NUMERICAL INVARIANTS OF PHANTOM MAPS
C. A. MCGIBBON AND JEFFREY STROM
Abstract Two numerical homotopy invariants of phantom
maps, the Gray index G(f) and the essential category weight E(f),
are studied. The possible values of these invariants are determined.
In certain cases bounds on these values are given in terms of ra-
tional homotopy data.
Examples are provided showing that the Gray index can take
any positive finite value. For certain cases it is shown that every
essential phantom f : X ! Y has finite Gray index. However
it is also shown that there exist spaces, e:g: CP 1, which are the
domains of essential phantoms with infinite index.
The same type of analysis is carried out on the essential category
weight of a phantom map. If the loop space of X is homotopy
equivalent to a finite complex, then every phantom f : X ! Y has
E(f) = 1. However, in certain other cases it is shown that E(f) is
strictly less than the rational Lusternik-Schnirelmann category of
the domain. A homotopy classification of phantoms f : K(Z; n) !
Sm is given along with the values of E(f).
The invariants G and E provide decreasing filtrations on the set
of homotopy classes of phantoms from X to Y . A third filtration
on this set is introduced for certain special targets. When the
rational cohomology of the domain X is finitely generated, this
filtration enables one to reduce the search for essential phantoms
(into finite type targets) to a finite list of spheres.
By a phantom map we mean a pointed map f from a CW -complex
X to another space Y with the property that the restriction of f to
the n-skeleton, f|Xn, is nullhomotopic for each natural number n. In
this paper we study certain numerical invariants associated with the
homotopy class of such a map, namely the Gray index and essential
category weight. These invariants give rise to decreasing filtrations on
the set Ph(X; Y ) of pointed homotopy classes of phantom maps from
X to Y .
____________
Date: January 25, 2000.
1
2 C. A. MCGIBBON AND JEFFREY STROM
1. The Gray Index
The first invariant is one we call the Gray index of a phantom map
f and we denote it by G(f). It was first defined by Brayton Gray in
his thesis [3]. Since f is a phantom map, one has a factorization of the
following sort for each natural number.
f
X FF_______________//Y< denotes the 2n-connective cover of
the sphere S2n. This connective cover has the rational homotopy type
of S4n-1.
6 C. A. MCGIBBON AND JEFFREY STROM
Example E Essential phantoms S2n<2n> ! S4n exist when n 2
and all of them have essential category weight at least 2. Uncountably
many of them have E(f) = 2.
For the moment let G = S2n<2n> . In the proof of Example E
we show that there are no essential phantoms from G to a finite type
target, so in this sense G resembles the finite dimensional groups in
Example C. This motivates the last open question mentioned at the
end of this paper.
3. A Filtration on Phantom maps into Spheres
The case where the target of a phantom map is a sphere might at
first glance seem bizarre to those used to thinking of Sn as a source.
However, it is known that for phantom maps out of finite type domains,
the spheres provide good test cases as targets. The following result of
McGibbon and Roitberg [10 ] makes this precise and motivates a closer
look at phantom maps into spheres.
Theorem 5 The following are equivalent for any finite type domain
X:
i) Ph(X; Y ) = 0 for every finite type target Y .
ii)Ph(X; Sn+1 ) = 0 for every n.
iii)There exists a map from X to a bouquet of spheres which induces
an isomorphism in rational homology.
It is reasonable to ask if, for a given X, it is really necessary to check
all spheres in part ii). A quick answer is no, in general it is only
necessary to check those dimensions n + 1 where Hn (X; Q) 6= 0. One
goal in this section is to improve on this answer-to reduce the search
to those n for which the module of indecomposables QHn (X; Q) is
nonzero. In doing this, we make use of the simple but useful filtration
described in the next result.
Theorem 6 Fix n 1 and let Y be a finite type space with the same
(n + 1)-type and the same rational type as K(Z; n + 1). Then there
is a decreasing filtration, with nth term zero, on the group Ph(X; Y ).
The filtration is induced by the cup product length filtration {Fk} on
the group
Hn (X; Z) [X; K(Z; n)] [Y; Y (n)]:
Phantom maps f : X ! Y in filtration k have have essential category
weight E(f) k. This filtration is natural with respect to maps between
NUMERICAL INVARIANTS OF PHANTOM MAPS 7
domains X ! X0 and maps between targets whose homotopy groups
satisfy the above hypothesis.
Of course this result is applicable when Y is an odd sphere or the
loops on an odd sphere. However, it does not apply to even spheres
which are, in a sense, more complicated. This prompted the question-
could the even spheres in Theorem 5 be replaced by the loops on odd
spheres? The next result shows that the answer is no.
Example F Let X = 2S5. Then Ph(X; Sn) = * and Ph(X; Sn) =
* for all odd n. However, there exist essential phantoms X ! S4.
On the positive side, we found the following unexpected stable result,
which says that the even spheres can be ignored if you are willing to
suspend the domain once.
Proposition 7 Assume X has finite type. Then Ph(X; S2n+1) =
Ph(X; S2n+1) = * for all n if and only if Ph(X; Y ) = * for all finite
type targets Y .
In particular Y could be the iterated loop space of another space,
and so the triviality of phantoms from X to odd spheres and loops on
odd spheres implies the triviality of phantoms from kX to any finite
type target for each k 1.
Of course, the filtration described in Theorem 6 might be trivial in
certain cases where the cup product length filtration is not. Indeed,
the whole group Ph(X; Y ) might be trivial in such cases (for example,
if X = Sn x Sm ). However the following result shows this filtration
is well behaved in the sense that the existence of nontrivial phantoms
ultimately depends on the indecomposable terms.
Theorem 8 Let X be a finite type domain. If Ph(X; Sn+1 ) = *
for all n such that QHn (X; Q) 6= 0, then Ph(X; Y ) = * for all finite
type targets Y .
4. Proofs
The proofs use basic facts about phantom maps and the towers that
determine them, see [1], Chapters 9 and 11, or [6]. In particular for
any two spaces X and Y there are bijections of pointed sets
Ph(X; Y ) lim1[X; Y (q)] lim1[Xn; Y ]:
Proof of Proposition 1 The proof of part a) amounts to showing
that the map Ph(X; Y ) Ph(X=X1;_Y ) induced by the quotient map
X ! X=X1 is surjective. Let Y denote the universal cover of Y .
8 C. A. MCGIBBON AND JEFFREY STROM
__
Zabrodsky shows in Section 2_of [16 ] that the covering ss : Y ! Y
induces a surjection Ph(X; Y) ! Ph(X; Y ).
__ __
We claim there is another surjection Ph(X=X1; Y) ! Ph(X; Y). It is
here we need the two homotopy groups to be finitely generated. Apply
__(q)
the functor [ ; Y ] to the cofiber sequence X1 ! X ! X=X1 to
obtain the exact sequence
__(q) ____//_ __(q) ____//_ __(q)
[X=X1; Y ] [X; Y ] [X1; Y ]
which we abbreviate Aq ! Bq ! Cq; write A = {Aq} and similarly for
B and C. We want to show that the map of towers A ! B induces a
surjection on lim1.
Since the fundamental group of X is finitely generated we may as-
sume that X1 is a finite bouquet of, say, k circles. The tower of groups
__(q)
Cq = [X1; Y ] is then constant; each nonzero term in it is isomorphic
to the direct sum of k copies of ss2Y and each nontrivial structure map
__(q)
is an isomorphism. We claim that the image of Bq = [X; Y ] in this
constant tower stabilizes_(becomes constant) at some finite stage q. To
see this note that Y F x T where T is a product of circles and ss1F
__(q)
is finite. Since the torus T is a retract of each Y , it follows that
Bq ! Cq is surjective modulo torsion. And since the torsion subgroup
of Cq is finite, it follows that the images of Bq cannot become ever
smaller as q increases; instead they become constant.
Let C0 denote this constant subtower of images. Then we have an
exact sequence of towers
A _____//B_____//C0____//_0
Since C0 is a constant tower, lim1C0 = *, and it follows, using the six
term lim - lim1 sequence, that_the induced_map lim1A ! lim1B is
surjective. Thus Ph(X=X1; Y) ! Ph(X; Y) is surjective.
The two surjections we have constructed fit into a commutative
square
__ __
Ph(X=X1; Y) _____//Ph(X; Y)
| |
| |
fflffl| fflffl|
Ph(X=X1; Y ) _____//Ph(X; Y )
which clearly shows that Ph(X=X1; Y ) ! Ph(X; Y ) is surjective, and
completes the proof of part a).
NUMERICAL INVARIANTS OF PHANTOM MAPS 9
The proof of part b) is easier. The conditions on X or on Y imply
that the quotient map X ! X=Xn induces a surjection of towers
[X=Xn; Y (q)] _____//[X; Y (q)]
which yields a surjection Ph(X=Xn; Y ) ! Ph(X; Y ).
Proof of Theorem 2 For each space X, we will construct a map
: X ! W which is universal with the property of being a phantom
map with infinite Gray index. We will then show that this map must
be essential under the given hypotheses.
A phantom map f : X ! Y has infinite Gray index if and only if it
factors as in the diagram
f
X _________//KKYOO
| KKKOEnKK |
| KKK |
fflffl|n K%%|
X [ CXn _____//Wn
for each n, where n : X [ CXn ! Wn is the universal phantom map
[4] out of X [ CXn.
Fix X, and let W be the pushout of the diagram consisting of all of
the maps X ! Wn. In other words, W is the space formed from the
disjoint union of X and each of the Wn by identifying OEn(x) with x for
each x 2 X. Let : X ! W (X) denote the natural inclusion.
It is easy to see that is a phantom map with infinite Gray index,
and that any phantom f : X ! Y with infinite Gray index out of X
must factor through . It follows that a space X is the domain for an
essential phantom map with infinite Gray index if and only if the map
: X ! W is essential.
Before we prove Theorem 2, we need a lemma.
Lemma 2.1 Let X be of finite type, and let u 2 Hn (X; G). Suppose
_
f : X ! Kff
W
and u = f*(v)Wfor some v 2 Hn ( Kff; G). Then there is aWfinite
subwedge ff2IKffsuch that (j O f)*(i*(v)) = u, where i : ff2IKffand
j is its left inverse.
Proof of Lemma 2.1 Since X is of finite type XmW is compact, so
f(Xm ) must be contained in a finite subwedge ff2IKff. It is easy to
verify the claim for this subwedge.
10 C. A. MCGIBBON AND JEFFREY STROM
Since H*(X; Z=p) is not locally finite as an Ap module, there is
a class u 2 H*(X; Z=p) and cohomology classes 1; 2; : : :such that
n . .1.(u) 6= 0 for each n. Choose such a u.
Extend the map X ! W to a cofiber sequence
_____// ____//_ __j__//
X W W=X X:
If is nullhomotopic, then the map j must have a section. We will
show that no map X ! W=X can have u in its image, and hence,
the map j cannot have a section.
Since W is formed from the Wn by gluing along X, we have the
following cofibration
W
X _____//W_____// nWn=X:
By Lemma 2.1, if u is in the image of a map induced by
W
X _____//n Wn=X
then we can restrict to a finite subwedge, so it suffices to show that u
cannot be in the image of a map induced by
fW
X _____//n2I Wn=X:
W
where I is a finite index set. Assume there is a v 2 H*( n2I Wn=X; G)
such that u = f*(v).
From the diagram
X ___________//X_________//*
| | |
| | |
fflffl| fflffl| W fflffl|
X [ CXn ______//_Wn______//_Kff
| | |
| | |
fflffl| fflffl| W fflffl|
Xn _______//_Wn=X_____// Kff
in which all the rows and columns are cofibrations, we find that there
are cofibration sequences
W
Xn ____//_Wn=X _____// Kff
in which each Kffis a finite complex. Wedging these together, we have
the following cofibration sequence
W W W W
n2IXn ____//_n2IWn=X _____// n2I Kff:
NUMERICAL INVARIANTS OF PHANTOM MAPS 11
Now let N be an upper bound for I and consider N . .1.(u)W=
f*(N . .1.(v)). Since N is larger than the dimension of n2IXn, we
see that N . .1.(v) is in the image of a map induced by
W W W
X ____//_n2IWn=X _____// n2I Kff:
by Lemma 2.1, we can restrict to a finite subwedge, which means that
N . .1.(u) is in the image of a map to a finite dimensional space, which
is clearly a contradiction.
Proof of Proposition 3 This proof uses a different description of
the Gray index-one that requires X and Y to be nilpotent and of finite
type. Let r : X ! Xo denote the rationalization map. By Theorem 5.1
of [6] the phantom maps X ! Y are precisely those maps which factor
through the rationalization of X and hence Ph(X; Y ) = r*[Xo; Y ].
The set [Xo; Y ] contains no essential phantom maps since Xo is a
rational space (and hence the universal phantom map out of it is trivial,
[4]). Therefore for each essential map OE : Xo ! Y there is a largest
natural number fl = fl(OE) such that the restriction OE to (Xn)o is null
homotopic (or equivalently that OE extends over (X=Xn)o) for n fl. It
follows that if f : X ! Y is an essential phantom map then its Gray
index can be defined as
G(f) = lub{fl(OE) | r*(OE) = f}:
This least upper bound will obviously be finite if the set over which
it is taken is finite. This happens when Map*(X; ^Y) is weakly con-
tractible because the function r* is then a bijection between Ph(X; Y )
and [Xo; Y ], by Theorem 5.4 of [6].
The second assertion has already been discussed in the paragraph
following Example B. We give a different proof here. We need to take
a closer look at [Xo; Y ]. By Theorem 5.2, ibid, there is a bijection of
pointed sets
Y
[Xo; Y ] Hk(X; ssk+1(Y ) R)
k
where R denotes a rational vector space whose cardinality equals that
of the real numbers.
Assume that Hk(X; ssk+1(Y ) Q) = 0 for k N. It is possible to
construct a subcomplex K X such that XN-1 K XN and the
natural map
Hk(X; Q) _____//Hk(K; Q)
12 C. A. MCGIBBON AND JEFFREY STROM
is an isomorphism for k N. If f : X ! Y is a phantom map with
G(f) N, then f factors through a phantom map f0 : X=K ! Y .
Since
Y
[(X=K)o; Y ] Hk(X=K; ssk+1(Y ) R) = 0;
k
this shows that f0 ' *, so f ' *.
Corollary 3.1 Suppose that X, Y and Y 0are finite type nilpotent
spaces such that the rationalization map r : X ! Xo induces bijections
Ph(X; Y ) [Xo; Y ] and Ph(X; Y 0) [Xo; Y 0]. If g : Y ! Y 0is a map
which induces a monomorphism on rational homotopy groups, then g* :
Ph(X; Y ) ! Ph(X; Y 0) is also one-to-one. Moreover G(f) = G(g O f)
for each phantom f : X ! Y .
Proof Since f is a phantom map, we can write f = OE O r for some
OE : Xo ! Y . Suppose G(f) = n, so OE|(Xn)o' *, but OE|(Xn+1)o6' *.
There is a cofibration
j
(Xn+1=Xn)o _i___//(X=Xn)o ____//_(X=Xn+1)o:
Since f does not factor through j, f Oi 6' *. Since g induces a monomor-
phism on rational homotopy groups and (Xn+1=Xn)o is a wedge of ra-
tional spheres, (gOf)Oi 6' *, and so gOf does not extend to (X=Xn+1)o.
Proof of Theorem 4 Let f : X ! Y be a phantom map, and
suppose cat(Xo) = n. To show that E(f) < n, we need to show that
f O j 6' * in the diagram
j f
Bn(X) ______//X_____//Y??
""
|rn| |r|""""
fflffl|jo fflffl|f""
Bn(X)o ____//_Xo
Since cat(Xo) = n the inclusion jo has a right inverse s; this a basic
fact due to Ganea. It follows that j*ois injective, and so f O jo 6' *.
Now f O j ' r*n(f O jo), so we will be done once we show that r*nis
bijective.
To this end it suffices to show that Map*(W; bY) is weakly contractible
by Theorem 5.4 of [6]. To simplify notation, let H = X. The proof
that Map(H; bY) = * implies that Map*(Bn(H); bY) = * goes back to
Zabrodsky, [16 ]. The weak contractibility of Map*(H?. .?.H; bY) follows
from the first assumption and the exponential rule. One then uses the
principal H-fibration
H _____//H ? . .?.H_____//Bn(H)
NUMERICAL INVARIANTS OF PHANTOM MAPS 13
and the Zabrodsky lemma (5.5 of [6]) to obtain the result for the base
space Bn(H).
Finally, suppose that cat(Xo) = 1. Let f : X ! Y be an essential
phantom and refer again to the diagram used in this proof. The ex-
tension f : Xo ! Y is not a phantom: it restricts nontrivially to some
finite dimensional skeleton of Xo. Since j : Bn(X) ! X is at least an
(n - 1)-equivalence, it follows that f O jo is essential for n sufficiently
large. Since r*nis still bijective, this implies that E(f) is bounded above
by such n. This completes the proof of Theorem 4.
Proof of Theorem 6 We again use the identification Ph(X; Y ) =
lim1[X; Y (q)]. Write Gq = [X; Y (q)] and let Hq denote the image of
Gq in Gn for q n and let Hq = 0 for smaller values of q. Notice
that Gn = [X; (Y )(n)] Hn (X; Z) since Y has the same (n + 1)-type
as K(Z; n + 1). If we let K be the kernel of the surjection of towers
G ! H, we obtain a short exact sequence of towers
0 _____//K_____//G____//_H____//_0:
The homotopy groups of Y are finite in dimensions above n because
Y has the same rational type as K(Z; n + 1). It follows that the kernels
Kq are finite for q n; hence lim1K = 0. From the six term lim- lim1
sequence applied to the short exact sequence of towers just displayed,
it follows that lim1G lim1H.
The filtration {Fk} on the groupTGn then induces a filtration on the
tower H; the kth stage being H Fk. Having identified Ph(X; Y )
with lim1H we define the kth stage of the filtration on Ph(X; Y ) to be
the image
"
lim1(H Fk) ! lim1H:
It is easy to check that the identifications used here are natural with
respect to the maps stated in the hypothesis.
It is well known that cup products of length k vanish in the co-
homology of a space of L-S category less than k. Since the category
of Bk(X) is at most k, the claim that phantoms in filtration k have
E(f) k then follows from Theorem 10 in [13 ] and naturality.
Proof of Proposition 7 To show that Ph(X; Y ) = * for all finite
type Y , it suffices to show that all phantom maps from X to spheres
are trivial. Since Ph(X; Y ) Ph(X; Y ) it follows from the hypoth-
esis that all phantoms from X to odd-dimensional spheres are trivial.
In the even dimensional case there exist maps S2n-1 x S4n-1 ! S2n
which are rational equivalences in general (and homotopy equivalences
14 C. A. MCGIBBON AND JEFFREY STROM
in the Hopf invariant one cases). By Theorem 7.2 of [6] these maps
induce surjections Ph(X; S2n-1 x S4n-1) ! Ph(X; S2n). The hy-
pothesis on X forces the first group to be trivial and the result follows.
Now assume that Ph(X; Y ) = * for all finite type Y . We have to
show that Ph(X; S2n-1) = * for all n. There are maps
S2n-1 ____//_S2n ____//_S2n-1
whose composite has degree 2. This composite, being a rational equiv-
alence, induces a surjection of Ph(X; S2n-1) to itself. This surjec-
tion must also be the trivial endomorphism, because Ph(X; S2n)
Ph(X; S2n) = *. This forces Ph(X; S2n-1) = *.
Proof of Theorem 8 The hypothesis is that Ph(X; Sn+1 ) = *
whenever QHn (X; Q) 6= 0. Assume then that uff2 Hn (X; Z) is a class
whose rational image projects to a nonzero indecomposable. We claim
that there is a map
gff: X ! Sn+1
such that some nonzero multiple of ufflies in the image of g*ff.
Assume for the moment that this claim is true. Choose a minimal
set of integral classes {uff} whose images span the graded vector space
QH*(X; Q). Let
Y
g : X ! Snff+1
ff
be given by gffin the ffth coordinate. This map induces a surjection
in rational cohomology since it is evidently surjective on the module of
rational indecomposables. Since the product has finite type, Theorem
2 of [10 ] shows that
iY j
g* : Ph Snff+1; Y ! Ph(X; Y )
is surjective for every finite type target Y . Now we are done, because
itQis proved in [4] that there are no essential phantom maps out of
Snff+1.
It remains to prove the claim made in the first paragraph. Let n =
nff. By hypothesis Ph(X; Sn+1 ) = * and so, since X also has finite
type, it follows that the tower {[X; (Sn+1 )(k)]} is Mittag-Leffler by
Theorem 2 of [8]. By Lemma 3.2, ibid, it follows that the image of
[X; Sn+1 ] in [X; (Sn+1 )(k)] has finite index for each k. Taking k = n,
we see that the image of
[X; Sn+1 ] ! Hn (X; Z)
has finite index, which proves the claim and completes the proof.
NUMERICAL INVARIANTS OF PHANTOM MAPS 15
Proof of Example A To establish the existence of essential phan-
tom maps CP 1 ! S2n+1, one could use Gray's proof for n = 1 (it
adapts easily for larger n) or one could compute, as in [6], with the aid
of Miller's proof of the Sullivan conjecture, that
Ph(CP 1; S2n+1) [CPo1; S2n+1] H2n(CP 1; ss2n+1(S2n+1) R)
where again R is a rational vector space with the cardinality of the real
numbers. Thus essential phantoms CP 1 ! S2n+1 exist. By Proposi-
tion 1 they have Gray index at least 2n - 1.
Now if X = CP 1=CP m where m n and Y = S2n+1 it is easy to
see that [Xo; Y ] = * and hence Ph(X; Y ) = *. Thus 2n-1 is also an
upper bound on the finite values of the Gray index in this case.
To check that the suspension operator here raises the Gray index by
exactly one, consider first the adjoint of the double suspension. Note
that E2 : Y ! 22Y is a rational equivalence when Y is an odd
sphere. Hence by Corollary 3.1, E2(f) has the same index as f. The
adjoint correspondence E2(f) 7! 2f clearly raises the Gray index
by 2. It follows that G(f) 2n. Since the inequality in the other
direction is obvious, the proof follows.
Proof of Example B By the Hilton-Milnor theorem
(S2_S2) ffSnff
where the set {nff} = {2; 3; 4; :::}: It follows that for each odd n 3
there is a map fn : Sn ! S2 _ S2 which induces a monomorphism on
integral (and hence on rational) homotopy groups. Hence Example B
follows from Example A and Corollary 3.1.
Proof of Example D The first part deals with phantoms into odd
dimensional spheres. Consider the cofiber sequence
Bk _____//K_____//Ck
where K = K(Z; 2n) and Bk = Bk(K). As in the proof of Theorem
4, we will once again exploit the equality [Xo; Y ] [X; Y ] = Ph(X; Y );
in other words, every map is phantom when X is any one of the terms
in this cofiber sequence and Y is a finite complex. The same is then
true of Ck, and this forces the induced sequence of phantom maps sets
to be exact.
The first map in the cofiber sequence induces an epimorphism in
rational cohomology with kernel . Thus, Ck is (2n(k + 1) - 1)-
connected, Ph(Ck; S2nq+1) = * for q k, which shows that the map
Ph(K; S2nq+1) ____//_Ph(Bk; S2nq+1)
16 C. A. MCGIBBON AND JEFFREY STROM
is a bijection for k q and is trivial for k < q. Hence, the category
weight of each essential phantom K ! S2nq+1 is exactly q.
The second part deals with phantoms into even dimensional spheres.
Since the rational cohomology of the domain is concentrated in even
degrees, the phantoms in this case must factor through a Whitehead
product for degree reasons. So each phantom in part ii) has the form
K(Z; 2a) _____//S4k-1____//_S2k
Thus, 2ab = 4k - 2 for some b, which means a and b must both be odd,
and 2k = ab + 1. The given formula comes from taking a = 2n - 1 and
b = 2q + 1. The essential category weight of these phantoms can be
calculated just as in part i).
Proof of Example E The existence of essential phantom maps
S2n<2n> ! S4n was shown in Example 6.1 of [9]. Let G = (S2n<2n> ).
Note that there is a map h : G ! S4n-1, which is a rational equiv-
alence. This map h can be taken to be the lift of the classical Hopf
invariant to (2n - 1)-connected covers. Since S4n-1 is homotopy
equivalent to a bouquet of spheres, it follows that Ph(G; Y ) = *,
for all finite type targets Y , by Theorem 5 iii). Thus all phantoms
f : BG ! S4n have E(f) 2.
To obtain the phantoms with E(f) = 2, we will study the "projective
plane" B3G. We will show that the natural map B3G ! BG ' S2n<2n>
induces an epimorphism Ph(S2n<2n> ; S4n) ! Ph(B3G; S4n) and that
the latter set is nontrivial.
To obtain the surjection recall that Ph(X; Y ) = r*[Xo; Y ] in the
nilpotent finite type case. For X = S2n<2n> and Y = S4n it was
shown that r* is a bijection in [9]. So it suffices to show that the
induced map [BGo; S4n] ! [B3Go; S4n] is surjective. The fiber of the
inclusion B3G ! BG is the 3-fold join G ? G ? G which is rationally
(12n - 5)-connected. Thus B3G ! BG induces an isomorphism in
rational homology in degrees 4n - 1 and 8n - 2 and so
[(B3G)o; S4n] [(BG)o; S4n] H4n-1(S4n-1; ss4n(S4n) R):
To obtain Ph(B3G; S4n) 6= * it suffices to show that tower Tq =
[B3G; (S4n)(q)]; q = 1; 2; 3; : : :is not Mittag-Leffler. Consider the
projection lim T ! T4n-1. We will show that the image of this homo-
morphism has infinite index in H4n-1(B3G; Z) and then invoke Lemma
3.2 of [8], as we did in the proof of Theorem 8. Since B3G has finite type
it is enough to show that there are no essential maps g : B3G ! S4n
which are nontrivial in H4n-1( ; Q).
NUMERICAL INVARIANTS OF PHANTOM MAPS 17
Suppose that g is such a map. There is a cofibration
_H___// _p3_//_
G ? G G B3G
in which p3 induces an isomorphism in H4n-1( ; Q). It follows that the
map f = g O i : G ! S4n is rationally a 4n-equivalence. Further-
more, the composite
__H__// __f__//
G?G G S4n
is nullhomotopic.
It follows that the adjoint f O eHof f O H is also nullhomotopic, and
so the composite
ixi q V eH f
S x S _____//G x G_____//G G ____//_G ____//_2S4n:
is nullhomotopic. Here i : S ! G denotesVthe (2n-1)-connective cover
of E : S2n-1 ! S2n and the map GxG ! G G is the usual quotient
map. >From this diagram we get a composite map OE : S x S ! 2S4n
which is evidently nullhomotopic.
Fix a prime p, and let Lp denote the Neisendorfer localization functor
([9],[2]). We will derive a contradiction by showing that Lp(OE) is an
essential map.
To prove this, we first decompose the map OE into more manageable
pieces. Recall that the adjoint of the Hopf construction has the form
eHq = E - Ess1 - Ess2
where : GxG ! G is the loop multiplication on G, and ss1 and ss2 are
the two projections. The last two terms on the right are "subtracted"
using the loop multiplication on G. Applying the loop map f
gives
f(Heq) = f(E) - f(Ess1) - f(Ess2):
Up to p-completion, the functor Lp is an inverse to the connective
cover functor in many important cases; in particular Lp(i : S ! G) is
the p-completion of E : S2n-1 ! S2n. This localization functor also
takes loop maps to loop maps (e.g. [2], [9]).
Since the map Lp(fEss1(i x i)) factors through the p-completion of
a map S2n-1 ! 2S4n it is evidently null homotopic. The same goes
for the term involving Ess2. It follows that the Neisendorfer localization
of the map OE is homotopic to the p-completion of the composite
ixi 2n 2n ef
S2n-1 x S2n-1 ____//_S x S ____//_S2n ____//_2S4n:
18 C. A. MCGIBBON AND JEFFREY STROM
We will show that this completed composite induces a nontrivial map
on H4n-2( ; Z=pr) for large enough r.
The key point is that if f : X ! Y induces a nontrivial map in
H4n-2( ; Z=pr), then so does the p-completed map. Since the original
map f : G ! S4n was assumed to induce a nontrivial map in
H4n-1( ; Q), the adjoint induces a nontrivial map in H4n-2( ; Q)
(this uses the fact that, rationally, G is a sphere). It follows that the
whole uncompleted composite is nontrivial on H4n-2( ; Q), and so is
nontrivial on H4n-2( ; Z=pr) for r large enough.
Thus, OE cannot be null-homotopic as our initial assumption implied.
In other words, no rationally essential map G ! S4n extends to
B3G, and we conclude that the tower Tq is not Mittag-Leffler. This
completes the proof for Example E.
Proof of Example F The existence of essential phantoms from
2S5 ! S4 is proved in [6], Proposition 8.2. Since 2S5 has the rational
homotopy type of S3, phantom maps into spheres of dimension n 6= 4
will necessarily be trivial. The evaluation map 22S5 ! S5 is a
rational equivalence and so, by Theorem 7.1c of [6], all phantoms from
2S5 into finite type targets are trivial.
5. Some open questions
Question 1 What is the Gray index of the universal phantom map
out of RP 1 ?
Question 2 Does every essential phantom X ! Y have finite Gray
index when both spaces are nilpotent and of finite type?
Question 3 Suppose that X is a space such that every phantom
from X to a sphere has infinite essential category weight. Does it follow
that every phantom from X to a finite complex (or a finite-type target)
also has infinite essential category weight?
Question 4 Does there exist a space whose universal phantom
map has essential category weight which is finite, but larger than 1?
Question 5 Suppose X is a space such that Ph(X; Y ) = * for
all spaces Y . Does E(f) = 1 for every phantom f out of X?
NUMERICAL INVARIANTS OF PHANTOM MAPS 19
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