Phantom maps and rational equivalences.

Given a CW-complex X, a phantom map f: X !approaches^ Y is, by definition, a map whose restriction to each n-skeleton, !X.sub.n^, is null homotopic. Let Ph (X, Y) denote the set of pointed homotopy classes of phantom maps from X to Y. The theorems in this paper deal with Ph (X, Y) when X and Y are spaces that satisfy certain finiteness conditions.

We call a space X a finite type domain if it is a pointed, connected CW-complex whose integral homology groups are finitely generated in each degree; a pointed space Y will be referred to as a finite type target if each of its homotopy groups is finitely generated.

Recently, Gray and McGibbon studied the universal phantom map out of a given space, !3^. For a finite type domain X, they showed that Ph (X, Y) = 0 for all spaces Y (not just those of finite type) if and only if !summation of^X is a retract of !V.sub.n^!summation of^!X.sub.n^. They also proved a p-local version of this that is somewhat sharper; it says that if X is the p-localization of a nilpotent, finite type domain, then Ph (X, Y) = 0 for all spaces Y if and only if !summation of^X has the homotopy type of a bouquet of finite dimensional spaces. They noted, however, that there are some spaces, such as X = !RP.sup.!infinity^^, for which the universal phantom map X !approaches^ !V.sub.n^!summation of^!X.sub.n^ is essential and yet Ph (X, Y) = 0 for every finite type target Y (see also !4^). The problem of characterizing those domains, X, that satisfy this weaker condition was left open in !3^. It is solved here.

THEOREM 1. Let X be a finite type domain. Then the following statements are equivalent:

(i) Ph (X, Y) = 0 for every finite type target Y.

(ii) Ph(X,!S.sup.n^) = 0 for every n.

(iii) There exists a map from !summation of^X to a bouquet of spheres V!S.sup.!n.sub.!Alpha^^^ that induces an isomorphism in rational homology.

The example where X = !RP.sup.!infinity^^, mentioned above, shows that the restriction to finite type targets is necessary in part (i). This same example also shows that sometimes there are no spheres in the bouquet in part (iii). In cases such as this, we define the empty bouquet to be a point. There is an Eckmann-Hilton dual result that goes as follows.

THEOREM 1'. Let Y be a finite type target. Then the following statements are equivalent:

(i) Ph (X, Y) = 0 for every finite type domain X.

(ii) Ph (K(Z, n), Y) = 0 for every n.

(iii) There exists a weak rational equivalence from a product of Eilenberg-MacLane spaces !Mathematical Expression Omitted^ to the basepoint component of !Omega^Y.

The direction of the rational equivalences in these theorems is important. In Theorem 1, for example, one can always construct a rational homology equivalence from an appropriate bouquet of spheres into !summation of^X, but one can not always find one going the other way. A case in point is X = !CP.sup.!infinity^^. Since there are essential phantoms from this space to other spaces of finite type, such as the 3-sphere, there is no rational equivalence from its suspension back to a bouquet of spheres. This lack of symmetry is truly an infinite dimensional phenomenon. With finite dimensional suspensions, or with loop spaces with only a finite number of nonzero homotopy groups, one can always get rational equivalences, in both directions, between these spaces and the models featured in Theorem 1.

One can, of course, regard Ph (X, Y) as a functor of two variables. The following results describe how it behaves when one variable is held constant and the other is allowed to vary, subject to certain rational conditions.

THEOREM 2.

(i) If f : X !approaches^ X!prime^ induces a monomorphism between the rational homology groups of two finite type domains, and Y is a finite type target, then

!Mathematical Expression Omitted^

is an epimorphism of pointed sets.

(ii) If g : Y !approaches^ Y!prime^ induces a rational epimorphism between the higher homotopy groups of two finite type targets, and X is a finite type domain, then

!Mathematical Expression Omitted^

is an epimorphism of pointed sets.

More precisely, we only require in (ii) that g#: !!Pi^.sub.n^Y !cross product^ Q !approaches^ !!Pi^.sub.n^Y!prime^ !cross product^ Q be an epimorphism for each n !is greater than or equal to^ 2. Thus !Omega^g, when rationalized, restricts to a retraction onto the basepoint component of !Omega^Y!prime^. Dually, the rationalization of !summation of^f, in part (i), has a left inverse.

Willi Meier was one of the first to notice a connection between phantom maps and rational equivalences. He announced in !7^ a special case of Theorem 2, wherein he required his targets to be !H.sub.0^-spaces; that is, nilpotent with the rational homotopy type of an H-space. He also assumed that Ph (X!prime^, Y) = 0 in the first part and that Ph (X, Y) = 0 in the second. In !3^, Gray and McGibbon gave a proof of the first half of his result without the !H.sub.0^ hypothesis on Y. Clearly, their version is still a special case of Theorem 2.

COROLLARY 2.1.

(i) If X and X!prime^ are finite type domains with rational homology equivalences between them in both directions, then Ph (X, Y) and Ph (X!prime^, Y) are isomorphic as pointed sets whenever Y is a finite type target.

(ii) If Y and Y!prime^ are nilpotent finite type targets with rational homotopy equivalences between them in both directions, then Ph (X, Y) and Ph (X, Y!prime^) are isomorphic as pointed sets for any finite type domain X.

The spaces !S.sup.3^ and K(Z, 3) have the same rational homotopy type but, of course, there is a rational equivalence between them in only one direction. In view of Corollary 2.1, this is as it should be since Ph (!S.sup.3^, Y) = 0 for all spaces Y, whereas there are essential phantoms from K(Z, 3) into many finite type targets Y--the simplest example being Y = !S.sup.4^, !10^.

Another example worth noting involves the iterated loops on a sphere. It was shown in !3^ that

Ph (!Omega^!S.sup.n^, Y) = 0

for all n and all spaces Y. In !5^ it was shown, for all integers k and n, and all primes p, that

Ph(!!Omega^.sup.k^!S.sup.n^, !Y.sub.(p)^) = 0

whenever Y is a nilpotent finite type target. These results are best possible because it was also shown in !5^, for each n !is less than^ 1, that

Ph(!!Omega^.sup.2^!S.sup.2n+1^, !S.sup.n^) !is not equal to^ 0.

Starting with this last example and using Theorem 2, we show

Example 3. If k !is greater than or equal to^1 and n !is less than^ 1, then

Ph(!!Omega^.sup.k+1^!S.sup.2n+k^, !S.sup.n^) !is not equal to^ 0.

This concludes the description of the main results in this paper. Their proofs are given in the next section, in the order the results were presented. The paper ends with a discussion of open questions related to this work. We want to thank Kay Magaard for some helpful discussions about the group theory in this paper.

Proofs.

Proof of Theorem 1. The case when X has the rational homology of a point is exceptional and we will handle it first. It was shown that statement (i) holds for all such X in Example 4.1 of !2^. This implies (ii) is true as well. The proof of (iii) is a triviality for such X. Having finished this trivial case, we will henceforth assume that X does not have the rational homology of a point. Let g: V!S.sup.!n.sub.!Alpha^^^ !approaches^ !summation of^X be a rational homology equivalence, and let !g.sub.!Alpha^^ denote its restriction to !S.sup.!n.sub.!Alpha^^^. As remarked earlier, there is always a rational homology equivalence in the direction of g, but not always one going in the opposite direction. Now assume that Ph (X, Y) = 0 for every finite type target Y. Then in particular,

Ph (X, !S.sup.!n.sub.!Alpha^^^) = 0 for each !n.sub.!Alpha^^.

This is equivalent, (!2^, Corollary 3.3, page 255), to saying that

!Mathematical Expression Omitted^

Since each group in this tower is countable, the tower is Mittag-Leffler by Theorem 2 of !6^. Recall that for an inverse tower of groups {!G.sub.k^}, the Mittag-Leffler property ensures that for each n, the images in !G.sub.n^ of the terms farther out in the sequence do not become smaller and smaller without end; instead they stablize at some point. That is, for some N sufficiently large,

image{!G.sub.n^ !left arrow^ !G.sub.N^} = image{!G.sub.n^ !left arrow^ !G.sub.N+K^},

for all k !is greater than or equal to^ 0. Moreover, the tower !G.sub.k^ = !(!summation of^X).sub.k^, !S.sup.!n.sub.!Alpha^^^ has the property that for each k, the image of !G.sub.k+1^ has finite index in !G.sub.k^. This follows from the finite order of the attaching maps in !summation of^X. According to Lemma 3.2 of !6^, if an inverse tower of countable groups has the Mittag-Leffler property and also the finite index property just mentioned, then for each n, the image of the canonical map !Mathematical Expression Omitted^ has finite index. Since !!summation of^X, !S.sup.!n.sub.!Alpha^^^^ maps onto !Mathematical Expression Omitted^, this implies that if

f : !(!summation of^X).sub.k^ !approaches^ !S.sup.!n.sub.!Alpha^^^

is a map, some nonzero multiple of which extends to the t-skeleton of !summation of^X for each t !is greater than or equal to^ k, then some nonzero multiple of f extends to all of !summation of^X. In particular, take k !is greater than or equal to^ !n.sub.!Alpha^^, and take f to a map such that the composite

!Mathematical Expression Omitted^

is nontrivial. Since suspensions have the rational homotopy type of a bouquet of spheres and finite suspensions are o-universal in the sense of Mimura and Toda, !8^, it is clear that maps f with this property exist out of each skeleton of !summation of^X. Consequently one of them has an extension, call it !f.sub.!Alpha^^, to all of !summation of^X. Thus, for each !Alpha^, there is a map !f.sub.!Alpha^^ such that the composite

!Mathematical Expression Omitted^

is nontrivial. Now sum up these !f.sub.!Alpha^^'s as follows: first let

!Mathematical Expression Omitted^

denote the adjoint of !f.sub.!Alpha^^ composed with the obvious inclusion. Then use the loop multiplication to form the finite product of maps into the nth Postnikov approximation,

!Mathematical Expression Omitted^

The !Mathematical Expression Omitted^'s form a coherent sequence and so there is a map, say

!Mathematical Expression Omitted^

that projects to !Mathematical Expression Omitted^ for each n. It is clear that the adjoint of !Mathematical Expression Omitted^ is the required rational equivalence.

The proof of (iii) !implies^ (i) amounts to showing that the tower !G.sub.n^ = !!(!summation of^X).sub.n^, Y^ has the Mittag-Leffler property. To this end, it is enough to show that the image of !!summation of^X, Y^ in !G.sub.n^, induced by restriction to the n-skeleton, has finite index for each n. The sufficiency here follows since this particular image is a lower bound for all the other images of the !G.sub.n+k^'s in !G.sub.n^. If it has finite index, then there can be at most a finite number of distinct subgroups of !G.sub.n^ that contain it and so the Mittag-Leffler property follows as a consequence.

Assume now that the rational equivalence f : !summation of^X !approaches^ V!S.sup.!n.sub.!Alpha^^^ exists and consider the following diagram,

!Mathematical Expression Omitted^,

in which K is a certain (k - 1)-dimensional subcomplex of X, chosen so that the restriction !f.sub.k^ is a rational equivalence from !summation of^K to the indicated subbouquet of spheres. This restriction would not necessarily be a rational equivalence if !summation of^K were the k-skeleton of !summation of^X; in that case !f.sub.k^ might have rational homology kernel in degree k. To get around this problem, let K be a complex of dimension k - 1 with the following properties:

(i) !K.sub.k-2^ = !X.sub.k-2^,

(ii) !H.sub.k-2^ (K;Z) !is approximately equal to^ !H.sub.k-2^ (X;Z),

(iii) !H.sub.k-1^ (K;Z) !is approximately equal to^ !H.sub.k-1^ (X;Z)/torsion.

Both homology isomorphisms are to be induced by a map of K into !X.sub.k-1^. Such a K, as well as the map, exist by Theorem 2.1 of !1^. For our homotopy theoretic purposes, we can (and will) regard K as a subcomplex of X by means of the mapping cylinder construction. It is then clear that f restricts, with the help of the cellular approximation theorem, to a rational homotopy equivalence from the subcomplex !summation of^K to the indicated subbouquet of spheres.

If the restriction, !f.sub.k^, were a suspension, it would then follow by Lemma 1.3 below that the image !f*.sub.k^ is a subgroup of finite index. This fact, together with a diagram chase, would imply that the image of !!summation of^X, Y^ also has finite index in the lower left corner. However, neither f nor !f.sub.k^ was assumed to be a co-H map in this result and so we have to work a little harder.

Take g !is an element of^ !!summation of^K, Y^. The group element !g.sup.n^ can be represented by the nth power map on !summation of^K followed by the map that represents g. The following lemma then shows that for some sufficiently large power, say !Lambda^, the element !g.sup.!Lambda^^ is in the image of !f*.sub.k^.

LEMMA 1.2. Let K be a connected complex whose suspension has the homotopy type of a finite complex and let

!Mathematical Expression Omitted^

be a rational equivalence. Then for some power map, !Lambda^, on !summation of^K, of sufficiently high power, there is a commutative diagram

!Mathematical Expression Omitted^

Assume for the moment that this lemma is true. The group !!summation of^K, Y^ is, of course, nilpotent and finitely generated. The image of !!summation of^X, Y^ in this group is easily seen to be a subgroup that contains the subset image (!f*.sub.k^). Therefore the image of !!summation of^X, Y^ has finite index in !!summation of^K, Y^ by the previous lemma and the following bit of group theory.

LEMMA 1.3. Let G be a finitely generated, nilpotent group and assume that H is a subgroup of G such that for each g in G, !g.sup.!Lambda^^ !is an element of^ H, where !Lambda^ is some nonzero integer that may depend on g. Then H has finite index in G.

The inclusions,

!(!summation of^X).sub.k-1^ !subset or is equal to^ !summation of^K !subset or is equal to^ !(!summation of^X).sub.k^,

induce maps,

!!(!summation of^X).sub.k-1^, Y^ !left arrow^ !!summation of^K, Y^ !left arrow^ !!(!summation of^X).sub.k^, Y^,

such that the image of the group on the right has finite index in the one on the left. It follows that the image of !!summation of^X, Y^ in the left group also has finite index. Since k was arbitrary here, the proof of the first half of Theorem 2 will be complete once we prove the two lemmas just used.

Proof of Lemma 1.2. First choose a set of classes {!g.sub.!Beta^^} !subset^ !Pi^* !summation of^K so that the map,

!Mathematical Expression Omitted^

is a rational homotopy equivalence and so that

f#(!g.sub.!Beta^^) = !m.sub.!Beta^^!!Iota^.sub.!Beta^^.

Here !!Iota^.sub.!Beta^^ denotes the standard inclusion of !S.sup.!n.sub.!Beta^^^ into the bouquet and each !m.sub.!Beta^^ is a nonzero integer. This is possible since f : !summation of^K !approaches^ V!S.sup.!n.sub.!Alpha^^^ is a rational equivalence, and so for each !Beta^, some nonzero multiple !m.sub.!Beta^^!!Iota^.sub.!Beta^^ lies in the image of f# and hence factors as a composition f!g.sub.!Beta^^.

Next recall that the rationalization of !summation of^K can be constructed as an infinite telescope using the power maps given by the suspension co-H-structure. The composition

!Mathematical Expression Omitted^

is certainly divisible by !m.sub.!Beta^^, say

r#(!g.sub.!Beta^^) = !m.sub.!Beta^^!g!prime^.sub.!Beta^^,

and it may be assumed that this equality holds at some finite stage of the telescope. In other words, for sufficiently large !Lambda^, the composition !Lambda^!g.sub.!Beta^^ is divisible by !m.sub.!Beta^^, say

!Lambda^#(!g.sub.!Beta^^) = !m.sub.!Beta^^ !h.sub.!Beta^^.

Define h : V!S.sup.!n.sub.!Beta^^^ !approaches^ !summation of^K by requiring

h#(!!Iota^.sub.!Beta^^) = !h.sub.!Beta^^.

Since evidently,

r!Lambda^ = rhf,

it follows from the infinite telescope description of !(!summation of^K).sub.0^ that for some nonzero power map, l, on !summation of^K,

l!Lambda^ = lhf .

Thus, by replacing !Lambda^ by l!Lambda^ and h by lh, the lemma follows.

Proof of Lemma 1.3. If H is a normal subgroup of G, the quotient G/H is then a finitely generated, nilpotent, torsion group. It must, therefore, be a finite group. Thus, the lemma is true if H is normal in G. If H is not normal, consider the sequence of subgroups

H !is less than or equal to^ !H.sub.1^ !is less than or equal to^ !H.sub.2^ !is less than or equal to^ ...

where each !H.sub.i+1^ is the normalizer in G of !H.sub.i^. Since !H.sub.i^ contains the ith term of the upper central series of G, it follows that G = !H.sub.n^ for some integer n. Any subgroup that contains H clearly satisfies the hypothesis of this lemma, and so as noted above, each quotient !H.sub.i^/!H.sub.i-1^ is finite. The result follows.

Proof of Theorem 1!prime^. For the most part, this will be the Eckmann-Hilton dual of the proof just given. Our treatment will therefore be somewhat sketchy. For simplicity, we henceforth assume, without loss of generality, that Y is simply connected. The assumption that Ph (X, Y) = 0 for all finite type domains implies, in particular, that the towers {!K(Z, n), !Omega^!Y.sub.(k)^^} are Mittag-Leffler. This leads to the existence of maps,

!Mathematical Expression Omitted^,

which induce epimorphisms in rational cohomology. Then let P denote the appropriate product of Eilenberg-MacLane spaces,

!Mathematical Expression Omitted^,

having the weak rational homotopy type of !Omega^Y and topologized as the direct limit of finite products. Use the maps !g.sub.!Alpha^^ obtained above to construct a coherent sequence

gn: P !approaches^ !Omega^!Y.sup.(n)^, n = 1,2,3 ...

of rational n-equivalences and let g: P !approaches^ !Omega^Y be a class that maps to !Mathematical Expression Omitted^. Clearly, the map g satisfies the requirements of part (iii) of Theorem 1!prime^.

The proof of (iii) !implies^ (i) in Theorem 1!prime^ involves the following commutative diagram,

!Mathematical Expression Omitted^,

in which P still denotes the weak product of Eilenberg-MacLane spaces and g is a weak rational equivalence. Again the strategy is to prove Ph (X, Y) = 0 by showing that the tower {!X, !Omega^!Y.sup.(n)^^} is Mittag-Leffler. To this end, it suffices to show that on the right side, the image of !X, !Omega^Y^ has finite index in !X, !Omega^!Y.sup.(n)^^. The image of the map down the left side has finite index since {!X, !P.sup.(n)^^} is a tower of epimorphisms. Thus it suffices to show that any subgroup containing the image of !g*.sup.(n)^ must have finite index in !X, !Omega^!Y.sup.(n)^^. Does this sound familiar? The following lemma is the Eckmann-Hilton dual of Lemma 1.2.

LEMMA 1.4. Let !P.sup.(n)^ denote a finite product of K(Z, i)'s and let

!Psi^ : !p.sup.(n)^ !approaches^ !Omega^!Y.sup.(n)^

be a rational equivalence. Then for some power map, !Lambda^, on !Omega^!Y.sup.(n)^, of sufficiently high power, there is a commutative diagram

!Mathematical Expression Omitted^.

Assume for the moment this result is true. Take a class !Phi^ !is an element of^ !X, !Omega^!Y.sup.(n)^^ and append it to the diagram above as follows,

!Mathematical Expression Omitted^.

The new diagram implies that !!Phi^.sup.!Lambda^^ is in the image of !Psi^*. Thus the rest of Theorem 1!prime^ follows from Lemmas 1.3 and 1.4.

Proof of Lemma 1.4. For each index !Beta^, let !!Pi^.sub.!Beta^^ : !approaches^ K(Z, !n.sub.!Beta^^) be the projection and let !!Iota^.sub.!Beta^^ !is an element of^ H*(!P.sup.(n)^; Z) denote the pullback by !!Pi^.sub.!Beta^^ of the fundamental class. Then choose classes {!z.sub.!Beta^^} !subset^ H*(!Omega^!Y.sup.(n)^; Z) so that

!Psi^* (!z.sub.!Beta^^) = !m.sub.!Beta^^ !!Iota^.sub.!Beta^^.

Let !Omega^ denote both a natural number and the power map on !Omega^!Y.sup.(n)^ of power !Lambda^. Choose it large enough that each !Lambda^*(!z.sub.!Beta^^) is divisible by !m.sub.!Beta^^. Again one verities this is possible by noting that the rationalization of !Omega^!Y.sup.(n)^ can be constructed as an infinite telescope using power maps. Thus in homology, the image of !Lambda^* is divisible by !m.sub.!Beta^^ for sufficiently large !Lambda^. The same is true in cohomology by the universal coefficient theorem. Now define a map

h : !Omega^!Y.sup.(n)^ !approaches^ !P.sup.(n)^

by requiring

h* (!!Iota^.sub.!Beta^^) = 1/!m.sub.!Beta^^ !Lambda^* (!z.sub.!Beta^^).

It follows that the maps !Lambda^ and !Psi^h are rationally homotopic since, for each !Beta^,

!Lambda^*(!z.sub.!Beta^^) = (!Psi^h)* (!z.sub.!Beta^^),

and since the !z.sub.!Beta^^'s determine a rational product decomposition of !Omega^!Y.sup.(n)^. We claim that for a suitably large power map, l, on !Omega^!Y.sup.(n)^,

!Lambda^l = !Psi^hl.

Notice that if we could compose with l on the other side, then this would almost be obvious; equality would follow since two elements, say x and y, in a nilpotent group have the same rational image if and only if !x.sup.l^ = !y.sup.l^ for some non-0 integer l.

We will use the Postnikov tower of !Omega^!Y.sup.(n)^ to prove the claim just made. To simplify notation, let Z denote !Omega^!Y.sup.(n)^ and assume that f and g are two self maps of Z which are rationally homotopic. Let !E.sub.m^ be the lowest stage in the Postnikov tower of Z for which the two compositions,

!Mathematical Expression Omitted^,

are not homotopic. Denote these compositions as !f.sub.m^ and !g.sub.m^. Apply !Z,^ to the principal loop fibration

!Mathematical Expression Omitted^

and note that there is a class !Sigma^ of finite order in !Z, K(!Pi^, m)^ such that !Mathematical Expression Omitted^. It follows again from the infinite telescope description of !Z.sub.0^ that l*(!Sigma^) = 0 for some power map, l, on Z, of sufficiently large power. This implies that l*(!f.sub.m^) = l*(!g.sub.m^). To see this, chase around the following diagram,

!Mathematical Expression Omitted^,

and use the fact that l* is a homomorphism. Now replace f and g by fl and gl and continue up the tower. The claim follows, as do Lemma 1.4 and Theorem 1!prime^.

Proof of Theorem 2. In this proof, the pointed set Ph (X, Y) will be identified with the term !Mathematical Expression Omitted^. To justify this, observe that if f : X !approaches^ Y is a phantom map, then for each n, the composition

!Mathematical Expression Omitted^

is null homotopic. This is so because in the following 3-fold composition,

!Mathematical Expression Omitted^,

the composition of the first two maps is null homotopic for all k, while for k !is grester than^ n, the restriction map induces a bijection between !!X.sub.k^, !Y.sup.(n)^^ and !X, !Y.sup.(n)^^. It follows that Ph (X, Y) is the kernel of the second map in the following short exact sequence of sets, (!2^, Corollary 3.2, page 254),

!Mathematical Expression Omitted^.

In part (i), the map f : X !approaches^ X!prime^ induces a monomorphism in rational homology. It therefore induces an epimorphism on rational cohomology. There are bijections

!Mathematical Expression Omitted^

and

!Mathematical Expression Omitted^,

and since f induces an epimorphism

!Mathematical Expression Omitted^

between each corresponding pair of factors, it follows that the induced homomorphism

!Mathematical Expression Omitted^

is a rational epimorphism. These are finitely generated, nilpotent groups, and so it follows by Lemma 1.3 that the image of f* has finite index in !X, !Omega^!Y.sup.(n)^^ for all n.

In part (ii), the map g: Y !approaches^ Y!prime^ induces an epimorphism on the higher rational homotopy groups. This implies that for any n, the basepoint component of !(!Omega^!Y!prime^.sup.(n)^).sub.0^ is a weak retract of !(!Omega^!Y.sup.(n)^).sub.0^, and that !(!Omega^!g.sup.(n)^).sub.0^ is a weak retraction. So again the induced homomorphism

!Mathematical Expression Omitted^

is a rational epimorphism, and again the index of the image of !Omega^!g*.sup.(n)^ is finite for each n.

The k-invariants of the loop space !Omega^Y have finite order and so it follows that for each n, the image of !X, !Omega^!Y.sup.(n+1)^^ has finite index in !X, !Omega^!Y.sup.(n)^^. Both parts of Theorem 2 are then a consequence of the following algebraic result.

LEMMA 2.2. Let {!H.sub.n^} and {!G.sub.n^} be two inverse towers of groups. Assume both towers have the following finite index property: for each n, the image of the (n+1)st term in the tower has finite index in the nth. Let !Phi^ be a tower map such that, for each n, the image of the homomorphism

!!Phi^.sub.n^ :!H.sub.n^ !approaches^ !G.sub.n^

has finite index in !G.sub.n^. Then the induced map

!Mathematical Expression Omitted^

is surjective.

Proof. Let !I.sub.n^ denote the image, !!Phi^.sub.n^(!H.sub.n^). Using the six term !Mathematical Expression Omitted^ sequence, it follows that the quotient map, {!H.sub.n^} !approaches^ {!I.sub.n^}, induces an epimorphism from !Mathematical Expression Omitted^ to !Mathematical Expression Omitted^. We claim that in the tower {!G.sub.n^}, there is a subtower, {!N.sub.n^}, with the properties that

(i) each !N.sub.n^, is normal and of finite index in !G.sub.n^, and

(ii) each !N.sub.n^ is contained in !I.sub.n^.

Assume for the moment this is true. The six term !Mathematical Expression Omitted^ sequence, applied to the short exact sequence of towers,

{!N.sub.n^} !approaches^ {!G.sub.n^} !approaches^ {!G.sub.n^/!N.sub.n^},

implies that the inclusion, !N.sub.n^ !approaches^ !G.sub.n^, induces an epimorphism

!Mathematical Expression Omitted^.

This uses the well known fact that a tower of finite groups (here !G.sub.n^/!N.sub.n^) has a trivial !Mathematical Expression Omitted^ term.

Since !N.sub.n^ !subset or is equal to^ !I.sub.n^ !subset or is equal to^ !G.sub.n^, the epimorphism just displayed factors through !Mathematical Expression Omitted^ !I.sub.n^. Consequently, the inclusion, !I.sub.n^ !approaches^ !G.sub.n^, must also induce an epimorphism on the !Mathematical Expression Omitted^ level. The conclusion of the lemma is evident; it only remains to justify the claims made about the subtower {!N.sub.n^}. To this end, let !Pi^ : !G.sub.n^ !approaches^ !G.sub.n-1^ denote the structure map in the tower and assume that !N.sub.n-1^ exists with the properties:

(i) !N.sub.n-1^ is normal and has finite index in !G.sub.n-1^, and

(ii) !N.sub.n-1^ is contained in !Pi^(!I.sub.n^).

Let !N.sub.n^ denote the largest normal subgroup in !G.sub.n^ that is contained in !Pi^(!I.sub.n+1^) !intersection^ !!Pi^.sub.-1^(!N.sub.n-1^). It is not difficult to check that this intersection has finite index in !G.sub.n^ and thus so does !N.sub.n^. The existence of the tower {!N.sub.n^} with the required properties then follows by induction.

There are some special cases of Theorem 2 that admit a simpler proof. In part (i), for example, if either f: X !approaches^ X!prime^ is a rational homotopy equivalence of nilpotent spaces, or if both X and X!prime^ are !H.sub.0^-spaces, then the map !f.sub.0^ admits a left inverse. One can then use a theorem of Zabrodsky, !10^, which asserts that the phantom maps from X to a space Y are precisely those maps that factor through a rationalization of X. In other words, if r: X !approaches^ !X.sub.0^ denotes a rationalization of X, then

Ph (X, Y) = r*!!X.sub.0^, Y^.

The proof of Theorem 2(i) then follows immediately from the following commutative diagram,

!Mathematical Expression Omitted^.

wherein the map on the right is surjective.

Proof of Example 3. Take the two fold loops on the double suspension,

!Mathematical Expression Omitted^,

and notice that it is a rational homology equivalence. By Theorem 2(i) and Example 2.1 of !5^, it follows that

Ph (!!Omega^.sup.4^!S.sup.2n+3^, !S.sup.2n^) !is not equal to^ 0.

This argument can be iterated to show

Ph (!!Omega^.sup.2t^!S.sup.2n+2t-1^, !S.sup.2n^) !is not equal to^ 0.

for all t !is greater than^ 1. So the example is verified when k = 2t. When k = 2t + 1, use

!!Omega^.sup.2t+1^ !S.sup.2n+2t^ and !!Omega^.sup.2t^ (!S.sup.2n+2t-1^ x !Omega^!S.sup.4n+4t-1^).

There are rational equivalences between these two spaces in both directions. Therefore the case k = 2t+1 follows from the case k = 2t and Corollary 2.1(i).

Some open questions.

(1) Is the duality expressed in Theorems 1, 1!prime^, and 2 as sharp as it appears to be? More precisely, we know that Theorem 1 and 2(i) are best possible in the sense that if one drops the finite type requirements on Y, they become false. We do not know if this is the case in the second parts. To put it another way, suppose Y is a finite type target such that Ph (X, Y) = 0 for all finite type domains X. Does it follow or not that Ph (X, Y) = 0 for all spaces X?

(2) In certain cases, there is a natural abelian group structure on Ph (X, Y). This is true, for example, if X and Y are nilpotent of finite type and either X has the rational homotopy type of a suspension or Y has the rational homotopy type of a loop space. In such cases, does the isomorphism of sets in Corollary 2.1 become an isomorphism of groups?

It is worth noting that there exist nonisomorphic groups, such as R and R!symmetry^Z/!p.sup.!infinity^^, with epimorphisms between them in both directions. Moreover, groups such as these are known to occur as Ph (X, Y) for suitable X and Y.

(3) Here is another possibility raised by Corollary 2.1. Following Wilkerson, let SNT(X) denote the pointed set of all homotopy types !Y^ having the same n-type as X for all n. In !9^, he showed that !Mathematical Expression Omitted^. If X and X!prime^ are two finite type spaces with rational equivalences between them in both directions, does it follow that the sets SNT(X) and SNT(X!prime^) are isomorphic?

(4) Notice that if X is a finite type domain such that Ph (X, Y) = 0 for all finite type targets, then it follows from Theorem 1 that for each natural number n, there is a self map of !summation of^X, say !!Phi^.sub.n^, which induces in rational homology,

(i) an isomorphism in degrees !is less than or equal to^ n, and

(ii) the zero map in degrees !is greater than^ n.

This seems to be a curious property. There are no maps like these on !CP.sup.!infinity^^, even stably. One might suspect that a space that has such self maps could never be atomic at some prime, but this isn't the case] A counterexample is the loops on the Lie group Sp(2). In his Northwestern thesis, Hopkins showed that !Omega^Sp(2) is stably atomic at p = 2, while in !3^, it is shown that Ph (!Omega^G, Y) = 0 for all 1-connected, compact Lie groups G and for all finite type targets Y. Here is the question: on a given !summation of^X, does the presence of the !!Phi^.sub.n^'s imply the existence of a rational homology equivalence from !summation of^X to a bouquet of spheres?

REFERENCES

!1^ I. Berstein and P. J. Hilton, On suspensions and comultiplications, Topology 2 (1963), 73-82.

!2^ A.K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Math. vol. 304, Springer-Verlag, New York, 1972.

!3^ B. Gray and C. A. McGibbon, Universal phantom maps, Topology 32 (1993), 371-394.

!4^ J. Lannes, Sur la cohomologie modulo p des p-groupes Abeliens elementaires, Homotopy Theory, London Math. Soc. Lecture Note Ser. vol. 117, (E. Rees and J.D.S. Jones eds.), London, 1987, pp. 97-116.

!5^ C.A. McGibbon, Loop spaces and phantom maps, Contemp. Math. 146 (1993), 297-308.

!6^ C. A. McGibbon and J. M. Moller, On spaces with the same n-type for all n, Topology 31 (1992), 177-201.

!7^ W. Meier, Determination de certains groupes d'applications fantomes, C. R. Acad. Sci. Paris Ser 1 Math. 282 (1976), 971-974.

!8^ M. Mimura and H. Toda, On p-equivalences and p-universal spaces, Comment Math. Helv. 46 (1971), 87-97.

!9^ C. W. Wilkerson, Classification of spaces of the same n-type for all n, Proc. Amer. Math. Soc. 60 (1976), 279-285.

!10^ A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. (2) 58 (1987), 129-143.
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