On Decompositions in Homotopy Theory
Brayton Gray
Dept. of Mathematics, Statistics
and Computer Science (M/C 249)
University of Illinois at Chicago
851 South Morgan Street
Chicago, IL 60607-7045
brayton@uic.edu
Abstract
We first describe Krull-Schmidt theorems decomposing H spaces and simpl*
*y con-
nected co-H spaces into atomic factors in the category of pointed nilpoten*
*t p-complete
spaces of finite type. We use this to construct a 1-1 correspondence betwe*
*en homotopy
types of atomic H spaces and homotopy types of atomic co-H spaces, and con*
*struct a
split fibration which connects them and illuminates the decomposition. Var*
*ious prop-
erties of these constructions are analyzed.
The Krull-Schmidt property first arose in the theory of R-modules, and when *
*valid, it
states that each object decomposes in a unique way into a sum of indecomposable*
* objects of
the same type. Numerous examples of decomposing the loop space on a co-H space *
*can be
found in the literature ([Hi], [M ], [CM ], [AG ], [G1 ], [G2 ]). Typically wh*
*at happens is that the
loop space of an atomic co-H space is a product of various factors, and the lea*
*st connected
factor is an H space of special interest, while the other factors, in some sens*
*e, represent
noise. The first general Krull-Schmidt type theorem in homotopy theory was prov*
*ed for the
p-localization of simply connected finite complexes which are either H spaces o*
*r co-H spaces
by Wilkerson [W ]. Various stable versions appear in [F ], [Ma ] and [H ].
We will eliminate the finite complex assumption at the expense of retreating*
* to the cate-
gory C^pof pointed connected nilpotent p-complete spaces with Hi(X; Z=p) finite*
*ly generated
for each i. Accordingly, we restrict ourselves to this category in the sequel. *
*All colimit con-
structions will be completed without further notice. In particular, co-H spaces*
* will be defined
in terms of the coproduct (which is the completion of the one point union), sus*
*pensions will
be completed and loop spaces will only be considered when the underlying space *
*is simply
connected.
In section 1 we will exploit the strengthened notion of atomicity in this ca*
*tegory developed
by Adams and Kuhn [AK ], and prove the following Krull-Scmidt theorem.
1
Theorem A Each H space in C^pis homotopy equivalent to the weak direct product*
* of
atomic H spaces unique up to order. Each simply connected co-H space in C^pis h*
*omotopy
equivalent to the coproduct of atomic complete co-H spaces unique up to order.
In section 2 we give a general correspondence between retracts of n-fold sus*
*pensions and
retracts of n-fold loop spaces (2.2). In particular, for n = 1 we get
Theorem B There is a 1-1 correspondence between homotopy types of atomic H spa*
*ces T
in C^pand homotopy types of 1-connected atomic co-H spaces G in C^p.1
We call such a pair (G, T ) a corresponding pair. In this corrrespondence, T*
* is a retract
of G and G is a retract of ST :
f g0 0
G -! ST -! G g f ~ 1
g h
T -! G -! T hg ~ 1.
In fact (2.5) it is possible to choose g and g0 so that they are adjoint. We w*
*ill call the
maps (f, g, g0, h) structure maps for the corresponding pair (G, T ). A choice*
* of structure
maps determines an H space structure on T and a co-H space structure on G via *
*the
compositions:
gxg h
m : T x T -! G x G -! G -! T
f g0_g0
n : G -! ST -! ST _ ST -! G _ G.
A surprising amount of the structure of known examples is found in the gener*
*al theory.
In particular, given structure maps (f, g0, h) there is a fibration:
T ---i! R --i-! G
with i null homotopic. This leads to 3.2 and 3.12 which we summarize as
Theorem C Suppose T is an atomic H space and G is the corresponding atomic co-H
space. Then there is a homotopy equivalence
G ' T x R
whereWR is a retract of the completed join T *T and hence a complete co-H space*
*. Furthermore
R = Rffwith each Rffatomic and
a) if G = SX, Rffis retract of SX(i)for some i 2
b) if G is homotopy co-associative, Sj-1Rffis a retract of G(j)for some j 2
________________________________
1Theorems A and B together imply that homotopy types of H spaces in C^pare i*
*n 1-1 correspondence with
homotopy types of simply connected co-H spaces in C^p. However we will not cons*
*ider this correspondence.
2
W
c)if G = SX and G ^ G ' ff2ASnffG, R ' G ^ W where W is a wedge of spheres*
* and
t(PT - 1)
PW = PT - _________+ 1
PG - 1
where PX is the Poincar'e series for X.
This result should be compared with the results of Selick and Wu [SW ]. The*
*y have a
similar decomposition in case G as the double suspension of a p tosion space:
S2X = A x B
where X is contained in A. A however is much larger than T . In fact, T is a re*
*tract of A.
In section 3 we construct the fibration which controls the splitting. Sectio*
*n 4 deals with
the possibility of dualizing the material in section 3 and gives a counter-exam*
*ple to a con-
jecture of Ganea. Section 5 discusses naturality and section 6 constructs a ref*
*inement in the
case that T has a homotopy associative multiplication. In section 7 we discuss *
*criteria which
are needed for the spece T to have a homotopy associative H space structure. Fi*
*nally, in an
appendix, we collect some results of a general nature regarding fibrations over*
* suspensions.
x1. Homology and cohomology groups will be with Z=p coefficients unless oth*
*erwise
noted. We will work in the category C^pof pointed connected nilpotent p-complet*
*e spaces of
the homotopy type of a CW complex X with Hi(X) finitely generated for each i. W*
*e need
to arrange our definitions so that p-completion preserves the usual operations *
*of homotopy
theory. All spaces we consider will either be simply connected or admit H space*
* structures,
so they will be Z=p good in the sense of [BK ]. All fibrations we consider will*
* have a simply
connected base, so p-completion preserves fibrations. The p-completion of a co-*
*H space is
often not a co-H space. However, the category of p-complete spaces does have a *
*co-product
- the p-completion of the one point union:
(X _ Y )^p
and we will write X _ Y for this co-product. With this in mind, we define a co-*
*H space in
C^pto be a space G together with a map
: G -! G _ G
such that the composition with the natural map G _ G -! G x G is homotopic to t*
*he
diagonal. Thus the p-completion of a co-H space is a co-H space in this sense.*
* If X is
simply connected, we will consider X and ( X)^p= (X^p). The functor has a *
*left
adjoint given by the completion of the suspension, and we write SX for the spac*
*e (SX)^p.
Then SX is a complete co-H space. Similarly we write X ^Y and X *Y for the p co*
*mpletions
of the smash and join of two spaces. Then all of these operations commute with *
*p-completion.
Adams and Kuhn discuss self maps of such spaces [AK ]. They show a close rel*
*ationship
between irreducibility of such spaces (having no nontrivial retracts) and atomi*
*city. We
summarize the results of [AK ] that we will need.
3
Definition 1.1 A based map f : X -! X is called topologically nilpotent if the *
*sequence
{fn} conveys to the constant map in the profinite topology on [X, X].
In particular, if ßk(X) is finite for each k, this is equivalent to the stat*
*ement that for all
k there is an n such that fn is null homotopic when restricted to the k-skeleto*
*n of X. On
f* k
the other hand, if for some k > 0 f* induces an isomorphism Hk(X) -! H (X) 6= *
*0 or f*
induces an isomorphism f* : ßk(X) -! ß*(X), then f is not topologically nilpote*
*nt.
Definition 1.2 A space is atomic iff every self map is either an equivalence or*
* is topologi-
cally nilpotent.
This seems stronger than the usual definition [CM ]. That it isn't (in the *
*case of p-complete
spaces of finite type) follows from:
Theorem 1.3 (Adams and Kuhn) If X is not atomic, there is a nontrivial ide*
*mpotent
e : X -! X; i.e., e2 = e, e 6= *, e not an equivalence.
As an immediate corollary we have a generalization of a result of Wilkerson *
*[W ] in the
p-complete case.
Corollary 1.4 Let X be an H space. Then there are atomic spaces Xi such that X*
* is
homotopy equivalent to the weak direct product of the Xi:
X ' Xi
Furthermore, the Xi are unique up to order and do not depend on the H space str*
*ucture of
X.
Corollary 1.5 Let X be a simply connected co-H space. Then there are atomic spa*
*ces Xi
such that X is homotopy equivalent to the co-product of the Xi in C^p:
i` j^
X ' Xi .
p
Furthermore the Xi are unique up to order and do not depend on the co-H structu*
*re on X.
Proof. We consider Corollary 1.4 first. If X is not atomic, choose a nontriv*
*ial idempotent
self map by Theorem 1.3; write Tel(e) for the telescope:
X -e! X -e! X -e! . ...
And similarly for Tel(1 - e) where the H space structure and inverse map are us*
*ed to
construct 1 - e : X -! X. Then the natural map:
X -! Tel(e) x Tel(1 - e)
induces isomorphisms in all homotopy groups. Thus both telescopes are retracts *
*of X and
so are themselves complete. Thus every non atomic H space can be split as a pr*
*oduct.
Since X is of finite type, iteration of this process converges, and the limit i*
*s complete. The
uniqueness assertion follows from the following argument essentially due to Wil*
*kerson [W;
1.6].
4
Lemma 1.6 Suppose X is atomic and X is a retract of X1 x X2 where X1 and X2 a*
*re H
spaces. Then either X is a retract of X1 or a retract of X2.
Proof. Let ei : X1 x X2 -! X1 x X2 be projection through Xi, i = 1, 2. Then
(e1)* + (e2)* = 1 in homotopy. Let fi= X -! X be the composition
ei
X -! X1 x X2 -! X1 x X2 -! X.
SoP(f1)* + (f2)* = 1. This implies that (f1)*(f2)* = (f2)*(f1)*. So 1 = [(f1)* *
*+ (f2)*]N =
N i N-i
i (f1)*(f2)* . If both f1 and f2 were topologically nilpotent, this sum wo*
*uld be trivial
for large N. Thus one of them is not topologically nilpotent and hence an equiv*
*alence.
The argument for Corollary 1.5 is similar. We replace homotopy by homology, *
*products
by coproducts and use the co-H structure to add and subtract.
x2. We will often require that the space T admits an H space structure and t*
*he space
G admits a co-H space structure, but will have no need to specify any particula*
*r structure.
This is equivalent to assuming that the inclusion ' : T -! ST has a left homot*
*opy inverse
and that the evaluation map ffl : S G -! G has a right homotopy inverse. For t*
*he later
result we use the fact that G is simply connected to conclude that the completi*
*on of the
pullback square [Ga ]:
S G ---! G _ G
? ?
? ?
y y
G ---! G x G
is again a pullback square. More generally we have (see [KSW ])
Definition 2.1 A space T is an Hn space if the inclusion ' : T -! nSnT has a *
*left
homotopy inverse and a space G is a co-Hn space if the evaluation ffl : Sn nG -*
*! G has a
right homotopy inverse.
Theorem 2.2 There is a 1-1 correspondence between homotopy types of connected*
* atomic
Hn spaces and homotopy types of n-connected atomic co-Hn spaces for n 1.
Proof. Both an n-fold suspension is a co-Hn space and retract of a co-Hn sp*
*ace is a
co-Hn space. Thus, for any space X we can apply Corollary 1.5 to SnX:
`
SnX ' Gi
and each Gi is a co-Hn space. Choose one of these Gi with the least connectivit*
*y. We shall
see that in case X is a connected atomic Hn space, there is only one such choic*
*e. Similarly,
if X is n-connected choose an atomic factor T of nX of least connectivity. We*
* shall see
that in case that X is an atomic co-Hn space, again, there is only one choice.
5
Now let us begin with an n-connected atomic co-Hn space G. Choose T atomic o*
*f least
connectivity as above. We have
g0 n h0
T -! G -! T
where h0g0' 1. Next choose G0atomic and of least connectivity:
f00 n g00 0
G0- ! S T -! G
with g00f00' 1. We now construct a homotopy equivalence between G and G0. Thi*
*s will
complete the proof. The maps are:
nh0 g00
ff :G -! Sn nG S-! SnT -! G0
f00 n Sng0 n n ffl
fi :G0- ! S T -! S G -! G
where ffl is the evaluation map and is any right inverse to ffl. We will show*
* that fffi is an
equivalence.
Suppose that G is (k + n - 1) connected (k 1) and ßk+n(G) 6= 0. Then * a*
*nd
ffl* are inverse isomorphisms between ßk+n(G) and ßk+n(Sn nG). Since (Snh0)(Sn*
*g0) ' 1
and g00f00' 1, it follows that (fffi)* : ßk+n(G0) -! ßk+n(G0) is an isomorphism*
*. nG is
k - 1 connected and ßk( nG) 6= 0. By choice of T having minimal connectivity, T*
* is k - 1
connected and ßk(T ) 6= 0. Hence SnT is k + n - 1 connected and ßk+n(SnT ) 6= *
*0. By
choice of G0having minimal connectivity, ßk+n(G0) 6= 0. Therefore fffi is not t*
*opologically
nilpotent. Since G0 is atomic, fffi is an equivalence. Let e = fi O (fffi)-1 *
*O ff : G -! G.
Then e is an idempotent. Since G is atomic and e induces an isomorphism in ßk+n*
*, e is an
equivalence. Consequently fi* is onto so fi and hence ff are equivalences.
Two choices were made in this proof: first we choose T from the factors of *
*nG which
had minimal connectivity. Then we choose G0from the factors of SnT which had mi*
*nimal
connectivity. However by choice, ßk+n(G0) ßk(T ) ßk+n(G). Since fi is an eq*
*uivalence,
these inclusions are equalities and G0is the only factor of SnT which is not k *
*+ n connected
and T is the only factor of nG which is not k connected. In fact, we have
Corollary 2.3 In the correspondence G ! T between atomic k + n - 1-connected co*
*-Hn
spaces G and atomic k - 1-connected Hn spaces T each of the maps in the commuta*
*tive
diagram:
ß`(T ) -h*-- ß`( nG)
? ?
ffn?y ?y~=
f*
ß`+n(SnT ) --- ß`+n(G)
is an isomorphism for ` 2k - 1.
6
Definition 2.4 We call a pair of connected atomic spaces (G, T ) a correspondin*
*g pair if
there are structure maps f, g, g0, h such that the composites:
g n h
T - ! G -! T
f n g0
G - ! S T -! G
are the identity.
Proposition 2.5 Given a corresponding pair (G, T ) we may choose maps f, g, g0*
*, h such
that g and g0 are adjoint.
Proof. Given f, g, g0, h, we will keep g, h and replace f with a map efso th*
*at g* . ef= 1
where g* is the adjoint of g. Since the composite:
f n g*
G -! S T -! G
induces an isomorphism in ßk+n and G is atomic, this composite is a homotopy eq*
*uivalence.
Now define ef= f O (g*f)-1. Clearly we could also prove this result retaining g*
*0 and f and
replacing h by a map ehwith the same effect.
Athough our main focus will be on the case n = 1, at this point we will disc*
*uss an
example in the case n = 2.
Example 2.6: 2S3<3> is an atomic H3 space, and the corresponding co-H3 spaces*
* is
P 2p+2= S2p+1[p'e2p+2; i.e., there are retractions:
g 3 2p+2 h 2 3
2S3<3> ---! P ---! S <3>
f 3 3 3 g0 2p+2
P 2p+2 ---! S S <3> ---! P .
In particular, h = 2h0where h0: P 2p+2-! S3<3> is onto in homotopy. In fact, *
*if p > 3,
it can be seen that h0factors through S2p+1{p}.
Proof: We first observe that S3<3> is 2p - 1 connected and ß2p(S3<3>) = Z=p. T*
*hus we
may construct a map P 2p+1-! S3<3> inducing an isomorphism in ß2p. We use the H*
* space
structure2 on S3<3> to extend this map to a map h0: P 2p+2-! S3<3>, and define*
* h = 2h0.
We construct g using a lifting H0of the loops on the Hopf invariant map 2S3 -H*
*! 2S2p+1:
0 2 2p+1
2S3 -H! S {p}
where S2p+1{p} is the fiber of the degree p map on S2p+1(see, for example, [G4,*
* x4]). There
is a natural map S2p+1{p} -L! P 2p+2obtained from the obvious fibrations, and *
*these maps
combine to define g:
0 2 2p+1 2L 3 2p+2
2S3<3> -! 2S3 -H! S {p} -! P
________________________________
2h0is actually a loop map since S3<3> is a loop space.
7
all of these maps induce isomorphism in ß2p-2 and 2S3<3> = BW (1) is atomic. *
*Since
2S3<3> is an H3 space and P 2p+2is a co-H3 space we have proven the correspond*
*ence.
Note that the corresponding map f an be defined as the composition:
3h 3 2 3
P 2p+2-! S3 3P 2p+2S-!S S <3>
so f is a triple suspension and h is a triple loop map.
One is tempted to generalize this. 2nS2n-1<2n + 1> is an H2n+1space, and on*
*e seeks to
understand the corresponding co-H2n+1 space G. Note that the transfer defines a*
* map
S2n+1Bnq -~!S2n+1<2n + 1>
where Bnq is the nq skeleton of the p localization of B p. We can extend this t*
*o a map
h0: S2n+2Bnq -! S2n+1<2n + 1>.
This gives a candidate for h:
h = 2nh0: 2n+1S2n+2Bnq -! 2nS2n+1<2n + 1>
and f = S2n+1f0
2n~ 2n 2n+1
f0 : SBnq -! 2nS2n+1Bnq -! S <2n + 1>.
Constructing G is not easy and what we can see is that the composite:
f 2n+1 2n 2n+1
S2n+2Bnq -! S S <2n + 1> -! G
is a monomorphism in homology. G may, however, be somewhat larger.
Another example is provided by Neisendorfers result [N ]: If P 2n+1(pr) = S2*
*n [pre2n+1
and r 2, then 2P 2n+1(pr) contains as an atomic factor, the fiber D(n, p) of*
* the map:
2S2n+1 -i! S2n-1 of degree pr constructed in [CMN ]. Thus D(n, r) $ P 2n+1(p*
*r) is a
correspondence of an atomic H2 space D(n, r) and an atomic co-H2 space P 2n+1(p*
*r) for
r 2.
x3. Let us return to the case n = 1. Let (G, T ) be a corresponding pair, *
*and choose
structure maps (f, g, h) as in section 2 with g0 adjoint to g. These maps dete*
*rmine an H
space structure on T and a co-H space structure on G as follows:
gxg h
m : T x T -! G x G -! G -! T
f g0_g0
n : G -! ST -! ST _ ST -! G _ G.
In the pull back diagram:
S G --u-! G _ G
? ?
? ?
y ffl y
p
G ---! G x G
8
u is the composition S G -! S G _ S G -ffl_ffl!G _ G. Consequently the right i*
*nverse
corresponding to n is the composition:
f sg
: G -! ST -! S G.
The case of the H space structure maps is more complicated. In the absence of h*
*omotopy
associativity, two non homotopic maps ~1, ~2 : ST - ! T can yield the same H s*
*pace
structure map m when composed with
T x T -! ST x ST -! ST.
However, one such choice is the composition:
g0 h
ST -! G -! T.
At this point we introduce a öH pf fibration" sequence for T . This can be d*
*one for any
connected H space either with the classical Hopf construction:
H(m) : T * T -! ST
(Sugawara [S]) or via a construction of Dold-Lashof construction ([DL ], [G2 ]):
Hm : T * T ' Em -! ST.
These constructions are different3 and we will find the Dold-Lashof constructio*
*n advanta-
geous. Specifically we will use the following corollary of Propositions A1 and*
* A2 of the
appendix.
Corollary 3.1 Suppose T is atomic. Then there is a 1-1 correspondence between f*
*iber ho-
motopy classes of fibrations:
T --i-! E - H--! ST
with i null homotopic and homotopy classes of maps
m : T x T -! T
such that:
a) m(*, t) = t
b) the map f : T -! T given by f(t) = m(t, *) is a homotopy equivalence.
This correspondence is given by H = Hm .
________________________________
3We wish to thank Yukata Hemmi and Norio Iwase for helpful e-mail notes at t*
*his point.
9
Proof. The only part that does not follow immediately from A1 and A2 of the *
*appendix
is that if i is null homotopic, f is a homotopy equivalence. f is the compositi*
*on:
T -! ST -@! T
where @ : ST -! T is from the fiber sequence. If i is null homotopic, @ has *
*a right
homotopy inverse. Hence f* : ßk(T ) -! ßk(T ) is onto. Now ßk is complete and*
* finitely
generated, and f* : ßk(T ) Z=pr ! ßk(T ) Z=pr is an isomorphism for each r.*
* Hence f* is
an isomorphism and f is a homotopy equivalence.
We now choose an arbitrary H space structure m : T x T - ! T and a correspon*
*ding
fibration using (3.1). We will see later that some improvements can be made if *
*we can choose
a homotopy associative H space structure. Choose a map f : G -! ST which induce*
*s an
isomorphism in ßk and choose h as the composite:
f @
G -! ST -! T,
where @ comes from the fibration sequence. It may not be possible to choose g0a*
*djoint to g
so that hg = 1 and g0f = 1; however any choice of g which induces an isomorphis*
*m in ßk-1
will yield that hg and g0f are homotopy equivalences. We now construct a commu*
*tative
diagram obtained by pulling back the fibration from (3.1) along the base.
f g0 f
G - --! ST ---! G ---! ST
? ? ? ?
? ? ? ?
y @0y hy @y
T - --! T ---! T ---! T
? ? ? ?
(A) ?y ?y '0?y i?y
R0 - --! Q ---! R ---! T * T
? ? ? ?
i0?y ?y i?y Hm ?y
f g0 f
G - --! ST ---! G ---! ST
By A1, Q is determined by the restriction of the action map
T x T -! ST x T -a! T.
0
Since @0 = h O g0, the composition T -! ST -@! T is a homotopy equivalence,*
* so
Q ' T * T by Corollary 3.1. Since g0f is a homotopy equivalence R0' R and R is *
*a retract
of T * T . Since h has a right homotopy inverse, i0is null homotopic and we have
Theorem 3.2 There is a homotopy equivalence
G ' T x R
W
where R is a retract of T * T and hence a co-H space. Write R = Rffwith Rffat*
*omic.
Then
10
a) if G = SX, Rffis a retract of SX(i)for some i 2
b) if G is homotopy is associative, Sj-1Rffis a retract of G(j)for some j 2.
We think of the exact sequence of spaces:
* -! R -! G -@! T -! *
as a minimal free presentation of T . Minimal since G is atomic and free since *
*the homology
of both R and G are tensor algebras. However @ is not an H map in general.
Proof. The only parts needing attention are the last twoWassertions. If G = *
*ST , R is a
retract of T * T which in turn is a retract of SX * SX ' SX(i)^ X(j). Part *
*b) follows
from the following theorem of Theriault.
Theorem 3.3 (Theriault [T ]) Let G be a simply connected homotopy co-associati*
*ve co-H
space. Then
1`
S G ' G(k)
k=1
where G(k) is a homotopy co-associative co-H space and Sk-1G(k) ' G(k).
Proof of 3.2b. WeWwrite R as a retract of G * G ' G ^ (S G). It follows t*
*hat
R is a retract of G ^ G(k). Thus each Rffis a retract of G ^ G(k) for some*
* k. But
k 1 W
G ^ G(k) is a retract of G ^ S G(k) ' G(`) ^ G(k); now
` 1
`
S G(k) ' G(k, j)
j 1
with Sj-1G(k, j) ' G(k)(j). Consequently SjRffis a retract of G(`) ^ G(k)(j)and*
* S`+kj-1Rff
is a retract of S(k-1)jG(`)^ G(k)(j)' G(`+kj).
There is another sense in which the H-space T is generated by G, namely:
Proposition 3.4 The image of the homomorphism:
f* : eHi+1(G) -! eHi+1(ST ) ' eHi(T )
generates the ring eH*(T ).
As an example, consider T02n+1 P 2n+1, the atomic factor of [CMN ]. Then H*(*
*T02n+1) is
generated by u 2 H2n-1(T02n+1) and r 2 H2n(T02n+1) as a non-associative ring.
Proof. Let R eH*(T ) be the subing generated by the image of f*. We firs*
*t observe
that the composition:
fx1 a
a0: G x T - --! ST x T ---! T
defines a H*( G) module structure on eH*(T ).
11
Lemma 3.5 R is a H*( G) submodule.
Proof. The map a0fits into a commutative diagram:
0
G x G x T -1xa--! G x T
? ?
? ?
ymx1 y a0
0
G x T - a--! T
which can be obtained from A3 of the appendix and the definition of a0. Thus
(a0)*(ff1ff2 t) = (a0)*(ff1 (a0)*(ff2 t))
for ff1, ff2 2 H*( G) and t 2 H*(T ). Iterating we get
(a0)*(ff1. .f.fk t) = (a0)*(ff1 (a0)*(a2, . .,.(a0)*(ffk t)*
* . .).).
It follows that it is sufficient to show that (a0)*(ff t) 2 R whenever t 2 R *
*and ff is indecom-
posable. Since indecomposable elements of H*( G) are in the image of the homomo*
*rphism
Hei(G) --*-! eHi(S G) ' eHi-1( G)
and ~ Sg O f, any indecomposable element ff can be written as g*(r) for some *
*r 2 R. Now
we have a commutative diagram:
T x T
gxg. & gx1
G x G - 1xh--! G x T
? ?
? ?
ym y a0
G - -h-! T
where the commutative square is a restriction of the above square to G x G x *
Gx GxT . Since the composition along the left and bottom is m, we have a*(g*(r*
*) t) =
m*(r t) = rt 2 R. This proves the lemma.
We now use this lemma to prove the proposition. Let , 2 H*(T ) be an element*
* of least
degree that is not in R. , = h*(ff) for some ff 2 H*( G). ff cannot be indecomp*
*osable since
then it would be in the image of * and hence , would be in the image of f*. Th*
*us ff = ff0iff00i
with ff0iand ff00iof positive degree. In particular , = h*( ff0iff00i) = (a0)**
*(ff0i h*(ff00i)). Since
deg h*(ff00i) < deg,, h*(ff00i) 2 R and hence, by the lemma , 2 R.
Proposition 3.6 If G is homotopy co-commutative, and co-associative each eleme*
*nt in imf*
is primitive (and hence H*(T ) is primitively generated). Conversely, if, in a*
*ddition T is
homotopy associative and homotopy commutative, each primitive element in H*(T )*
* is a sum
of prth the powers of elements in imf*.
12
This depends on the following lemma which has been noted by [T ] as a coroll*
*ary of the
results of Berstein [B ]. The proof in [B ] is not direct, and we offer here a *
*direct proof.
Lemma 3.7 If G is co-commutative co-associative co-H space, the image of
* : eHi(G) -! eHi-1( G)
consists of primitive elements.
Proof. We first examine the case that G = SX for some X. Consider the compos*
*ite:
X : SX -S--! SX ^ X - W--!SX _ SX
where is the diagonal and W is the Whitehead product. By definition, the Whi*
*tehead
product is '1 + '2 - '1 - '2 where '1, '2 : SX -! SX _ SX are the inclusions. *
*X can be
written as ('1+'2)-('2+'1) = OE-øOE where OE : SX -! SX _SX is the usual co-H s*
*tructure
map. In particular, the composite X ~ * iff the usual co-H structure is co-com*
*mutative.
Now consider the composite:
G ffl_ffl
G ---! S G ---! S G _ S G ---! G _ G.
Since G is co-associative, is a co H map and hence
(ffl _ ffl) ~G (ffl _ ffl)(OE G - øOE G )
~ (ffl _ ffl)OE G - ø(ffl _ ffl)OE G
~ OEG - øOEG
~ *.
However we further factor this as
G ---! S G -S--! S( G ^ G) - W--!S G _ S G - ffl_ffl--!G _ G.
The composite (ffl _ ffl)W is the inclusion of the fiber in the fibering
S( G ^ G) ---! G _ G - --! G x G;
since this inclusion has a null homotopic fiber, we conclude that the composite
G ---! S G -S--! S( G ^ G)
is null homotopic from which the result follows.
Proof of 3.6. Since f : G ! ST is the composition G -! S G -Sh!ST1 the fir*
*st
part is immediate. However, if , 2 H*(T ) is primitive, so is g*(,) 2 H*( G). S*
*ince H*( G)
is a primitively generated tensor algebra, g*(,) is a sum of prth powers of com*
*mutators
(r 0) and prth powers of elements in the image of *. But if T is homotopy as*
*sociative
and homotopy commutative , = h*g*(,) is a sum of prth pwers of elements in imf*.
We now derive a result from the following theorem of Theriault [T, Theorem 8*
*.4].
13
Theorem 3.8 (Theriault) If G is a homotopy co-commutative and co-associative *
*co-H
space and (l, p) = 1, there is a decomposition:
G = Ulx Fl
induced by a map OEl: Ul- ! G, where the homology of Ulis the subalgebra of th*
*e homology
of G generated by the commutators of length l.
Corollary 3.9 For some H space structure on T , the image in homology of ß : *
*R ! G
contains all commutators of length l where (l, p) = 1, for l > 1.
Proof. Ulis homotopy equivalent to a product of some of the factors of G. *
*Since Ul
is lk - 1 connected, T is not among them. These factors lie in R, and we repla*
*ce them by
Ul, for each l with (l, p) = 1, and take T to be the fiber of this new map. Th*
*is new T will
still be the bottom atomic factor of G, but the map from G to T and hence the*
* H space
structure may be different. (See example 3.13 below.)
G is a generator for T in another sense as well:
Proposition 3.10 Let X be any H space and _i: T ! X be H maps for i = 1, 2. Th*
*en
a) _1 ~ _2 () (S_1)f ~ (S_2)f
f S_i
G -! ST -! SX.
__ __
b) If, in addition, G = SK and f = Sf for f : K ! T , then
__ __
_1 ~ _2 iff_1f ~ _2f.
Proof. From the commutative diagram:
f S_i
G -! ST - ! SX
h& #~ #
_i
T - ! X
__
we see that if (S_1)f ~ (S_2)f,__1h ~ _2h so _1 ~ _1hg ~ _2hg ~ _2. If f = Sf ,
(S_1)f ~ (S_2)f iff S(_1f) ~ S(_2f). Since X is an H space, this is true iff _1*
*f ~ _2f.
There are numerous examples in the literature of spaces K with this property*
* ([CMN ],
[G1 ], [Se]).
Write PX (t) for the Poincar'e series for X.
Proposition 3.11 PR = PTPG - (t + 1)(PT - 1).
14
Proof. Since G = R x T , we have
______PT______ 1
= ______________
1 - t-1(PR - 1) 1 - t-1(PG - 1)
which implies the result.
W
Corollary 3.12 If G = SX and G ^ G ~= SnffG, then R ' G ^ W where W is a wed*
*ge
ff2A
of spheres and
t(PT - 1)
PW = PT - _________+ 1.
PG - 1
Q
Furthermore if Tn corresponds to SnX for eachWn 1, SX ' Tnifor some sequen*
*ce ni.
Proof. The hypothesis implies that R = Rffwhere each Rffis a suspension of*
* G by
3.2. Thus R ' G ^ W where W is a wedge of spheres. The formula for PW follow*
*s from
3.11. Applying the Hilton-Milnor theorem and induction decomposes the loop spac*
*e of the
wedge into a product.
In the next example we see that the map ß does depend on the H space structu*
*re.
Example 3.13 Let T = S2p-3{p}, the fiberWof the degree p map on S2p-3. We wil*
*l write
P n=WSn-1 [p'en. Then G = P 2p-2, and R = Rffwhere each Rff= P nffby 3.12. In*
* fact,
R = P 2p-1+k(2p-4).
k 1
One way of obtaining this decomposition is given in [CMN ], where*
* the maps
P 2p-1+k(2p-4)-! P 2p-2are given by iterated Whitehead products. In particular *
*ß is stably
inessential. On the other hand, let B = (B p)(p). There is a fibration sequence
p 2p-3
B -! S2p-3- ! S - ! B.
Since B ' S2p-3{p}, this determines another H space structure on S2p-3{p}. The*
*re is a
pull back diagram
W i
S2p-3{p}- --! P 2p-1+k(2p-4)---!P 2p-2
? k 1 ? ?
'?y ?y ?y
B - --! P B ---! B
from which we can see that the composition
`
P 2p-1+k(2p-4)-! P 2p-2-! B
k 1
15
is null homotopic. In particular, with k = 1, the restirction of ß to P 4p-5is *
*the attaching
map for the next two cells of B. But the 4p - 4 skeleton of B is P 2p-2[ CP 4p-*
*5and the
Steenrod operation P1 is non zero. Thus P 4p-5-! P 2p-2is stably essential. In *
*particular,
the Whitehead product map P 4p-5-! P 2p-2does not factor through R and the comm*
*utator
P 4p-6-! P 2p-2-! S2p-3{p} is non trivial.
We now consider the special case when there is a homotopy associative multip*
*lication
on T .
Proposition 3.14 Suppose there is a homotopy associative multiplication on T .*
* Then in
diagram (A), @ = @0: ST ! T is an H map, h is an H map, the fiberings Q -! ST *
*and
R -! ST are fiber homotopy equivalent, g can be chosen so that hg ~ 1 and g0f ~*
* 1 and ß
is the composition
g0
R - --! T * T- Hm--!ST - --! G.
Proof. We show that there are structure maps (f, g, h) so that the composit*
*ion
Sg0 h
ST -! G -! T factors the given map ~ : ST -! T extending the multiplicatio*
*n. To
do this, we choose any structure maps (f0, g0, h0). We begin with the compositi*
*on:
g0 f0 ~
e : T---! G - --! ST ---! T
e induces an isomorphism in ßk, so it is a homotopy equivalence. Define g = g0*
*(e-1) and
h = ~( f0). Then hg = 1. Now the composition:
f0 g0
e0: G---! ST ---! G
is a homotopy equivalence where g0is the adjoint of g. Let f = f0(e0)-1. Then g*
*0f ~ 1. To
show that these structure maps determine ~, define __~as the indicated composit*
*e:
__~: ST -!g0 G -f0! ST -~! T.
*
*fi
Since ~ is an H map, __~is as well. To see that ~ ~ __~we need only calculate _*
*_~fiT= ~( f0)g =
hg ~ 1. By A3 of the appendix, @ ~ ~ and @0 = h( g0) ~ ~. The fibering Q -! S*
*T is
determined by the restruction of the action map:
0
T x T -! ST x T -a! T
a0is the composite:
fx1 g0x1 a
ST x T - ! G x T - ! ST x T -! T.
Since T is homotopy associative, the formula in A3 implies that a is the compos*
*ite
~x1
ST x T ---! T x T ---! T ;
16
combining these we see that a0is the composite:
fx1 hx1 m
ST x T ---! G x T ---! T x T - --! T
and restricting to T x T is thus m.
x4. Given the strong duality involved here, it is natural to ask whether a d*
*ual discussion
can be obtained. This would require a dual to the Hopf fibering. The following *
*Conjecture
is due to Ganea [Ga ].
Conjecture 4.1 Given a co-H space G there is a cofibration sequence:
G -ff!X -! G -! S G.
This is certainly the case when G is a suspension, for if G = SA we can take*
* X = SA=A.
This is false in general.
Example 4.2 Let ff1 : S2p -! S3 be the first element of order p with p > 3 and
G = S3 [ffe2p+1. Harper [H ] has shown that ff is a co-H map and hence G is a c*
*o-H space.
Suppose such a space X does exist. H*( G) through dimension 2p has classes u, u*
*2, . .,.up
and v with |u| = 2 and |v| = 2p. In p-local cohomology, the class up- pv transg*
*resses to the
class in dimension 2p + 1 in the base. Since p > 3, the classes u2 and up-2 are*
* in the image
of oe*. Hence up is in the image of oe*. This is impossible since, mod p, up tr*
*ansgresses.
The requirement that p > 3 is essential in this example.
f 2n+1
Proposition 4.3 Suppose S2Y - ! S is a co-H map localized at 3 and n 1, *
*and set
G = S2n+1[f CS2Y . Then there is a Hopf co-fibration:
G --ff-!X ---! G ---! S G.
The proof of 4.3 depends on the following
Lemma 4.4 Let bS2n= S2n[ e4n[ . .[.e2n(p-1) (S2n)1 . Then the loops on the p*
*rojection
map:
Sb2n ---! S2n(p-1)
is null homotopic.
Proof. Let K = S2n-1 [wn e2np-2where wn is the first element in the kernel *
*of the
double suspension localized at p. Then there is a loop map SK -! Sb2nwhich h*
*as a
right homotopy inverse [G1 ], so it suffices to show that the composite:
SK ---! Sb2n ---! S2n(p-1)
17
is null homotopic. We will show that the composite SK -! Sb2n -! S2n(p-1)is n*
*ull
homotopic. The degree k map on S2n induces a map [k] : Sb2n- ! Sb2nand there a*
*re
commutative diagrams:
SK ---! bS2n---! S2n(p-1)
? ? ?
? ? ?
y ffi y[k] ykp-1
SK ---! bS2n---! S2n(p-1)
S2n ---! SK ---! S2np-1
? ? ?
? ? ? p
y k yffi yk
S2n ---! SK ---! S2np-1.
Since S2wn ~ 0, the composite SK -! bS2n-! S2n(p-1)factors uniquely over S2n*
*p-1:
SK ---! bS2n
? ?
? ?
y y
S2np-1--ffi-!S2n(p-1)
and these diagrams, together with the uniqueness imply that kp-1ffi = ffikp. Si*
*nce ffi must be
a suspension, we conclude that (kp - kp-1)ffi = 0. Let k = -1 to get -2ffi = 0 *
*or ffi = 0.
Proof of 4.3. According to Harper [H ], the Hopf invariant of f is the co-H *
*deviation
and hence is null homotopic. Therefore we can factor:
f0 2n+1
SY - ! S
& "
f~ Sb2n= S2n [[',']e4n.
Consequently (S2n+1[f CS2Y ) contains as a subcomplex bS2n[_fCSY . Then
ffi2n
(S2n+1[f CS2Y ) S
__
contains S4n[feCSY where efis the projection of f onto S4n. However efis null h*
*omotopic
ffi
since j : Sb2n-! S4n is nullfhomotopic.fThusif(S2n+1[ffCS2Yi) S2n contains *
*(S4n_
S2Y ). Let X = (S2n+1[fCS2Y ) S2n S2Y and oe be the projection (S2n+1[fCS2Y )*
* -!
X. In homology oe* is onto and it's kernel consists of all tensors of length 1.*
* Let C be the
cofiber of oe. Then H*(C) -! H*(S G) consists of the suspension of the tensors *
*of length
1 and hence the composite
C -! S G -ffl!G
is a homotopy equivalence and we are done.
18
x5. We need to say something about naturality. Given corresponding pairs (G1*
*, T1), (G2, T2)
with structure maps (f1, g1, h1) and (f2, g2, h2), a map from (G1, T1) to (G2, *
*T2) should be
a pair of maps (OE, _) so that certain diagrams commute. In many situations we *
*may begin
with an H map _ : T1 -! T2 and it is not possible to find an appropriate co-H m*
*ap OE. For
example, _ = fl where fl : S2n+1- ! S2m+1 is not a co-H map.
The following is the strongest result that seems reasonable, and we offer th*
*is as a definition
of a map from (G1, T1) to (G2, T2).
Proposition 5.1 Suppose _ : T1 ! T2 is an H map and OE : G1 ! G2 is a co-H map*
*. Then
the following are equivalent:
f1 S_ g02
a) OE is the composition: G1 -! ST1 -! ST2 -! G2
b) The following square commutes:
ffi
G1 ---! G2
? ?
h1?y ?yh2
_
T1 ---! T2
g1 ffi h2
c)_ is the composition: T1 -! G1 -! G2 -! T2
d) The following square commutes:
ffi
G1 ---! G2
? ?
f1?y ?yf2
Sffi
ST1 ---! ST2.
Proof. To see that a) implies b) note that ~i = hiO g0i. So hi ~ ~iO fi;*
* thus
_ O h1 ~ _ O ~1 O f1 ~ ~2 O S_ O f1 ~ h2 O g02O S_ O f1 = h2 O OE. To se*
*e that b)
g1
implies c), compose the square with the map T1 -! G1. The last two parts are s*
*imilar to
the first two parts.
On the other hand if T2 is homotopy associative b) implies that _ is an H ma*
*p, and if
G1 is homotopy co-associative d) implies that OE is a co-H map.
x6. In this section we discuss a further refinement in the determination of*
* R in case
that T has a homotopy associative H-space structure and p > 2. The results are *
*based on
the following well known result.
Lemma 6.1 Suppose p > 2 andføi: X ! X is a map such that ø2 ~ 1. Then SX ' X+*
* _X-
where eH*(X ) = {, 2 bH*(SX)fiø*(,) = ,}.
19
Proof. Let e : SX -! SX be given by e = ø 1 and
e e
X = lim{SX -! SX -! SX -! . .}..
-!
Then the composition SX - ! SX _ SX - ! X+ _ X- induces homology isomorphisms
between simply connected spaces.
We will apply this lemma in two cases. We first consider the transposition *
*map ø :
T ^ T - ! T ^ T , from which we write T * T ' R+ _ R-. The second application *
*deals
with the inverse map for a homotopy associative H-space. For any connected H-sp*
*ace, it is
standard to construct left and right slicing maps:
T x T ---! T
(a, b)---! a=b
(a, b)---! a\b
by choosing homotopy inverses for the maps:
T x T - --! T x T
(a, b)- --! (m(a, b), b)
(a, b)- --! (a, m(a, b)).
These maps define operations on the set of homotopy classes [(X, *), (T1e)] whi*
*ch satisfy the
identities:
ffi
(ff . fi) fi=ff
(ff=fi) . fi=ff
ff\(fffi)= fi
ff(ff\fi)= fi.
In case T is homotopy associative one can see that e=ff = ff\e and write ff-1 f*
*or this homotopy
class; then we have (ff-1)-1 = ff and (fffi)-1 = fi-1ff-1. Write fl : T -! T fo*
*r the inverse of
the identify map. Then fl2 = 1, so we have ST ' T+ _ T-.
If , 2 eH*(T ) is primitive, fl*(,) = -,. Since T is k - 1 connected, all cl*
*asses in eHk(T )
are primitive. Consequently, the composition:
f
f0 : G---! ST ---! T- ---! ST
is an isomorphism in dimension k. Now choose g0, h0 so that g00f0 ~ 1 and h0g0 *
*~ 1. We
then apply the modification of 3.8 to obtain a triple (f, g, h) with ~ ~ h( g0)*
*, and observe
that f : G -! ST factors through T-.
20
Proposition 6.2 The diagram:
f
G -! ST
& # -Sfl
f ST.
commutes up to homotopy.
fi
Proof. It suffices to show that Sfl(T ) T and (-Sfl)fiT-= 1. This follows*
* from the
commutative ladders:
1-fl 1-fl
ST - --! ST ---! ST ---! . . .
? ? ?
-Sfl?y ?y1 ?y1
1-fl 1-fl
ST - --! ST ---! ST ---! . . .
1+fl 1+fl
ST - --! ST ---! ST ---! . . .
? ? ?
-Sfl?y ?y-1 ?y-Sfl
1+fl 1+fl
ST - --! ST ---! ST ---! . ...
Theorem 6.3 The following diagram commutes up to homotopy
T * T- --! ST
? ?
-S(fi)?y -Sfl?y
T * T- --! ST.
Corollary 6.4 R is a retract of R- T * T .
Proof of 6.4. The diagram:
T * T
% ?
?
R ?y-S(fi)
&
T * T
commutes after projection to ST by 6.3. However the inclusion of the fiber T -*
* ! T * T
is null homotopic, so this diagram actually commutes up to homotopy. As in the *
*previous
case, this implies that the projection onto R+ is null homotopic.
21
Proof of 6.3. We begin by considering the equivalence Em ' T * T from A2 of*
* the
appendix (where A = F = T and ` = m). From this we see that corresponding to ß1*
* in the
push out diagram for Em is the map (a, b) -! a=b in the push out diagram for T *
** T . In
other words, the map Em -! ST corresponds to the map of push out diagrams:
T -i1 T?* T -i2! T
?
?ff
y
* - T - ! *
where oe(a, b) = a=b. This is, in fact, the classical Hopf construction on the *
*map oe:
H(ff)
T * T- --! ST ;
here we use the reduced join X*Y , which is the quotient of XxIxY under the ide*
*ntifications:
(x, 0, y)~ (x, 0, y0)
(x, 1, y)~ (x0, 1, y)
(*, t, *)~ *,
and the reduced suspension. We introduce maps ff : S(X x Y ) -! X * Y and fi : *
*X * Y - !
S(X ^ Y ) whose composite is homotopic to the quotient map S(X x Y ) -! S(X ^ Y*
* ). We
define ff by
8
< (x, 1 - 3s, *)0 s 1=3
ff(s, x, y) = (x, 3s - 1, y)1=3 s 2=3
: (*, 3 - 3s, y)2=3 s 1
and fi(x, t, y) = (t, x, y). fi collapses the subspace (*, t, y) [ (x, t, *) to*
* a point. Since the join
is reduced, this subspace is contractible. Consequently ff has a right homotopy*
* inverse, and
we can study H(oe) by considering H(oe)ff:
8
< (x, 1 - 3t) 0 t 1=3
H(oe)(ff(t, x, y)) = (x=y, 3t - 1)1=3 t 2=3
: (*=y, 3 - 3t)2=3 t 1
Thus
H(oe)ff = -S(ß1) + S(m(ß1, flß2)) - S(flß2)
here we use associativity to write *=y = fl(y) and x=y = x . fl(y). Now observe*
* that
(-S(fl)) . H(oe)ff = S(flß2) - S(flm(ß1, flß2)) + Sß2.
Note that flm(ß1, flß2) = m(ß2, flß1), so
H(oe)ff(-Sø) = (-S(ß2) + S(m(ß2, flß1)) - S(flß1))(-1)
= S(flß1) - Sm(ß2, flß1) + S(flß2)
22
which completes the proof.
Finally we observe that in case p = 2 it is possible to have R = T * T . For*
* an example let
T = RP 1 and note that G = SRP 1 is atomic, even as a module over {Sq1, Sq2}. I*
*t follows
that R = RP 1 * RP 1.
x7. In this section we will show that in many cases, an associative H space *
*structure
on T is not possible. The model for this is the case G = P 2n+1(pr) of [CMN ]*
*. In this
case Neisenderfer has pointed out that the space T02n+1which is a retract of P*
* 2n+1(pr)
does not admit an associative H space structure. The key fact here is that ther*
*e is a class
v 2 H2n+1(P 2n+1(pr)) with fi(r)(v) 6= 0. We will consider generalized Bockste*
*in homol-
ogy operations fi* defined and natural on some full subcategory Cfiof the categ*
*ory of CW
complexes (for example, spaces on which fi(1), . .f.i(r-1)are all zero, and hen*
*ce fi(r)is well
defined homology operation). We assume that Cfiis closed under finite products*
* colimits
and retracts. Consequently it is closed under suspension and the James construc*
*tion.
The operations we consider include both the Bockstein operations and the Mil*
*nor oper-
ations Qi as well as possible higher order Milnor operations. We assume a stabl*
*e homology
operation:
fi : Hi(X) -! Hi-2d-1(X)
defined and natural for X 2 Cfi. We assume fi2 = 0, fi(oe(x)) = oe(fi(x)) wher*
*e oe is the
homology suspensions, and
fi(x x y) = fi(x) x y + (-1)|x|x x fi(y).
Note that if T 2 Cfi, ST 2 Cfiand hence G 2 Cfi. Conversely, if G 2 Cfiand G is*
* homotopy
associative and homotopy commutative, G 2 Cfiby 3.3. (Here we see the fact th*
*at fi is
stable so fi can be defined on X iff it can be defined on SX.) Consequently T 2*
* Cfi.
Proposition 7.1 Suppose G admits a homotopy co-associative and co-commutative *
*struc-
ture, and H*(G) is fi-acyclic. Suppose that T admits a homotopy associative and*
* commutative
structure. Then 0 = fi : H2n+1(G) -! H2n-2d(G).
One may apply this result, for example, to the space
_(1)n = P n-2p+1[A CP n-1 n 2p + 4
where A : P n-1-! P n-2p+1is the Adams map. We apply Q0 in case n is odd and Q1
in case n is even to see that the corresponding space T (v1)n-1 does not carry *
*a homotopy
associative and homotopy commutative H space structure.
Proof: Suppose v 2 H2n+1(G) is such that fiv06= 0. Let v = f*(v0) 2 H2n(T ) *
*and u = fiv.
By 3.6, u and v are both primitive. Since T is homotopy associative and commuta*
*tive, u2 = 0
23
while uvi 6= 0 for i < p by induction and applying * : H*(T ) -! H*(T ) H*(T*
* ). Also
fi(uvp-1) = -u2vp-1 = 0. Since H*(G) is fi-acyclic H*( G) and H*(T ) are also f*
*i acyclic.
Consequently there is a class w 2 H2np(T ) with fiw = uvp-1. Define e!2 H*(T ) *
* H*(T ) by
p-1X` '
1 p i p-i
e!= ! 1 + __ v v + 1 !.
i=1p i
Then
p-1X` '
p - 1 i p-i-1 i p-i-1
*(uvp-1) = {uv v + v uv }
i=1 i
= fi(e!).
Consequently fi( *(!) - e!) = 0.
Claim: if z 2 H*(T ) H*(T ) is a cycle its does not contain the term v v*
*p-1.
Proof: write for the subspace spanned by x. Then
H2n(T ) = A
H2n-2d-1(T )= __ B
and we can arrange A and B so that fi(a) 2 B for each a 2 A. Likewise write
H2n(p-1)(T )= C
H2n(p-1)-2d-1(T )= D
with fi(c) 2 D for all c 2 C. Write z = ziwith zi2 Hi(T ) Hm-i(T ). If fiz =*
* 0, it follows
that
(fi 1)zi6= (1 fi)zi-1= 0
for all i and hence (fi fi)zi= 0 for all i. Now
H2n(T ) H2n(p-1)(T ) ~= CA
A C.
Applying fi fi we obtain an element of
____ ____ D B B D.
Since fi fi(v vp-1) = <-u uvp-2>, the term v vp-1 is not present in any*
* cycle. It is
present in e!, so it must also be present in *(!). Since T is primitively gene*
*rated by 3.6,
each term in ! is a product of primitives; now
X
*(!1. .!.s) = cfffi!ff !fi
where cfffiare coefficients and !ff, !fiare products of the !i in such a way th*
*at !ff!fi=
!1. .!.s. Consequently if *(!) contains the term v vp-1, it must come from a*
* term vp in
!. But *(vp) = vp 1 + 1 vp so this is impossible.
24
Appendix
In this section we collect some general facts about Hopf map constructions. We*
* wish to
thank Yukata Hemmi and Norio Iwase for some helpful e-mail conversations.
Proposition A1 There is a 1-1 correspondence between homotopy classes of maps*
* ` : A x
F -! F with `(*, f) = f and fiber homotopy classes of fibrations:
F -! E -! SA
where ` is the restriction to A x F of the action map
SA x F -a! F
defined by the homotopy lifting property.
Proof. For each such ` : A x F -! F we define a quasifibering:
E` = F [` (CA) x F -! SA
where the subspace A x F (CA) x F is identified with F via `. Both the cone a*
*nd the
suspension are reduced. Thus construction is due to Dold and Lashof [DL ] and i*
*t is shown
in [G2 ] that each Hurewicz fibering of the form considered here is homotopy eq*
*uivalent to
such_a construction. Given a_homotopy `t : A x F - ! F with `t(*, f) = f we co*
*nstruct
` : A x (F x I) -! F x I by `(a, f, t) = (`t(a, f), t). From this we construct *
*a quasi fibering
F x I -! E_`-! SA
and the inclusion of E`0and E`1into E_`are clearly homotopy equivalences.
To recover the map ` : A x F -! F from an arbitrary Hurewicz fibering we fir*
*st discuss
the action map a : B x F -! F defined for each Hurewicz fibering
F -! E -! B.
This is constructed in a standard way by choosing a lifting in the diagram:
B x F x 0 -i! E
L
# % # ß
B x F x I -H! B
where i(w, f, o) = f and H(w, f, t) = w(t). Then L(w, f, 1) 2 F = ß-1(*) so by *
*restriction
L defines the action map:
fi
a = Lfi BxFx1: B x F -! F.
25
For each map _ : X -! B we can, by restriction, construct a lifting L0in the d*
*iagram:
iO(ffix1)
X x F?x 0 -! ?E
? L0 ?
? % ? ß
y y
HO(ffix1)
X x F x I -! B.
We now assert that given any two choices L00and L01of liftings, the associated *
*ä ction maps"
a0, a1 : X x F -! F
(defined by restricting to X x F x 1) are homotopic. To see this we define a map
I : X x F x (I x 0 [ 0 x I [ 1 x I) [ * x F x I x I -! E
by applying L0fflon X x F x ffl x I (for ffl = 0, 1) and projecting X x F x I x*
* 0 [ * x F x I x I
onto F E. Define J : X x F x I x I -! B by J(x, f, s, t) = _(x)(t). We then e*
*xtend I to
a homotopy covering J, and this homotopy, when restricted to X x F x I x 1 is a*
* homotopy
between a0 and a1.
It follows that in the case of a fibering
F -! E -! SA
0
we have constructed a well defined homotopy class of maps SAxF -`! F with `0(**
*, f) = f.
By restriction to A x F we obtain a class `. We only need to see that we can ch*
*oose ` as a
lifting for E`:
A x F?x 0 -'! E`?
? ?
? L% ?ß
y y
A x F x I -H! SA
with '(a, f) = f 2 F E`, H(a, f, t) = (a, t) 2 SA and L(a, f, t) = ((a, t), f*
*) 2 CA x F .
Here CA = A x I=A x 0 [ * x I and L(a, f, 1) = `(a, f) 2 F E.
Proposition A2 If the composition A x * A x F -`! F is a homotopy equivale*
*nce,
i : F -! E` is null homotopic and E` ' F * F .
fi fi
Proof. Since `fiAx*is a homotopy equivalence, i factors through `fiAx*up to *
*homotopy
and hence it factors through (CA) x *, so i is null homotopic. E` is the reduce*
*d homotopy
push out of the diagram:
F -` A x F -i2!F.
26
We use the commutative diagram:
Ffl- ` A x?F -i2! Ffl
fl ? fl
fl ? fl
fl y fl
F -i1 F x F -i2! F
where (a, f) = (`(a, f), f). By hypothesis, is a homotopy equivalence so E` *
*is homotopy
equivalent to the reduced homotopy push out of
F -i1 F x F -i2!F
which is the reduced join F * F .
Proposition A3 Replacing SA by the James construction A1 , the action map
a : A1 x F -! F
is given by the formula:
a((a1, . .,.aff), f) = `(a1, `(a2, . .,.`(ak, f) . .)..
Proof. One way to construct the action map is to replace the projection map*
* by the
canonical construction:
__ fi
E = (w, e) 2 P B x Efiß(e) = w(0)
__ fi
F = (w, e) 2 P B x Efiß(e) = w(0), w(1) = * .
__ __
Then the action map a : B x F -! F is given by
a(w, (w0, e)) = (w00, e)
where
(
w0(2s) 0 s 1_
w00(s) = 1 2
w(2s - 1) _2 s 1.
It is clear from this that the following diagram commutes up to homotopy:
B x B x F - 1xa--! B x F
? ?
? ?
y mx1 y a
B x F - -a-! F
The result follows immediately since A1 is generated by A as a monoid.
27
References
[AG] D. Anick and B. Gray, Small H spaces related to Moore spaces, Topology v*
*ol. 34,
no. 4, (1995), 859-881.
[AK] J.F. Adams and N.J. Kuhn, Atomic Spaces and Spectra, Proc. Edinborough M*
*ath.
Soc. (2), 32, no. 3, (1989), 473-481.
[B] I. Berstein, On co-groups in the category of graded algebras, Trans. Ame*
*r. Math.
Soc. 115 (1965), 257-269.
[BK] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizati*
*ons, LNM
304 (1972), Springer, Verlag.
[CM] F.R. Cohen, and M.E. Mahowald, A remark on the self maps of 2S2n+1, Ind*
*iana
Univ. Math. J. 30(1981), no.4, 583-588.
[CMN] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, Torsion in homotopy group*
*s, Ann.
of Math. (2)109 (1979) no. 1, 121-168.
[DL] A. Dold and R. Lashof, Principal quasifibrations and fiber homotopy equi*
*valence of
bundles, Illinois Journal of Mathematics, vol. 3, no. 2, (1959), 285-305.
[F] P. Freyd, Stable Homotopy II, Proc. Symp. Pure Math XVII (1970), 161-183.
[Ga] T. Ganea, Cogroups and Suspensions, Invent. Math. 9 (1970), 185-197.
[G1] B. Gray, On Toda's fibrations, Math. Proc. Camb. Phil. Soc. (1985), 97, *
*289-298.
[G2] B. Gray, On the Iterated Suspension, Topology, vol. 27, no. 3, (1988), 3*
*01-310.
[G3] B. Gray, On the homotopy type of the loops on a 2-cell complex, Contempo*
*rary
Mathematics, vol. 271, (2001), 77-98.
[G4] B. Gray, Unstable Families related to the image of J, Math. Proc. Comb. *
*Phil. Soc.
(1984), 96, 95-113.
[H] J.R. Harper, Co-H-Maps to spheres, Israel Journal of Mathematics, vol. 6*
*6, nos. 1-3,
(1989), 223-237.
[He] H.W. Henn, Finiteness properties of injective resolutions of certain uns*
*table modules
and applications, Math. Ann. 291 (1991), 191-203.
[Hi] P.J. Hilton, On the homotopy groups of the union of spheres, J. London M*
*ath. Soc.
30 (1955), 154-172.
[KSW] N.J. Kuhn, M. Slack, and F. Williams, Hopf constructions and higher proj*
*ective
planes for iterated loop spaces, TAMS 347 (1995) no. 4, 1201-1238.
28
[M] J. Milnor, The Construction FK, Algebraic Topology: A Student's Guide, b*
*y J.F.
Adams, Cambridge University Press, 1972.
[Ma] H. R. Margolis, Spectra and the Steemrod Algebra, North-Holland, 1983.
[N] J. Neisendorfer, Product Decompositions of the double loops on odd prima*
*ry Moore
spaces, Topology, Vol. 38, no. 6, (1999), 1293-1311.
[S] M. Sugawara, On a condition that a space is an H-space, Math J. Okayama *
*Univ. 6
(1957), 109-129.
[Se] P. Selick, A spectral sequence concerning the double suspension, Invent.*
* Math 64
(1981) no. 1, 15-24.
[SW] P. Selick and J. Wu, On Natural Coalgebra Decompositions of Tensor Algeb*
*ras and
Loop Suspensions, Memoirs AMS, Vol. 148, No. 701 (2000).
[T] S.D. Theriault, Homotopy decompositions involving the loops of coassocia*
*tive Co-H
spaces, Canad. J. Math., 55 (2003), no. 1, 181-203.
[W] C. Wilkerson, Genus and Cancelation, Topology, vol. 14, (1975), 29-36.
29
__