ON GENERALIZED WHITEHEAD PRODUCTS
BRAYTON GRAY
Whitehead products have played an important role in unstable homotopy.
They were originally introduced [Whi41 ] as a bilinear pairing of homotopy
groups:
ssm (X) ssn(X) ! ssm+n1 (X) m, n > 1.
This was generalized ([Ark62 ],[Coh57 ],[Hil59]) by constructing a map:
W :S(A ^ B) ! SA _ SB.
Precomposition with W defines a function on based homotopy classes:
[SA, X] x [SB, X] ! [S(A ^ B), X]
which is bilinear in case A and B are suspensions.
The case where A and B are Moore spaces was central to the work of
Cohen, Moore and Neisendorfer ([CMN79 ]). In [Ani93 ] and in particular
[AG95 ], this work was generalized. Much of this has since been simplified
in [GT10 ], but further understanding will require a generalization from sus
pensions to coH spaces.
The purpose of this work is to carry out and study such a generalization.
Let CO be the category of simply connected coH spaces and coH maps.
We define a functor:
CO x CO ! CO
(G, H) ! G O H
and a natural transformation:
(1) W :G O H ! G _ H
generalizing the Whitehead product map. The existence of GOH generalizes
a result of Theriault [The03 ] who showed that the smash product of two
simply connected coassociative coH spaces is the suspension of a coH
space. We do not need the coH spaces to be coassociative and require only
one of them to be simply connected.* We call G O H the Theriault product
of G and H.
We summarize our results in the following theorems.
____________
*In fact we can define G O H for any two coH spaces but require at least on*
*e of them
to be either simply connected or a suspension in order to obtain the coH space*
* structure
map on G O H. Recently Grbi'c, Theriault and Wu have shown that the smash produ*
*ct
of any two coH spaces is the suspension of a coH space, but their constructio*
*n cannot
satisfy theorem 1 part a below [GTW ].
1
2 BRAYTON GRAY
Theorem 1. There is a functor: CO x CO ! CO given by
(G, H) ! G O H
and equivalences in CO
(a) (SX) O H ' X ^ H
(b) S(G O H) ' G ^ H
(c) (G1 _ G2) O H ' G1 O H _ G2 O H
and homotopy equivalences:
(d) G O H ' H O G
(e) (G O H) O K ' G O (H O K)
Theorem 2. There is a natural transformation:
W :G O H ! G _ H
which is the Whitehead product map (1) in case G and H are both suspen
sions. Furthermore, there is a homotopy equivalence:
G x H ' G _ H [W C(G O H).
The next theorem concerns the inclusion of the fiber in certain standard
fibration sequences [Gra71 ]:
G o H '1!G _ H ss2!H
G * H '2!G _ H ! G x H.
Define adn(H)(G) inductively by ad0(H)(G) = G and
ad n(H)(G) = [adn1 (H)(G)] O H
and an iterated Whitehead product
ad n:adn (H)(G) ! G _ H
as the composition:
n1_1
adn(H)(G) W!ad n1(H)(G) _ H ad!G _ H _ H ! G _ H.
Theorem 3. Suppose G and H are simply connected coH spaces. Then
there are homotopy equivalences:
W n n
(a) G o H ' ad (H)(G) where '1 corresponds to ad on the appro
n>0
priate factorW
(b) G * H ' adj(H)(adi(G)(G)) where '2 corresponds to adj(adi)
i>0
j>1
on to theWappropriate factor
(c) S G ' adn (G)(G) where the composition S G ! S G_S G !
n>0
G _ G corresponds to the appropriate iterated Whitehead product on
each factor.
ON GENERALIZED WHITEHEAD PRODUCTS 3
It should be pointed out that the equivalence (c) generalizes the result
of Theriault [The03 , 1.1] where it is shown that a simply connected co
associative coH space decomposes
`
G ' Mn
n>1
for some spaces Mn, which are not further decomposed.
Theorem 4. Suppose X is finite dimensional and f: SX ! G _ H then f
is the sum of the projections onto G and H and a finite sum of iterated
Whitehead products.
Throughout this work we will assume that all spaces are of the homotopy
type of a CW complex. All homology and cohomology will be with a field
of coefficients. We will often show that a map between simply connected
CW complexes is a homotopy equivalence by showing that it induces an
isomorphism in homology with an arbitrary field of coefficients, without
further comment.
Section 1 will be devoted to some general remarks about telescopes and
we will construct the Theriault product in section 2. Theorem 1 will follow
from 2.3, 2.5 and 2.7. The functor in Theorem 2 is defined after 3.2 and the
equivalence follows from 3.8. The proof of the first part of Theorem 3 occurs
just prior to 3.5 and ithe rest follows from 3.5 Theorem 4 follows from 3.7.
1
In this section we will discuss some general properties of telescopes of a
self may e: G ! G where G is a coH space. We do not assume that e is
idempotent. We will call e a quasiidempotent if the induced homomorphism
in homology satisfies the equation:
(e*)2 = e*
where u is a unit. We construct two telescopes:
T (e):G e!G e!G ! . . .
T (1 + e):G 1+e!G 1+e!G ! . . .
and a map:
: G ! G _ G 1_2!T (e) _ T (1 + e).
Proposition 1.1. If G is simply connected and e is a quasiidempotent, is
a homotopy equivalence. Furthermore H*(T (e)) = im e* and eH*(T (1+e)) ~=
kere*.
Proof.Suppose *(,) = 0. Since ( 1)*(,) = 0 (e*)k(,) = 0 for some k, so
e*(,) = 0. Since ( 2)*(,) = 0, (1 + e)*k(,) = 0. But (1 + e)2*= (1 + e)*,
so , = e*(,) = 0. Clearly * is onto. Moreover, H*(T (e)) ~= im e* and
eH*(T (1 + e)) ~=im (1 + e*) = kere*.
4 BRAYTON GRAY
Corollary 1.2. Suppose G is a simply connected coH space and A G is
a retract of G. Let e be the composition:
G r!A i!G.
Then T (e) ' A, T (1  e) ' G=A, and the identity map of T (e) can be
factored:
T (e) ,!A i!G r!A j!T (e)
where , and j are inverse homotopy equivalences.
Proof.The telescope T (e) and A are both simply connected and there are
maps T (e) ! A and A ! T (e) making A a retract of T (,), and these maps
are homotopy equivalences. By the Van Kampen theorem G=A ' G [ CA
is simply connected. Since 1  e: G ! G factors through the projection
ss :G ! G=A, we can factor the identity map up to homotopy
G ! G _ G r_ss!A _ G=A ! G
and hence G ' A _ G=A. The factorization of the identity map of T (e)
is obtained by replacing each space by a telescope where the three in the
center are constant.
Now consider two maps f1, f2:X ! X.
Proposition 1.3. T (f1f2) ' T (f2f1).
Proof.We define maps between the telescopes:
f1f2 f1f2
X _____//X_____//X_____//_. . .
f2  f2 f2
fflfflfflfflffflffl2f1f2f1
X _____//X_____//X_____//_. . .
f1  f1 f1
fflfflfflfflffflffl1f2f1f2
X _____//X_____//X____//_... .
The composition is the shift map which is a homotopy equivalence.
2
In this section we will consider a pair of coH spaces in which at least
one is simply connected. Let G and H be two such coH spaces with their
structure determined by maps 1: G ! S G and 2: H ! S H each of
which is a right inverse to the respective evaluation maps (See [Gan70 ]),
which we label as ffl1, ffl2. We define self maps of S( G ^ H) as follows:
e1: S( G ^ H) ffl1^1!G ^ H 1^1!S( G ^ H)
e2: S( G ^ H) 1^ffl2! G ^ H 1^2!S( G ^ H);
here we freely move the suspension coordinate to wherever it is needed.
Clearly e1 and e2 are idempotents but e1e2 is not an idempotent; however
ON GENERALIZED WHITEHEAD PRODUCTS 5
it is a quasiidempotent. In fact by swapping coordinates, one can see that
S(e1e2) is homotopic to the negative of the composition:
_e:S2( G ^ H) ffl1^ffl2!G ^ H 1^2!S2( G ^ H).
Since _eis clearly an idempotent S(e1e2) O S(e1e2) ~ _e2= _e~ S(e1e2), so
(e1e2)2*= (e1e2)*.
Now assuming that one of G, H is simply connected, it follows that G ^
H is connected, so S( G ^ H) is simply connected. Consequently
Proposition 2.1. If one of G and H is simply connected, there is a homo
topy equivalence:
S( G ^ H) ' T (e1e2) _ T (1 + e1e2).
Let
` :S( G ^ H) ! T (e1e2)
be the projection and
_ :T (e1e2) ! S( G ^ H)
be the unique right inverse to ` which projects trivially onto T (1 + e1e2).
These maps determine a coH space structure on T (e1e2).
Definition 2.2. G O H = T (e1e2).
Proposition 2.3. Given coH maps f :G ! G0 and g :H ! H0, there is
an induced coH map
f O g :G O H ! G0O H0
making G O H a functor of two variables. In addition there are equivalences
of coH spaces.
(a) SX O H ' X ^ H
(b) S(G O H) ' G ^ H
(c) S1 O H ' H
and there is a homotopy equivalence G O H ' H O G.
Proof.Since f and g are coH maps, the squares
1 2
G _____//_S G H _____//_S H
f  S f  g  S g
fflffl 0fflffl1 fflffl 0fflffl2
G0 _____//S G0 H0 _____//S H0
commute up to homotopy. It follows that f and g induce maps that commute
with e1 and e2 and hence with the equivalences of 2.1, ` and _. For part a,
observe that the composition e2e1 factors:
S( SX ^ H)ffl1^1//_SXO^S'H^1//_S SX ^ 1H^ffl2//_7SX1^^H/2/_S(7SX ^ H)
OO ooo
OOOO oooo
1^GOOOO''O ooooL^1o
X ^ H
6 BRAYTON GRAY
where (1 ^ ffl)(ffl1 ^ 1) is a right universe to (1 ^ 2)(' ^ 1). Thus we can
apply 1.2 to see that SX O H ' X ^ H with coH structure given by the
composite (1 ^ 2)(' ^ 1). This is precisely the coH structure induced by 2.
Part b follows since S(G O H) is the telescope of _ewith coH structure given
by 1 ^ 2. Part c is a special case of part a: The last statement follows
directly from 1.3.
For the associativity assertion in Theorem 1, it will be convenient to
describe an alternative definition of G O H. For this we assume that G is a
retract of a space SX and H a retract of SY :
G 1!SX ffl1!G H !2SY ffl2!H.
We can then replace the telescope in the definition by the telescope of the
composition:
T :SX ^ Y 1^ffl2!X ^ H 1^2!SX ^ Y ffl1^1!G ^ Y 1^1!SX ^ Y.
The coH structures defined by these maps are equivalent to the structures
defined by
e2:G 1!SX Seffl1!S X e 1:H !2SY Seffl2!S H
and we have a homotopy commutative ladder:
1^ffl2 1^ 2 ffl1^1 2^1
SX ^ Y ______//_X ^_H____//_SX ^_Y_____//G ^_Y_____//SX ^ Y
Seffl1^effl2bffl1^1 Seffl1^effl21^effl2 Seffl1^effl2
fflffl1^ffl fflffl1^e 2 fflfflffl^1 fflffle^11 fflffl
S G ^ H _____// G ^ H___//_S G ^ _H___//G ^ H____//S G ^ H.
Hence we have a commutative diagram
__
SX ^ Y ______________//T _ T
Seffl1^effl2 ff_fi
fflffl fflffl
S G ^ H ____//_G O H _ Tel(1 + e1e2)
__
where T is the telescope defined by 1 + ( 1 ^ 1)(ffl1 ^ 1)(1 ^ 2)(1 ^ ffl2). T*
*he
map ff: T ! G O H is compatible with the homotopy equivalence ST '
G ^ H ' S(G O H) so is itself a homotopy equivalence. Now choose a right_
homotopy inverse for the map SX ^ Y ! T which projects trivially to T .
It's composite with Seffl1^ effl2will then project trivially to T (1 + e1e2). H*
*ence
we have a homotopy commutative diagram:
T ________//_SX ^ Y________//_T
ff Sfefl1^effl2 ff
fflffl fflffl fflffl
G O H _____//S G ^ H _____//G O H
and consequently the coH structure on T is compatible under ff with the
coH structure on G O H. We have proven
ON GENERALIZED WHITEHEAD PRODUCTS 7
Proposition 2.4. Suppose G is represented as a retract of SX and H a
retract of SY . Then G O H is homotopy equivalent to the telescope T of the
composition.
SX ^ Y ! G ^ Y ! SX ^ Y ! X ^ H ! SX ^ Y
as coH spaces,_where the coH structure on T is given by the equivalence
SX ^ Y ' T _ T.
Proposition 2.5. (G1 _ G2) O H ' G1 O H _ G2 O H as coH spaces.
Proof.Write Gi as a retract of SXi, i = 1, 2. Then G1 _ G2 is a retract of
S(X1_ X2). Thus the telescope for (G1_ G2) O H is at each point the wedge
of the telescopes for G1 O H and G2 O H.
At this point we will apply 2.4 to prove theorem 1 part e, the associativity
formula. We will make repeated use of
Lemma 2.6. There is a homotopy commutative square
S_
S(G O H) _____//_S2X8^8Y
ppp
'  1^p2ppppp S`
fflfflppp fflffl
G ^ H ___'__//S(G O H).
Proof.G ^ H is a retract of S2X ^ Y , so we may apply 1.2.
Proposition 2.7. (G O H) O K ' G O (H O K).
Proof.We suppose that G, H, and K are presented by retractions
G _1!SX `1!G
H _2!SY `2!H
K _3!SZ `3!K
and we then construct retractions for G O H and H O K:
G O H _3!SX ^ Y `3!G O H
H O K _4!SY ^ Z `4!H O K.
Using these we construct retractions for G O (H O K) and (G O H) O K
(G O H) O K _!S(X ^ Y ^ Z) `!(G O H) O K
0 `0
G O (H O K) _!S(X ^ Y ^ Z) ! (G O (H O K)).
8 BRAYTON GRAY
We will show that `0_ :(GOH)OK ! GO(H OK) is a homotopy equivalence.
By 2.6, we have homotopy commutative diagrams:
(GOH)8^8KL G ^ (H OK)M
'qqqqq LL_4^_3LL _1^_5ssss MMMM'M
qqq LLL ssss MMM
qq S_ LL%% yyss S`0 M&&
S((GOK)OK) ______________//_S2X ^Y ^Z______________//_S(GO(H OK)).
Suspending and applying 2.6 again we obtain a homotopy commutative di
agram:
Gl^ H ^ KRR
'llllllll  RRRR'RR
RRR
uulllll  RRR))
S(G O H) ^ K _1^_2^_3  SG ^ (H O K)
RRR  ll
RRRR  llll
RRRRR  llll
S(_4^_3) R(( fflfflS(_1^_5)vvlll
S3(X ^ Y ^ Z)
from these diagrams it follows that S2(`0_) is a homotopy equivalence and
hence `0_ is as well.
3
In this section we generalize the clutching construction [Gra88 , Proposi
tion 1] for fibrations over a suspension to fibrations over a coH space. This
allows for the decomposition results in theorems 2 and 3.
Proposition 3.1. Suppose F ! E ! G is a fibration where G is a coH
space. Then E=F ' G o F .
Proof.In the case G = SX, we have by [Gra88 , Proposition 1]
E ' F [` (CX) x F.
So E=F ' SX o F . It is easy to construct a map G o F ! E=F in general.
Consider the sequence of pull backs:
F ________F________F
  
  
fflffl fflffl fflffl
E ______//E0_____//_E
  
  
fflffl fflfflfflfflffl
G ____//_S G_____//G.
Then we consider the composite:
E=F ! E0=F ' S G o F ! G o F
where the middle equivalence follows since the base is a suspension. Showing
that the composite is a homotopy equivalence will take some work.
ON GENERALIZED WHITEHEAD PRODUCTS 9
Since ffl: S G ! S G is an idempotent, we can decompose S G:
S G ' G _ G0.
We now observe that we can construct a quasifibration model for a fibration
over a one point union. Suppose we have such a fibration
EA _______//eEoo____EB
  
  
fflffl fflffl fflffl
A _____//_A _ Boo___B
with pull backs EA and EB and fiber F . Then we can construct
EA [F EB
the union of EA and EB with the subspace F identified. Then
OE
EA [F EB _______//eE
 
 
fflffl fflffl
A _ B ________A _ B
OE is a homotopy equivalence. In our case S G ' G _ G0 and EG = E,
EG0 = G0x F , so
E0' E [F G0x F


fflffl
G _ G0
is a quasifibering by [DT58 , 2.10]. On the other hand E0=F ' S G o F '
G o F _ G0o F while E [F G0o F ' E=F _ G0x F . Since the map between
E0 and E [F G0o F is a map over G _ G0we see that E=F ' G x F .
Corollary 3.2. S G ' G o G.
Proof.Apply 3.1 to the path space fibration over G.
Proof of theorem 2.Construction: We now describe our generalization of
the Whitehead product. Suppose G and H are coH spaces and one of them
is simply connected. The Whitehead product:
W : G O H ! G _ H.
is then defined as the composition:
G O H _!S G ^ H ' G * H !!G _ H
where ! is the inclusion of the fiber in the fibration sequence:
G * H !!G _ H ! G x H.
10 BRAYTON GRAY
Clearly _ and ! are natural transformations, so W is as well.
Before we prove the homotopy equivalence in theorem 2, we need to es
tablish some results in theorem 3. We begin by constructing maps:
ad n:adn (H)(G) ! G _ H
inductively. For n = 0 this is just the inclusion of G in G _ H. For n > 0
we define adn as the composition:
n1
adn(H)(G) = ad (H)(G) O H
! (adn1 (H)(G)) _ H ! (G _ H) _ H = G _ H.
Next we calculate the effect of adn in loop space homology:
(adn)*: H*( (adn(H)(G))) ! H*( (G _ H))
To do this we need some notation. For each coH space G, write oe1 :Her(G) !
eHr1( G) for the composition:
Her(G) +! eHr(S G) ' eHr1( G).
Let {xi} be a basis for He*(G). Then H*( G) is the tensor algebra on the
classes {oe1 (xi)}. Given two classes x 2 eHr(G), y 2 bHs(H) we will write
x O y 2 eHr+s1(G O H)
for the class that corresponds to x ^y 2 Her+s(G ^ H) under the isomor
phism:
eHr1(G O H) ~=Her(S(G O H)) ' eHr(G ^ H).
Then the classes {xiO yj} form a basis for eH*(G O H) where {xi} and {yi}
respectively are bases for eH*(G) and eH*(H).
Proposition 3.3. ( W )*(oe1 (x O y)) = oe1 x, oe1wyhere
( W )*: H*( (G O H)) ! H*( (G _ H)).
Proof.By lemma 2.6
_*(x O y) = e ^oe1 (x) ^oe1 (y) 2 H*(S G ^ H)
so
( _)*(oe1 (x O y)) = oe1 (x) ^oe1 (y) 2 H*( G ^ H)
regarded as a submodule of H*(S G ^ H). It now suffices to evaluate the
composition:
G ^ H ! (S G ^ H) ,! ( G * H) ! (G _ H)
where , is the standard homotopy equivalence SX ^ Y ' X * Y .
Lemma 3.4. The composition:
G ^ H ! S( G ^ H) ,! ( G * H) ! (G _ H)
carries oe1 (x) ^oe1 (y) 2 H*( G ^ H) to oe1 (x), oe1 (y).
ON GENERALIZED WHITEHEAD PRODUCTS 11
Proof.We first need to describe the homotopy equivalence
SX ^ Y ,!X * Y.
Here we write points of the join as tx + (1  t)y, 0 t 1, So X * Y is the
quotient of X x I x Y given by the identifications (x, 0, y) ~ (x0, 0, y) and
(x, 1, y) ~ (x, 1, y0). Then , is given by the formula:
8
><(*, 1  3t, y)0 3t 1
,(t, x, y) = (x, 3t  1, y)1 3t 2.
>:
(x, 3  3t, *)2 3t 3
The map X * Y !!X _ Y is given by
(
!1(2t) 0 2t 1
(!1, t, !2) ! .
!2(2  2t) 1 2t 2
Combining these we get
S X * Y ,! X * Y !!X _ Y
with a 6part formula:
8
>>>(*, !2(6t)) 0 6t 1
>>>
>><* 1 6t 2
(!1(6t  2), *)2 6t 3
(t, !1, !2) =
>>>(*, !2(4  6t))3 6t 4
>>>
>>:* 4 6t 5
(!1(6  6t), *)5 6t 6
so the adjoint takes the pair (!1, !2) to the product of loops !11!12!1!2.
The effect of this on a primitive element is the graded commutator.
Now the iterated circle product adn(H)(G) has homology generated by
classes of the form
(. .(.(x O y1)O y2).O.y.n)
where x 2 eH*(G) and yi2 bH*(H). By 3.3
1
( W )* oe (. .(.x O y1)O y2).O.y.n
is the graded commutator
1 1 1 1
. . .oe (x), oe (y1), oe . (y2).o.e (y*)
where the classes x and yi are thought of as classes in eH*(G _ H).
Proof of theorem 3 part (a):Now let G* = He*(G) and H* = He*(H). Let
L(G* H*) be the free Lie algebra generated by G* and H*, and L(H*)
the free Lie algebra generated by H*. Then Neisendorfer has analyzed the
kernel
L(G* _ H*) ! L(H*)
12 BRAYTON GRAY
([Nei09, 8.7.4]). He has shown that this is the free Lie algebra
0 1
M
L @ adn(H*)(G*)A
n>0
The universal enveloping algebra is thus the tensor algebra generated by the
elements adn(H*)(G*) for n > 0. Consequently the fiber of the projection
(G _ H) ! H is
0 1
`
@ adn(H)(G)A
n>0
and this is homotopy equivalent to (G o H) and the map
`
adn(H)(G) ! G _ H
n>0
which factors through G o H establishes the homotopy equivalence in
theorem 3.
W n
Corollary 3.5. (a) If G is simply connected, S G ' ad (G)(G)
n>0
(b) if both G and H are simply connected
`
G * H ' adj(H) adi(G)(G) .
i>0
j>1
Proof.For part (a) apply 3.2 and theorem 3 part a. For part (b), we expand
G * H ' (S G) ^ H
2 3
`
' 4 adi(G)(G) 5^ H
i>0
`
' adi(G)(G) ^ H
i>0
`
' adj(H) adi(G)(G)
i>0
j>1
using 2.5 and theorem 3 part a.
Given a Theriault product P = G1 O . .O.Gs with some fixed association,
let us write `(P ) = s for the length of P .
Theorem 3.6. Suppose G and H are both simply connected coH spaces and
k > 1. Then there is a locally finite collection of iterated Theriault products
{Pff(k)} of length `ffand iterated Whitehead product maps:
!ff(k): Pff(k) ! G _ H
ON GENERALIZED WHITEHEAD PRODUCTS 13
such that 0 1 0 1
` Y
(G _ H) ' @ Pff(k)Ax @ Pff(k)A
`ff>k `ff6k
and the factors of the righthand side are mapped to the lefthand side by
the !ff(k).
Proof.For k = 1 we use the decomposition:
(G _ H) ' ( G * H) x (G x H)
where G * H is a boquet of iterated Theriault products of length at
least 2 by 3.5(b). Now we proceed by induction on k. Among the finite list
of products Pff(k) of length k + 1, choose one which we label P . Then
` `
( Pff)' (P _ Pff)
`ff>k `ff>k
Pff6=P
`
' P x ( Pffo P ).
`ff>k
Pff6=P
The second factor has one less product of length k + 1. If we repeat this
process once for each Pff(k) of length k + 1, we obtain:
` i ` j
( Pff) = Pj x . .x. Pm x Pff(k + 1) .
`(Pff)>k `(Pff)>k+1
Now add the P1 . .P.mto the list of Pffwith `(Pff) k to obtain all Pff(k+1)
with length k + 1.
Corollary 3.7. Suppose X is a finite dimensional coassociative coH space
and f: X ! G_H where G and H are simply connected coH spaces. Then f
is a sum of iterated Whitehead products.
Proof.Suppose dim X = k and f: X ! G_H is given. Decompose (G_H)
as in theorem 3.6 and note that any product Pffof length k is at least k
connected. Consequently the restriction of f:
( X)k1 f! (G _ H)
Q
factors through the product ( Pff(k)) and the adjoint:
`ff k
h i
S ( X)k1 ! G _ H
is a sum of iterated Whitehead products. However f is the composition of
this map with the coH space structure map:
h i
X ! S ( X)k1
which is a coH map, so f is such a sum as well.
14 BRAYTON GRAY
Proposition 3.8. If G and H are simply connected, then there is a homo
topy equivalence
OE: (G _ H) [W C(G O H) ! G x H.
Proof.Since the composition
G O H ! G * H ! G _ H ! G x H
is null homotopic, there is an extension
C = (G _ H) [ C(G O H) OE!G x H.
The problem is to show that this map is a homotopy equivalence. We begin
by observing that we can construct a right inverse i to OE as the sum of
the loops on the inclusions of G and H into C:
G x H i! C OE! G x H
so ( OE)*: H*( C) ! H*( G x H) is an epimorphism. We will complete
the proof by showing that the rank of Hk( C) is less than or equal to the
rank of Hk( G x H). We will need several lemmas.
Lemma 3.9. Write G * H ' G O H _ SQ. Then the restriction:
SQ ! G * H !!G _ H ! C
is null homotopic
Proof.We first look at the homotopy commutative diagram
_!__//_ _____//
G * H G _ H G x H
  
  ss2
fflffl  fflffl
G o H _____//_G _ H______//H.
Applying theorem 3a we see that the composition
SQ ! G * H ! G o H ! G _ H
is a sum of maps fli factoring through adi(H)(G) ! G _ H for i 6 2. By
induction on i we see that
adi(H)(G) ! G _ H ! C
is null homotopic for i > 1. This follows since adifactors
i1_1
ad i(H)(G) ! adi1(H)(G) _ H ad!G _ H _ H ! G _ H ! C.
It follows from 3.9 that the mapping cone of ! is homotopy equivalent to
C _ S2Q. Recall that Ganea proved [Gan70 ] that given a fibration sequence
F ! E ! B. One can construct a fibration sequence
F * B ! E [ CF ss!B
ON GENERALIZED WHITEHEAD PRODUCTS 15
where ss pinches the cone on F to a point. Apply this to the fibration
sequence:
G * H !!G _ H ! G x H,
to obtain:
G * H * (G * H) ! C _ S2Q ss!G x H.
It is possible that the map ssS2Q is nontrivial. However ssS2Q is the sum
of the projections onto G and H, so it factors through C up to homotopy.
Using such a factorization we can construct a homotopy equivalence:
: C _ S2Q ! C _ S2Q
such that ss S2Q is null homotopic. Replacing ss with ss does not alter the
homotopy type of the fiber of ss, so we can form the following diagram of
fibrations:
S2Q o C _________//_S2Q o C
 
 
fflffl fflffl
G * H * (G * H) _____//_C _ S2Q_____//_G x H
 ss 
  1 
fflffl fflfflOE 
K _________________//_C________//G x H.
The lefthand verticle fibration has a cross section since ss1 does, hence K is
a coH space and we have a splitting:
( G * H * (G x H)) ' (S2Q o C) x K.
Each space here is the loops on a coH space, so the homologies are all tensor
algebras. It follows that for each i > 0
rank Hi( G * H * (G x H)) > rankHi(S2Q o C).
Suppose now that we have two power series f(t) = antn and g(t) = bntn
with ai, bi non negative integers. We will say that f > g iff ai > bi for each
i. Write X (X) for the Poincar'e series of a space X. In these terms, we have
shown that
X ( G * H * (G x H)) > X (S2Q o C)
We now calculate the Poincar'e series for each of these spaces. Suppose
X (G) = 1 + gt and X (H) = 1 + ht where g and h are polynomial in t with
16 BRAYTON GRAY
positive integral coefficient. We then have the following consequences:
g
X ( G) = 1 + _____
1  g
h
X ( H) = 1 + _____
1  h
ght
X ( G * H) = 1 + _____________
(1  g)(1  h)
X (G O H)= 1 + ght
g + h  gh
X (SQ) = 1 + ght_____________
(1  g)(1  h)
g + h  gh
X (S2Q o C) = 1 + ght2_____________ . X ( C).
(1  g)(1  h)
On the other hand:
g + h  gh
X ( G x H) = 1 + _____________
(1  g)(1  h)
Consequently
ght2(g + h  gh)
X ( G * H * ( G x H)) = 1 + ________________.
(1  g)2(1  h)2
Observe that if h(t) = cntn 6= 0 also has non negative coefficients, f(t) >
1
g(t) iff h(t)f(t) > h(t)g(t). It follows that X ( C) _____________ =
(1  g)(1  h)
X ( G x H) so rankHi( C) rankHi( G x H) for all i and thus ( OE)*
is an isomorphism and hence OE is as well.
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Department of Mathematics, Statistics and Computer Science, University
of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL, 606077045, USA
Email address: brayton@uic.edu