Pub
.
Ma
.
UAB
Vol
.
26
Nó
3
Des
.
1982
1
.
INTRODUCTION
COHOMOLOGYOPERATIONS
AND
H-SPACES
A
.
Zab odsky
The
heo y
o
cohomology
ope a ions
and he heo y
o
H-spaces
we e
in e locked
h cuqhou
hei a ious
s ages
o
de elopmen
:
The
. i s
sys ema icapp oach
o
he
heo y
o
(high
o de )
cohomology
.
ope a ions
is
due
o
J
.
F
.
Adams
([Adams])
.
In
ha celab a ed
pape
a
solu ion
was
gi en
o
a
ques ionwhose
one
o mula ion
is
he
ollowing
:
Wha
sphe es
suppo
con inuous
mul iplica ions
wi h
uni s
(i
.e
.
H-s uc u es)?
The
cohomologyope a ions
o
he
ope a ions
.
These
we e
ied oge he
Bocks ein
spec al
sequence
which
was
dimensional
H-spaces
.
[Zab odsky]
1,2,3'
[
Kane
]1
,
[Li ]1,2,3
and
o he sused
high
o de
ope a ions
o
. a he
analyze
he
cohomology
o ini eH-spaces
.
In
pa icula ,
[Lin]1,2p o ed
he
classical
"loop
space
congec u e"
:
The
homologyo
he
loop
space
o
a
ini e
dimensional
H-space
is
o sion
ee
.
[Hubbuck]
1
~
2,3
used k- heo yope a ions
o
s udy
he
cohomology
and
opologyo ini e H-spaces
.
He
ound
es ie ions
on
hei possible
ypes
and
hei
Pon iagin
ings
.
Amongo he
heo ems
he
p o ed
([Hubbuck]
1 )
ha
a
simples ype
a e
he
Bocks ein
by
B owde
([B owde ]
1
2
,3
)
o
o m
he
used
o
s udy
he
cohomology
o ini e
215
homo opycommu a i e
ini e
H-space
has he
homo opy
ype
o a
o us
.
Finally
[Kane12,3
ecen ly
used
BP
ope a ions o
s udy
he
cohomology
o
H-spaces
.
Going
in
he
o he
di ec ion,
he
heo y o H-spaces
was
used
in he
cbns uc ións
and-e alua ions
o high
o de ope a ions
.
in
he
ollowing
lec u es
I
shall
y
o
demons a e
by
some
examples
hese
ela ions
be ween
he
wo
heo ies
.
2
.
BASIC
DEFINITIONS
Wle
usually
assume spaces
o be
o
. he
homo opy
ype
o
CW
complexes
wi h
a
(non-degene a e)
base
poin
.
Maps
and
homo opies
a e
base
poin
p ese -
ing
.
Thus,
an
H-space
could
be
assumed
o
be
a
space
X
wi h
a
mul iplica ion
so
ha
he
base
poin
*xo
is
an
ac ual'uni
:
u(x,x
0
)
=
x =
u(xo,x)
.
The
de ini iono
a
cohomology
ope a ion
has
a ious
deg ees
o
abs ac ions
. .
One
o
he
mos
gene al
o m
is
he
ollowing
:
A
cohomology
o
e a ion
¿
consis so
h ee
.spaces
and wo
maps
_
<K
o
,
E, K
1 ,
, h>
:
:
E
-
KoY
h
:
E-
"
K
1
:
de ines
a
"na u al
ans o ma ion"
om
im([
,
E]
-
"[
,
Ko
])
o he
amily o
subse s
o
[
,
K
1
]
.
In a
mo e di ec
e ms
:
Fo
any
space
Xm
de ines
a
unc ion
om
a
subse o
he se
[X, Ko
]
o
homo opy
classes
o
maps
X
-
"K
o o
he se
o
subse s
o
[X, K1]
in
he
ollowingway
:
The
domain o
0
is
he se
.im(
.,
:
[X, E]
->
[X, Ko
])
whe e
* is
he
le
composi ion
wi h

:
*
([ ]
=
[
. ]

([u]

he
homo opy
class
o

u)
.

Hence,
[ ]
E
[(,,K0]
.
is in
he
domain o
o
i
and
only
i
[ ]
"li s"
o
[ ]
.
E [X, E],
o
i'
.
(see
diag am
D1)
.

The
alue
~D([ ])

is
hen
he se
{[h o ]1 o % }
c
[X,
K
1
]
(Dl)
In
case
E = Ko
,
= 1
he
ope a ion
is
called
ima
and
is
simply
he
igh
composi ion
wi h
h
.
The
domain o
o
is
hen
all
o [X,K
0
]
and
i s alues
a e
single ons,
i
.e
.
:
elemen s
o
[X, K
1
]
.
This
is a
gene al
o mula ion
which
is
no
e y
use ul
i
one
does
no
es ic
onesel
.
o
some
speciaL
cases
.

No mallywe
conside cohomology
ope a ions
eTa ed
o
.(gene alized)
cohomology heo ies
(hence he
name)
.
Al]
cohomologyope a ions
he e will
be
gi en
in
e ms
o
si-spec a
:
An
.

9
.

spec um
is a
sequence
.
.E
*
=
{E
n
,
`pn}n=o

whe e

E n

a e spaces and
a e
homo opy
equi alences,
(p
n
:
En
a

n+l'
The
cohomology
heo y
E*
associa ed
wi h
he
sa-spec um
E
is
a
.
L,
.,

sequence

oi

u

. .
nco

s
a
. .
. .
. .

l
.
L
n
,
I

_

L

,

E
n
l
J

a
1 .
.
.
a
he

-

~

wia

i s
:

Fo
á

space
X - E
n(X) = [X,
E '
.
Fo
a
.map
:
X
-
"
Y
E
n
( )
:
E
n
(Y)
-
E
n
(X)
is
he
igh
composi ion
wi li
:
E
n
( )[u]
= .[u
o ]
(u
:
Y
-
E
n
) .
As E
n
a e
double loop
spaces
(and
much
mo e)
.
E
n
(X)
a e
.abelian
g oups and
E
n
( )
a e
homomo phisms
.
3
.
PRIMARY
OPERATIONS
.
STABLE OPERATIONS
An
elemen a y
p ima y
ope a ion
o
ype
m,n in
he
cohomology
heo y
E*
is
an
elemen

[a]
E

E
n
-
E
n
.

I
de ines
a
p ima y
ope a ion
m
=
<Ko
=
E
m = E, K
1
= En
,
= l,
h
=
a>
which
is
ob iously
he
le
compesi -
ion
wi h
a
.
The se
o
all
p ima y
ope a ionso
ype
m,n
is
he se
[E m
,
E
n
]
.

As an
ope a ion
a
is
a
unc ion
E
m
(X)
-
.
En(X)
.
A
s able
elemen a y
p ima y
ope a ion
o deg ee
k
is
a
sequence
a
=
{a
n
E
[
E
n'

En+k]ln=o

( o
k
<
o we
conside
E =
poin
o
<
o)
.

These
a e
ela ed
by
he
ollowing
(homo opy)
commu a i e
diag am
:
i

a
n
En
En+k
(D2)

(D
nl

1
O
n+k
Sia
.n+l
í2
E
n+l

--,
í?
E
n+l+k
In
his case
a
n
:
E
n
(X)
-
.
E
n+k
(X)
a e
homomo phisms
.
The se
o s able
cohomology
ope a ions
in
he
heo y
E

o ms
a
g aded
. ing
:

One can
add
any
wo
ope a ions
o
he
same
deg ee
as

[E
n
,
E
n
+
is an
abelian

g oup
.

The
p oduc

is
gi en
by
:

(n"
"
a')
n
=
a~
+k
o
a~
i
a'
is o
deg ee
k
.
The
deg ee o
a"
"
a' is he
sum
o
he
deg ees
o
a'

and
a"
.

These
de ini ions
a e
consis en
wi h
he
de ining
ela ions
o
a
s able
ope a ion
(D2)
.
Exam
le
:

The
S een od
Algeb a
.

Le

E
n
=
K(Z/pZ,n)
-
he
Eilenbe g
MacLanespaces,
p-a
p ime,(E,,
is
hen called
he
Eilenbe g
MacLane
spec um
K(Z/pZ))
.
The
ing o
elemen a y
s able
cohomology
ope a ions
is
called
he
S een od
algeb a
a(p)
.
Fo
p =
2
a(2)
is
gene a ed
by
ope a ions
Sq~
o
deg ee
i

(Sq
o
.=
1)
subjec
o
ela ions
known
as
he
Adem
ela ions
.
Re e ence
:
[S een od-Eps ein]
.
219

A
non
elemen a y
p ima y
cohomology
ope a ion
in E

is
a
map
a
:
E(0)
-
E(1)
whe e
E(i)
=
n
E(')
i
= 0,1
.
One
can
easily
see how
o
de ine
a
non
elemen a y
s able
p ima y
ope a ion
.
Such
an
ope a ion
is
gi en
by
a
ma ix
whose
en ies
(a
ij
)
a e
elemen a y
s able
ope a ions
wi h
he
p ope y
:
deg ee
a
.

-
deg ee
a
.

.
is
independen
o
j
.
*
Fix he
cohomology
heo y
E
.
By
E(i)
we
always
deno e
a
p oduc
o
e ms

ni
_Ek~
)
.
(E(i,j)
will
deno e
o he
ypes
o
spaces
as
will be seen
j =
1
j
in
he
sequel)
.
Gi en
(non
elemen a y
and
no
necessa ily
s able)
ope a ions
01
0
:
E(0)
~
E(1),
a
l :
E(1)
-
"
É(2)
.
"
A
ela ion
among
p ima y
ope a ions
is
a
ela ion
o
he
ype
a
1
,
no
i
*
.
(*- he cons an
ma
P)
"
(I a
l
a e
s able,
and
he e o e
gi en
by
ma ices,
his
ela ion
desc ibes
o dina y
ela ions
in
he
ing o
s able
ope a ions)
.
The
ela ion
a
l
o
no
%
*
induces
a
commu a i e
diag am
:
4
.
SECONDARY
OPERATIONS
ASSOCIATED
WITH
A
RELATION
n
E(2)
whe e
E(i,i+1)
is' he homo opy
ibe
o a
i
,
i
=0,1
.
j1
is
he
inclusion
o
he
ibe
o
1 . 01
0,1
exis s
since
a
l
ono % *
.
ao,1,2
is
induced
by
el
oj
.

no,1

and

a
o,1,2

a e
uniquelyde e mined
by
he
choice o
he
homo opy
E(0)
x
I
+
E(2),
*
i
a
l
o
01
0
.
The
ope a ion
o
=< E(0),
E(0,1),
n E
(2),
0, n
o,1,2
>

is
called
a
seconda y
ope a ion
associa edwi h
he
ela ion
The
abo e
ope a ion

o
depends
on
he
choice
o
a
o,l
,
o as
ema ked,
on
he
choice o
he
homo opy
*
i
a
l
o
no
.
The
di e ence
be ween
choices
o such homo opies
is
gi en
by
a
map

w
:
E(0)
--

2
E(2)
.

The
di e ence
be ween
he wo
maps
ao,l
.2
and
ao,l,2
induced
by
he
wo
choices
o
homo-
opies
is
hen
gi en
by
ao,l,2
-a
o,l,2
=
w
,
o
.
Gi en
a
space
X
and
a
cohomologyclass

x E [X,
E(0)]
.
(x is
n
.
ac ually
a
" ec o "
o
cohomology
classes
x
n
.
(0)E
E
J(X))
.
x is in
he
do-
J
_
main
o
o
( o
any
o,
induced
by
any
null
homo opy
* La
l
.
a
o
)
i
and
only
i

a
ox
=
0
.

The

alue

o(x)

is

hen

[a
o,1,2

°
x]

whe e

x

:

X

,
E(0,1)
is a
"li ing"
o x
:
X
-
E(0),
o
o
x
i
x
.

l a', <D"
co espond
o
wo
di e en
homo opies
*
i
a
l
,
ao
whose
di e ence,
as
abo e,
is
w
:
E(0)
,
9
E(2)

hen

[ao,l,2
o
x]
-
[ao~,]
,
20
X]
=
[w
o
o
o
X]
=
Lw
o
x]
.
Hence,
D"(x)
is
ob ained
by
ansla ing
<D'(x)
by
w
.
x
whe e
w
E [E
o
,
2
E(2)]
is a
p ima y
ope a ion
.
This
could
be
o mula ed
as
ollows
:
A
ela ion

a
lo
a
o
i
*
among
p ima y
ope a ions
induces
seconda y
ope a ions
{m}
.
Any
wo
.such
ope a ions
di e
by a
p imá y
ope a ion
.
5
.
MASSEY PRODUCTS
-
TODA BRACKETS
.
HIGH
ORDER
OPERATIONS
Le
a
o
:
E(0)
--
E(1),
a
l
:
E(1)

E(2),
a2
:
E(2)
y
E(3)
be
p ima y
ope a ions
and
suppose
a
l
o
ao
i
*, a
2 o
a
l
i
*
.
Again,
E(i)
a e
p oduc s
ni
Eñ
e)
.
Ex end
diag am
(D3)
o ob ain
:
J=1
E(i,i+1)
-
he
homo opy
ibe
o
a
i ,
3
i
-
he
inclusion
o
he
ib e
.
ao,l,
a
l,2'
a
l,2,3'
ao,1,2
exis
as a
l
o
ao
a
2
o
a
l
*
.
They
a e
uniquely
de e mined
by
choices
o
`
homo opies
*
i
a
l
o
a
o
,
*
i
a
2 a
a
l
.
The
class
[(1
1,2,3
-
ao,l
]
E
[E(0),
2
E(3)]
is
a
p ima y
ope a ion
.
Two
di e en
choices
o
he
homo opy
*
i
a
l
oao will
yield' wo
maps ao
,l
,
a~
,l
.
These
maps
a e
ela ed
by [a
l,2,3
-
a
o,l]
-[x
1,2,3

[í2a
2
°
wo]'
whe e
w
o
E
[E(0),
o
E(2)]
measu es
he
di e ence
be ween
he
wo
choices
o
homo opies

*
i
a
l

o
ao
.

(No e
ha
he
di e ence

(1
1,2,3
0
a
o,1

-
a
]

2,3
°
ao,l
is
independen
o
he
choice o
he
homo opy

*
i
a2
a a
l
and
i s
induced
map
(11,2,3')
(R)

implies

[a
l

o n
o
] =
Sq
4n+1
Sq
1

+ Sq2Sq4n =
Sq4n+2
=
0

( he
la e
anishes
by
he non
s abili ycondi ion

Sg4n+2[1
E(o)
] =
0

as
1
E(o)
:

E(0)
->
E(0)

is an
elemen
o

H
4n+l
(E(0),

Z/2Z))
.
We
shall
in es iga e
he
alue
o
a
seconda yope a ion
i
associa ed
wi h
*
i
a
l
~
no
on
a
p imi i e

class

x
E
H
4n+l
(X,
Z/2Z)
in
he
domain
o
(We
shall
conclude ha
he e
is no
such
class
and
he e o e
Sg
4n
x ~ 0
o
.any
p imi i e
elemen
o
dimension
4n+l
.)
H
de ia ions
:
Le
X,p,
Y,
_
p
'
be
H-space
.
Gi en
a
map
:
X
}
Y
he e
exis s
a
map
D
:
X n X
-
.
Y
called
he
H-de ia ion
o
wi h
he
ollowing
p ope íes
.
(Compa e
wi h
[Zab odsky]
4
,
Chap e
1
whe e
D is
deno ed
by H
D( ,
, p'))
:
i)

The wo
maps
X

X:Y
gi en
by
x,
y
-
(x
.y)
and
x,
y
- D
(x,y)
"
[ (x)
"
(y)]
a e
homo opic
(he e
( ) "
(
)
deno es
bo h
p oduc s
u
and
u') o in a
unc ional
no a ion
:
o
u
i
po{D n
x
[u
.( - )]1
-,
áxxx
whe e
á
X-X
(x,y)
`
(x,y,
x,y)
is
he
.diagonal
map
and
n
:
.X
x
.X
y
X
n
X -
he
p ojec ion
.
ii)
is
an
H-map
i
and
only
i D
i
*
.
iii)
Le
X
o
"
X
i
be
H-spaces
and
X2
-
a
loop
space
.
Gi en
maps
0
:
X
o
-
X
1 ,
:
X
1
~
X
2
.
Suppose
D
lo
(D 0
Al)
:
X0 n
.
X
0
n X
l
~X2
is
null

homo opic
.

Then

[D
1
o
0
]

=
[D
1o
( 0
n
o
)]
+
[
l
,D
]
.
0
In
pa icula
:
I
l
is an
H-map
D a
%
l
o D
and
i
1
0

0
o
is
an
H-map
D
,
i
D
,
( on o)
.
1
0
1

Now
conside
again
he
ope a ion
1
associa ed
wi h
(p)
and
he
ollowingcommu a i e
diag ams
:
K(Z/2Z,
8n_+3)
Bj
(D7)

B
E(1,2)
B

1
B 1

g
K(Z/2Z, 4n+2)
-
o-,
K(Z/2Z,
4n+3)
x
K(Z/2Z,
Sn+2)
--
1
-
K(Z/2Z,
8n
;l4)
Bao
is
gi en
by
p
l
°
Bao
=
Sq
l :
K(Z/2Z,
4n+2)
-
K(Z/2Z,
4n+3),
p2
°
Bao
=
Sg4n
:
K(Z/2Z, 4n+2)
-
.
K(Z/2Z,
8n+2)
.
Ba
1
=
Sg4nó1p,

+
Sq
2
°

p2
,

hence

2
Ba
o
%
CC
o
,

s2
Ba
l
 i
a
l .

E -

he
homo opy
ibe
o
S
g
4
n+
2
,
B
E(1,2)
-
he
homo opy
ibe
o Ba
l
,
sz
B
E(1,2)
=
E(1,2)
as in
he
(D4)
diag am
de ining
m
.
Loop
he
abo e
diag am
and
obse e
ha
9
E
_
K(Z/2Z, 4n+1)
x
K(Z/2Z,
8n+2)
and
he e o e
=s2 B
:
9
É

K(Z/2Z, 4n+1)
admi s
a
le
in e se
x
:
K(Z/2Z,
4n+1)
°
x
i
1
.
Oné
can see
ha
he
choices
o such
in e ses
(also
called
c oss
sec ions)
a e
in
1-1
co espondence
wi h
li ings
ao,l
:
E(0)
=
K(Z/2Z,
4n+1)
-
E(1,2)
o
diag am
04
o
D
.
Thus,
looping
(D7)
one
ob ains
:
(D8)
RBj=j
Sa
É
SZ
&
1 112
E(O)=
K(Z/2Z, 4n+1)
1
;
1
E(1,2)
1
.
1
E(1)
I 1
(any
choice!)
is
an
H-map
one can use
some
obs uc ion
heo y
o
show ha hen
x
is
indeed
a
loop
map
.
Hence,
B
admi s
a
c oss
sec ion
B2,

B
,
Bl

1
BE(o)'

This
will

imply
ha

Sg4n+2
:

K(Z/2Z, 4n+2)
--
K(Z/2Z, 8n+4)
is null
homo opic
which
is
alse
.
I
ollows
ha
x
is
no
an
H-map
and
D
:
X
E(0)
A
E(0)
-"
2E
is
no
null
homo opic
.
Now,
[E(0)
n
E(0)
=
K(Z/2Z, 4n+1)
A
K(Z/2Z,
4n+1),
2
E
a
K(Z/2Z,
4n+1)%K(Z/2Z,
8n+2)]
a
H
4n+1
(K(Z/2Z,
4n+1)
A
K(Z/2Z,
4n+1),
Z/2Z)
+
H
8n+2
(K(Z/2Z,
4n+1)
A
K(Z/2Z,
4n+1),
Z/2Z)
.
The
i s
summand
is
ze o
(E(0)
A
E(0)
is 8n+1
connec ed)
he
second
equals
Z/2Z
.
.
Hence,
he
only
non
i ial
map
in

[E(0)
A
E(0),
2
E]

is
gi en
by
w
K(Z/2Z, 4n+1)
A
K(Z/2Z, 4n+1)

-
K(Z/2Z, 8n+2)
-~-,
n
E
and
DX

wd
(w
d.is
also
deno ed
by
'4n+1
(x)
`4n+1)'

By
he
p ope ies
o
H-de ia ions
[p ope y
(iii)]
[Da

]_
[s2
a
o
D
ic ]
,
_
En
a
o J o
w
o ] =
[J
l
owo
]
0,1
And
again,
his
is
ue
o
any
choice
o

a
o
~ l .
Now
conside
he
(D4)
diag am
o
D
and i s
e alua ion
on
a
p imi i e
class

x E H4n+1 (X,
Z/2Z),
xE
ke
Sg4n
,

(x
E
ke
-
Sg
l
by
ema k
a))
.
(D9)
P
E(1)

s
~-+
2
E(2)
CL
o,1,2
~1
Sg4n
x=0,
Sg
l
x=0
implies
a
0
.
x
i
*
and
x

exis s
.
Now,
x, 0
.
a e
H-maps
hence
:
0

=

ED
x]

=

ID
.

0
X]

=

E 0

0

DX],

hence,

.
0
DX
:
X
,E0,1
li s
o
a
map
w
:
X n X
- .
9
E(1),
DX
i
jo
0
w
.
Ej
l

0

Da
]

=

ID
.

a

]

=

IDa

0
(
0

A

0)]

=

Ej
1

0

w
0

.(
o

n

o
]
.
0,1
,2

~l°
0,1
.2

0,1
Now, E(0)
n
E(0)
is
8n+l
connec ed,
n
i
(sz
E(1))=
0
o
i
hence
EE(0)
n
E(0),
si
E(1)]
=
0
and
consequen ly
j
l*
:
EE(0)
n
E(0),
sZ
E(2)]
-
.
EE(0)
n
E(0),
E(1,2)]
is
injec i e
and
0
(
n )
.
e¿
o
.1
.2

w
0

0

0
Now,
Da

0
(Dx
A
1)
,,
w
o0( 0
^
0
)0(Dx
n
1)
_
o,1,2
On
he
o he
hand,
conside

w
'
E EX
n
X,
2
E(])]
a
1 .
j0
.
w
o0( o
0
DX
n
0
)
-n
.
*as
o
0
DX
i
D
x
Hence
he
condi ions
in
p ope y
(iii)
o
H-de ia ion
hold
and
8n,
ED
"o,1,2
0x]
-
Eao
.l
.2
0Dx] +
ED
ao
l
2°
(x
n
x)]
=
[ao,1,2
0 j
o
w]
+
.
.
+Ew0
0
( 0 ^ 0
)
0
(x
n
x)]
=
Esza
l
0
w]
+
N
oo
(x
n x)]
.
Ew
0]
-
1
4n+l
0
'4n+1'
Ew0
0
(x n
x)]
=x
i
.
x
E H*
(X
n
X,
Z/2Z),
*
Now,
he
image
o

x
O
x

in

H (X x
X,
Z/2Z)

is
o
algeb a
il a ion
2
(and
no
o
il a ion
>
2) as

x
©
x = (x
0
1)
"

(1
U
x), and

x

is in-
decomposable
.
H4n+1 (X ^X,
Z/2Z)
+
H8n(X
n
X,
Z/2Z)
.

Fo
dimension
easons
he
image
o

w
in EX
x
X,
sa
E(1)]
mus ha e
algeb a
il a ion
a
leas
.4
:
The
image
o
H
*
(X
n X,

Z/2Z)
- .
H*
(X
x
X,

Z/2Z)

has
il a ion

>
2
.

As
all

gene a o s
in
H*(X
x
X,
Z/2Z)

a e
o
cong uency

s
1

(mod 4)

elemen s
o
dimension
mod
4
ha e
il a ion
>
4
.
Elemen so
dimension
il a ion
>
1
mus
ha e
il a ion
%5
.
-
H
*
(X
x
X,

Z/2Z)

is

injec i e)
sequen ly

[D

n,]
$
x
0
x
D

i
~
*
and
a
o,l
,2
'
x
he e
a e
no
algeb a
gene a o s
in
hese
dimensions
.
*
.
As
his
holds,
o
all
choices
o
a
o,l
,
0 1
o(x)
.
Mo eo e ,
one can use
Hop
algeb a
p ope ies
o
The
conclusion
is
he e o e
ha
he e
a e
no
p imi i e
elemen s
o
1
mod
4
and
o
As
he
S een od
algeb a
p ese e il a ion
(and
H
*(X
A
X,
Z/2Z)
[ga
l
ow]
has
il a ion
>
4
and
con-
mod
F4H
*
(X xX,
Z/2Z))
.
In
pa icula ,
*
H
(X,
Z/2Z)
and he
abo e
e alua ion
o
D

i
o
conclude
ha
he
elemen s
in
e,
o,1,2

-

x
m(x)

a e
al]

gene a o s
.

This
is
impossible
o

P(x)
c
H
8n+2
(X,

Z/2Z)

and
in

H
*(X
.

Z/2Z)

in
he
domain
o

o
.

As

Sq
l
x = 0

o
e e y

x
E
H*
(X,

Z/2Z)
Sg
4n
x
#
0

o
e e y
p imi i e
el
.emen

x

in

H
4n+1
(X,
Z/2Z)
.
Rema k
:
The e
a e
H-spaceswi h
his ype
o
cohomology
:
I Sp is
he
simplec ic
g oup
hen Sp
p
52
2X
.

Bo h
X
and he
uni e sal
co e ing
space
o
9
2
Sp
a e
H-spaceswi h
.
.cohomology
o
he
ype
desc ibed
in
he
heo em
.
We
shall
show he e
how he
heo yo
cohomology
ope a ionsuses
H-space
heo y
.
Conside
he
Adem ela ion
(R
1 )

Sq2Sq
2
+
S
g
lS
q 2
Sq
1
=
0
R
1
induces
a
seconda y
ope a ion
o
1
desc ibed
by
he
(D4)
ype
diag am
as
ollows
:
a
PE(2)
=
K(Z/2Z,N+3)
K(Z/2Z,N)
=E(O)
-
O
~
E(1)
=
K(Z/2Z,N+2)
-
K(Z/2Z,N+3)
-~
E(2) =
K(Z/2Z,N+4)
o¡
0is
gi en
by
p
l
e
a
o= Sq2
,
P2
,
ao
a
l
is
gi en
by
[a
l
]
=
[Sg2
"
p
l
]
+
[Sql
a
l

o

lo
i

*

by

(R
l )
.
7
.
H-SPACES
AND
COHOMOLOGYOPERATIONS
Sg2Sg
1
P2]
he
.p e ious
chap e
[P
l
a
a
o
o
Sg
4n
]
=[Pl
o
ao
o
.Sg4n
o
p]
=
[Sq
2
o
Sg4n
o
p]
_
_
[Sg
4n+
o
p +
Sg4n+l
.e
Sq
l
o
p]
.
(Di
0)
Conside
he
composi ion
Sg
4n
as in
he
las
chap e
°
4n
K(Z,
4n+1)
p
-
.
K(Z/2Z, 4n+1)
-
S
g---~
K(Z/2Z,
8n+1)
whe e
p
is
induced
by
he
educ ion
Z -
Z/2Z
.
Sg
4n
is
in
he
domain o
i
( o
N =
8n+1)
.
Indeed,
by
(R)
o

loop maps
.
Now,
Sq
4n+2
[p]
=
0 as
p
is
o
dimension
4n+1
(using he
non
s abili ycondi ion
o
he
S een od
algeb a)
.
[Sg
l
op]
=
0 as
H
4n+2
(K(Z,4n+1),M)
=
0
o
any
coe icien s
module
M
.
Consequen ly,
[p
l
oa
o
oSg
4n
]
=
0,
[p
2
oa°
oSg4
n
]
=
[Sg2SgluSq4noa]
.
Using
Adem
ela ions
one has
Sg2Sg1Sq4n
=
Sg4n+2Sg1 and
as Sq
l
[
p]
=
0,
[P2oao,Sg4n]
=0
and [Sg
4n
]
E
Ke
a
o
.
We
shall
e alua e o[[Sg
4n
]
:
As
in
he
p e ious
chap e
:
D&
_
[n&~
°)
0x
]
_
[a&~
°) oj o
ow
o
]
=
[j
ooi
l
ow
o]
0

0
K(Z,4h+1)
Sg
~

K(Z/2Z
8n+1)

a°

.
E
( l
j

a

--
.
E(2)
w
o=
p
0
p
E
[K(Z,4n+1)
n
K(Z,4n+1),
K(Z/2Z,8n+2)]
.
¡Pl
SQ4n+2
1
°
-----------------
.
K(j/2Z,8n+3)
d
o'
o
,
xo
,
nE
o
a e
analogous
o
j, , x,
siE
in (D8)
and
sha e
simila
p ope ies
.
All
spáces
and
maps
excep o
xo
and
&°
a e
loop
spaces
and
4n
_
whe e
As

a
o,1,2

is
an
H-map

Da
o,1,2'
a0
=

[a
o,l,2
]
`

Da0
=
_[a
o11,2

j
o
~i
1
1w
0
1 =
[na
l
~
i
l
ow
0
]
=
Sq
2
o
w0
(2al`i1
=
Sq
2
*
K(Z/2Z,8n+2)
.,
K(Z/2Z,8n+4))
.
Using
he
Ca an
o mula
([S een od
Eps ein])
one
ob ains
o
any
li ing
a0
o
Sg04n
:
Da

a

=
Sq
2
w0
=
Sq2 (pop)
=
Sg
2 p
op
+
p
©
Sg2
p

(Sq
'
p=D)
.
0,1,2`
o
s
u
:
K(Z,4n+1)
->
K(Z/2Z,8n+4)
is
being
gi en
algeb aically
by
[u]
=
[p]-S9~[p]
(o
"geome ically"
by
he
composi ion
K(Z,4n+1)
A
y
K(Z,4n+l)
-
K(Z,4n+1)
-px
-
P
-->
K(Z/2Z,4n+1)
X
K(Z/2Z,4n+1)
Sq
2
xl

_
K(Z/2Z,4n+3)
i
K(Z/2Z,4n+1)
-"-+
K(Z/2Z,4n+3)
A
K(Z/2Z,4n+1)
K(Z/2Z,8n+3)]whe e
®
ep esen s
he
nene a o
o
H
8n+3
(K(Z/2Z,4n+3)
A
K(Z/2Z,4n+1),
Z/2Z
=
Z/2Z)
.

Then

D
u = Sg2
p
0p
+ p
®
Sg2p
o
a
cohomology
class
u
o
an
H-space
X
is
he
educed
cop oduc
in
(D
u
he
Hop
algeb a
H
(X,Z/2Z))
.
l
ollows
easily ha
i

=
[
a
0,1,2'
ao
]
EP,(Sg4n
)

is
any
elemen
-u is
p imi i e
.
Now,
one can
show
ha
á
0
can
be
chosen
so
ha
=u
and
[p]
"
Sq2 [p] E o
l
(Sg4n
) .
(Ou ]ine
1oopino
o
p oo
:,,(Dll)
wice
one
ob ains
22(Sg0on)
i
*
and
o
some
z
:
K(Z/2Z,4n-1)
->
n3
E(1)
=
K(Z/2Z,8n)
x
K(Z/2Z,8n+1)
jo
H-map
as

92
%
% n2

o
z

is

an

H-map,

0 =

[92
jo]

o

Dz
,
n
K(Z/2Z,4n-1),
23
E(o)
=
K(Z/2Z,8n-2)]
=
0
(2
2
jo
)
*
on
3
A
K(Z/2Z,4n-1),
0
E(1)]

is
injec i e,
D
z= 0
.

Any
H-map
be ween
2
2
6,
0%
.
22
j
o ,z
z
mus be
an
[K(Z/2Z,4n-1)
[K(Z/2Z,4n-1)
Eilenbe g
MacLane
spaces
is
an -loop
map o
any


and

z
i
n
2
z

o
some
and
as
z
:
K(Z/2Z,4n+1)
-,-
s2E(1)
.
Use
z
o change
he
homo opy
i
ao
o
Sq
o
and
hen,
o he new
ao
one has
2
[no,1,2oáo]
= 0,
*
i
22
i
áa
2
( -u)
(as
P2u
i
*,
since
51
2
A
`,
*
in he
"geome ic"
de ini ion
o
u)
.

Bu

one

can see
ha

o
2
:
H
8n+4
(K/Z,4n+1),
Z/2Z)

+
H8n+2(K(Z,4n-1),Z/2Z)
is
injec i e
on
p imi i es
(Eilenbe g
Moo e
spec al
sequence)
and
u
. i
)
.
Consequen ly
:
01(Sgo
n
)
_
[p]
oSq2[p]
+
im Sq
2
+
Co olla y
I
:

Le

x E H
4
n
+1
(X,Z)
be
any
class
>
X
-
any
space
.
I
4n
4n

22 1

8n+2
Sqo x = Sq

px
= 0

hen

px
"
Sq px
=
Sq yl

+Sq
y2

o
some

yl

E H

(X
,Z/2Z),
y
2EH
8n+3
(X,Z/2Z)
.
P oo
o
I :
Conside
he
ollowing
i

J
oI
,
a
o,1,2
1-1
E(0,1)
im Sq
l
.
X
-
X
"
K(Z,4n+l)-
o
--
"
K(Z/2Z,8n+1)
2)
a
o' °`o,1,2 as in
D11,
d
o
chosen
so
ha
a
o,1,2
,ao
=
[p]
Sq2[p]
.
As
Sg
4n
x = 0 ao

x
= j
o
,
y
o
some
y
:
X
,
iE(1)
d
K(Z/2Z,8n+2)
-
K(Z/2Z,8n+3)
.
Pu
yi = p
i
y
and
hen
px
"
Sg2px = [a
o,l,2o
aoox] _
[pa
lo
y]
=
Sg2yl
+
Sg1y2
.
Co olla y
II
:
The e
is no
space
X
wi h
H*
(X,Z/2Z)
being
he
ex e io
algeb a
on-
x

and

Sq
2
x,

dim
x
=
4n+1
.

(I .e
.

X
sa is ies
:

H
i
(X,Z/2Z)

~
0
only
i
i=0,
4n+1, 4n+3, 8n+4,
and
in
hese
dimensions
H
i
(X,Z/2Z)
s
:d
Z/2Z wi h
non
ze o
elemen s
1,
x, Sg
2
x
and
x
"
Sg 2x
o
i=0,
4n+l,
4n+3
and
8n+4
espec i ely1
P oo
o
Co olla y
II
:
In
such
a
space
Sgó
nx =
0
(as
H
8n+1
(X,Z/2Z)
=
0)
bu
0 ~ x
o
Sg2x
=
Sg2
y
1
+ Sg
l
y
2
is
impossible
o
he e
a e
no
elemen s
in he
dimensions
o
yl
and
y
2
.