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and
i>2p-2. This follows by induction on the stages of the Postnikov s y s t e m since
the only non-trivial k-invariants are in the range w h e r e an isomorphism.
the following result.
Corollary
H-space
rr.(X) = 0 for sented by an
An i
and
A -map" n
is
W e go into this in greater detail in the next chapter w h e r e w e
will be able to verify 9.10.
Hi(y)--~ Hi-I(~Y)
X
i>3p-1.
has the homotopy Todo
in cohomological
t y p e of a l o o p s p a c e
so, we will interpret terms.
if
"ct i s r e p r e -
DIFFERENTIALS
Throughout particularly
this chapter,
nice class
as a cohomology
~K(w,q+l).
w. : K × K - ~ K 1
generator
of
Hq(K;~r).
Alternatively,
the usual addition
in
Hq(;1~)
Hi(K/. as
K; Ir.(K)) 1
u
class,
is a multiplica-
Any other multiplicato the
We shall see later
that
group
Yr.
is represented
u E Hq(x;yr)
by an H-map
if and only if
the equation
into a homotopy
between
repre-
maps.
is primitive
10.2.
of
E 1 in the bar construction
if and only if
of the higher Let
dl[U ] = 0.
Theorem
spectral
sequence
10.1 g e n e r a l i z e s
to an
differentials.
X be an associative
following are equivalent:
q
a
i . e . , m S u = ~rlSu + ~ 2 ~ u .
interpretation Theorem
+ wZ St
u ¢ Hq(X,~r).
Our description shows
admits
~r
One need only translate sentative
q
since the obstructions
which are all zero.
the
K(w, q ) ,
~rl~ t
is the fundamental
A
BB.
Now consider
is primitive,
q
to this one for q >1
K(~r, 0) b e i n g t h e a b e l i a n
A class
the class
Wl~% q + ~Z~t q
"B
10.1.
consider
K(~,q)
by regarding
of
is homotopic
Theorem
for example,
That
#
K(vr, q)
with
ways,
u E Hq(x;~).
AnT representative
tion on
K(zr, n )
class
H-space.
in which case
L
and induces
we can regard
Y = K(lr,q)
and
K{w,q),
belong to
with
are the projections
tion on
homotopy
will denote an associative
occur
can be seen in several
up to homotopy, as where
X
of A - m a p s n
map can be interpreted a multiplication
IN THE BAR CONSTRUCTION
H-space
and
u • Hq(X;~v), t h e n t h e
u
45
l)
d [u] = 0 f o r
r < n
r
2)
Su
3)
u
pulls back to
XP(n)
is represented
by anA
-map. n
The equivalence differential
in a spectral
of
1) a n d
sequence
e x a c t c o u p l e of t h e f i l t r a t i o n . d [u] = 0 r class
is easy.
in
r < n, since A -maps
is a standard
u
proof that
as
f*(L
f being an An-map respect
interpretation
which can be seen most
A direct
If w e r e g a r d
K(lr, q ) , t h e n
2)
u
implies
the differentials
from
being an A -map n
) where
q
easily
of t h e
L
q
the implies
is the fundamental
dr[U ] = (fx...xf)*
dr[L q]
which are defined in terms
for
of
n
XP(s),
s < n. --
resents
L
Now
X-* ~XP(n)
Corollary
if
Su
If X
v ~ Hq+I(Bx;w)
r
since
extends
if a n d o n l y if
reasons,
L
q-r+l
Special
XP(n)-*
survives
q
cases
K(lr, q + l )
in which all maps and
dr[U ] = 0 for
of course,
i.e. , dr[U ] ~
which is zero for
to
is (p-1)-connected
The point is,
Corollary
] for all
-* ~ B K ( y r , q) -* K
10.3.
nectivity
q
to
E
where
it rep-
q+l" Finally
as
dr[t_
then
can be factored
are at least A -maps. n
u ~ Hq(x;~r), then
r + 1 < ¢L+2 -- p+l
that higher
El+r, q- r+l r
u
for
"
differentials
and
u = ~v
vanish
for con-
l+r, q=H ~ E1 r+l q-r+l
r+l
~X)
< (r+l)p. are well known.
10.4.
~ : H q + l (Y;~r) -* H q ( ~ Y ; ~ r )
is onto for
10.5.
~ : H q + l ( y ; l r ) -* H q ( ~ Y ; l r )
maps
q < 2p
where
Y
is p-
connected. Corollary q < 3p
where
Y
is p-connected.
We write "loop class"
greater these
~
rather
(Corollary than
a
content than the over-worked corollaries
yield Corollaries
9.10 is now established.
and refer
rather than as a " s u s p e n s i o n "
onto the primitive
to a class
subspace )
in the image
as a
as w e prefer terminology of
"suspension".
Applied
9. 9 - 1 0 of t h e l a s t c h a p t e r .
to k - i n v a r i a n t s ,
for
46
For class
Y = K(G,n),
it so happens that primitive
implies being a loop
if w = Z , b u t t h i s i s n o t t r u e in g e n e r a l . P
Example
10. 6. [ S t a s h e f f ] . --
a loop class.
In f a c t
Let
~ ~ 0 { HZn(z
d2(~P) ~ 0, s o
, 2n-1;Z).
p
~P
~P is n o t r e p r e s e n t e d
is p r i m i t i v e
but not
by ahomotopy
associ-
ative map. To obtain examples ferent
of m a p s w i t h
d
r
~ 0 for
r > 2, w e h a v e t w o d i f -
sources.
Example
10.7.
An-map
A non-zero
class
a
in
H2n(~cP(n))
is r e p r e s e n t e d
by an
b u t n o t b y a n A n + l - m a P. The class
cannot be a loop class
since
H2n+l(cP(n))
= 0.
Recalling
that ~CP(n) ~ S 1 X ~ S zn+l, the only possible'non-zero differential is d lot] = k[ul.., lu] where n
u ¢ HI(~cP(n)).
It follows that S I-~ COP(n)
is not an
An+l-mapE x a m p l e I0.8. [Zabrodsky]. _
Let X ~ K(Z --
, 2n-l) )< K(Z p
ative multiplication obtained as the loops on the space so that B x ~ Y. Ap_l-map,
The class
, Znp-g) have the associP
u = ~ 2np-Z c HZnP'Z(X;Zp)
Y
with k-invariant
(~ Zn )P,
is represented by an
not an A p - m a p . B y comparing the spectral sequence with that of the product structure
on K(Z
, gn-l) X K(Z P
, 2-np-Z) w e see the only w a y the class
(L Zn )P =
P
it Zn_ll... [~ gn_l ] can be killed is by E x a m p l e I0. 9. [Zabrodsky]. and k-invariant ~ l WZp_z(Y)
restricts to ~ p - 2 in K ~
and
k-invariant ~ l
= 0, there is a class
2p-3
to ~ P - 2 L 4p-6"
be the space with
WZp- 3
Now
u
4p-5"
~Y
where
2p-2 + k(L 2p-2 )2 for any
u c H2P(P'I)-3(X)
(We have -~p-2~pl
k ~ Z p"
which restricts
is not a loop class since there is no class in Y
which
+ ks 2) = -2s P + k z ~ P P - J ~ P J - Z ~
, 2p-2). ) B y the s a m e token, for k = 0, L P goes to P
W4p- 6 ~ Zp
Zp-3' which can be regarded as the loop space
w4 p_5(Y) ~ Zp
Since ]PP-Z~IL
Let X
dp_l[~ 2np_2].
zero
in Y
and
0
47
thus u
[~ I ' ' "
I t ] must
is the first
class
assasin.
Thus
Example
10.10.
u
be killed in the Eilenberg-Moore
in
X
which exists
is represented
[Gheng].
Let
for unstable
spectral
reasons,
by an Ap_l-rna pbut
Y be the space with
~
With respect
back to the fundamental
not an Azi-rna p.
class
is represented
Again
be the
k-invari-
" 2 u ~ H z l - (f~Y) w h i c h p u l l s
a class
K(Zz, zi-z)
must
w2 = w 2 i _ l = Z 2 a n d
to loop multiplication, of
u
Since
not by an Ap-map.
2 i+l ant
sequence.
by an A . -map but 21_i
u kills [~ ii... IL i] in the spectral sequence.
For
i=g,
v ~ H 7 {Y) w h i c h restricts to S g S IL is represented by an A ) - m a p q q 7
the class
but not an A4-rna p for any A4-structure on
Y.
It is possible to give chain formula for using a spectral sequence is to avoid such work.
d , although a m a j o r point of r T h e case
d g is quite m a n a g e -
able and illuminating in t e r m s of our next topic. If
dl[U ] = 0, t h e n
any representative Gq'l{x~:
X;w).
u ~ -For
u
u
we have
any choice
c = (1 × m ) ~F b - {m × 1)~
is primitive.
b
- w1 ~ -u - w/ u
m~
of b , t h e c o m p o n e n t
represents
by a coboundary
On the chain level,
dg[u].
u, w e a l t e r
b
a n d if w e a l t e r
we alter
c
by
dg[u ] is
[(1 X m ) * - ( m X 1)*] H q ' l ( x ~ x ; w )
[(1 X m ) ~: - ( m X 1 ~ ]
= 5b{u)
in
this means
where
our choice
of a c o c y c l e .
b •
G q - l { x ~b X ~ - X ; w )
Notice by altering of b
of
our choice
for a given
of u,
Thus the indeterminacy
which is
dl(Hq-l(x
for
~=X;w))
in
just as
it should be. Example
10. 6 i s w o r k e d
In C h a p t e r sented
by H-maps.
associative
out this way in [Stasheff].
8 we saw that the k-invariants
Similarly
of an H-space
one can show the k-invariants
H - s p a c e are r e p r e s e n t e d
by A3-rnaps,
are repre-
of a h o m o t o p y
so e x a m p l e s I0. 6 and 10. 8
and I0. 9 for p = 3 provide e x a m p l e s of H - s p a c e s w h i c h are not h o m o t o p y associative.
In order
associativity
to generalize
more
fully from
these
results
our homotopy
to A -maps n
we need to study
p o i n t of v i e w .
A
-SPACES n
We h a v e s e e n t h a t t h e e x i s t e n c e H-space
is e q u i v a l e n t to h o m o t o p y a s s o c i a t i v i t y ;
significance
of p r o j e c t i v e
induced by A3-maps natural
of a p r o j e c t i v e
n-space.
In b o t h c a s e s w e a r e l e d to c o n s i d e r
invariant
for an
to i n q u i r e a s to t h e
On the other hand, we have seen that fibrings
to a s k a b o u t t h e s i g n i f i c a n c e
homotopy
it is n a t u r a l
admit homotop7 associative
equation but as a conjery
three-space
again,
it i s
of a f i b r a t i o n b e i n g i n d u c e d b y a n A - m a p . n the associative
of n - v a r i a b l e
characterization
multiplications;
equations.
of s p a c e s
law not as a three variable T h i s in t u r n l e a d s to a
of t h e h o m o t o p y t y p e of a s s o c i a t i v e
H-spaces. Consider determine
five maps
a single application topy as a map
the various
w a y s of a s s o c i a t i n g
of X 4
X, e a c h of w h i c h is h o m o t o p i c
into
of h o m o t o p y a s s o c i a t i v i t y .
h : I-~ X X3, we can construct
Regarding a map
S1 as a pentagon with the five maps as vertices
If t h i s m a p c a n b e e x t e n d e d to a t w o c e l l of p r o j e c t i v e case for
~Ix,
the associating
spaces
four factors.
These
to t w o o t h e r s b y
the associating
homo-
S 1-* X X 4 b y r e p r e s e n t i n g
and the five homotopies
as edges.
,iwx e 2 with boundary
can be extended one stage further.
t h e s p a c e of l o o p s p a r a m e t e r i z e d homotopy can be represented
S 1, t h e c o n s t r u c t i o n
T h i s i s of c o u r s e
by the unit interval.
schematically
by
the
In ~ I x ,
49
s o t h e m a p of
S1 we are looking at is represented
which can be extended to
e 2 by deforming
by
all paths to
(wx) (yz)
in the obvious
way.
To proceed volving maps morphic
to
m.
1
K 2 = ~.
(K r × K s ) k
K . X X i -~ X
:
1
where
w e n e e d a f a m i l y of c o n d i t i o n s
K.
is a s p e c i a l
1
in-
cell complex borneo-
Ii-2.
D e f i n i t i o n U . 1. Let
with this approach
K. d e n o t e s a c o m p l e x 1
Let of
K. = C L . , 1
1
(Kr × K s )
symbols,
e.g.,1
Z ...
responds
to i n s e r t i n g
constructed
t h e c o n e on
L.
(k k + l . . . two pairs
w h i c h i s t h e u n i o n of v a r i o u s
1
corresponding
inductively as follows:
to inserting
k+s-1) ...
i.
of p a r e n t h e s e s
a p a i r of p a r e n t h e s e s
The intersection with no overlap
copies in
of c o p i e s c o r or with one as a
subset of the other: I ...
(k...k+s-l)
...
I...
(k... (j...j+t-l)
(j...j+t-l)
...
r
× K
s
-~ K.
1
(An a l t e r n a t i v e
sense,
is t h e i n c l u s i o n of the c o p y i n d e x e d by indexing by trees
i
k+s+t-Z) ...
Thus the foUowing definition makes K
...
or i.
where
a k ( r , s) :
I . . . (k... k+s-1) . ..
i s g i v e n a t t h e e n d of t h i s c h a p t e r . )
i.
i
50
Definition maps
11.2.
An An-space
(X;{Mi})
M . : K. X X . - ~ X , i < n
x i) = M r (p, x [ . . . . .
M.
exist and satisfy
and a family
of
M s (or, x k . . . .
Xk+ s _ l ) . . . . .
xi)
p ~ K , ~ c K . r
If t h e
X
with unit and
M i ( ~ k ( r , s) (p, a ) , x I . . . . . for
of a s p a c e
such that
1) M 2 i s a m u l t i p l i c a t i o n 2)
consists
these
S
conditions
for all
i > 2, w e s p e a k
of
{X, { M i } )
1
as anAl-space.
Where
Conditions
necessary,
we refer
approximating
The complexes
K.
to the
these were
{Mi}
as an An-fOrm.
first presented
a r e a l s o of i m p o r t a n c e
in [Sugawara].
in category
theory
in
1
relation
to coherence
morphic
to
exhibited
Ii-2
the
of f u n c t o r s
is not obvious.
K.
1
as specific
[MacLane]. Several
That the complexes
are homeo-
ways to see it are available.
convex subsets
of
Ii-2
which are
clearly
I have homeo-
K3 morphic
to the whole cube,
e.g. ,
K2 = *
~0
1~2
/"
1
/
K4
\ Adams
has computed
the homology
and fundamental
group
of
L.
and
1
thus shown
L. f o r 1
i>
shown the cell complex Z~i - 2 .
[Boardman]
5 has the homotopy
t y p e of a s p h e r e .
L . i s t h e d u a l of a c e r t a i n z
has given a cubical
decomposition
subdivision of
K. 1
idea first
suggested
by Adams.
Stallings
has
of the boundary
indexed by trees,
of an
51
Associative fined to have constant
H-spaces value
are
of c o u r s e
Before
description
proving
this,
-spaces
o0
since
M.
can be de-
1
x 1. . . x i.
T h e m a i n p o i n t of t h e d e f i n i t i o n invariant
A
of A
CO
of a s p a c e
of t h e h o m o t o p y
we present
the main theorem
-space
is that it is a homotopy
t y p e of a n a s s o c i a t i v e about A -spaces,
H-space. which is
n
w,h a t o n e s h o u l d e x p e c t . Theorem
11.3.
A connected
CW
admits
X
the structure
of a n A - s p a c e
if a n d
n
only if there
exists
a sequence
of quasLfibrations
E 0 =X-*
1-*
,., with
E.1 c o n t r a c t i b l e
in
En_ 1
. . .
B I-~ ...
Bn_ 1
E i + 1.
The construction
is not iterative,
En
although
En_l~
be
~
inductive.
We let
Kn+ z X X n+l
n
Pn Bn
Bn_l~
Kn+ z X X n n
The attaching ~n{0k(r' s)(P'c)'Xl with the
M
term
s
.....
map for
if
the first
x
factor,
. M. s (. ~ ' X. k . . . .X k. +.s -.1 ). .
k + s - 1 = n + 2.
x
coordinate
By induction we prove first
is given by
n
Xn+l) = ~ r ( P ' X l ' .
deleted
obtained by dropping
E
Pn'
is a quasifibration.
The attaching
map for
B
n
is
consistently.
induced by projection
This
Xn+l)
time
we break
onto all but the
B
into two
over-
n
lapping pieces
by considering
1
crucial
condition
weak homotopy one
onto
in proving
equivalence
and breaking
K. = CL.
Pn
is a quasifibration
occurs
as the fibre
~ -~ ~
occurs
over
over
(~,x_ .....
x ) where n
L..
This map
can be identified with mapping
1
up a cone as before.
The
1
in showing that a
(T,x z .....
is a deformation x
Xn )
is mapped
of a n e i g h b o r h o o d into
x
by
of
to L. 1
52
x-~ M r(p,x,x 2 ..... M
r
(p,x,e .....
e)
As for the limit, Theorem o n l y if
l l . 4. X
Xn) since
for fixed X
p
and
is c o n n e c t e d ,
the arguments A connected
X
admits
the structure
admitting
an associative
before,
an A
00
for some
invariant
multiplication
is not.
associating
if a n d
a s o p p o s e d to
We h a v e m a d e t h i s l a t t e r
follow.
quasifibrations
Thus
multiplicative
as in T h e o r e m
S3 with these particular
the unit. A d a m s
has given m e
-form.
remark
on
S3
chosen
Actually he works by
exotic multiplications
cannot be deformed
T h e p r o o f of T h e o r e m
o0
11. 3 f r o m w h i c h t h e A
h o m o t o p y t y p e of an a s s o c i a t i v e
these multiplications
of m u l t i p l i c a t i o n s
[Slifker] shows that a properly
h o m o t o p y c a n b e e x t e n d e d to a n A
constructing
H-space
=o
-forms
has the homotopy
t h o u g h on t h e s t a n d a r d
to be a s s o c i a t i v e .
11. 4 w h i c h i s i m p l i e d b y o u r e x p o s i t i o n u s e s
an alternative proof of a stronger result w h i c h
no use of units.
Theorem {Mi} ~
-space
b u t l e t us e x p a n d on it now.
only eight are homotopy associative.
makes
00
is a homotopy invariant while
R e c a l l t h a t of t h e t w e l v e h o m o t o p y c l a s s e s
S3
of a n A
case.
Y.
statement
-structure
to
to t h e i d e n t i t y .
t o t h o s e in t h e a s s o c i a t i v e
CW
since admitting
and hence homotopic
similar
N o t i c e t h i s is a h o m o t o p y 4.3,
T h i s in t u r n i s h o m o t o p i c
are
h a s t h e h o m o t o p y t y p e of ~ Y
Theorem
x..1
II. 5. {Adams). satisfying
If X
admits a m a p
2) of II. 2, then X
with an associative multiplication n
M z : X X X-~ X
and a family
is a deformation retract of a space
such that n IX X X
is h o m o t o p i c in Y
Y to
m. T h e proof has b e e n simplified by [Boardman]. while for A
n
-spaces with n
Definition U. 6. Bn_iVKn+
defer the proof
finite, w e again look at projective spaces.
If (X, { M i } ) is an A
Z X Xn
We
n
-space, XP(n)
constructed in proving T h e o r e m
will denote the space
II. 4.
53
Theorem
Ii. 7.
If Y
is a M o o r e
space of type
(G, Zp+l) w h e r e
abelian group in w h i c h division is possible for all p r i m e s prime
p, t h e n
Y admits
The maps
the structure
M.
for
i< p
of a n A p . l - s p a c e
are constructed
G
is an
q less than the b u t n o t of a n A p - s p a c e .
a s in t h e c a s e
p = 5
1
(Example M
P
7.6) by deforming
t h e t r i v i a l o n e s in ~ 2 ~ Z y .
follows from the decomposability
p-fold cup products
in
YP(p)
of ~ p + l
The nonexistence
contrasted
of
with the non-trivial
if i t w e r e to e x i s t .
Given two A n - s p a c e s ,
w e can again consider m a p s
which respect the
structure. Definition II. 8.
If (X;(Mi})
a homomorphism
if
and
(Y, {Ni})
are A n - S p a c e s ,
a map
f : X -~ Y
f M i (7, x 1 ..... xi) = N i (~, fx I..... fx.1)" It is also possible to consider m a p s
of A
-spaces which respect the n
structure
up to homotopy,
pletely here.
but the details are too complicated
F o r example,
to m e n t i o n
respecting a n associating h o m o t o p y
corn-
involves a 2-
cell subdivided as a hexagon, while respecting an A4-structure involves a c o m plex w h i c h looks like
7ill
z_&L
k\ \
is
54
However,
maps
of an A
n
-space into an associative H - s p a c e
or vice v e r s a are
manage able. Definition A map
U . 9.
Let
f : X -~ Y
(X, { M i } ) b e a n A n - s p a c e
is an A -map
if t h e r e
exists
and
Y
an associative
a family
H-space.
of m a p s
n
h i:
Ki+ 1 × X i-~ Y
such that
h 1 = f and h i (Ok ( r , s ) ( p , a ) ,
x 1. . . . .
= h r ( 9 , x 1. . . . .
x .1) =
Ms(a,x k .....
= h r _ l ( p , x 1. . . . .
Xk+s.1) .....
X r . 1 ) h s _ 1 (¢r, X r . . . . .
It is easy to see that an A -map
xi)
x,)l
of a s s o c i a t i v e
for
for
r + s = i+2 k < r
k = r.
H-spaces
is an A -map
n
sense
with respect
Theorem
11.10.
to the trivial If
(X;{Mi})
higher
homotopies
is an An-space,
The proof is a generalization h i : Ki+ 1 X X i-~ ~XP(n)
then
X -~ ~ X P ( n )
conveniently
by defining some
cO
-form.
is an An-map.
8.6.
The maps
in terms
of
reasonable
homeomorphisms
K i + 2. M a n y of o u r r e m a r k s
associative H - s p a c e s
Theorem
used as the A
of t h a t of T h e o r e m
can be described
K i + 1 × X i × I "*- K i + 2 × X i ' * X P ( i ) Ki+ 1 X I~
in this
n
11.11.
X
about H-maps
c a r r y over to A
admits
n
-maps
of H - s p a c e
in this m o r e
and A -maps n
of
general sense.
of a n A - s p a c e if a n d o n l y if e a c h s t a g e
the structure
n
of a n y P o s t n i k o v
system
for
homomorphisms
and the k- invariants
Corollary ll. lZ. (cf. 9. I0). space provided
X
does in such a way that the projections are represented
An An_l-space
~.(X) = 0 for
i< p
and
X
Pn
are
by An-maps.
has the h o m o t o p y
type of a loop
i > np+n-4.
I
Example
11.12.
class
of ExampIe
u
Let
W be the space 10.8
or
constructed
10. 9, t h e n
W
by using as k-invariant
admits
an Ap_i-form
the
but not an
A -form. P These
examples
used the bar
construction
spectral
sequence.
More
55
generally,
for an A -space
X
we have the spectral
sequence
derived
from
the
n
finite filtration Theorem
of
11.14.
XP(n) Let
by
XP(i),
i < n.
(X, { M i ) ) b e a n A - s p a c e
and
u e Hq(x;Tr).
Then
n
If
3)
u
is represented
holds for
1)
d
2)
Su
r
[u] = 0 f o r
r < i
pulls back to
by an A.-map
then
1
i f a n d o n l y if
XP(i)
1) a n d
2)
follow.
The converse
i < n.
The converse know to prove
~XP(i)
is stated
-* X
in this limited
is an A.-map
way because
is to use
the only way I
XP(i+I).
1
Our analysis
of A - m a p s
in terms
of c o h o m o l o g y
classes
also applies
n
to the maps
inducing the succession
Zabrodsky'
s technique.
Zabrodsky's
method
of f i b r a t i o n s
used to construct
T h u s w e f i n d if t h e i n g r e d i e n t s
are A -spaces
and maps,
first
the result
is anA
used the technique
to construct
a homotopy
b u t n o t of t h e h o m o t o p y
sky].
was not an A5-space.
Example
the example
11.15.
Let
P1 = {2,3),
That the resulting
X
~ 1 : H3 ( X z ; Z 5 ) -~ s l l ( X z ; Z 5 ) essentially A3-space marks,
the same if
G
the example
not A -spaces P
is trivial.
can he adapted
several
i d e a of i n d e x i n g b y p l a n a r in the plane,
H-space
t y p e of a l o o p s p a c e
[Zabrod-
follows from Xz
is an A3-space
Adams
divisible
by
the fact that
to show
Z and
3.
to give finite complexes
follows from
Y(G;Zn+I)
is an
As Zabrodsky
re-
which are Ap_ 1 but
p.
Boardman' are
associative
X 2 = (S 3 × S 5 X S 7 × S 9 × s l l ) ( P 2 ) .
That
used by Frank
of r a t i o n a l s
for any prime
There
X 1 = SU(5),
is not an A5-space
argument
consists
-space. n
which was a finite complex In fact,
in
to be mixed by
n
Zabrodsky
X(]t~ 1)
s Proof clever
trees,
so as to keep track
of T h e o r e m
ideas
11. 5.
in the proof.
i.e. , directed
connected
of w a y s of i n s e r t i n g
First, acyclic
parentheses.
there
is Adam's
finite graphs For
example,
56
w{(xy)z)
corresponds Second,
an associative
to t h e t r e e there
operation
m ( m × 1) = m(1 X m )
D e f i n i t i o n 11.16.
~
o
is an idea from categorical not by a multiplication
algebra
of c h a r a c t e r i z i n g
m : X × X -* Y
and a relation
but as follows:
An A-structure
on a s p a c e
X
i s a f a m i l y of c o n t i n u o u s m a p s
n
{k i : X i - ~ X , i > 2} Usually, The other trees
k.
1
s u c h t h a t if k
i
~ m. = m then k o(vk ) = k . 1 n rn. m 1 1 is t o b e t h o u g h t of a s t h e m u l t i v a r i a b l e m a p
can be indexed by the trees
can be obtained by composites The complex
where
T
K.
1
is a tree with
without disconnecting
n
the tree).
For
-//
m
.
and all the
of t h e s e .
will now be represented
branches
edge is subdivided needlessly.
~kk
x 1. . . x
a s a u n i o n of c u b e s
C(T)
(= i n p u t s = e d g e s w h i c h c a n b e r e m o v e d our present
The cube
C(T)
purposes
we will assume
will have parameters
no
indexed by
t h e e d g e s of t h e t r e e w h i c h a r e n o t b r a n c h e s .
Definition II. 17.
WA
with n-branches.
(n, I) is the union with identifications of C(T) over all trees
T h e identifications are that a face
t. = 0
of
C(T)
is to be
1
identified with edge indexing
C(T' ) w h e r e
T'
is obtained f r o m
t.. 1
F o r example, WA(Z,I) =
=
C(V)
W A ( 3 , 1) = ~ ( ~ / ) G ( ~ )
G(h) ~)
T
b y shrinking to 0 the
57
It c a n b e s h o w n t h a t Definition
U. 18.
A
WA(n,1)
WA-structure
on
is a cubical
X
is an A
subdivision
-structure
of
K . n
without units,
i.e. ,
~C
a family
of maps
l~l : W A ( n , 1 ) × X n - ~ X n
Mn(~, x I .....
x n) = lVir (p, x 1 . . . . .
if ~ = (tI ..... tn_Z)
•
C(T)
with
such that
Xk_ 1, M s (c;, x k . . . . .
t. = 0 w h e r e ,
Xk+ s) . . . . .
x n)
if the e d g e indexing
t. is
1
deleted, same
T
as
decomposes
T' [J T "
values to the c o r r e s p o n d i n g
1
while
p e C(T' ), a e C ( T " )
e d g e s as does
7.
T h e f i n a l i d e a of t h e p r o o f i s to u s e t h e nective
tissue
to build something
like a tensor
associate the
WA(n,1)
algebra
complexes
of w h i c h
X
as con-
will be a
retract. First
we let
that now we permit
For
example,
description as
WA(n,1)
trees
WA(Z,I)
be the complex
with an extended
= C(y) = ;
root,
: and
constructed
except
i.e. , "'~'"
WA(n,I)
gives a useful parameterization.
as before
= WA(n,l)
We also let
× I but the tree
WA(1,1) ~ •
regarded
C(1).
Definition MX
If. 19.
Given a
is defined by taking
(T,x I..... • ,p,a
x n)
with
(~)
WA-structure WA(n,1) × Xn (p,x I .....
on
X, the associated
for each
n
Xk_l, Ms(~,x k .....
associative
space
and identifying Xk+s_l) .....
x n)
where
are as above. If
t. = 0
on the edge corresponding
to the extended
root,
then
(•)
1
means
Mn(t I .....
~i .....
The operation
t k - l ' Xl . . . . . on
MX
= ((p,a),x I ..... regarded
as being in
tive involves combinatories
essentially
only the
aptly called tree
is given by
Xr+ s)
C ( T 1 v TZ).
Xn)"
where
(p,x I .....
now for
That the operation WA-parameters,
surgery.
x r ) • (a, X r + I . . . . .
p c C(TI),
ae
X r + s)
C(Tz), (p,a)
is well defined and associaan exercise
in parameterized
is
MASSEY PRODUCTS The differentials were
usefu~ in analyzing
the homology
spectral
AND GENERALIZED in the cohomology
k - invariants
sequence,
BAR (31)NSTRUCTION
Eilenberg-Moore
in terms
of
spectral
A -maps. n
we have in particular
sequence
If we turn to
differentials
of the form
dr[all .. l Ur+ l] represented These
by homology
are closely
ucts originally introduced,
related
of t h e a s s o c i a t i v e
to the Pontrjagin
defined in the cohomology
these homology
the duality; however, algebra,
classes
products
Massey'
ring analogues of a n a r b i t r a r y
were
s procedure
so we will use the term
"Massey
should now expect,
strict
homotopy
we save such generality
analogue;
associativity
As for the differentials
H-space
called
in question.
of t h e [ M a s s e ~ p r o d space.
Yessam
When first
products
product"
generically.
as one
by an appropriate
f o r t h e e n d of t h i s c h a p t e r .
more
generally,
it turns
products
out [ May]
by appropriate
Massey
allow matrices
classes
than single homology
rather
differential
In fact,
that they are all determined of h o m o l o g y
to emphasize
is valid in any associative
can be replaced
dr
X
if w e a r e w i l l i n g t o classes
as
arguments. Until further
notice,
with differential
d.
Definition
Let. u,v,w
product
12.1. ~
Remark.
If defined,
has the larger remarks higher
where
.
u
E H(A)
.
indeterminacy
Massey
be an associative s original
such that
of H ( A )
represents
like this continue order
A
We start with Massey'
is the coset
B
ux(.1)deg u
let
by
uH(A) + H(A)w
.
.
[Uehara
.
triple
by
dy = uv.
Notice that
d2[ulvlw]
this situation,
The differentials are usually
and Massey]-
The Massey
As we generalize
but the latter
algebra
determined
dx = vw,
d2[ulvlw]
represents
to be applicable.
products,
product
uv = 0 = vw.
u, e t c . , a n d
H(A). H(A).
differential
more
are determined delicate,
less
by
60
often defined and with s m a l l e r i n d e t e r m i n a c y .
choice, for of course
T h u s in g e n e r a l i z i n g we h a v e s o m e
d r [ . 1 1 . . , l ur+ 1] can itself be regarded as a generalization.
We w i s h to d e f i n e h i g h e r o r d e r M a s s e y p r o d u c t s < u l. . . . . , u i + s >
a r e defined and z e r o for
l<_i
and
s < r-1.
< u 1. . . . .
Ur> w h e r e
[May] h a s
d e v e l o p e d the f o l l o w i n g n o t a t i o n .
D e f i n i t i o n 12.2.
aij ~ A
The M a s s e y p r o d u c t < a 1. . . . , a t >
for l<_i_< j<_r
excluding the case
is d e f i n e d if t h e r e e x i s t
ij = ir such that a..11 represents
a. ~ H(A) a n d 1
daij = /i) k=i where
aik ak+l. J
aik has signs altered appropriately.
The M a s s e y p r o d u c t < a 1. . . . .
ar>
is the set of all homology classes represented by r-I 1
a_ik a k ÷ l r
for a n y s u c h s y s t e m .
T h e o r e m 12.3.
represent
If < a 1, . . . , a r > is d e f i n e d , t h e n a n y of its r e p r e s e n t a t i v e s
dr'l[all...
]ar].
The p r o o f is s t r a i g h t f o r w a r d , though t e d i o u s .
The d e f i n i n g s y s t e m
aij is used directly to show d s [all " ' " l a r ] = 0 for s < r - 1 a n d to o b t a i n a representative
X i n B'{A) of
[ a l l . . . J a r ] , w h i c h is a c y c l e u n d e r the t o t a l
d i f f e r e n t i a l in B(A).
E x a m p l e lZ. 4.
In H . ( ~ C
P ( n ) ) , if u g e n e r a t e s
Hi, t h e n < u , . ~ n+l
generates
HZn.
S i n c e n - f o l d M a s s e y o p e r a t i o n s a r e e a s i l y s e e n to be n a t u r a l
w i t h r e s p e c t to A - m a p s , t h i s a g a i n shows n
S1-~ ~ C
P(u)
i s - n o t a n A n + l - m a p.
61
T h e c o m p u t a t i o n m a y be done by o b s e r v i n g
H2n+I(CP{n)) = 0 w h i c h c a n b e
a c h i e v e d o n l y if the g e n e r a t o r of H Z n ( ~ C P(n)) The m a t r i c M a s s e y product,
is k i l l e d by
dn.
i n t r o d u c e d b y M a y , is a f a i r l y
s t r a i g h t f o r w a r d g e n e r a l i z a t i o n in w h i c h
a.
V..
is r e p l a c e d by a m a t r i x
l
obtain a reasonable definition, certain conventions about matrices
To
1
will be
observed. If V is a m a t r i x
(v..), t h e n ~
w i l l b e the m a t r i x
((-I) l + d e g v i i v . . ) .
U
An o r d e r e d p a i r of m a t r i c e s n × q and for each
i,j,
matric Massay product Vn
is
q × 1 and
U
(X,Y)
is m u l t i p l i c a b l e if X is
deg Xik + deg Ykj
is c o n s t a n t
for
m × n
and
1 < k < n.
The
V > w i l l be c o n s i d e r e d only if V 1 is n
V 1 . . . V . , j V j+ 1 . . .
Vk
is m u l t i p l i a b l e f o r e a c h
G i v e n the a b o v e c o n v e n t i o n s , D e f i n i t i o n 12. g c a r r i e s d e f i n e the m a t r i c M a s s e y p r o d u c t
Y is
1 × P,
j , k _ < n. o v e r v e r b a t i m to
T h e i n d e t e r m i n a c y is the s e t
of a l l p o s s i b l e d i f f e r e n c e s c o r r e s p o n d i n g to d i f f e r e n t c h o i c e s of t h e s y s t e m
A... U
M a y (in p a r t f o l l o w i n g [ K r a i n e s ] ) g i v e s b o u n d s on the i n d e t e r m i n a c y , linearity formulas,
associativity formulas,
Massey products, permutation rules. As b e f o r e ,
" s l i d e " r u l e s and for o r d i n a r y
He a l s o d i s c u s s e s n a t u r a l i t y .
t h e s e M a s s e y p r o d u c t s c a n b e r e l a t e d to the d i f f e r e n t i a l s
in the s p e c t r a l s e q u e n c e .
Matric products are particularly relevant when
dr
is
d e f i n e d on a c o m b i n a t i o n of t e r m s w i t h o u t b e i n g d e f i n e d on the i n d i v i d u a l t e r m s . F i n a l l y M a y s h o w s t h a t m a t r i c M a s s e y p r o d u c t s d e t e r m i n e the s p e c t r a l s e q u e n c e in t h e f o l l o w i n g s e n s e : such that for each element
x
if x
d x P
s u r v i v e s to
<W 0 . . . . .
E , then P
Wp_I,V>.
all
q there are matrices
of E p ' q t h e r e is a c o l u m n m a t r i x is r e p r e s e n t e d
The differentials
of p a r t i a l d e f i n i n g s y s t e m s f o r
For each
drX f o r
<W0 . . . .
x ; on the o t h e r h a n d , t h e y a r e h u g e .
'
W
p-l'
V
such that
by a suitable r e p r e s e n t a t i v e r <_p V>.
W.
of
c a n b e e v a l u a t e d in t e r m s
N o t e t h a t the
W.
i
work for
62
Of course
A -spaces
can be mimiced
on t h e c h a i n l e v e l ,
just as
n
associative
differential
algebras
Definition 12.5 [Stasheff]:
mimic
topological monoids.
( A , m . , 1 < i < n)
is a n A - a l g e b r a
1
--
if m . : A x -* A
n
i
such that 0 = T
(-I)
m r {l@. . . @ m s @ ' ' ' @ I )
k,s
k-1 where for
al@...@a.x
the
(s+l)k + s (i + ~
~ is
d i m a.) 2
/
I
m 1 p l a y s t h e r o l e of d i f f e r e n t i a l . )
(Note:
For an A
-algebra,
there
is a g e n e r a l i z a t i o n
of t h e b a r c o n s t r u c t i o n .
00
Given an A
Definition 12.6 [Stasheff]:
-algebra
(A,m.),
o0
B'(A)
is
@ i=0
A i with the differential
~[all-.-lanl : where
the tilde construction
1
(-1)x [all... [ms(ak®...®%+s_l)l'-" I%1.
k-1 )~ = (s+l)(k+l) + s(i + >
dim aj).
1 A l l of o u r m a c h i n e r y and Massey
products,
is d e f i n e d b y Finally,
connected
C QC P q
uy+xw+_m3(u,v,w)
differential (1~)
Let
including the spectral
for
coalgebra,
JC = C/A
~
Let
x e C
P
For example,
Let
~
P,
and define
E@...®C
n>O ~ --
be a simply
graded,
denote the component
> 0 that
•
(C, ~)
i . e . , if p o s i t i v e l y
%, q
7-Cp
~(C) = T(~) =
sequence
[Stasheff].
the dual situation.
= (&®l)~.
and we shall assume
£~0, p(X) = x@l.
over,
albeit with additional complication.
l e t us c o n s i d e r
associative
C O = A, C 1 = 0 and
carries
n
o(X) = l ~ x
and
of
~
in
63
with
@c = - d c =~'---- (-l)P A
Z__
p,n-q
c for c e C
and e x t e n d m u l t i p l i c a t i v e l y to
n
l
r e g a r d e d as the f r e e a s s o c i a t i v e a l g e b r a g e n e r a t e d by ~ . [Adams] shows
H,(~(C~(X))
is isomorphic to H,(~X)
b u t the p r o o f is a t o u r de f o r c e u s i n g s i m p l i c i a l c h a i n s f o r
as an algebra,
X and c u b i c a l c h a i n s
for f~X. Following a suggestion of Peter May, w e outline a proof which follows m o r e n a t u r a l l y f r o m the p o i n t of v i e w we h a v e e s t a b l i s h e d .
w
T h e o r e m 12.7.
Proof.
T h e r e is an
shin
equivalence
~ C , ( X ) ~--~ C , ( ~ X ) .
We w i l l c o n s t r u c t m u l t i p l i c a t i v e e q u i v a l e n c e s .
B C, (X) ~-~ C, (Bf~X) -- ~ B C , (~X) -~ C, (f~X) The left hand m a p is induced by the h o m o t o p y equivalence multiplicative by naturality of ~ . the standard inclusion
B~X-~
B
and is
The s a m e applies to the middle m a p which is
BC~(Y) -~ C$(BY).
The fact that ~ B
is naturally
equivalent to the identity via shin m a p s should be a basic bit of general nonsense in the appropriate category; at present only a laboriously detailed proof is conceivable. Since ~ B A construct
: BA-~ A
homotopy equivalence.
is the f r e e a s s o c i a t i v e a l g e b r a on B'A, it is e n o u g h to s u c h t h a t its m u l t i p l i c a t i v e e x t e n s i o n We d e f i n e [a] = a = 0 on
A n inverse for given by L
and @ A ®n . n>l
is the obvious (but not multiplicative m a p
L(a) = [a] for which
_~ I ~ A
:~] B A -~ A is a
L = IA.
i : A-~BA
The details of the h o m o t o p y
are given in [Halperin and Stasheff].
It is worth remarking that if A
modules are of finite type then letting
64
(
)
denote
Horn(
, A ) we h a v e 8 ( A )
= (B'A) .
Thus
H~,(~X)
can be
c a l c u l a t e d a s t h e h o m o l o g y of [BC ~ (X)] ~. The spectral sequence derived from filtering n u m b e r of b a r s " h a s
E 1 = BH (X) a n d
(X) b y
E Z = T o r H * (X) (A, A).
a r e now g i v e n in t e r m s of M a s s e y p r o d u c t s w i t h Massey' s original
BC
dz[ul v]w]
The differentials
being represented
.
In general
B and ~
are too large for complete computation but
s p e c i f i c e x a m p l e s in l o w d i m e n s i o n s c a n o f t e n b e done n e a t l y in t e r m s m
"the
of B ~
m
o r ~]B, e . g . ,
Example I0.7.
I t i s t i m e we l e a v e a b s t r a c t d i f f e r e n t i a l a l g e b r a a n d r e t u r n to s o m e geometry.
by
HOMOTOPY We have analyzed view.
associativity
We could do the same
expressed length;
diagramatically.
other examples
X 3 -* X
commutativity
Every
{x,y,z)-~
direct
homotopic
to
An H-space
mT
where
The easiest H-space k + /a
homotopy =/~(s-t)
/~ + k
t<
s<
evaluation
on a space
is homotopy
Moufang.
of t h e o b s t r u c t i o n s .
I, a homotopy
on a sphere
is homotopy
disassociative,
commutativity.
{X, m )
is homotopy
commutative
~X
of a n H - s p a c e . to pointwise
{Here to have
if t
is determined
is
commutative
The point is that loop addition
/~ : [ O , r ] - ~ runs from
by
but homotopy
multiplication
t E I. ) F o r
if m
T{x,y) = yx.
of a n o n c o m m u t a t i v e
k-/~ t w h e r e
t+r.
but we could normalize izedby
example
is homotopic
is given by for
{x(yz))x
T : X × X-* X × X by
is the loop space or
M o u f a n g if t h e f o l l o w i n g
(xy)x.
We now turn to homotopy
13.4.
is homotopy
multiplication
multiplication
x(yx) ~
Definition
X
y = e, we also see
13.3.
we have
at some
{ x , y , z) -~ ( x y ) ( z x )
Every
The proof involves
i.e.,
p o i n t of
We will study homotopy
An H-space
13.2 [Norman].
Theorem
a homotopy
condition which can be
are homotopic:
Taking
from
for any other algebraic
and
Theorem
extensively
have also been handled.
Definition 13.1 [Norman]. maps
COMMUTATIVITY
~Ix
X 0
to
then q
of l o o p s v t (s) = *
where
consisting
12 × ~ X X ~ X - ~
X
k"
k./~ .
The
for
O< s<
[O,q]-~ X
of loops parametergiven by
t
66
(t, s, k,/~) -~ k(t)./~(s).
There
are very few examples
of finite dimensional
homotopy
commuta-
tive H-spaces.
Theorem
13.5 [Hubbuck].
If X
is a connected finite c o m p l e x and
h o m o t o p y c o m m u t a t i v e H - s p a c e , then torus
X
(X,m)
is a
has the h o m o t o p y type of a point or a
S 1 X ... X S I. H u b b u c k ' s proof involves applying c o m p l e x K - t h e o r y to the projective
plane
XP(2).
ways.
T h e h o m o t o p y c o m m u t a t i v i t y of m
H u b b u c k uses the existence of a m a p
is reflected in XP(2)
f : XP(Z) -~ XP(Z)
in two
such that there is
a homotopy commutative diagram S X - ~ SP(2)-~ S X / / S X
tT S X -~ X P (2) -~ S X ~ Z S X where
SX-~ SX
r e v e r s e s the p a r a m e t e r .
T h e ~ e is another reflection of h o m o t o p y c o m m u t a t i v i t y in X P (Z).
Theorem
13.6 [Stasheff].
SX × SX-~ XP(2) associative a map
If
(X, m )
which is the inclusion
and right translation
S X × S X -~ X P ( 2 ) Proof.
is homotopy
on each factor.
is a homotopy
implies
The obstruction
commutative,
the homotopy to extending
If
equivalence commutativity
there
(X,m)
is a map
is homotopy
the existence of m .
S X v S X -~ X P ( 2 )
to the
of s u c h
67
SX X SX
is a generalized
out explicitly,
one sees
a : X-* ~XP(2).
class
Whitehead
product
it is adjoint to
The obstruction
[Pl + P2 ] - [P2 + Pl ] and
Pl
in
[SX z, X P ( Z ) ] .
(a} + P2
to homotopy
(~) - P l
Writing
(a)-
commutativity
a, : iX 2,x]-* ix 2,axp(2)]
P2
in
the map
(~) w h e r e
iX Z,X] is the
takes the one ob-
struction to the other. Now, homotopy
if X
is h o m o t o p y associative and right translation is a
equivalence, w e have the retraction
r : ~ X P ( 2 ) -* X
and hence
a~
is a monomorphism.
Corollary
13.7.
~CP(3)
Proof.
is homotopy
We know
type of a w e d g e o s s p h e r e s . t o a l l of
~Clm(3) ~ S1 ~ ~S 7 so
S~CP(3)
has the homotopy
Thus for
to extend
SX v SX-~ XP(Z)
SX × SX
involves
obstructions
Now all Whitehead
products
in
we at least
have
CP(3)
S X × S X - * B X.
tivity since the obstruction
commutative.
X = ~CP(3),
a l l of w h i c h a r e are trivial
to zero
products.
[Barratt-James-Stein]
This is sufficient
is mapped
Whitehead
in
so
to imply homotopy
[XZ,~Bx]
commuta-
but the map is an
is omorphism. Note u
4
CP(3}
is not an H-space,
= 0 which is not a primitive
further. Since
The familiar f~CP(3)
maps
is homotopy
S I X ~ S 7-* ~ C P ( 3 )
relation.
is again an H - m a p .
ue
HZ(CP(3)}, we have
The discrepancy
S 1 -~ ~ C P ( 3 ) commutative
since for
and
is worth
~ S 7 -~ f ~ C P ( 3 )
considering
are H-maps.
it is not hard to show the product
Now
S 1 )< ~ S 7 ~
~(CP(o0) X S 7)
m
and hence is the loop space of an H - s p a c e so be an A
n
- m a p for s o m e
n.
S l X ~ S 7-* ~ C P ( 3 )
Indeed w e have seen
Higher homotopies related to h o m o t o p y
S I-* ~ C P ( 3 )
m u s t fail to
is not an
A 4 - m a p.
commutativity are of interest (cf.
[Kudo and Araki]) but, as w e shall see, the approximations
do not converge to
68
strict commutativity,
Definition 13.8.
w h i c h is b e t t e r s t u d i e d d i r e c t l y .
An H - s p a c e
(X,m)
is a b e l i a n if m
c o m m u t a t i v e and a s s o c i a t i v e with E x a m p l e s abound.
For each abelian group (v,n).
is
X, d e f i n e d as t h e l i m i t of x n ] ~ { n )
t i o n g r o u p a c t i n g by i n t e r c h a n g i n g c o o r d i n a t e s . t h e b a s e p o i n t and u n i t of SP °°{X).
~r a n d n a t u r a l n u m b e r
n,
S o m e w h a t m o r e g e n e r a l l y , for any
X we c a n c o n s t r u c t the a b e l i a n m o n o i d
p r o d u c t on
If m
u n i t , we s p e a k of an a b e l i a n m o n o i d .
t h e r e is an a b e l i a n m o n o i d of t y p e space
is c o m m u t a t i v e .
SP°°(X), t h e i n f i n i t e s y m m e t r i c where ~(n)
is the f u l l p e r m u t a -
T h e b a s e p o i n t of X b e c o m e s
A n y o t h e r p o i n t in S P °° {X) c a n be r e p r e -
n
sented as a f o r m a l sum
~
/
m(x.)x, 1
1
where
m{x.) 1
is a p o s i t i v e i n t e g e r and
x.
1
i=l i s n o t t h e b a s e p o i n t of X.
Ir.(SP °°{X)) is n a t u r a l l y i s o m o r p h i c to
T h e o r e m 13.9.
1
T h e p r o o f c o n s i s t s in verify~ing t h a t Eilenberg-Steenrod
C o r o l l a r ~ r 13.10.
Ir.(SP°°())
H.(X) 1
if X is
CW.
s a t i s f i e s the
1
axioms.
If X is a M o o r e s p a c e
Y(G,n), then
SP°°{X) h a s the
h o m o t o p y t y p e of K(G, n}.
T h e o r e m 13. iI.
A n a b e l i a n m o n o i d h a s the w e a k h o m o t o p y t y p e of a p r o d u c t of
Eilenberg-MacLane
Proof.
spaces
Let
i n d u c e s an i s o m o r p h i s m
[D__old a n d Thorn].
Y. = Y{H.{X), i). 1 1 in h o m o l o g y .
T h e r e is a m a p
Since
VY. -* X w h i c h i 1
X is an a b e l i a n m o n o i d , t h i s m a p
69
e x t e n d s to
SP°°(VYi)
or
T~SPC°(Yi)
s o a s to r e d u c e
an isomorphism
of h o m o t o p y
1
groups. Leaving aside associativity, ized by a retraction
S P 2 ( X ) -~ X.
e n o u g h to r u l e o u t m a n y s p a c e s multiplication
o n l y in t h e 2 - p r i m a r y
then
Here
strictly
a
of S P Z ( X )
[ e . g . ,if X = S n, t h e n
X
admits
is r i c h
an abelian
Adams
indicates that the obstruction
is e s s e n t i a l l y
If G
is a g r o u p of r a t i o n a l s
with even denominators,
is a n a b e l i a n H - s p a c e .
Associatlvity
illustrated
rood 2 structure
The
is c h a r a c t e r -
part.
13.12 [ A d a m s ] .
Y(G, Zn+l)
either
(X,m)
o n l y if n = 1].
On the other hand,
Theorem
an abelian H-space
and commutativit7
o r up t o h o m o t o p y .
There
can occur independently are
10 p o s s i b i l i t i e s
of e a c h o t h e r ,
to c o n s i d e r
in t h e f o l l o w i n g c h a r t .
and
c
na
ha
ea
nc
1
2
3
hc
4
5
6
ec
7
8
~ ~ / . /[u
abbreviate "associative " a n d c o m m u t a t i v e ,
"equivalent ,Ia n d "not h o m o t o p y " a n d T h e subdivision
9-10
h
e
and
n
abbreviate
stands for " h o m o t o p y but not equivalent".
refers to the alternatives:
9.
equivalent to an associative
multiplication a n d equivalent to a c o m m u t a t i v e
one but not simultaneously;
i0.
multiplication.
equivalent to an associative, c o m m u t a t i v e
means
via a n H - m a p
homotopy
same homotop7 type. indicates
X with
equivalence to a multiplication o n a s p a c e of the
[Adams] provides
~.
H e r e equivalent
the following examples;
killed above dimension
where
X[0, n]
n.
1
Theorem
as
13.13.
There
are examples
of t h e
10 t y p e s of H - s p a c e
above:
70
I.
S7
2.
S 7 w i t h c e r t a i n p - c o m p o n e n t s of lr. (S7) k i l l e d for
i > Zl
1
3.
Q u a t e r n i o n i c m u l t i p l i c a t i o n on S 3
4.
S 7 with the Z - c o m p o n e n t s of ~.(S 7) k i l l e d f o r
i > 14
1
5.
S 7 with c e r t a i n p - c o m p o n e n t s of ~. (S7) k i l l e d for
i > Zl and Z-components
1
k i l l e d for
i > 14
6.
s7
7.
AMoore
space
Y(Z[I/Z],7)
8.
A Moore
space
Y{Z[I/Z,I/3],5)
9.
Y(Z[1/ Z], 5)
10.
[o, lz]
[0,8]
S1 If X is a f i n i t e c o m p l e x , the s i t u a t i o n c h a n g e s r a d i c a l l y .
a r e p o s s i b l e only for
1, Z, 3, a n d 10, of w h i c h a l l b u t
Examples
Z are given above.
For
a n e x a m p l e is p r o v i d e d b y [ Z a b r o d s k y ] : the H - s p a c e w h i c h m o d 2 a n d 3 is SU(6) b u t m o d
a l l o t h e r p r i m e s is
S 3 X S 5 X S 7 X S 9 X S 11.
A t t e m p t s h a v e b e e n m a d e to c h a r a c t e r i z e h o m o t o p y c o m m u t a t i v i t y in t e r m s of a u n i v e r s a l e x a m p l e .
C o n j e c t u r e 13.14.
X a d m i t s a h o m o t o p y c o m m u t a t i v e m u l t i p l i c a t i o n if a n d o n l y
if X is a r e t r a c t of ~ZSZX. [ W i l l i a m s ] h a s g i v e n c o n d i t i o n s w h i c h a r e e q u i v a l e n t to X b e i n g a n A ° ° - r e t r a c t of ~ Z s Z x .
STRUCTURE Since an associative on
BX
implies
homotopy
homotopy
commutative
H-space
X
commutativity
and homotopy
ON
Bx
is essentially
of
X.
~B x,
a multiplication
O n t h e o t h e r h a n d if m
associative,
then
m
is at least
is
an H-map,
since we have
(wx) (yz) _~ w(x(yz)) _~ w((xy)z) _~ w((yx)z) _~ w(y(xz)) _~ (wy)(xz).
Theorem
1 4 . 1 [Su_._gawara].
multiplication
(X,m)
if a n d o n l y i f m
Notice that for x y ..~ y x
If
by taking
m
is strongly
Similarly, regarded
xyz
yzx
Proof B X × By
and
of T h e o r e m . B X X Y"
homotopy
to be an H-map
w = z = e.
fill in the following triangles
is a n a s s o c i a t i v e
m
H-space,
Bx
implies
being an A3-ma p implies of
we can
S 1 X X 3 -* X :
xyz
zxy
The key to the proof is the equivalence
A specific
a
multiplicative.
(i. e . , w x y z _~ w y x z )
as maps
admits
equivalence
called the shuffle map
of is
induced by : A p × (X X e)p × Ziq )< (e × Y)q-~ A p + q )< Of × Y)P+q w h i c h triangulates (e X Y)q
~P X ~q
and shuffles
(X)< e) p
and
together according to w h i c h s i m p l e x of the triangulation is involved. Specific f o r m u l a s are e a s y to write d o w n [Sugawara, Iv[ilgram,
Steenrod] if w e r e p a r a m e t e r i z e • .. < s
< I.
We
An
by n-tuples
(sI ..... s ) s/t
set up the c o r r e s p o n d e n c e so that the face
0 < sI < s 2 <
t. = 0 c o r r e s p o n d s
72
to the f a c e
s i = si+ I.
The m a p ~
can then be w r i t t e n as
(sI..... Sp, x I ..... Xp) (Sp+ I..... Sp+q,
Yl' " " " Yq)
= (s (i)..... s (p+q), z (i)..... z (p+q)) where
z.x = (x.,e)x for
tion such that Although this
i<_p
s (1). . . . .
and z i =
( e , y i p) for
i > p, a n d = is a n y p e r m u t a -
s (p+q) is c o r r e c t l y o r d e r e d ,
i . e . , is i n AP+q.
~ is n o t w e l l - d e f i n e d , it d o e s i n d u c e a w e U - d e f i n e d m a p
XP(p) X YP(q)--
(X × Y) P(p+q).
T h u s if m
is s t r o n g l y h o m o t o p y m u l t i p l i c a t i v e , we h a v e B X X B X_~B
x × X-" B×
w h i c h can easily be checked to be a multiplication.
T h e converse is straight-
forward. N o t i c e t h a t if m
is a n
s h r n m a p , the m u l t i p l i c a t i o n r e s t r i c t s to
X P (p) × X P (q) -~ X P (p+q) just as does map
CP(p) × CP(q) -~ CP(p+q).
S X X S X - ~ XP(2)
(X X X) P(2)
How homotopy
can be seen by going through the above m a p
SX X SX-~
explicitly.
T h e condition involved in constructing reduced to the following.
(in X x 3 ) :
S X X XP(Z) -~ XP(3)
Let h(t,x, y) be a c o m m u t i n g
h(0,x,y) = xy, h(l,x,y) = yx.
triangle
commutativity gives a
T o construct the m a p
homotopy,
can be
i.e. ,
w e m u s t fill in the following
, ~ ~ s ~
h ( t , x , yz) If m
w e r e o n l y h o m o t o p y a s s o c i a t i v e , the f i g u r e w o u l d b e a h e x a g o n .
A
c o r r e s p o n d i n g f i g u r e a p p e a r s i n M a c L a n e ' s s t u d y of c o h e r e n t f u n c t o r s [MacLane]. The symmetric
condition can he c o m b i n e d with this one to f o r m a
73
f i g u r e i n v a r i a n t u n d e r the s y m m e t r i c
group.
y~z
T h e r e c t a n g l e c a n b e f i l l e d in xyz ~
y
z
× by h(t,x, h(s, y, z)).
x z y ~ , z y x
zxy T h i s h e x a g o n a p p e a r s a l s o in M i l g r a m ~ s s t u d y of ~ Z S 2 X
[Milgram].
T h i n g s a r e m u c h s i m p l e r if X is a n a b e l i a n m o n o i d .
The multiplica-
tion B x X B x-~ B X can be described directly as
(s 1 . . . . .
Sp, x 1. . . . .
Xp) (Sp+ 1. . . . .
Sp+q, Xp+ 1. . . . .
Xp+q)
= (sa (1) . . . . # a (p+q)' Xa (1) . . . . . x a (p+q))" One can check directly that this multiplication is again associative and commutative.
By i n d u c t i o n d e f i n e
Theorem 14.3.
B(n)x
as
BB(n-I)X
with
If X is a n a b e l i a n m o n o i d , B ( n ) x
a d i s c r e t e a b e l i a n g r o u p , B(n)~r is a s p a c e of t y p e One c a n a l s o n o t i c e t h a t Now f o r a m o n o i d morphism. group
BSp~(x)
B(1)(X) = B X.
e x i s t s f o r a l l n.
If ~ is
(~r,n).
= SP°o(SX).
X, b e i n g a b e l i a n is e q u i v a l e n t to m
being a homo-
On t h e o t h e r h a n d , t h e r e a r e s p a c e s s u c h a s t h e i n f i n i t e u n i t a r y
U for which
Eilenberg-MacLane of s i g n i f i c a n c e .
Theorem 14.3.
B (n) e x i s t s , spaces.
e v e n t h o u g h the s p a c e s a r e n o t p r o d u c t s of
T h u s the u s e of s h i n
There is, however,
Let
a compromise
m a p s in T h e o r e m 14.1 is c o n d i t i o n of s o m e r e l e v a n c e .
(X, m) b e a n a s s o c i a t i v e H - s p a c e .
t i o n if t h e r e is a h o m o m o r p h i s m
B X admits a multiplica-
n : X X X - * X s u c h t h a t the u n i t of m
is a
74
h o m o t o p y u n i t of n . For a multiplication
example, on
if ~ l X
X
induces a homomorphism
The multiplication is of c o u r s e we have
homotopy
while with
then so is In g e n e r a l ,
structure
Theorem
some
Y
B
n
BX
of l o o p s ~ln
is induced by
on
Bn.
x = y = e
we have
An associative
then
The multiplication so with
n ( w , z ) "-~wz.
and hence the multiplication
on
)~ : [0, 1] -~ X,
~Ix.
n(wx, yz) = n(wy)n(xz)
we can look for additional structure
"up to homotopy"
14.4.
on
commutative:
n ( x , y ) -~ y x
associative,
denotes the space
on
BX on
m
w = z = e If n
is
is also.
BX
in t e r m s
of
X.
H-space
X
h a s t h e h o m o t o p y t y p e of ~ 2 y for K. if there exists a family of shin m a p s M . : X i -~ X 1 satisfying the
conditions for an A - f o r m as shin m a p s . K. i structure and X t h a t i n d u c e d f r o m X. )
{Here
X i has the product H - s p a c e
T h e t h e o r e m is trivial at this point, though it is tediously difficult to write out explicitly w h a t the c o m b i n e d compatibility conditions on higher h o m o t o p i e s are. the f o r m StiU m o r e
C. × X i-~ X J
It is clear, h o w e v e r , where
spaces,
that they can stiU be written in
C. is an appropriate (if foreboding) cell c o m p l e x . J
elaborate conditions are m a n a g e a b l e
homotopy everything
M .i and its
manageable
in B o a r d m a n ' s theory of
because
t h e y do n o t n e e d to be e x p l i c i t .
INFINITE
There
are spaces
are not only loop spaces
D e f i n i t i o n 15.1. i = 1,2 . . . .
such that Of course
X
X
O,
MacLane'
s
have this property,
U, S p
and their analogues
such spaces
is a f o r m i d a b l e PACTs
task.
In a c a t e g o r y
(a)
the objects are
(b)
the morphisms
such as
in t e r m s
[Boardman]
0 , 1 , Z. . . . from
of h i g h e r h o m o t o p i e s
invents a new gadget (modelled on we need.
We follow him
that
has permutations,
morphism
S
n
to
n
form a topological
-~
associative
lre S
m
m
• n= m
+ n ;
we are also given for each
(n,n), S
we impose and
~(m,n),
continuous functor
n
the symmetric
g r o u p on
(We o m i t a n y s y m b o l f o r t h i s h o m o m o r p h i s m . permutations
space
is continuous;
:~..~such
if
on t h e
;
m
(c) w e a r e g i v e n a s t r i c t l y
(i)
F = lira H(sn).
of operators
and composition
if ~
but less trivially we
[B_.oardman a n d V o g t ] .
Definition i5.2.
(d)
X. f o r z
QX = lira f~nsnx ~ f~(lim~n-lsnx).
[MacLane]) to index the structure
throughout this chapter
topology which
X. h a s t h e h o m o t o p y t y p e of f~Xi+ 1. z
the infinite loop space
To characterize
in a l g e b r a i c
loop spaces.
abelian monoids
We a l s o h a v e f o r a n y
importance
is an i n f i n i t e l o o p s p a c e if t h e r e a r e s p a c e s
X = X1 and
have the infinite Lie groups
multiplication
of c e n t r a l
but iterated
A space
LOOP SPACES
p e S
n
two further then
lr @ p
n
a homon
letters.
) In t h e c a s e w i t h
axioms: l i e s in
S
m+n
az~d i s t h e u s u a l
sum permutation; (ii)
given any
r
morphisms
a . : m . -* n. a n d z z L
~r e S , w e h a v e r
76
~r(n)o(aI ~
a z • ... • Ur ) = Ir(al @ aZ • ... ~ a r )
where
:
m
~,m.,1 n = r.n.,1 Tr p e r m u t e s
and the permutation
w(n) E S
o Tr(m),
the factors
is obtained
from
of a 1 @ a 2 @)" " " • a r , lr b y r e p l a c i n g
i by
n
a b l o c k of
n.
elements.
We require
functors
to preserve
all this
1
structure.
Example maps
15.3.
E n d x, f o r a b a s e d
X m -~ X n , w h e r e
This example
Definition
Xn
space
X.
is the n th p o w e r
Endx(m,n) of X.
is the space
T h e functor
of a l l ( b a s e d )
@) is just
X.
has permutations.
15.4.
The category
if we are given a functor
~-~
~
of o p e r a t o r s
End
acts on
X, or
X
is a~-space,
. X
A particularly
Example
A(m,n)
15.5.
A
{I, Z} -~ {I}
element,
A
m
example
encodes
the associative
law.
be the category of operators described as follows:
i s t h e s e t of a l l o r d e r If
map
Let
important
acts on
preserving
X, then
corresponds
is associative.
X
maps
admits
{1,2 .....
the structure
to a multiplication
rn.
m}--> {1,2 .....
n}.
of a m o n o i d .
The
Since
A(3, I) has only one
Since the single e l e m e n t of A(I, I) can be r e g a r d e d
as the composition of l-~ I c {1,2}-~ I or as the composition of I--~ 2 E {l,Z} -~ I, the multiplication has a unit. B e i n g a rnonoid is not a hornotopy invariant. a category action by
WA WA
to construct
of the s a m e
homotopy
is a h o m o t o p y WB
for any
Consider
B
A(m,l).
W e w i s h to replace
A
by
type (as a category of operators) so that
invariant.
Essentially the s a m e
that m a p s
nicely e n o u g h into A.
It consists of precisely one m a p
w h i c h can be r e g a r d e d as factorized in a variety of ways.
method
may
be u s e d
{I ..... n} -~ {I}
T h e s e factorizations
77
c a n be d e s c r i b e d f a i t h f u l l y b y f i n i t e d i r e c t e d p l a n a r t r e e s w i t h a s i n g l e r o o t (no v e r t e x of the f o r m
~ being p e r m i t t e d and I
b e l o n g s to o n l y one edge). unique m a p
(V)
ponding to~
The i d e a i~ c o n s t r u c t i n g and ~
WA(m, n)
~
i.e.,
or
(V) ~ ( V / )
is a c u b i c a l c e l l c o m p l e x i n w h i c h the c e l l s a r e i n d e x e d by
a I ..... ar
such that ~I .... err E A(m,n). less than the n u m b e r
(for n = I) or of the corresponding such a set a copse.
WA is to h a v e d i s t i n c t m a p s c o r r e s -
a n d to h a v e a s p e c i f i c h o m o t o p y b e t w e e n t h e m .
d i m e n s i o n equal to m
For
WA-structure
C(a I ..... ar)
n = I w e have the cell c o m p l e x of K
m
is a cell of
of edges of the corresponding tree
o r d e r e d set of trees for
Chapter II, a cubical d e c o m p o s i t i o n has a
or
d e s c r i b e s the
T h a t t h e s e a r e two f a c t o r i z a t i o n s of the s a m e m a p e x p r e s s e s the
a s s o c i a t i v e law.
sequences
F o r e x a m p l e , in A(3,1), the t r e e ~
w h i c h can be factored as ~
a (~V).
is the o n l y t r e e for w h i c h the r o o t
.
WA(m,I)
T h e category
if Definition Ii. 18 is satisfied,
n > I.
WA
Boardman
calls
described in
acts on
X
or
X
i.e. , w e have compatible m a p s
WA(n,I) )< X n-~ X.
Theorem X
15.6.
If X
if and only if W A
and
Y
acts on
have the s a m e Y.
If W A
a n d a deformation retraction of M X T h e construction of M X deformation retraction. WA-structure
onto
homotopy
acts on
type, then
X, there is a m o n o i d
acts on MX~X
X.
w a s given in Chapter 11 along with the
A s one should expect, the m a p
up to h o m o t o p 7 ,
WA
X-~ MX
respects the
at least in the sense w e n o w define.
Definition 15.7.
Let
L
be the category with two objects
Definition 15.8.
Let
X
and
Y
be W B - s p a c e s .
A map
I and
2 and one m a p
f : X -~ Y
is a h o m o t o p y
79
B-map Y.
if W ( B X L)
By
Endf (re,n}
acts on
Endf
s o a s to i n d u c e t h e g i v e n W B - a c t i o n s
we m e a n the s p a c e of c o m m u t a t i v e
Xm ~fm
example,
Theorem
I
. T so that
is a homotopy A-map
uniquely up to homotopy through a homomorphism
Corollary 15. I0. Bf : B E - ~ B y .
If f : X - ~ Y (Take
f must be an H-map.
into a monoid,
it f a c t o r s
M X ~ Y.
is a h o m o t o p y A - m a p ,
B X = BMX,
diagrams
yn
Vo.ex. V
If f : X ~ Y
and
~fn -~
W ( A × L) (2,1) = !
15.9.
X
_~ X n
ym For
on
then there exists
Bf = BMf.)
T h e real point of B o a r d m a n ' s a p p r o a c h is to be able to iterate the construction of B E
without a s s u m i n g an abelian m o n o l d structure.
will n e e d lots of homotopies. Boardman)
of a c a t e g o r y
contractible
Theorem
X B
for all
15.12.
is an E - s p a c e of o p e r a t o r s
An E-space
map being described
15.13.
if i t if g i v e n a n E - s t r u c t u r e ;
with permutations
such that
i. e . , a n a c t i o n o n B(n,1)
is
n.
The contractibility
Theorem
Rather than give all the details (we await t h e m f r o m
w e will try to outline the theory conceptually.
D e f i n i t i o n 15.11. X
Clearly w e
is a WA-space. of
B(n,1)
is u s e d to m a p
WA~
B, t h e
0-skeleton
specifically.
If X
is an E - s p a c e ,
the operations
A-maps. Thus there are induced operations
on
B X.
: X m ~ Xn
are homotopy
79
Corollar~r 15.14.
is a n E - s p a c e .
BE
T h e a i m of B o a r d r n a n ' examples
the category
{algebraic) dimension
and linear
R °° w i t h o r t h o n o r m a l
{el, e z .....
is to g e t s e v e r a l
~
of r e a l i n n e r - p r o d u c t
isometric
significant
base
finite topology, dimensional
from
A to
of c o u n t a b l e
As examples
{ e l , e2, e 3. . . }, a n d i t s s u b s p a c e
linear maps
which makes
spaces
maps between them.
e n } , w h i c h is a l l t h e r e a r e up to i s o m o r p h i s m .
t h e s e t of a l l i s o m e t r i c
Lemma
s approach
exists.
BB...BX
of E - s p a c e s . Consider
have
By iteration,
R n with base
We topologize~
B, by first giving
we
A
and
(A,B), B
the
e a c h t h e t o p o l o g i c a l d i r e c t l i m i t of i t s f i n i t e -
subspaces.
15.15.
T h e s p a c e ~ (A, R °°)
This is a consequence
(a) ~ i 2 (b) i l
is c o n t r a c t i b l e .
of t w o e a s i l y c o n s t r u c t e d
homotopies:
: A-~ A@A, u :R
°°
-~
R °o
O R °°, for s o m e
S u p p o s e w e have a functor
T
isomorphism
u.
defined on the c a t e g o r y ~
, taking topo-
logical spaces as values, and a continuous natural transformation : T A X T B -~ T { A ~) B) (a) Tf
called W h i t n e y s u m ,
such that:
is a continuous function of f E ~ (A, B);
(b) T R 0 consists of one point; (c) co p r e s e r v e s associativity, c o m m u t a t i v i t y and units;
(d)
Theorem classifying
T R °° is t h e d i r e c t l i m i t of t h e s p a c e s
15.16. space
T R °° is a n E - s p a c e . B T R °° a g r e e s
If
T
h a p p e n s to b e r n o n o i d - v a l u e d ,
with that from Theorem
As a (noncanonical) multiplication T R °° X T R ° ° ~
T R n.
on
15. lZ.
T R °° w e t a k e
T ( R ~ O IR.~ ) - ~ T R °°, Tf
the
80
where
f : R °° @ i~ °° -~ R °°
provides
is any linear
homotopy-associativity,
commutativity,
and all higher
In t h e e x a m p l e s sional
since
the maps
Tf
The Lemma
homotopies.
below we define (d)
embedding.
f o (f $ 1) ~ f " (1 • f), h o m o t o p y -
coherence
A, and note that axiom
each case
isometric
TA
extends
explicitiy
only for finite-dimen-
t h e d e f i n i t i o n t o t h e w h o l e of ~ .
and the Whitney sum
~
are obvious,
In
i n v i e w of t h e i n n e r
products.
Example
1 5 . 1 7 . 1 . T A = O (/%), t h e o r t h o g o n a l Z. T A = U ( A ®
group
C), the unitary
3. T A = B O ( A ) ,
a suitable
of
group
classifying
A.
of
Then
T R °° = O.
A®
C.
Then
space
for
O(A).
T R °° -- U. Then
TRam= B0. 4. T A
= F(A), the space of b a s e d h o m o t o p y
sphere with
Theorem
15.18.
SA, which is the one-point o0 a s b a s e p o i n t .
The following spaces
Then
equivalences of the
compactification
and maps
admit the structure
U-~ 0-~ F and
Top
of A ,
T R o° = F .
loop spaces :
The semi-simplicial spaces PL
A~)oo
c a n also be handled.
of i n f i n i t e
OPERATIONS
For order
IN I T E R A T E D
LOOP
SPACES
X to be an infinite loop space required
homotopies.
Certain
subfamilies
g i v e r i s e to h o m o l o g y o p e r a t i o n s
a r e of i n t e r e s t
of g r e a t s i g n i f i c a n c e .
a whole congery in t h e m s e l v e s Historically
t h e s e o c c u r in t h e [ K u d o a n d A r a k i ] d e f i n i t i o n of H - s q u a r i n g Steenrod
for they t h e f i r s t of
operations,
analogs
operations.
D e f i n i t i o n 16.1. maps
of h i g h e r
An H -space n
consists
of a s p a c e
X together
w i t h a f a m i l y of
0. : i i X X 2 -*" X, i < n, s/t 1
D
Oi(t I . . . . .
ti,x,y ) = Oj.l(t I .....
tj_ I ,
x,y)
t.j : 0
= 8j_l(1-t I..... l-tj_l,Y,X)
[More efficiently,
the family
=x
if y = e
=y
if x = e .
O. c a n b e r e p l a c e d 1 O
n
such that O (T,x,e) = O (T,e,x) = x n n
by switching factors
in
Thus for
where
T ~ 0 E Zz
on
commutative
H-spaces
map
acts antipodally on
is a h o m o t o p y c o m m u t a t i v e
are H
X
-spaces
H-space.
Notice the
n e e d not be a loop space.
Corollary
16.2.
If X
16.3.
If X
Proof~of Theorem. notation,
i.e.,
is a n H n - s p a c e ,
is
~ny,
Represent
then
Sn
a p o i n t is r e p r e s e n t e d
~X
X
as
is a n H n + l - s p a c e .
is a n H n . l - s p a c e .
Z 2 * Z 2 '~ . . . as
t0a 0 •
For
e v e n if n o t h o m o t o p y a s s o c i a t i v e .
cO
Theorem
S n,
X. ]
condition b e a r s no relation to associativity so
example,
by an equivariant
: Sn × X z -"X
X 2 and trivially
n > 0 X
t.j = I
;~ Z 2 w i t h M i l n o r ' s
. . . {~ t a n
n
where
of
82
a i• Z z, (to ..... tn) • A n.
Define
O n + l ( t 0 a 0 • . . . • t n a n • t n + l l ; k 1, kZ) (t) to b e tn o to ~n+l (t)) whe r e k ~ ( t ) = k z ( t - (l-s )(r2)) e n ( l _ - ~ n +1 0 0 e . . . e n l _ t n +1 ; k l ( t ) ' k
w i t h the u n d e r s t a n d i n g
kz(t) = kZ(0)
for
t < 0
= k2(r Z) f o r
Extend by equivariance. an H-space
For
t >r z .
n = 0, this is a standard proof that the loop space of
is homotopy commutative.
Definition 16.4.
Given an H -space
(X, O ), the K u d o - A r a k i
n
Qi : I_Iq(X;Z2) -* H N + i (X;Z z) is d e f i n e d f o r where
square
n
i-n < q _< i by oi([u]) = Or~ (ei_q®U®U)
e. is an i-dimensional cell in the standard equivariant decomposition of I
S n" Before going further w e introduce the rood p analogs [Dyer and Lashof ].
Defintion 16.5.
An Hn-space P
X consists
o n : ~(p)*.
s/t on(~,e ..... e,x,e ..... e) = x
Theorem
Proof.
l6. 6.
If X
We p r o c e e d
tn no
x xP-
map
x
and on(o • ... $ O @
v e r y m u c h a s in t h e c a s e
l'l;xI..... Xn) = x l...x n.
p = 2..
If k i : [0, r i ] - ~ X, d e f i n e
f o r On+ 1 t h e n b e c o m e s to t @ s.id; k I..... k p ) (t) = SniFfs a0~).., l@~.~an;kl(t) ..... kS(t)). p
Now consider TJe.,, 0_< j < p
.. *J(p~
X and an equivariant
is H n, f~X is ~I-F "+I. P P
k s (t) = k i ( t - ( l - s ) ( r 1 + . . . + r i . 1 ) ) .
en+l(t0a 0 @ . . . ®
of a m o n o i d
such that
The formula
W , the standard P 8eZi+l = (T-1)ezi
resolution
of
7.
@ezi -- (1 + T +
P
with generators + T p-l) eZi_l.
83
The i n c l u s i o n "n-skeleton"
Z -~ ~ ( p ) as c y c l i c p e r m u t a t i o n s i n d u c e s c o n s i s t e n t m a p s of the P n of W into C ( ~ ( p ) * . . . * ~ ( p ) ) . Thus we think of 0 . a c t i n g o n P
W ® C (X) @p. P
Definition 16.7.
where
~tO,q)
For
(2j-q)(p-1) <_n and x E H (X), define q OJ(x) = ~ / ( j , q ) ( ~ ) l 8 n, (e(2j_q)(p_l) @ x ® . . .
@ x)
p-12
is the s i g n of q + [j + ( Z - ~ ]
The o p e r a t i o n s defined have the following p r o p e r t i e s ,
s o m e of w h i c h
d e p e n d on this p r e c i s e c h o i c e of ~ (j, q). QJ : Hq(X;Zp)-~
Hq+zj(p_l)(XiZp)
QJ is a h o m o m o r p h i s m
if
for j _> q / Z
and
(3j-q)(p-l)_< n
(gj-q)(p-l) < n-I
QJ is natural with respect to H n - m a p s P QI
is trivial if q = 0
Q q / Z l x ) = xP
if q is e v e n
a . Q j = QJ~, w h e n b o t h a r e d e f i n e d and a , : Hq_I(~X) -~ Hq(X) is the h o m o l o g y " s u s p e n s i o n " . J u s t as S t e e n r o d o p e r a t i o n s give a n i c e d e s c r i p t i o n of H ( Z , n ; Z ) P t h e s e o p e r a t i o n s give a n i c e d e s c r i p t i o n o f i i denote by QI the c o m p o s i t i o n Q 1 . . . Q r. if<2i2-- .... 'ir-i
excess,
is defined on is < q+n.
r
H
of an H n - s p a c e
if the
If e(I) = i = q, then QI
--
r
zr-th power.
T h e o r e m 16.8. ~Kudo and Araki].
For
h a v i n g as g e n e r a t o r s all a d m i s s i b l e Similarly for
k > 0, H , ( ~ n s n + k ; z Z) is a p o l y n o m i a l r i n g
Q~uk of e x c e s s
1.
> k and l a s t d e g r e e
p > 2, we c o n s i d e r s e q u e n c e s £
1
lr), we
q
--
with e. = 0 or
I = (i 1. . . . .
We s a y QI is a d m i s s i b l e if
r
excess of I is > q and the last degree i raises to the
For
e(I) = il - Zi2 + i Z - Zi3 + ... + i r = 2iI- ~i"3
e(I) < i . Notice that QI --
H . (~nsn+k).
so
We let QI denote
S
¢'_
~ 1Q 16 ~ . . . ~
£
I = (e 1, s 1. . . . . S
kQ k, w h e r e
< n+k.
e k ' Sk)
84
: Hq (X) -~ Hq_l(X) 0-~ Z p - ~ Z p 2 - ~ Z the r e l a t i o n
is the B o c k s t e i n b o u n d a r y f o r the s e q u e n c e -~ 0.
[For
~ = 2, ~ n e e d not be u s e d e x p l i c i t l y s i n c e we have
~Qgi+l = oZi. ] [ D y e r and L a s h o f ] d e s c r i b e g e n e r a t o r s f o r
H , {"Ansn+k ~ ; Z p ) in t e r m s of a l l o w a b l e s e q u e n c e s .
May has translated" allowable"
into the following.
k Definition 16.8.
QI is m - a d m i s s i b l e
if PSi+ 1 " ¢ i+l >-- s i > [ m + >
Zsj(p-l)-¢j]l 2.
i+l The following two t h e o r e m s a r e then t r a n s l a t i o n s of t h o s e g i v e n by D y e r and L a s h o f .
T h e o r e m 16. 9.
F o r any
connected
X we have that H.(l~m 6 ~ n s n x ; z ) is f r e e P commutative on {x, Olx I x , basis of H,(X;Zp), Ol is d i m x-admissible}. N o w let O X
denote the base point component of lirn onsnx.
T h e o r e m 16.10.
H , ( Q S 0 ; Z ) is f r e e c o m m u t a t i v e on g e n e r a t o r s y(I) of d i m e n s i o n P e q u a l to deg I w h e r e I r u n s o v e r all 0 - a d m i s s i b l e s e q u e n c e s . [ F o r p = Z, k I = (s 1. . . . . Sk) is 0 - a d m i s s i b l e if 2si+ 1 > s i > ~ ' - - sj. ] The D y e r - L a s h o f o p e r a t i o n s b e h a v e as f o l l o w s : J = (¢l, Sl . . . . .
If
i+l
~k_l, Sk_l), then Q J ~ k y ( s k )
Y(S k) e HSk(QS0;Z z) and if J = (s 1. . . . . The g e n e r a t o r QI
=y(£1, Sl . . . . .
Sk_l)
then
Ck, Sk).
[For
Q J y ( s k) = y(s 1. . . . .
y(I) c a n not be i n t e r p r e t e d as
p = Z, Sk).]
QI(x) f o r a n y x s i n c e
is t r i v i a l on H0(QS 0) a l t h o u g h an i n t e r p r e t a t i o n in t e r m s of QI is p o s s i b l e
if we c o n s i d e r all the c o m p o n e n t s of
lira 0nsn.
Kudo and Araki prove their result by mimicing H*(Z, n;Zz). Theorem
for
The crucial machine is a dual t o ~ o r e ~ s transgression theorem.
16. U.
If H,(X;Z2)
Serre
Let X
be a simply connected, homotopy associative
has a simple system of transgressive generators
x. then I-I,(~X;Z 2) 1
is a polynomial ring on generators
Yi such that Yi £ Tx..1
H-space.
85
Dyer and Lashof proceed somewhat differently. H.(~ksix;Zp)
in the range < 3i- Zk which determines
They compute
H~(lim..f~nsn+kx;Zp).
They do the computation by analyzing the homology as a tensor product of monogenic Hopf algebras, mapping the corresponding tensor product of model spectral sequences into the Serre spectral sequence for ~ k + I s i x - J ~ k s i x - ~ ~ksix
and applying the comparison theorem.
Finally, they identify the various
classes at hand in terms of the operations QJ. ~nsn
The final stage from
~n-Isn
to
requires s o m e special effort. N o w let us return to a space of m o r e geometric interest.
The space
Q S 0 has the s a m e homotopy type as S F = lira SH(Sn), S denoting m a p s of degree I, but the equivalence is not one of infinite loop spaces. multiplications are not equivalent.
In fact even the
A major accomplishment of the last few years
has been the determination of the Hopf algebra H $ ~ ; Z z) and the algebra # H ( B F ; Z ) for p >_.Z. The operations Qi have played an important role in this P development. That H * 0BSF) was not the s a m e as H *(BQS 0) has tong been k n o w n for H ( B S F ; Z 2 ) D Z 2 [w i I i [ 2] w h e r e H~'(BQS0;Zz)
{wi}
is a n e x t e r i o r a l g e b r a .
a r e the S t i e f e l - W h i t n e y c l a s s e s .
While
The c o m p l e t e r e s u l t s c a n n o w b e s t a t e d ,
a l t h o u g h the p r o o f s a r e s o i n v o l v e d a l g e b r a i c a l l y a s to b e i n a p p r o p r i a t e f o r p r e sentation here. Let
o denote composition,
l o o p a d d i t i o n in QS °.
the m u l t i p l i c a t i o n in S F
We h a v e c o r r e s p o n d i n g
operations
t h e f i l t r a t i o n of H~ (SF) b y p o w e r s of t h e a r g u m e n t a t i o n
and l e t
Q~ and Q I. i d e a l and l e t
# denote Consider
E"
denote
the associated graded.
T h e o r e m 16.12. [ M i l g r a m ] . of H~(QS °)
For
p = 2, l e t
o r the i s o m o r p h i c c l a s s e s
I Q,~y(k) b e the D y e r - L a s h o f
in H ~ ( S F ) .
generators
86
y(k)
o y(k) ~ 0 ,
y~)o
y(k) o y e ) .
y(k) : 0
Q y(k) ° Q,y(k) = 0 in E ° (y(k) * y(k)) ° (y(k) * y(k)) : 0
Corollar~" 16.13.
H
(BSF;Z2) ~. Zz[Wi] Q C
E°
in
where
C
is isomorphic to
E(ezi+t Ii > I) ® r(g([)i I is l-admissible of length > 1). Here
eZi+l is dual to sly(i) • y(i)] and g([) is dual to ~y(1). The spectral sequence f r o m
ExtH~(SF)
to H
(BSF) has no choice but
to collapse since, being a spectral sequence of Hopf algebras, only primitive relations can be added.
This m e a n s only 2i-th powers of primitive classes could
be killed, but the only nontrivial ones present in E 2 are in ZZ[wi] which w e k n o w survives untouched. For
p > Z, there is a striking difference which is really a subtle
s i m i l a r ity.
T h e o r e m 16.14. [May].
For
T h e o r e m 16.15. [May].
In the E i ! e n b e r g - M o o r e
E2 ~ EXtH ( S F ; Z ) p If I = (1, j , J )
we h a v e
is 0 - a d m i s s i b l e ,
dp_l[Y(J) I . . . I
p > 2, H , ( S F ; Z
E 2 ~ E p . 1 with
P
)~--H(QS0;Z
P
) as Hopf a l g e b r a s .
spectral sequence for
BSF
with
dp_ t g i v e n as follows:
J odd of d e g r e e Zj-1 and l e n g t h > 1, then
Y(J)] =y(I)-
J.
C o r o l l a r y , 16.16. 1) 2)
H (BSF;Zp) ~ Zp[qi ] O E(~q i) O E O F w h e r e
{qi } a r e the
Wu c l a s s e s
E is a n e x t e r i o r a l g e b r a on p r i m i t i v e g e n e r a t o r s
e. dual to 1
~y(1, Z(p-1) 1, i) in H Z P i ( p - 1 ) - I ( B s F ) a n d
g(I) dual to ~y(I) w h e r e
I r u n s o v e r all 1 - a d m i s s i b l e s e q u e n c e s of e v e n d e g r e e and length
>
1.
87
3)
1~ is a divided polynomial algebra on primitive generators
~e. i
and g(J) where
J runs over all 1-admissible sequences of odd
degree and length> I. [The W u
class qi is dual to ~(l,i) and
~qi to a(0, i). ] Milgram' s and May' s proofs are rather unusual exercises in manipulating Hopf algebras over the Steenrod algebra or its dual. the C a f t a n f o r m u l a e and A d e m r e l a t i o n s for the t h e [ N i s h i d a ] r e l a t i o n s b e t w e e n the operations. structures
Theorem
Qi
Qi
It is important to have
[ D y e r - L a s h o f ] and e s p e c i a l l y
a n d t h e h o m o l o g y d u a l s of t h e S t e e n r o d
An old f a c t in h o m o t o p y t h e o r y c r u c i a l in r e l a t i n g the v a r i o u s a l g e b r a i c _ S n-l, •
involved. 16.17.
(~nsnx, e) is a module over
Corollary 16.18.
He{~nsnx)
Theorem
(~nsnx, e) is a module over
(sn-l,e
is a Hope algebra over
,°).
H e(s n'l,eS
n-I • , ).
S n-l, • 16.17.
(Sn-l,e
, .).
S n-l, Corollary 16.18.
I-Ie {~nsnx)
is a Hopf algebra over
T h e t h e o r e m is a r e s t a t e m e n t
(leg) . ~.a = f • r.c~eg . r.~. ~nsnx
is commutative, corresponding
e).
of the r i g h t d i s t r i b u t i v i t y of c o m p o s i t i o n
if f , g : S n -~ s n x
over track addition, i.e.,
He (Sn'l, •
and
a : S n ' l -~ S n-1
D i a g r a m a t i c a U y we h a v e w i t h
X ~ n s n X × F(n-1) -~ ~ n s n x
then
F ( n ) = Sn , e S n ' e , t h a t
X ~ n ~ n x X F(n)
~ n s n x ~'~F(n)
~ n S n "X - -X ~ n S n~X X F(n) X F(n)
~-~ n x
~'~nsnx X F (n) X ~ n s n x X F (n)
~
at least up to homotopy.
The diagram helps in describing the
c o n d i t i o n in h o m o l o g y .
T h e h i g h e r o r d e r p h e n o m e n a i n v o l v e d in h a n d l i n g t h e o p e r a t i o n s n e c e s sitate studying this distributive t e r n a t e d e s c r i p t i o n s of H n
or
l a w up to h i g h e r h o m o t o p i e s . Hn-structures P
on BO
and
BF
There are also al[Boardman,
T s u c h i y a , M i l g r a m ] w h i c h s h o u l d g i v e the s a m e h o m o l o g y o p e r a t i o n s b u t at the
88
m o m e n t a r e n o t k n o w n to do s o . characterizeable problems,
FinaUT, infinite loop spaces should be
in t e r m s of t h e m a p s J ( n )
*...*
~ ( n ) X X n - ~ X.
In a l l t h e s e
we a r e f a c e d w i t h a n a l T z i n g a f a m i l i a r a l g e b r a i c s t r u c t u r e f r o m a
h o m o t o p 7 p o i n t of v i e w , b u t p e r h a p s t h e s p i r i t of t h a t p o i n t of v i e w is by n o w sufficientl 7 clear.
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