On stable homotopy equivalences
by
R. R. Bruner, F. R. Cohen1, and C. A. McGibbon
A fundamental construction in the study of stable homotopy is the free infinite
loop space generated by a space X. This is the colimit QX = lim-!nnX.
The ith homotopy group of QX is canonically isomorphic to the ith stable
homotopy group of X. Thus, one may obtain stable information about X by
obtaining topological results about QX. One such result is the Kahn-Priddy
theorem [7]. In another direction, Kuhn conjectured in [8] that the homotopy
type of QX determines the stable homotopy type of X. In this note we prove
his conjecture for a finite CW-complex X; that is, we prove the following.
Theorem If X and Y are finite CW -complexes, then QX and QY are ho-
motopy equivalent if and only if nX and nY are homotopy equivalent for
some sufficiently large integer n. 2
Of course, one direction of our theorem is obvious. A homotopy equiv-
alence nX ! nY clearly induces a homotopy equivalence QX ! QY .
The proof in the other direction has three steps. Here is a brief outline of
it. We begin with the case where X and Y are connected and we prove a
p-local version of our result for each prime p. In doing so, we use results of
Wilkerson, [11], to first express the stable p-local homotopy types of X and
Y as bouquets of prime retracts. The same sort of analysis is then carried
out, p-locally, on QX and QY , through an appropriate range of dimensions.
The assumption that QX and QY are homotopy equivalent then forces the
stable prime retracts of X(p)to match up with those of Y(p).
______________________________
1Partially supported by NSF
1
In the second step the spaces are still assumed to be connected. Here we
assemble the p-local results of step 1 using Mislin's notion of the stable genus
of a space. The key ingredient in this step is Zabrodsky's presentation of the
genus of certain spaces in terms of their self maps. The third step handles the
case when the spaces are not connected. Here the Segal conjecture is used to
show that X and Y must have the same number of path components.
In the final section we consider the related question of when {QnX} is
the only infinite delooping of QX among connective spectra. Examples are
provided showing that QX does not deloop uniquely in general. We also give
a result which implies 2-local uniqueness of deloopings in certain cases.
We thank Clarence Wilkerson for helpful comments on this project. Re-
sults of his, from [11], play a crucial role in our proof. We also thank the
referee for improvements in the exposition.
The connected p-local case
In this section we collect the results needed to prove the following.
Theorem 1 Let X and Y be connected finite CW -complexes. If QX and
QY are homotopy equivalent at a prime p then X and Y are stably homotopy
equivalent at p. 2
We start with a well known observation. Let and ffl be used generically
to denote the unit and counit of the adjunction between n and n, for
1 n 1.
Lemma 2 If a space B is a retract of an n-fold suspension, where n 1,
then there is a lift , of over ffl.
2
nnQB
j3
j j ||
j j ||ffl
jj |
|?
B _____________- QB 2
When splitting the suspensions of certain p-local spaces, we will need the
following definition from [11], wherein it is referred to as "H*-prime".
Definition Let X be a 1-connected p-local space of finite type. Then X is
said to be prime if for every self-map f : X ! X, either
i) f induces an isomorphism in mod-p homology, or
ii) for every n, there exists an m such that the m-fold iterate of f induces
the zero map on Hi(X; Zp) for 0 < i n. 2
We will use the following facts from [11].
Theorem 3 (Wilkerson)
i) Any finite dimensional 1-connected p-local co-H-space is equivalent to a
wedge of prime spaces.
ii) If a 1-connected p-local space of finite type is equivalent to a wedge of
primes, then the prime wedge summands are unique up to order.
iii) A prime space which is a retract of a wedge of 1-connected p-local spaces
of finite type is a retract of one of the wedge summands. 2
Proofs of the p-local results
f n g
Proof of Lemma 2 Let B -! W - ! B be maps such that gf ' 1.
The map = (nnQg)(n)f in the following diagram provides the needed
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factorization = ffl.
n nn n ffl n
nW ______- Q W ______- Q W
| | |
|6 |nn |
f | | Qg |Qg
| | |
| |? |?
nn ffl
B ______- QB ______- QB 2
Proof of Theorem 1 Let : QX -! QY be a homotopy equivalence where
X and Y are connected finite complexes. Fix a prime p and henceforth assume
that all nilpotent spaces in this proof have been localized at p. (Of course,
X and Y are not necessarily nilpotent, but they will become so after one
suspension.) We will not burden the notation with this p-local assumption.
Let n denote a fixed integer greater than both dim (X) and dim (Y ). By
Theorem 3, nX and nY decompose uniquely into wedges of prime retracts.
Let W denote the subbouquet of all prime retracts common to both spaces
and let X0 and Y 0denote the rest. That is, write
nX ' W _ X0 and nY ' W _ Y 0
where X0and Y 0are assumed to have no nontrivial prime retracts in common.
Now consider the composition
n()n n n( )n n r n
nX ____________- (QX)n ____________- (QY )n ______- Y
where ( )n denotes the n-skeleton or restriction to it. The last map is
the restriction of a retraction of 1 QY onto 1 Y , which exists by Kahn's
theorem [5]. In the stable range this map exists on the space level. Since X
and Y are connected, each of the maps in this sequence induces a homology
isomorphism in the lowest degree in which reduced homology for these spaces
does not vanish. Applying the same argument to the inverse of , it follows
that the prime wedge summands of nX and nY of lowest connectivity
must have their homology mapped isomorphically by these composites, and
hence be common to both nX and nY . Assuming that X and Y are not
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contractible, it follows then that the subbouquet W is not empty, and that
the connectivity of X0 and of Y 0is strictly greater than that of W . We may
assume conn(X0) conn(Y 0). Note that
nQX ' nnQnX
' nnQ(W _ X0)
' nn(QW x QX0)
' nnQW _ nnQX0 _ n(nQW ^ nQX0):
Similarly,
nQY ' nnQW _ nnQY 0_ n(nQW ^ nQY 0):
Now compare the 2n-skeletons of nQX and nQY . By Theorem 3,
they each split uniquely as a wedge of primes. Among these prime retracts
consider only those whose dimension is less than 2n. Since the restriction
of n and its inverse are at least (2n - 1)-equivalences on the skeleta in
question it follows that there is a one-to-one correspondence between the
primes of dimension less than 2n in n(QX)n and those in n(QY )n. From
the decomposition of nQX and nQY , obtained in terms of the summand
W , it follows that the primes of dimension less than 2n in
n(nQX0)n _ n(nQW ^ nQX0)n (1)
must coincide, up to order, with those in
n(nQY 0)n _ n(nQW ^ nQY 0)n (2)
Observe that the 2n-skeleton of QX0 is just X0 because X0 is at least
n + 1-connected. Also, dim (X0) < 2n since dim (X) < n. Thus, the map
of Lemma 2 factors through n(nQX0)n and provides a splitting of
n(nQX0)n into X0 and another factor whose connectivity is higher than
that of X0. (The latter claim follows by examining homology.) Similarly for
Y 0.
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Now let P be a prime in X0 of minimal connectivity. It has dimension less
than 2n because X0 does. It occurs in (1), and so it must also occur in (2). It
is not a retract of Y 0, by construction. It does not occur in the complement of
Y 0in n(nQY 0)n because that complement has connectivity higher than
that of Y 0and hence higher than that of P . The connectivity of the second
wedge summand in (2) is easily seen to be
1 + conn(W ) - n + conn(Y 0) > conn(Y 0) conn(X0) = conn(P )
so P cannot occur here either. Thus, no such prime P exists and we have
shown that X0 ' *. Since the connectivity of Y 0is no lower than that of
X0, we must have Y 0' * as well. 2
The connected integral case
The goal in this section is to prove the following.
Theorem 4 Let X and Y be connected finite CW -complexes. If QX is
homotopy equivalent to QY then X is stably homotopy equivalent to Y .
2
The proof of this theorem involves the genus G(X), of a space X, as
defined by Mislin [10]. Recall that if X is a nilpotent space of finite type
then G(X) is the set of all homotopy types [Y ] where Y is a nilpotent space
with finite type and Y(p)' X(p)for each prime p. According to Wilkerson
[13], G(X) is a finite set when X is a simply connected finite complex. Since
localization commutes with suspension, there is an obvious map from G(X)
to G(X). Define the stable genus of a finite complex X to be
Gs(X) = lim-!G(nX):
It is not difficult to see that the stable genus of a finite complex X can be
identified with G(nX) for n sufficiently large.
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Suppose now that QX ' QY . Then, of course, QX(p)' QY(p)for
every prime p. By Theorem 1, X(p)is stably homotopy equivalent to Y(p)for
every prime p, and so Y 2 Gs(X). Therefore to prove Theorem 4 it suffices
to prove the following lemma.
Lemma 5 If X is a finite complex, then the function
: Gs(X) - ! G(QX)
that sends [Z] in G(nX) to [nQZ] in G(QX) is one-to-one. 2
Proof: To show that is one-to-one, it suffices to show that the composite
Gs(X) - ! G(QX) - ! G(QX(n))
is injective for some n. Here QX(n)denotes the Postnikov approximation of
QX through dimension n. It may be obtained by attaching cells to kill off
the homotopy groups of QX in dimensions greater than n. The second map
here sends a homotopy type [Z] in G(QX) to its Postnikov approximation
[Z(n)] in G(QX(n)).
Let W denote a connected H-space which has finite type and only finitely
many nonzero homotopy groups. The main result in Zabrodsky's paper [14],
is an exact sequence
Et(W ) -d! (Z*t= 1)` -! G(W ) - ! *
which is defined as follows. In the middle term, Z*t denotes the group
of units in the ring of integers modulo t. The exponent ` is the number
of positive degrees k, in which the module of indecomposables in rational
cohomology,
QHk(W ; Q) 6= 0:
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If there is more than one such degree k, order them k1 < . . .< k`. The
number t depends upon the space W . The prime divisors of t include those
primes p, for which there is p-torsion in either the homotopy groups of W , or
in the integral homology groups of W through degree k`, or in the cokernel
of the Hurewicz homomorphism, through the same range. Zabrodsky gives a
description of the smallest possible exponents p(t), as well. However, for our
purposes the exact value of t is unimportant; indeed given any one choice of
t, any integer multiple of it works equally well in this sequence.
The first term in the sequence, Et(W ), denotes the monoid (under
composition) of homotopy classes of those self-maps of W , which are local
equivalences at each prime divisor of t. The function d then assigns to each
such map a sequence of determinants - or rather the image of such a sequence
in the middle group. The jth determinant here is that of the linear transfor-
mation on QHkj(W ; Q) induced by the map f. Zabrodsky shows that this
image is a subgroup and that the quotient is isomorphic as an abelian group
to G(W ).
If W is a finite complex in the stable range (and hence a co-H-space),
there is a similar presentation of G(W ). This was first proved by Davis in
[4]; a much shorter proof of this was subsequently given in [9]. Take n to be
larger than dim(X) so that nX is in the stable range.
The map fits into a commutative diagram
0
Et(nX) -d! (Z*t= 1)` -! Gs(X) -! *
| | |
| | |
| |= |
| | |
|? |? |?
Et(QX(n)) -d! (Z*t= 1)` -! G(QX(n)) -! *
whose left hand side sends a self-map f of nX to the self-map (nQf)(n)
of QX(n). Recall that QX(n) denotes the Postnikov approximation for
QX through dimension n. To show is one-to-one it suffices to show that
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the image of d is contained in the image of d0. Thus, given a self map f, of
QX(n), it suffices to produce a self map of nX with the same determinant
sequence as f, up to sign. As mentioned earlier, we may view QX(n) as a
cell complex obtained by attaching to QX cells of dimension n + 2 and higher
and so there is an equivalence of skeleta
(QX(n))n ' (QX)n:
There is also a retraction
n(QX)n -r! nX:
Therefore, given a self-map
f (n)
QX(n) -! QX
we can take the composition
f (n) r n
n X - ! QX - ! QX(n) -! QX -! X
n
It is straightforward to check that this map has, up to signs, the same deter-
minant sequence as f. This completes the proof of Lemma 5.
The non-connected case
Suppose now that X has path components X0; : :;:Xn, where n 1. It
is known, (e.g., see [1]), that in this case,
QX ' QX0 x . .x.QXn x (QS0)n:
Let B denote the path component of QS0 that contains the constant loop. It
is well known that QS0 ' Z x B. Thus each path component of QX has
the homotopy type of
QX0 x . .x.QXn x Bn:
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Assume now that QX ' QY where Y is another finite complex. Obviously
then, QY , and hence Y , are not connected either. Each path component of
QX must be homotopy equivalent to a path component of QY . Thus we may
assume that
QX0 x. .x.QXn x Bn ' QY0 x. .x.QYm x Bm (3)
where each Xi and Yi is a connected finite complex. We first claim that
n = m. To see this, assume m < n and take the p-completion of both sides
of (3). Completions commute with products. Moreover, after completing
we can cancel the p-completion of Bm from both sides. This follows using
the unique factorization results of [11] for p-local H-spaces with only finitely
many nonzero homotopy groups and the results of [12] which imply that a
p-complete homotopy type is determined by its sequence of n-types. Thus we
have
(QX0 x. .x.QXn x Bn-m )bp ' (QY0 x. .x.QYm )bp (4)
Since each Yiis a finite connected complex, the Segal conjecture (or rather its
affirmation, [3]) implies that there are no essential maps from BZp into QYi
and hence there are no essential maps of BZp into the right hand side. The
Segal conjecture 2also asserts that there are many essential maps from BZp
into B, and hence into the left hand side of (4). From this contradiction we
conclude that m = n.
Return now to equation (3), localize at p, and take the Postnikov approx-
imation of both sides, through dimension d, where d exceeds the dimensions
of X and Y . Results from [11], essentially the Eckmann-Hilton dual of The-
orem 3, quoted earlier, allow us to cancel (Bn)(d) from both sides. We are
left with
(QX0 x. .x.QXn)(d) ' (QY0 x. .x.QYn)(d): (5)
______________________________
2Of course, the Kahn-Priddy theorem could have been used here just as well.
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Notice that the left side has the form (Q(X0 _ . ._.Xn))(d)and similarly
for the right side. Thus both sides have the form (QW )(d)where W is a
connected complex. If we restrict to the d-skeleton of both sides of (5) and
suspend d times, it follows from Theorem 1 that _iXi and _iYi are stably
p-equivalent. Hence these two bouquets are in the same stable genus and, by
Lemma 5, it follows that they are stably homotopy equivalent. Since
_n _n
kX ' ( kXi) _ ( Sk);
i=0
when k 1, and a similar expression holds for kY , it follows that X and Y
are stably homotopy equivalent. 2
When does QX deloop uniquely ?
The results obtained so far could be regarded as a first step toward answering
the question just raised. This general question seems harder to answer.
Example 6 The space QS1 has at least two distinct connective deloopings;
namely {QSn} and {K(Z; n)xBn-1F }, where F is the fiber of the retraction
QS1 ! S1.
The key property here is that the circle S1 is an infinite loop space. However,
this method will not produce many more examples, since Hubbuck has shown
that a finite complex which is a homotopy commutative H-space, for example,
an infinite loop space, must be a torus [6].
If we consider QX for infinite complexes X, the Kahn-Priddy theorem,
[7], provides some further examples of non-uniqueness of deloopings.
Example 7 Let Sp be the symmetric group on p letters. The space QBSp
has at least two distinct connective deloopings, because it is p-equivalent (as
spaces but not as infinite loop spaces) to the product QgS1 x F 0, where QgS1
denotes the universal cover of QS1 and F 0is the fiber of a p-local infinite lo*
*op
map QBSp ! QgS1.
The next theorem shows, on the other hand, that there are many examples
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where QX admits precisely one infinite loop structure, provided X is a double
suspension. We say that a connected space is atomic if any self map of it which
induces an isomorphism on its first nonzero reduced integral homology group
is a homotopy equivalence.
Theorem 8 Let X be a 1-connected, 2-local, stably atomic finite CW -
complex. Then Q2X has only one connective infinite delooping. 2
Proof: It was shown in [2] that Q2X is atomic when X satisfies the
hypothesis of this theorem. Now assume that Y is a connected infinite loop-
space and that
g : Q2X -! Y
is a homotopy equivalence. We need to produce an infinite loop map from
Q2X to Y that is also a homotopy equivalence. Extend the composition
2 g
2X _______-Q X _______-Y
to an infinite loop map : Q2X -! Y . The composite
g-1 2
Q2X ________-Y ________-Q X
is an isomorphism on bottom dimensional homology, and hence an equivalence
since Q2X is atomic. Since g and g-1 are homotopy equivalences, so
is . 2
In fact, calculations suggest the following conjecture.
Conjecture 9 Let X be a finite CW -complex. Then Q2X has only
one connective infinite delooping. 2
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R. R. Bruner and C.A. McGibbon
Wayne State University
Detroit, MI, 48202
rrb@math.wayne.edu and mcgibbon@math.wayne.edu
F. R. Cohen
University of Rochester
Rochester, NY, 14620
cohf@db1.cc.rochester.edu
14