional CW complex
or such that H*(P, Zp) is locally finite over Ap for each prime p and
be of type F0 . Assume further that ßnBaut(P ) is reduced group for
n 2. Assume further that P(0)satisfying one of the following.
o P is rationally equivalent to Kähler manifold
o H*(P ; Q) as an algebra has at most 3 generators
o P is rationally equivalent to G=U where G is a compact Lie
group and U is a closed subgroup of maximal rank
PHANTOM ELEMENTS AND ITS APPLICATIONS 3
Then for all m 1 , H and every principal K(H, 2m)-bundle with total
space homotopy equivalent to P , Forgetful map is injective.
The organization of this paper is as follows. In section2 Theorem1.2
will be proved. The applications to Forgetful map will be discussed in
section3. In this paper, We will use the following notations:
o H will denote a finitely generated abelian group
o map(X, Y ) is the space of continuous mappings from X to Y
o map*(X, Y ) is the subspace of pointed mappings from (X, x0)
to (Y, y0)
o l : X ! X(0)is the rationalization
o Let ø : Xfi! X be the homotopy fiber of l. Then Xfi!fiX !
X(0)is a cofibration up to homotopy
o ep : Y ! ^YZp1is Bousfield-Kan's p-completion. Let Y^ =
Q ^
YZp1 and e = (e2, e3, . .).: Y ! Y^. Let Yj be the homo-
p
topy fiber of e
The readers should refer to [11 ] for all the other notations which have
not been explained here.
In concluding the Introduction , we 'd like to give the following
Conjecture 1.4. The condition that ßnY is reduced group in this paper
can be removed.
2. Phantom elements
Let's begin with definition.
Definition 2.1. Let spaces X be a CW complex, Y be a space and
g : X ! Y any map. Then an element ff 2 ßj(map*(X, Y ); g) is
called a g-phantom element if (i*n)# (ff) = 0 for all n 0 where (i*n)# :
ßjmap*(X, Y ) ! ßjmap*(Xn , Y ) is the homomorphism induced by the
inclusion in : Xn ! X. Denoted by
P hgj(X, Y ) = {ff 2 ßj(map*(X, Y ); g)|ff is a g-phantom element }
Obviously if g =constant and j = 0 , then ff is a g-phantom element
iff it represents the homotopy class of a map which is a phantom map.
4 JIANZHONG PAN AND MOO HA WOO
The main aim of this section is to prove Theorem1.2. Before that ,
let's give some results necessary to the proof .
Lemma 2.2. Let Y be 1-connected such that ßn(Y ) is reduced group
for n 2. Then
Q
o ßn(Y^) = Ext(Zp1 , ßn(Y ))
p
o For W a finite CW complex, e* : ßjmap*(W, Y )f ! ßjmap*(W, ^Y)f^
is injective
Proof. The first statement follows from the fact that Hom(Zp1 , B) = 0
for a reduced group since otherwise there will be nontrivial divisible
subgroup in B.
To prove the second statement , note that the induced map ßn(Yj) !
ßn(Y ) is trivial since ßn(Y ) is reduced group and ßn(Yj) is rational
thus divisible by the arithmetic square Theorem[3]. It follows that
e* : ßn(Y ) ! ßn(Y^) is injective and an easy induction argument shows
what we want for j 1. For j = 0, it can be proved by an argument
similar to that of Theorem 2.5.3 of [6].
Proposition 2.3. Let X be nilpotent space and Y be 1-connected such
that ßn(Y ) is reduced group for n 2. Then the followings hold:
o [ nX(0), ^Y] = *, ~Hn(X(0), ßi(Y^)) = 0 for all n, i 0
o [ nXfi, Yj] = *, ~Hn(Xfi, ßi(Yj)) = 0 for all n, i 0
Proof. That H~n(X(0), ßi(Y^)) = 0 follows from the fact that
Hom(A, B) = 0, Ext(A, B) = 0
Q
for ratinal group A and B = Ext(Zp1 , B0)
p
Then the equation
[ nX(0), ^Y] = lim [ nX(0), ^Y (n)]
n
implies the first statement.
The equation about cohomology in second statement is true since
ßn(Yj) is rational while the proof of another equation is similar to that
as in the first statement.
PHANTOM ELEMENTS AND ITS APPLICATIONS 5
Proposition 2.4. Let X be nilpotent space and Y be 1-connected such
that ßn(Y ) is reduced group for n 2. Then the followings hold :
w
o ø *: map*(X, ^Y) ' map*(Xfi, ^Y)
w
o æ* : map*(X(0), Yj) ' map*(X(0), Y )
w
o e* : map*(Xfi, Y ) ' map*(Xfi, ^Y)
w
o l* : map*(X(0), Yj) ' map*(X, Yj)
Proof. The first and the last statements follow from the last Propo-
sition and the well known Zabrodsky Lemma. The second and third
statements follow from a lim1 argument for j 0
* ! lim1ßj+1map*(Z, En) ! ßjmap*(Z, E) ! lim ßjmap*(Z, En) ! *
n n
where in the second statement, Z = X(0)and En is the n-th term in the
Postnikov-Moore tower of the map æ : Yj ! Y while in the third state-
ment , Z = Xfiand En is the n-th term in the Postnikov-Moore tower
of the map e : Y ! ^Y. In both case the sequence {ßjmap*(Z, En)} is
a sequence consisting of isomorphisms and thus the lim1 is trivial and
the wanted isomorphisms follows immediately.
Proposition 2.5. Let X be nilpotent space and Y 1-connected such
that ßn(Y ) is reduced group for n 2. Let g : X ! ^Y be any map.
Then
P hgj(X, ^Y) = *
Proof. P hgj(X, ^Y) is the lim 1 of a sequence of compact groups and
continuous homomorphisms which is will known to be trivial.
Proposition 2.6. Let X, Y be two nilpotent spaces with Y 1-connected
such that ßn(Y ) is reduced group for n 2. Then the followings hold :
w
o map*(Xfi, Y ) ' map*(X, ^Y)
w
o map*(X(0), Y ) ' map*(X, Yj)
Proof. This is an easy consequence of the Proposition above.
Proof of Theorem1.2. The equivalence between the last two statements
follows directly from the following commutative diagram where the
6 JIANZHONG PAN AND MOO HA WOO
bottom horizontal homomorphism and the right side vertical homo-
morphism are isomorphisms by Proposition2.4 .
(fi*)#
ßjmap*(X, Y ) -- - ! ßjmap*(Xfi, Y )
? ?
(e*)#?y (e*)#?y
(fi*)#
ßjmap*(X, ^Y) -- - ! ßjmap*(Xfi, ^Y)
Now assume the first statement, then we have (i*n)# (e*(ff)) = 0 for all
n 0. It follows from Proposition 2.5 that e*(ff)) = 0. The proof
of another direction is similar to that in [11 ] using Lemma2.2 instead
of Sullivan's origional result which is stated only for space of finite
type.
Remark 2.7. It is easy to see that the above proof follows the same
pattern as that given by Oda and Shitanda[9]. We give a prove here
because Oda informed us that there were gaps in their proof and he
don't know if the result is true or not. The similar proof applies also
to the equivariant case which will be discussed in future publication.
As noted in [11 ] , the natural question related to the application of
phantom element to the forgetful map is
Question 2.8. For two maps f, g : X ! Y ,what is the relation between
P hgj(X, Y ) and P hfj(X, Y )?
Proposition 2.9. Let X, Y be nilpotent CW complexes such that
[ jXfi, Y ] = [ j+1Xfi, Y ] = 0
If g : X ! Y is a phantom map,then we have
P hgj(X, Y ) = ßj(map*(X, Y ); g)
Proof. The proof is the same as that in [11 ] .
In our application we have to be able to compute P hgj(X, Y ). Before
giving this kind of result, recall that a CW complex is called unstable
if all the attaching maps vanish under suspension. It is Baues [2] who
noted the following which is dual to Zabrodsky's integral approxima-
tion.
PHANTOM ELEMENTS AND ITS APPLICATIONS 7
Theorem 2.10. Let X be 1-connected CW complex. Then there is an
unstable complex and a rational equivalence h : X~ ! X.
Remark 2.11. Let X be an unstable CW complex. Then it is easy to
prove that P hgj(X, Y ) = * for any map g : X ! Y .
Proposition 2.12. Let X be a 1-connected CW complex and Y 1-
connected such that ßn(Y ) is reduced group for n 2. Suppose further
that the component of map*(X, ^Y) consisting constant map is weakly
contractible and g : X ! Y is a phantom map. Then
Y
P hgj(X, Y ) = ßj(map*(X, Y ); g) = Hk(X, ßk+j+1(Yj))
k>0
Proof. As first noted by Oda and Shitanda , similar proof as in that of
Theorem B of [15 ] leads to the following homotopy fibration
[
map*(X, Y )g ! map*(X~, Y )* ! map*(X~, ^Y)*
g
where the union is over phantom maps g. On the other hand , dif-
S
ferent components of map*(X, Y )g are homotopy equivalent since
S g
map*(X, Y )g is the homotopy fiber of a map between two connected
g
spaces. It follows that
P hgj(X, Y ) = ßj(map*(X, Y ); const) = ßj(map*(X, Yj); const) =
Y
= [ j-1X, Yj] = Hk(X, ßj+k+1(Yj))
k>0
We are ready to state results related to the applications. Before that
we have another definition
Definition 2.13. Let Ap be the modp Steenrod algebra. An unstable
module M over Ap is called locally finite iff , for any x 2 M, only finite
elements of Ap can acts nontrivially on M.
Example 2.14. Let P be a space such that H*(P, Zp) is locally finite
over Ap . Then so is P . In particular, if P is finite CW , then
H*( P, Zp) is locally finite over Ap.
8 JIANZHONG PAN AND MOO HA WOO
Theorem 2.15. Let X = K(H, m + 2) , Y = Baut(P ) such that
ßn(Y ) is reduced group for n 2 and g : X ! Y is any map where P
is 1-connected finite dimensional CW complex or such that H*(P, Zp)
is locally finite over Ap for each prime p and m 1. Then for j 1
P hgj(X, Y ) = ßj(map*(X, Y ); g) = [ jX, Yj]
Proof. The proof is the same as that of the corresponding result in [11 ]
using results of Zabrodsky and Miller[8] or Theorem 8.8 in [14 ].
Similarly we have
Theorem 2.16. Let X = BG, Y = Baut(P ) such that ßn(Y ) is re-
duced group for n 2. and g : X ! Y is a phantom map where G is a
connected compact Lie group and P is 1-connected finite dimensional
CW complex or such that H*(P, Zp) is locally finite over Ap for each
prime p. Then for j 1 we have
P hgj(X, Y ) = ßj(map*(X, Y ); g) = [ jX, Yj]
3. Application to the forgetful map
Given a principal G-bundle ß : P ! B, Let
autG (P ) = {g|g : P ! P is a G-equivariant homotopy equivalence }
and
aut(P ) = {g|g : P ! P is a homotopy equivalence }
There is a natural map f : autG (P ) ! aut(P ). Let
AutG (P ) = ß0(autG (P ))
and
Aut(P ) = ß0(aut(P ))
Then the map f induces a map
F : AutG (P ) ! Aut(P )
which is called a Forgetful map by Tsukiyama. The question posed by
Tsukiyama in [7] is the following
Question 3.1. Is the forgetting map F injective?
PHANTOM ELEMENTS AND ITS APPLICATIONS 9
One of the main results in [11 ] is the following
Theorem 3.2. Let ß : P ! B be a principal G-bundle. Then there is
an exact sequence
ß1aut(P ) !ffiß1(map*(BG, Baut(P )), c) ! AutG (P ) F! Aut(P )
where c : BG ! Baut(P ) is determined by the principal bundle.
Combined with results in [11 ] , we have
Proposition 3.3. Let P be as in Theorem2.16 . If
M
ß2i(map(P(0), P(0)); id) = 0
i>1
then for all m 1 , finitely generated abelian group H and every
principal K(H, 2m)-bundle with total space homotopy equivalent to P
,the associated Forgetful map is injective .
We have also similar results for K(H, 2m + 1) or G bundle where G
is a connected compact Lie group which will be omitted.
Unlike that in [11 ], there are no complete results if group
ß1(map*(BG, Baut(P )), c) is nontrivial although we know that it is still
uncountable since group ß1aut(P ) itself may be uncountable too. Thus
same results as in [11 ] can be obtained if ß1aut(P ) is countable. This
is so if P = P 0where P 0is finite complex. An interesting question is
Question 3.4. Study the map ffi in the exact sequence of Theorem 3.2.
Is it possible that Image(ffi) is always countable group.
_____________________-
References
[1]J.F.Adams and J.Walker,An example in homotopy theory,Proc. Camb. Phil.
Soc., 60 (1960), pp. 699-700
[2]H.J.Baues, Rationale Homotopietypen, Manus.Math. 20(1977), pp.119-131
[3]E.Dror, W.G. Dwyer, D.M.Kan, An arithmetic square for virtually nilpotent
spaces, Ill. J. Math., 21(1977), pp.242-254
[4]N.Dupont Problems and conjectures in rational homotopy theory, Expos.
Math., 12(1994), pp. 323-352
[5]J.R.Harper, J.Roitberg, Phantom maps and spaces of the same n-type for all
n , J. of Pure and Applied Algebra 80(1992), pp.123-137
10 JIANZHONG PAN AND MOO HA WOO
[6]P.Hilton, G.Mislin, J.Roitberg, Localization of nilpotent groups and spaces,
North Holland Math. Series 15, 1975
[7]D.W.Kahn, Some Research Problems on Homotopy-Self-Equivalences ,LN in
Math. v.1425 (1990),pp.204-207
[8]H.Miller, The Sullivan conjecture on maps from classifying spaces, Ann.Math*
*.,
120(1984) , pp.39-87
[9]N.Oda and Y.Shitanda,Localization,Completion and Detecting Equivariant
Maps on Skeletons , Manuscript Math.,65(1989), pp.1-18
[10]Pan Jianzhong, Having H-space structure is not a generic property, submitted
[11]Pan Jianzhong, Woo Mooha , Phantom maps and Forgetable maps, to appear
in J.Japan. Math. Soc.
[12]J.Roitberg, Note on phantom phenomena and group of self-homotopy equiva-
lences, Comment. Math. Helv. 66 (1991), pp. 448-457
[13]J.Roitberg, The Lusternik-Schnirelmann category of certain infiniteCW
complex-complexes, Topology 39(2000), pp.95-101
[14]L.Schwartz,Unstable modules over the Steenrod algebra and Sullivan fixed po*
*int
set conjecture, Chicago Lectures in Mathematics, 1994
[15]A.Zabrodsky, On phantom maps and a theorem of H.Miller, Israel J. Math.
58(1987), pp.129-143
_____________________-
Institute of Math.,Academia Sinica ,Beijing 100080, China
E-mail address: pjz@math03.math.ac.cn
Department of Mathematics Education , Korea University , Seoul ,
Korea