HIGHER ORDER PHANTOM MAPS
JEFFREY STROM
Abstract For each ordinal number ff, we define phantom
maps of order ff. We construct universal phantom maps out
of X with order ff, and show that under easily verifiable condi-
tions, every one of these universal phantom maps is essential.
Math. Subject Classifications: 55P05, 55S36
Keywords: Phantom Maps, Gray Index
1. The Main Result
A map f : X -! Y is a phantom map if it can be factored
f
X ____________________//JJY::t
JJJ tttt
JJJ ttt
J$$J tt fn
X [ CXn
for each n. The Gray index of a phantom map, G(f), is the least integer
n such that the map fn cannot be chosen to be a phantom map. The
Gray index was introduced by Brayton Gray in his thesis [1]. There he
claimed that every essential phantom map has finite index. However,
there is a flaw in his argument; recently McGibbon and Strom [4] have
shown that there are essential phantom maps with infinite Gray index.
This might seem like the end of the story, but that is far from the
case. In this paper, we show that the phantom maps with infinite Gray
index are just the second step in a filtration on [X; Y ] whose length can
take on any ordinal number value.
To begin, we define the order of a phantom map.
Definition A phantom map of order at least 0 is simply a map. Let
ff be an ordinal number. If ff = fi + 1, then f : X -! Y is a phantom
of order at least ff if, for each n, it can be factored
f
X ____________________//JJY::t
JJJ tttt
JJJ ttt
J$$J tt fn
X [ CXn
1
2 JEFFREY STROM
in such a way that fn is a phantom map of order at least fi. If ff is a
limit ordinal, then a phantom map is of order at least ff if it has order
at least fi for every fi < ff.
Thus, a phantom map of order 1 is simply an ordinary phantom
map with finite Gray index, and a phantom map has order at least 2
if and only if it is a phantom map with infinite Gray index. There is a
generalized Gray index that is defined for all phantom maps of order ff;
it is the least integer n such that f does not factor through a phantom
of order ff out of X [ CXn.
In the spirit of Gray and McGibbon [2], we inductively construct
maps
ff: X -! W ff(X)
out of X which are universal with the property of having order at least
ff. Define 0 to be the identity map. Now let ff be an ordinal number.
If ff = fi + 1, then ffis the natural map from X to the colimit of the
diagram of maps
fi fi
X -! X [ CXn -! W (X [ CXn):
If ff is a limit ordinal, then ffis the colimit of the diagram of the
fi: X -! W fi(X) for fi < ff. It is not hard to see that 1 is the
universal phantom map of Gray and McGibbon [2].
Proposition 1. The map ff : X -! W ff(X) has order at least ff.
Furthermore, any essential phantom f : X -! Y with order at least ff
must factor through ff.
Proof
The order of ffis at least ff by construction. The factorization prop-
erty follows by induction and the definition of the colimit. ||
It follows that a space X is the domain for an essential phantom with
order at least ff if and only if the map ff: X -! W ff(X) is essential.
Now we can state our main result.
Theorem 2. Let X be of finite type, and suppose there is a class u 2
H*(X; G) and cohomology operations 1; 2; : :o:f positive degree such
that nn-1 . .1.(u) 6= 0 for each n. Then the map
ff: X -! W ff(X)
is essential for each ordinal number ff.
It is a consequence of Miller's solution of the Sullivan conjecture [5]
that a Postnikov section of a nilpotent space of finite type satisfies the
condition in the theorem. But the condition is easily verified for many
familiar spaces X without such heavy machinery.
HIGHER ORDER PHANTOM MAPS 3
Examplen3. For example, take X = CP1 , u 6= 0 2 H3(X; Z=2) and
n = Sq2 .
There is a dual version of the order of a phantom map, and a dual
construction of a universal map into Y with order at least ff. Unfor-
tunately, the dual to Theorem 2 does not seem to be true; at least,
our proof does not dualize. The problem lies in dualizing Lemma B
below-there is no reasonable concept dual to compactness.
We end this section with a nagging question.
Question Does every phantom map have an order? In other words,
is it possible that an essential phantom map has order at least ff for
every ordinal number ff? If we could form colimits indexed on the class
of all ordinal numbers, then our proof would show that the answer is
yes!
2. Proof of the Main Result
Our proof depends on two Lemmas. The proofs of the Lemmas are
postponed to the end of the section.
Lemma A Let f : X -! W be a map, and construct the cofiber
sequence
f @ j
X -! W -! W=X -! X:
Then f ' * if and only if there is a map s : X -! C such that
j O s ' 1X .
Lemma B Let X be of finite type, and let u 2 Hn (X; G). Suppose
_
f : X -! Kff
W
and u =Wf*(v) for some v 2 Hn ( Kff; G). Then thereWis a finite sub-W
wedge ff2IKffsuch that (jOf)*(i*(v)) = u; here i : ff2IKff-! Kff
is the inclusion and j is its left inverse.
In view of Lemma A, it suffices to show that if X is as in the Theorem,
then u cannot be in the image of a map induced by
X -! W ff(X)=X:
To prove this, we actually prove the following stronger statement.
Proposition 2.1 Let X and u be as in Theorem 2. Then u is not in
the image of any map induced by a map
_
X -! W ff()(Z )=Z
for any set of spaces {Z }.
4 JEFFREY STROM
Proof
We work by transfinite induction. We know that the Proposition is
true for all Z and ff 0, because W 0(Z)=Z = * for all Z. Assume that
the result is known for all Z and all ff < A. By way of contradiction,
assume that u is in the image of a map induced by
_
X -! W ff()(Z )=Z
in which each ff() A (from now on, we will suppress the dependence
of ff on ). We will show that this implies that u is in the image of a
map induced by a map to a wedge of this kind with each ff < A, which
will contradict the inductive hypothesis.
The case in which A is a limit ordinal is easiest, for in this case we
have
_
W A(Z)=Z ' W ff(Z)=Z;
ff