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 rational 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^{∞}, 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) = ∞. 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) → S^{m} 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|X_{n}, 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. 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.

\xymatrix{X \ar[dr]_{q_{n}} \ar[rr]^{f} &  & Y \\  & X/X_{n} \ar[ur]_{f_{n}}}

Here q_{n} is the obvious quotient map. The extension f_{n} is not necessarily unique since it depends on a nullhomotopy of f|X_{n}.

Definition The Gray index G(f) is defined to be the greatest integer n for which the extension f_{n} can be chosen to be a phantom map.

We say that G(f) = ∞ if the extension f_{n} can be chosen to be a phantom map for every n. Obviously, the Gray index of the trivial map is infinity. The following example shows that this index can take on any positive integer value.

Example A For each n ≥ 1, there exist essential phantom maps f : ℂP^{∞} → S^{2n+1} and each has Gray index 2n − 1. The suspension, Σf, of such a map is essential and has index 2n .

The following result provides lower bounds on the Gray index.

Proposition 1

a) G(f) ≥ 1 for all phantoms f : X → Y if the groups π_{1}X and π_{2}Y are finitely generated.

b) G(f) ≥ n for every phantom f : X → Y if X is n-connected or if Y is (n + 1)-connected.

The proofs will be given later. While Example A might suggest that all essential phantoms between two given spaces have the same Gray index, the next example shows this is not always the case. In general, there is a natural filtration on Ph(X, Y), where f is in filtration n if G(f) ≥ n. The next example shows this Gray index filtration can be highly nontrivial.

Example B For each n ≥ 1 there are phantom maps ℂP^{∞} → S^{2}∨S^{2} of Gray index 2n − 1.

There is a simple but useful fact about phantom maps between finite type spaces that we first learned from Brayton Gray: essential phantoms X → Y can exist only if H^{n}(X; ℚ) and π_{n+1}Y ⊗ ℚ are both nonzero for some n ≥ 1. Using this fact it follows that if either the rational homology of X or the rational homotopy of Y vanish above a certain dimension, then there is a finite N such that all essential phantoms from X to Y have index at most N. This observation is recorded below in Proposition 3. Clearly this applies in the first example but not in the second.

Is the Gray index of every essential phantom map necessarily finite? In his thesis, Gray claimed the answer is yes, with no restrictions on X and Y. However, the following result contradicts this claim.

Theorem 2 Let X be a connected finite type CW -complex such that for some prime H^{∗}(X; ℤ/p) is not locally finite as a module over the Steenrod algebra. Then there exists an essential phantom map out of X with infinite Gray index.

The best known examples of spaces which satisfy this hypothesis are Bℤ/p and ℂP^{∞}. The proof of Theorem 2 involves constructing a universal example of a phantom map with infinite index out of X. The target for such a map does not have finite type. It remains an open question if there exists an essential phantom map with infinite Gray index between two spaces of finite type. Here is a partial result on this problem.

Proposition 3 If X and Y are nilpotent spaces of finite type, then every essential phantom X → Y has finite Gray index if either the function space Map_{∗}(X, Ŷ) is weakly contractible or the group H^{n}(X, π_{n+1}(Y) ⊗ ℚ) = 0 for n sufficiently large.

Here Ŷ denotes the profinite completion of Y. The hypothesis that π_{∗} Map_{∗}(X, Ŷ) = 0 is satisfied, for example, when X has only finitely nonzero homotopy groups and Y is a finite complex. This fact is a consequence of the Sullivan conjecture. See [6] and [7] for other examples which satisfy this hypothesis.

Lê Minh Hà and the second author have established many further properties of the Gray index [5]. They show, for example, that for phantom maps between spaces of finite type, the finite values of the Gray index are limited to those integers n − 1 for which H^{n}(X; ℚ) and π_{n+1}Y ⊗ ℚ are both nonzero. They also give a tower theoretic interpretation of the Gray index which is novel and useful.

2. Essential Category Weight

Another invariant one might consider for phantom maps is cat(f)– the Lusternik-Schnirelmann category of a map. Recall that this is the minimum number of closed subsets needed to cover the domain of f in such a way that the restriction of f to each subset is nullhomotopic. If f is an essential phantom map, then cat(f) = 2; the reason is that f factors through a reduced suspension via the universal phantom map out of X, [4].

\xymatrix{X \ar[dr]_{Θ} \ar[rr]^{f} &  & Y \\  & ⋁_{n} ΣX_{n} \ar@{-->}[ur]_{f̄}}

Thus cat(f) is not very interesting for the maps considered here. We therefore leave it and turn to the related notion of essential category weight, which turns out to be an interesting invariant of phantom maps.

Definition The essential category weight of a map f : X → Y is the least upper bound on those integers n such that the composition

\xymatrix{W \ar[r]^{g} & X \ar[r]^{f} & Y}

is nullhomotopic whenever cat(g) ≤ n. We write E(f) = n if the essential category weight of f is n (n = ∞ is allowed).

This notion was developed independently by Rudyak, [11] and Strom, [12]. If a map f has infinite essential category weight, then it must be a phantom map. Of course the trivial map has infinite essential category weight, but some essential phantom maps also have this property.

Example C Every phantom map out of a classifying space BG has infinite essential category weight when G is a finite dimensional topological group.

According to Theorem 10 in [13], E(f) ≥ n if and only if the composition

\xymatrix{B_{n}(G) \ar[r] & BG \ar[r]^{f} & Y}

is nullhomotopic, where the first map denotes the inclusion of the nth stage in the Milnor construction of BG (the L-S category of B_{n}(G) is at most n). When G is finite dimensional, so is B_{n}(G) for each finite n. Consequently, the above composition is trivial if f is a phantom map. For a specific example, each of the phantom maps out of ℂP^{∞} = BS^{1} described in Examples A and B has infinite essential category weight.

We mentioned earlier some cases where the Gray index of a phantom map is constrained by rational invariants of its domain and range. The next result reveals a similar phenomenon for the essential category weight of a phantom map.

Theorem 4 Assume that X and Y are finite type nilpotent spaces such that Map_{∗}(ΩX, Ŷ) is weakly contractible. Then E(f) < cat(X_{o}) for each essential phantom map f : X → Y. In particular, if cat(X_{o}) = ∞, then E(f) is finite for each essential phantom f : X → Y.

An important class of examples to which this theorem applies is when X is a 2-connected Eilenberg-MacLane space and Y is a finite complex. In this case every map X → Y is a phantom, [6]. In particular, a K(ℤ, 2n+1) has rational category 2, and so Theorem 4 implies that each essential f : K(ℤ, 2n + 1) → Y has E(f) = 1. For X = K(ℤ, 2n) the situation is more complicated. The rational category of this space is infinite for each n ≥ 1. When n = 1 it was noted in Example C that every phantom out of K(ℤ, 2) has infinite essential category weight. For larger values of n, Theorem 4 implies the essential phantoms have finite category weight. The following example gives the exact value and completes the classification of phantoms from a K(ℤ, 2m) to a sphere.

Example D Let n ≥ 2. For each q ≥ 1,

i) essential phantom maps f : K(ℤ, 2n) → S^{2nq+1} exist and each has E(f) = q.

ii) essential phantom maps g : K(ℤ, 4n − 2) → S^{(2n−1)(2q+1)+1} exist and each has E(g) = 2q + 1.

According to Theorem 9 of [13] the essential category weight of a composite of two maps is at least that of each of the factors. Since we have produced essential phantom maps out of K(ℤ, n) with essential category weight 1 for each n ≥ 3, we have the following result.

Corollary 4.1 If Θ_{n} denotes the universal phantom map out of K(ℤ, n), then E(Θ_{2}) = ∞ and E(Θ_{n}) = 1 for n ≥ 3.

Since the essential category weight of a map is bounded above by the category of its domain, the results in Theorem 4 and Example D led us to wonder if the mapping space hypothesis in Theorem 4 was really necessary. The following example shows it cannot be removed entirely. In other words, the rational category of the domain does not always give an upper bound on the essential category weight of phantom maps out of it. In this example S^{2n}⟨2n⟩ denotes the 2n-connective cover of the sphere S^{2n}. This connective cover has the rational homotopy type of S^{4n−1}.

Example E Essential phantoms S^{2n}⟨2n⟩ → S^{4n} 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 = ΩS^{2n}⟨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 S^{n} 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, S^{n+1}) = 0 for every n. iii) There exists a map from ΣX 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 H^{n}(X; ℚ) ≠ 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 QH^{n}(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 n^{th} term zero, on the group Ph(X, Y). The filtration is induced by the cup product length filtration {F_{k}} on the group

H^{n}(X; ℤ) ≈ [X, K(ℤ, n)] ≈ [Y, ΩY^{(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 domains X → X′ 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 = Ω^{2}S^{5}. Then Ph(X, S^{n}) = ∗ and Ph(X, ΩS^{n}) = ∗ for all odd n. However, there exist essential phantoms X → S^{4}.

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, S^{2n+1}) = Ph(ΣX, S^{2n+1}) = ∗ for all n if and only if Ph(ΣX, 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 Σ^{k}X 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 = S^{n} × S^{m}). 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, S^{n+1}) = ∗ for all n such that QH^{n}(X; ℚ) ≠ 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) ≈ lim^{1}[X, ΩY^{(q)}] ≈ lim^{1}[ΣX_{n}, Y].

Proof of Proposition 1 The proof of part a) amounts to showing that the map Ph(X, Y) ← Ph(X/X_{1}, Y) induced by the quotient map X → X/X_{1} is surjective. Let \overline{Y} denote the universal cover of Y.

Zabrodsky shows in Section 2 of [16] that the covering π : \overline{Y} → Y induces a surjection Ph(X, \overline{Y}) → Ph(X, Y).

We claim there is another surjection Ph(X/X_{1}, \overline{Y}) → Ph(X, \overline{Y}). It is here we need the two homotopy groups to be finitely generated. Apply the functor [ , Ω\overline{Y}^{(q)}] to the cofiber sequence X_{1} → X → X/X_{1} to obtain the exact sequence

\xymatrix{[X/X_{1}, Ω\overline{Y}^{(q)}] \ar[r] & [X, Ω\overline{Y}^{(q)}] \ar[r] & [X_{1}, Ω\overline{Y}^{(q)}]}

which we abbreviate A_{q} → B_{q} → C_{q}; write A = {A_{q}} and similarly for B and C. We want to show that the map of towers A → B induces a surjection on lim^{1}.

Since the fundamental group of X is finitely generated we may assume that X_{1} is a finite bouquet of, say, k circles. The tower of groups C_{q} = [X_{1}, Ω\overline{Y}^{(q)}] is then constant; each nonzero term in it is isomorphic to the direct sum of k copies of π_{2}Y and each nontrivial structure map is an isomorphism. We claim that the image of B_{q} = [X, Ω\overline{Y}^{(q)}] in this constant tower stabilizes (becomes constant) at some finite stage q. To see this note that Ω\overline{Y} ≈ F × T where T is a product of circles and π_{1}F is finite. Since the torus T is a retract of each Ω\overline{Y}^{(q)}, it follows that B_{q} → C_{q} is surjective modulo torsion. And since the torsion subgroup of C_{q} is finite, it follows that the images of B_{q} cannot become ever smaller as q increases; instead they become constant.

Let C′ denote this constant subtower of images. Then we have an exact sequence of towers

\xymatrix{A \ar[r] & B \ar[r] & C′ \ar[r] & 0}

Since C′ is a constant tower, lim^{1}C′ = ∗, and it follows, using the six term lim − lim^{1} sequence, that the induced map lim^{1}A → lim^{1}B is surjective. Thus Ph(X/X_{1}, \overline{Y}) → Ph(X, \overline{Y}) is surjective.

The two surjections we have constructed fit into a commutative square

\xymatrix{Ph(X/X_{1}, \overline{Y}) \ar[r] \ar[d] & Ph(X, \overline{Y}) \ar[d] \\ Ph(X/X_{1}, Y) \ar[r] & Ph(X, Y)}

which clearly shows that Ph(X/X_{1}, Y) → Ph(X, Y) is surjective, and completes the proof of part a).

The proof of part b) is easier. The conditions on X or on Y imply that the quotient map X → X/X_{n} induces a surjection of towers

\xymatrix{[X/X_{n}, ΩY^{(q)}] \ar[r] & [X, ΩY^{(q)}]}

which yields a surjection Ph(X/X_{n}, Y) → Ph(X, Y).

Proof of Theorem 2 For each space X, we will construct a map Φ : 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

\xymatrix{X \ar[r]^{f} \ar[d] \ar[dr]^{ϕ_{n}} & Y \\ X ∪ CX_{n} \ar[r]^{Θ_{n}} & W_{n} \ar[u]}

for each n, where Θ_{n} : X ∪ CX_{n} → W_{n} is the universal phantom map [4] out of X ∪ CX_{n}.

Fix X, and let W be the pushout of the diagram consisting of all of the maps X → W_{n}. In other words, W is the space formed from the disjoint union of X and each of the W_{n} by identifying ϕ_{n}(x) with x for each x ∈ X. Let Φ : X → W (X) denote the natural inclusion.

It is easy to see that Φ 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 Φ. It follows that a space X is the domain for an essential phantom map with infinite Gray index if and only if the map Φ : 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 ∈ H^{n}(X; G). Suppose

f : X → ⋁ K_{α}

and u = f^{∗}(v) for some v ∈ H^{n}(⋁ K_{α}; G). Then there is a finite subwedge ⋁_{α∈I} K_{α} such that (j ∘ f)^{∗}(i^{∗}(v)) = u, where i : ⋁_{α∈I} K_{α} and j is its left inverse. Proof of Lemma 2.1 Since X is of finite type X_{m} is compact, so f(X_{m}) must be contained in a finite subwedge ⋁_{α∈I} K_{α}. It is easy to verify the claim for this subwedge.

Since H^{∗}(ΣX; ℤ/p) is not locally finite as an A_{p} module, there is a class u ∈ H^{∗}(ΣX; ℤ/p) and cohomology classes θ_{1}, θ_{2}, … such that θ_{n} ⋯ θ_{1}(u) ≠ 0 for each n. Choose such a u.

Extend the map X → W to a cofiber sequence

\xymatrix{X \ar[r]^{Φ} & W \ar[r] & W/X \ar[r]^{j} & ΣX.}

If Φ is nullhomotopic, then the map j must have a section. We will show that no map ΣX → W/X can have u in its image, and hence, the map j cannot have a section.

Since W is formed from the W_{n} by gluing along X, we have the following cofibration

\xymatrix{X \ar[r] & W \ar[r] & ⋁_{n} W_{n}/X.}

By Lemma 2.1, if u is in the image of a map induced by

\xymatrix{ΣX \ar[r] & ⋁_{n} W_{n}/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

\xymatrix{ΣX \ar[r]^{f} & ⋁_{n∈I} W_{n}/X.}

where I is a finite index set. Assume there is a v ∈ H^{∗}(⋁_{n∈I} W_{n}/X; G) such that u = f^{∗}(v).

From the diagram

\xymatrix{X \ar[r] \ar[d] & X \ar[r] \ar[d] & ∗ \ar[d] \\ X ∪ CX_{n} \ar[r] \ar[d] & W_{n} \ar[r] \ar[d] & ⋁ K_{α} \ar[d] \\ ΣX_{n} \ar[r] & W_{n}/X \ar[r] & ⋁ K_{α}}

in which all the rows and columns are cofibrations, we find that there are cofibration sequences

\xymatrix{ΣX_{n} \ar[r] & W_{n}/X \ar[r] & ⋁ K_{α}}

in which each K_{α} is a finite complex. Wedging these together, we have the following cofibration sequence

\xymatrix{⋁_{n∈I} ΣX_{n} \ar[r] & ⋁_{n∈I} W_{n}/X \ar[r] & ⋁_{n∈I} ⋁ K_{α}.}

Now let N be an upper bound for I and consider θ_{N} ⋯ θ_{1}(u) = f^{∗}(θ_{N} ⋯ θ_{1}(v)). Since N is larger than the dimension of ⋁_{n∈I} ΣX_{n}, we see that θ_{N} ⋯ θ_{1}(v) is in the image of a map induced by

\xymatrix{X \ar[r] & ⋁_{n∈I} W_{n}/X \ar[r] & ⋁_{n∈I} ⋁ K_{α}.}

by Lemma 2.1, we can restrict to a finite subwedge, which means that θ_{N} ⋯ θ_{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 → X_{o} 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^{∗}[X_{o}, Y].

The set [X_{o}, Y] contains no essential phantom maps since X_{o} is a rational space (and hence the universal phantom map out of it is trivial, [4]). Therefore for each essential map ϕ : X_{o} → Y there is a largest natural number γ = γ(ϕ) such that the restriction ϕ to (X_{n})_{o} is null homotopic (or equivalently that ϕ extends over (X/X_{n})_{o}) for n ≤ γ. It follows that if f : X → Y is an essential phantom map then its Gray index can be defined as

G(f) = lub{γ(ϕ) | r^{∗}(ϕ) = f}.

This least upper bound will obviously be finite if the set over which it is taken is finite. This happens when Map_{∗}(X, Ŷ) is weakly contractible because the function r^{∗} is then a bijection between Ph(X, Y) and [X_{o}, 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 [X_{o}, Y]. By Theorem 5.2, ibid, there is a bijection of pointed sets

[X_{o}, Y] ≈ ∏_{k} H^{k}(X; π_{k+1}(Y) ⊗ ℝ)

where ℝ denotes a rational vector space whose cardinality equals that of the real numbers.

Assume that H^{k}(X; π_{k+1}(Y) ⊗ ℚ) = 0 for k ≥ N. It is possible to construct a subcomplex K ⊆ X such that X_{N−1} ⊆ K ⊆ X_{N} and the natural map

\xymatrix{H^{k}(X; ℚ) \ar[r] & H^{k}(K; ℚ)}

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 f′ : X/K → Y. Since

[(X/K)_{o}, Y] ≈ ∏_{k} H^{k}(X/K; π_{k+1}(Y) ⊗ ℝ) = 0,

this shows that f′ ≃ ∗, so f ≃ ∗.

Corollary 3.1 Suppose that X, Y and Y′ are finite type nilpotent spaces such that the rationalization map r : X → X_{o} induces bijections Ph(X, Y) ≈ [X_{o}, Y] and Ph(X, Y′) ≈ [X_{o}, Y′]. If g : Y → Y′ is a map which induces a monomorphism on rational homotopy groups, then g_{∗} : Ph(X, Y) → Ph(X, Y′) is also one-to-one. Moreover G(f) = G(g ∘ f) for each phantom f : X → Y. Proof Since f is a phantom map, we can write f = ϕ ∘ r for some ϕ : X_{o} → Y. Suppose G(f) = n, so ϕ|_{(X_{n})_{o}} ≃ ∗, but ϕ|_{(X_{n+1})_{o}} ≄ ∗.

There is a cofibration

\xymatrix{(X_{n+1}/X_{n})_{o} \ar[r]^{i} & (X/X_{n})_{o} \ar[r]^{j} & (X/X_{n+1})_{o}.}

Since f does not factor through j, f ∘i ≄ ∗. Since g induces a monomorphism on rational homotopy groups and (X_{n+1}/X_{n})_{o} is a wedge of rational spheres, (g∘f)∘i ≄ ∗, and so g∘f does not extend to (X/X_{n+1})_{o}.

Proof of Theorem 4 Let f : X → Y be a phantom map, and suppose cat(X_{o}) = n. To show that E(f) < n, we need to show that f ∘ j ≄ ∗ in the diagram

\xymatrix{B_{n}(ΩX) \ar[d]^{r_{n}} \ar[r]^{j} & X \ar[d]^{r} \ar[r]^{f} & Y \\ B_{n}(ΩX)_{o} \ar[r]^{j_{o}} & X_{o} \ar[ur]_{f̄}}

Since cat(X_{o}) = n the inclusion j_{o} has a right inverse s; this a basic fact due to Ganea. It follows that j_{o}^{∗} is injective, and so f̄ ∘ j_{o} ≄ ∗. Now f ∘ j ≃ r_{n}^{∗}(f̄ ∘ j_{o}), so we will be done once we show that r_{n}^{∗} is bijective.

To this end it suffices to show that Map_{∗}(W, Ŷ) is weakly contractible by Theorem 5.4 of [6]. To simplify notation, let H = ΩX. The proof that Map(H, Ŷ) = ∗ implies that Map_{∗}(B_{n}(H), Ŷ) = ∗ goes back to Zabrodsky, [16]. The weak contractibility of Map_{∗}(H⋆⋯⋆H, Ŷ) follows from the first assumption and the exponential rule. One then uses the principal H-fibration

\xymatrix{H \ar[r] & H ⋆ ⋯ ⋆ H \ar[r] & B_{n}(H)}

and the Zabrodsky lemma (5.5 of [6]) to obtain the result for the base space B_{n}(H).

Finally, suppose that cat(X_{o}) = ∞. Let f : X → Y be an essential phantom and refer again to the diagram used in this proof. The extension f̄: X_{o} → Y is not a phantom: it restricts nontrivially to some finite dimensional skeleton of X_{o}. Since j : B_{n}(ΩX) → X is at least an (n − 1)-equivalence, it follows that f̄∘ j_{o} is essential for n sufficiently large. Since r_{n}^{∗} is 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) = lim^{1}[X, ΩY^{(q)}]. Write G_{q} = [X, ΩY^{(q)}] and let H_{q} denote the image of G_{q} in G_{n} for q ≥ n and let H_{q} = 0 for smaller values of q. Notice that G_{n} = [X, (ΩY)^{(n)}] ≈ H^{n}(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

\xymatrix{0 \ar[r] & K \ar[r] & G \ar[r] & H \ar[r] & 0.}

The homotopy groups of ΩY are finite in dimensions above n because Y has the same rational type as K(ℤ, n+1). It follows that the kernels K_{q} are finite for q ≥ n; hence lim^{1}K = 0. From the six term lim − lim^{1} sequence applied to the short exact sequence of towers just displayed, it follows that lim^{1}G ≈ lim^{1}H.

The filtration {F_{k}} on the group G_{n} then induces a filtration on the tower H; the kth stage being H ⋂ F_{k}. Having identified Ph(X, Y) with lim^{1}H we define the kth stage of the filtration on Ph(X, Y) to be the image

lim^{1}(H ⋂ F_{k}) → lim^{1}H.

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 cohomology of a space of L-S category less than k. Since the category of B_{k}(ΩX) 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(ΣX, Y) = ∗ for all finite type Y, it suffices to show that all phantom maps from ΣX to spheres are trivial. Since Ph(ΣX, Y) ≈ Ph(X, ΩY) it follows from the hypothesis that all phantoms from X to odd-dimensional spheres are trivial. In the even dimensional case there exist maps S^{2n−1} × ΩS^{4n−1} → ΩS^{2n} which are rational equivalences in general (and homotopy equivalences in the Hopf invariant one cases). By Theorem 7.2 of [6] these maps induce surjections Ph(X, S^{2n−1} × ΩS^{4n−1}) → Ph(X, ΩS^{2n}). The hypothesis on X forces the first group to be trivial and the result follows.

Now assume that Ph(ΣX, Y) = ∗ for all finite type Y. We have to show that Ph(X, S^{2n−1}) = ∗ for all n. There are maps

\xymatrix{S^{2n−1} \ar[r] & ΩS^{2n} \ar[r] & S^{2n−1}}

whose composite has degree 2. This composite, being a rational equivalence, induces a surjection of Ph(X, S^{2n−1}) to itself. This surjection must also be the trivial endomorphism, because Ph(X, ΩS^{2n}) ≈ Ph(ΣX, S^{2n}) = ∗. This forces Ph(X, S^{2n−1}) = ∗.

Proof of Theorem 8 The hypothesis is that Ph(X, S^{n+1}) = ∗ whenever QH^{n}(X; ℚ) ≠ 0. Assume then that u_{α} ∈ H^{n}(X; ℤ) is a class whose rational image projects to a nonzero indecomposable. We claim that there is a map

g_{α} : X → ΩS^{n+1}

such that some nonzero multiple of u_{α} lies in the image of g_{α}^{∗}.

Assume for the moment that this claim is true. Choose a minimal set of integral classes {u_{α}} whose images span the graded vector space QH^{∗}(X; ℚ). Let

g : X → ∏_{α} ΩS^{n_{α}+1}

be given by g_{α} in the αth 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

g^{∗} : Ph (∏ ΩS^{n_{α}+1}, Y ) → Ph(X, Y)

is surjective for every finite type target Y. Now we are done, because it is proved in [4] that there are no essential phantom maps out of ∏ ΩS^{n_{α}+1}.

It remains to prove the claim made in the first paragraph. Let n = n_{α}. By hypothesis Ph(X, S^{n+1}) = ∗ and so, since X also has finite type, it follows that the tower {[X, (ΩS^{n+1})^{(k)}]} is Mittag-Leffler by Theorem 2 of [8]. By Lemma 3.2, ibid, it follows that the image of [X, ΩS^{n+1}] in [X, (ΩS^{n+1})^{(k)}] has finite index for each k. Taking k = n, we see that the image of

[X, ΩS^{n+1}] → H^{n}(X; ℤ)

has finite index, which proves the claim and completes the proof.

Proof of Example A To establish the existence of essential phantom maps ℂP^{∞} → S^{2n+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^{∞}, S^{2n+1}) ≈ [ℂP_{o}^{∞}, S^{2n+1}] ≈ H^{2n}(ℂP^{∞}; π_{2n+1}(S^{2n+1}) ⊗ ℝ)

where again ℝ is a rational vector space with the cardinality of the real numbers. Thus essential phantoms ℂP^{∞} → S^{2n+1} exist. By Proposition 1 they have Gray index at least 2n − 1.

Now if X = ℂP^{∞}/ℂP^{m} where m ≥ n and Y = S^{2n+1} it is easy to see that [X_{o}, 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 E^{2} : Y → Ω^{2}Σ^{2}Y is a rational equivalence when Y is an odd sphere. Hence by Corollary 3.1, E^{2}(f) has the same index as f. The adjoint correspondence E^{2}(f) ↦ Σ^{2}f clearly raises the Gray index by 2. It follows that G(Σf) ≤ 2n. Since the inequality in the other direction is obvious, the proof follows.

Proof of Example B By the Hilton-Milnor theorem

Ω(S^{2}∨S^{2}) ≈ Π_{α}ΩS^{n_{α}}

where the set {n_{α}} = {2, 3, 4, ...}. It follows that for each odd n ≥ 3 there is a map f_{n} : S^{n} → S^{2} ∨ S^{2} 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

\xymatrix{B_{k} \ar[r] & K \ar[r] & C_{k}}

where K = K(ℤ, 2n) and B_{k} = B_{k}(ΩK). As in the proof of Theorem 4, we will once again exploit the equality [X_{o}, 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 C_{k}, 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 ⟨x_{2n}^{k}⟩. Thus, C_{k} is (2n(k + 1) − 1)- connected, Ph(C_{k}, S^{2nq+1}) = ∗ for q ≤ k, which shows that the map

\xymatrix{Ph(K, S^{2nq+1}) \ar[r] & Ph(B_{k}, S^{2nq+1})}

is a bijection for k ≥ q and is trivial for k < q. Hence, the category weight of each essential phantom K → S^{2nq+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

\xymatrix{K(ℤ, 2a) \ar[r] & S^{4k−1} \ar[r] & S^{2k}}

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 S^{2n}⟨2n⟩ → S^{4n} was shown in Example 6.1 of [9]. Let G = Ω(S^{2n}⟨2n⟩). Note that there is a map h : G → ΩS^{4n−1}, which is a rational equivalence. This map h can be taken to be the lift of the classical Hopf invariant to (2n − 1)-connected covers. Since ΣΩS^{4n−1} is homotopy equivalent to a bouquet of spheres, it follows that Ph(ΣG, Y) = ∗, for all finite type targets Y, by Theorem 5 iii). Thus all phantoms f : BG → S^{4n} have E(f) ≥ 2.

To obtain the phantoms with E(f) = 2, we will study the “projective plane” B_{3}G. We will show that the natural map B_{3}G → BG ≃ S^{2n}⟨2n⟩ induces an epimorphism Ph(S^{2n}⟨2n⟩, S^{4n}) → Ph(B_{3}G, S^{4n}) and that the latter set is nontrivial.

To obtain the surjection recall that Ph(X, Y) = r^{∗}[X_{o}, Y] in the nilpotent finite type case. For X = S^{2n}⟨2n⟩ and Y = S^{4n} it was shown that r^{∗} is a bijection in [9]. So it suffices to show that the induced map [BG_{o}, S^{4n}] → [B_{3}G_{o}, S^{4n}] is surjective. The fiber of the inclusion B_{3}G → BG is the 3-fold join G⋆G⋆G which is rationally (12n − 5)-connected. Thus B_{3}G → BG induces an isomorphism in rational homology in degrees 4n − 1 and 8n − 2 and so

[(B_{3}G)_{o}, S^{4n}] ≈ [(BG)_{o}, S^{4n}] ≈ H^{4n−1}(S^{4n−1}; π_{4n}(S^{4n}) ⊗ ℝ).

To obtain Ph(B_{3}G, S^{4n}) ≠ ∗ it suffices to show that tower T_{q} = [B_{3}G, (ΩS^{4n})^{(q)}], q = 1, 2, 3, … is not Mittag-Leffler. Consider the projection lim T → T_{4n−1}. We will show that the image of this homomorphism has infinite index in H^{4n−1}(B_{3}G; ℤ) and then invoke Lemma 3.2 of [8], as we did in the proof of Theorem 8. Since B_{3}G has finite type it is enough to show that there are no essential maps g : B_{3}G → ΩS^{4n} which are nontrivial in H^{4n−1}( ; ℚ).

Suppose that g is such a map. There is a cofibration

\xymatrix{G ⋆ G \ar[r]^{H} & ΣG \ar[r]^{p_{3}} & B_{3}G}

in which p_{3} induces an isomorphism in H^{4n−1}( ; ℚ). It follows that the map f = g ∘ i : ΣG → ΩS^{4n} is rationally a 4n-equivalence. Furthermore, the composite

\xymatrix{G⋆G \ar[r]^{H} & ΣG \ar[r]^{f} & ΩS^{4n}}

is nullhomotopic.

It follows that the adjoint Ωf ∘H̃ of f ∘H is also nullhomotopic, and so the composite

\xymatrix{S × S \ar[r]^{i×i} & G × G \ar[r]^{q} & G ⋀ G \ar[r]^{H̃} & ΩΣG \ar[r]^{Ωf} & Ω^{2}S^{4n}.}

is nullhomotopic. Here i : S → G denotes the (2n−1)-connective cover of E : S^{2n−1} → ΩS^{2n} and the map G×G → G ⋀ G is the usual quotient map. ¿From this diagram we get a composite map ϕ : S × S → Ω^{2}S^{4n} which is evidently nullhomotopic.

Fix a prime p, and let L_{p} denote the Neisendorfer localization functor ([9],[2]). We will derive a contradiction by showing that L_{p}(ϕ) is an essential map.

To prove this, we first decompose the map ϕ into more manageable pieces. Recall that the adjoint of the Hopf construction has the form

H̃q = Eμ − Eπ_{1} − Eπ_{2}

where μ : G×G → G is the loop multiplication on G, and π_{1} and π_{2} are the two projections. The last two terms on the right are “subtracted” using the loop multiplication on ΩΣG. Applying the loop map Ωf gives

Ωf(H̃q) = Ωf(Eμ) − Ωf(Eπ_{1}) − Ωf(Eπ_{2}).

Up to p-completion, the functor L_{p} is an inverse to the connective cover functor in many important cases; in particular L_{p}(i : S → G) is the p-completion of E : S^{2n−1} → ΩS^{2n}. This localization functor also takes loop maps to loop maps (e.g. [2], [9]).

Since the map L_{p}(ΩfEπ_{1}(i × i)) factors through the p-completion of a map S^{2n−1} → Ω^{2}S^{4n} it is evidently null homotopic. The same goes for the term involving Eπ_{2}. It follows that the Neisendorfer localization of the map ϕ is homotopic to the p-completion of the composite

\xymatrix{S^{2n−1} × S^{2n−1} \ar[r]^{i×i} & ΩS^{2n} × ΩS^{2n} \ar[r]^{μ} & ΩS^{2n} \ar[r]^{f̃} & Ω^{2}S^{4n}.}

We will show that this completed composite induces a nontrivial map on H^{4n−2}( ; ℤ/p^{r}) for large enough r.

The key point is that if f : X → Y induces a nontrivial map in H^{4n−2}( ; ℤ/p^{r}), then so does the p-completed map. Since the original map f : ΣG → ΩS^{4n} was assumed to induce a nontrivial map in H^{4n−1}( ; ℚ), the adjoint induces a nontrivial map in H^{4n−2}( ; ℚ) (this uses the fact that, rationally, G is a sphere). It follows that the whole uncompleted composite is nontrivial on H^{4n−2}( ; ℚ), and so is nontrivial on H^{4n−2}( ; ℤ/p^{r}) for r large enough.

Thus, ϕ cannot be null-homotopic as our initial assumption implied. In other words, no rationally essential map ΣG → ΩS^{4n} extends to B_{3}G, and we conclude that the tower T_{q} is not Mittag-Leffler. This completes the proof for Example E.

Proof of Example F The existence of essential phantoms from Ω^{2}S^{5} → S^{4} is proved in [6], Proposition 8.2. Since Ω^{2}S^{5} has the rational homotopy type of S^{3}, phantom maps into spheres of dimension n ≠ 4 will necessarily be trivial. The evaluation map Σ^{2}Ω^{2}S^{5} → S^{5} is a rational equivalence and so, by Theorem 7.1c of [6], all phantoms from ΣΩ^{2}S^{5} 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^{∞} ?

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(ΩX, Y) = ∗ for all spaces Y. Does E(f) = ∞ for every phantom f out of X?

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