Annals of Mathematics
The Steenrod Algebra and Its Dual
Author(s): John Milnor
Reviewed work(s):
Source: The Annals of Mathematics, Second Series, Vol. 67, No. 1 (Jan., 1958), pp. 150-171
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1969932 .
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ANNALS
OF
M
ATLIEMATICS
Vol. 67, No. 1, January, 1958
Printed in Japan
THE STEENROD ALGEBRA AND ITS DUAL1
BY
JOHN MILNOR
(Received May 15, 1957)
1. Summary
to an odd prime
Let 57* denotethe Steenrodalgebra corrresponding
p. (See ? 2 for definitions.)Our basic results(? 3) is that 5i* is a Hopf
algebra. That is in additionto the productoperation
c* ?
An*
*
n
thereis a homomorphism
A
9*
satisfyingcertainconditions.This homomorphism
sb*relatesthecup product structurein any cohomologyringH*(K, Zp) withthe action of A*
denotesa Steenrodreduced
on H*(K, Zr). For exampleif n e 922n(p-1)
pth powerthen
{g(n)
=
+
n
-9
nl I
I
+
***+
i
n
The Hopf algebra
92*
09*?9*
9*
has a dual Hopf algebra
The maintoolin the studyof this dual algebra is a homomorphism
A*: H*(K, Zp) -+H*(K, Zp) (g 9*
which takes the place of the actionof 91* on H*(K, Z.). (See ?4.) The
simplestructure.In
dual Hopfalgebraturnsout to have a comparatively
p*) it has the
fact as an algebra (ignoringthe "diagonal homomorphism"
form
E(ro, 1) (g)E(rj, 2p -1) (g ... (g P($,, 2p -2) OM P2, 2p2 -2) @* *
whereE(z-, 2pi - 1) denotesthe Grassmannalgebra generatedby a ceralgebra
tain elementri e R l, and P($j, 2pi - 2) denotesthepolynomial
generatedby $j e -~pt-2.
The author holds an Alfred P. Sloan fellowship.
150
151
THE STEENROD ALGEBRA AND ITS DUAL
In ? 6 the above information
about a is used to give a new description
of the Steenrodalgebra 9 *. An additivebasis is givenconsistingof elements
QO OQle .el
rir2
with E, = 0, 1; ri _ 0. Here the elementsQi can be definedinductively
by
QO
P Q - QJ) p
Qi+1
while each rl ... r7 is a certainpolynomialin the Steenrod operations,2
of dimension
rl(2p - 2) + r.2(2p2-2) + * + r,(2p -2)
The productoperationand the diagonal homomorphism
in j/* are explicitlycomputedwithrespectto this basis.
The Steenrod algebra has a canonical anti-automorphism
which was
firststudied by R. Thom. This anti-automorphism
is computedin ?7.
Section 8 is devoted to miscellaneous remarks. The equation 0 01 0 is
studied; and a proofis given that Q<* is nil-potent.
A brief appendix is devotedto the case p = 2. Since the sign conventionsused in thispaperare nottheusual ones (see ? 2), a secondappendix
is concernedwith the changes necessaryin order to use standardsign
conventions.
-
2. Prerequisites: sign conventions, Hopf algebras,
the Steenrod algebra
If a and b are any two objects to which dimensionscan be assigned,
then whenevera and b are interchangedthe sign (_ )dima dimb will be
introduced.For example the formulafor the relationshipbetween the
homologycross productand the cohomologycross productbecomes
( 1)
<u x1X) as xt
d > = (-
1)dimadmc < o,! at > <
A~>
This contradictsthe usual usage in whichno sign is introduced.In the
same spiritwe will call a gradedalgebra commutative
if
ab =
(-
1)dimadim
baba
Let A =(A-1, Ao,Al, ... ) be a graded vectorspace overa fieldF.
The dual A' is definedby A' = Hom (A-n, F). The value of a homomorphism a' on a e A will be denoted by <a', a>. It is understoodthat
< a', a > = 0 unless dima' + dima = 0. (By an elementof A we mean an
elementof some An.) Similarlywe can definethe dual A" ofA'. Identify
2
This has no relation to the generalized Steenrod operations
,/9I
defined by Adem.
152
JOHN MILNOR
each a e A withthe elementa" e A" whichsatisfies
< a,1,at> =( - V~imall dim a' < a', a >
( 2)
for each a' e A'. Thus every graded vector space A is containedin its
doubledual A". If A is of finitetype(that is if each Anis a finitedimensionalvectorspace) thenA is equal to A".
Now if f: A -? B is a homomorphismof degree zero then f': B' -+ A'
and f": A" -- B" are definedin the usual way. If A and B are bothof
finitetypeit is clear thatf = f".
ThetensorproductA 0 B is definedby (A 0 B). = Ei+ j=nA
Bj,
type
where" E " standsfor" directsum". If A and B are bothoffinite
large
small i (or for all sufficiently
and if Ai = Bi = 0 forall sufficiently
i) then the productA 0 B is also of finitetype. In this case the dual
(A 0 B)' can be identifiedwithA' 0 B' underthe rule
(3)
<a'&bf, a (, b>
= (
)dimadim
bK <a',a
><b',
b>.
In practice we will use the notationA, fora graded vectorspace A
satisfyingthe conditionAi = 0 fori < 0. The dual will then be denoted
by A* where A = A' n= Hom(An,F). A similarnotationwill be used
forhomomorphisms.
By a gradedalgebra(A*, V') is meant a graded vector space A* togetherwitha homomorphism
sP*:A* 0 A* A*.*
It is usually required that Sb, be associative and have a unit element
1 e A.. The algebra is connectedif the vectorspace A. is generatedby 1.
By a connectedHopf algebra(A*, o*,q)*) is meanta connectedgraded
algebra withunit(A*,ys*), togetherwitha homomorphism
qb*:A* -*A* 0A*
satisfyingthe followingtwo conditions.
of algebraswithunit. Here we referto the
2.1. 0* is a homomorphism
productoperationsP*in A* and the product
(a,
0 a.)-(a3 ( a) =(
l)dima2dim
a3 (a, a3) 0
(a,,a)
in A* (g A.
2.2. For dima > 0, the element +*(a) has the forma 1 + 1 (a a +
E bi0 ci withdimbi, dimci > 0.
are defined,
Appropriateconceptsof associativityand commutativity
, butalso forthe diagonalhomomornot onlyforthe productoperationsV*
phismsq,*. (See Milnorand Moore[3]).
To everyconnectedHopf algebra (A*,y P*) of finitetypethereis as-
THE STEENROD ALGEBRA AND ITS DUAL
153
where the homomorphisms
sociatedthe dual Hopf algebra(A*, g*, Ab*),
A*
A*
are the duals in the sense explainedabove. For the proofthatthe dual is
again a Hopf algebra see [3].
d
GxG
(As an example,forany connectedLie groupG the maps G
P G give rise to a Hopf algebra (H*(G), p*, d*). The dual algebra
(H*(G), I, p*) is essentiallythe example which was originallystudied
by Hopf.)
For any complexK the SteenrodoperationSA is a homomorphism
'~i: HJ(K~Zv) *HJ+2i
(.V-1)(K Zv) .
The basic propertiesof these operationsare the following.(See Steenrod
[4].)
2.3. Naturality. If f maps K into L then f *5j/ i = ao if
2.4. For afe Hj(K, Zr), if i > j/2 then jict = 0. If i = j/2 then 5 ica
.
ca =
cf. If i = 0 then
?c
ia .'\ 9 JA
'
2.5. 9 n(Ca A) = i+ j=n
We will also make use of the coboundaryoperationa: HJ(K,Zp)
sequence
H'l+(K, Z.) associated withthe coefficient
-
-
0
-Z
-*
Z2
-* Zp-
0
.
The mostimportantpropertieshere are
2.6. 3a = O and
2.7. a(ac P) = (d&) d + (- l)dimla 3d, as well as the naturality
condition.
FollowingAdem [1] the Steenrodalgebra 7' is definedas follows.
The free associative graded algebra f* generated by the symbolsa,
.-evade,
)!>)
(6162
...
Ok)-
acts on any cohomology ring H*(K, Z.) by the rule
= (61(62
(Oka) ... )). (It is understoodthat a has dimen...
sion 1 in _j7* and that ~i7 has dimension2i(p - 1).) Let jr* denote
the ideal consistingof all f e * such thatfca= 0 for all complexesK
and all cohomologyclasses afe H*(K, Z.). Then 9* is definedas the
It is clear that * is a connectedgradedasquotientalgebra *
sociativealgebraof finitetypeoverZ.. However&>* is notcommutative.
(For an alternativedefinitionof the Steenrodalgebra see Cartan [2].
is that Cartan adds a sign to the operation
The mostimportantdifference
aT)
Howeverit has been shown
The above definitionis non-constructive.
154
JOHN MILNOR
is generated additivelyby the "basic
by Adem and Cartan that &*
"
monomials
S1l81 . .. .
205)
/
Ska'k
whereeach 8j is zero or 1 and
S1>
PS2
+ '1,
82,,
s2 >_ PS3 +
Sk-1
>
P~k
+ 8Ek-1,
Sk >_l
FurthermoreCartan has shown that these elements forman additive
basis for
9*
3. The homomorphism b
LEMMA 1. For each element6 of 5 * thereis a uniqueelementP*(O)=
0A,?3 6' of A* ? J7* suchthattheidentity
ti~ot ^
=
E
(-)dim0,'
)
(a
a_
dime
is satisfiedfor all complexesK and all elementsa, P e II*(K).
more
?
j%
>9*
Further-
si*
is a ring homomorphism.
(By an " element"ofa gradedmodulewe meana homogeneouselement.
The coefficient
groupZ, is to be understood.)
'
* act on H*(X) ? H*(X) by the
It will be convenientto let _97
rule
(6' ? 6")(a
? P)
(-_)dim0"
=
Let c: H*(X) ? H*(X) - H*(X)
identitycan now be writtenas
Oc(a ? js)
dimco
06(a)
? 6"(9)
denote the cup product.The required
cWb*(6)(? )
consistingof
PROOF OF EXISTENCE. Let P denote the subset of C
all 6 such that forsome p e 92* ? -92* the requiredidentity
=
6c(a ? ,2) = cp(a
-
is satisfied.We mustshow that _ =
The identities
and
-a
-(a
()
(
#A)
A
X
=
E
i
9)
+ (_
1)di,',
aj=n
t
a
a:
-
clearlyshow that the operationsa and Sca belongto .X. If (I,
to 5 thenthe identity
6t2belong
THE STEENROD
?
6162c(ja)
155
ALGEBRA AND ITS DUAL
? j)
) = Cpop2(oa
?
6c
= Op2(ic
?s is closedunderaddition.Thus
showthat6162belongsto i9. Similarly
of S* whichcontainsthegenerators
X is a subalgebra
8(,?n of $*
Thisprovesthat?s =
PROOF OF UNIQUENESS. From the definitionof the Steenrod algebra we
see that given an integern we can choosea complex Y and an element
r e H*(Y) so that the correspondence
6 -far
of Vi into Hk+!(Y) fori < n. (For example take
definesan isomorphism
k
Y = K(Z,, k) with > n.) It followsthat the correspondence
(_
of
of hi,
V~imo dimy Of(I-) x 0 r
_5 intoH2k+i(Yx Y) fori _ n.
? J7*)
j of (
definesan isomorphism
* bothsatisfythe identityOc(a ? @
Now supposethatPi, P2e 57* ?
Taking X = Y x Y, a=
cp~i(a? A) for the same element 0 of Jn.
r x 1, jI= 1 x r,we have cpi(a ? j) = j(pi). But the equality j(p1) = j(p,)
with dim Pi = dim p2= n implies that Pi = P2* This completes the uniquefollows
ness proof. Since the assertionthat s/*is a ring homomorphism
easily fromthe proofused in the existenceargument,this completesthe
proof.
As a biproductof the proofwe have the followingexplicitformulas:
-
a? 1 + 1 ? a
0*()=
S}*(&Dn)
=
AXD n
?1 +
9
nl-1 (?
d1 +
***+
1?
nd
THEOREM1. The homomorphisms
C *
A
*
>
-*
~*
9*
9
A
*
>
*
give 9* thestructureof a Hopf algebra. Furthermorethe productA* is
associativeand the "diagonal homomorphism"/* is bothassociativeand
commutative.
PROOF. It is knownthat (SW*,A*) is a connectedalgebra withunit; and
is a ringhomomorphism.
Hence to show that &<`* is a Hopf althat V,*
gebra it is only necessaryto verifyCondition2.2. But this conditionis
clearlysatisfiedforthe generators3, and J n of 9Q*, whichimpliesthat
it is satisfiedforall positivedimensionalelementsof A
It is also knownthat the productA* is associative. The assertionsthat
are expressedby the identities
b* is associativeand commutative
(1)
(* X 1)0*0 = (1?X *)0*O ,
156
JOHN MILNOR
= s*
Tk*Od
(2)
for all 0, where T(6' ? 0") is definedas (- 1)dimOdimO' 9" ?'.
Both
n of
identitiesare clearlysatisfiedif 0 is one of the generators3 or
t *. But since each of the homomorphisms
in questionis a ringhomomorphism,thiscompletesthe proof.
As an immediateconsequencewe have:
COROLLARY1. Thereis a dual Hopf algebra
withassociative,commutative
productoperation.
4. The homomorphism A*
Let H*, H* denotethe homologyand cohomology,
withcoefficients
ZP,
of a finitecomplex. The actionof &* on H* gives rise to an action of
on H* whichis definedby the rule:
<P0, a> = <Ay, oa>
forall 1ue H*, 0 e
morphism
* a e H*. This actioncan be consideredas a homoH*:H*?
*H*.
The dual homomorphism
A*: H* -H* ?*
will be the subject of this section.
Alternatively,the restrictedhomomorphism
Hn+ ?
dual whichwe will denoteby
Stat
Hn has a
AHn -,Hn+ 3f
In this terminology
we have
A*= so + 21+ 22+
4. The conditionthat H* be thecohomolocarryingHn into i Hn+i
gy of a finitecomplexis essentialhere, since otherwiseA* would be an
infinitesum.
The identity
P(0102)
=
(i1)O2
can easilybe derivedfromtheidentity
(010)a =
01(02a)
whichis usedto
definethe productoperationin JX*. In otherwordsthe diagram
157
THE STEENROD ALGEBRA AND ITS DUAL
1A*(S) 1
H
_-*
1A*
AH
H**
H* (D
is commutative.Thereforethe dual diagram
H* @(D
H*
(g1
H* (2
*
*
H*
is also commutative.Thus we have proved:
LEMMA 2. The identity
(W*? 1)2*(a) = (1 ? qb*)2*(a)
holdsfor everya e H*.
The cup productin H* and the0* productin 3* inducea productopera-
tionin H* ()9*/'.
LEMMA
3. Thehomomorphism
A*:H* - H* ? S* is a ringhomomorphism.
PROOF. Let K and L be finite
complexes,
let 0 be an elementof 9*,
"
and let b*(O)=
Of? 67. Thenforanya e H*(K), ,Be H*(L) we have
O.(a x j) =
(- 1)dim0'dimc Oa x 7d'.Usingtherule
K
<(,
<fP x V, Mo(ax P)>
x )., a x
we easilyariveat theidentity
(p2
X
=
S).O}
E
( _)dim d
dim
f
9X
In otherwordsthediagram
14(K) 0 H*(L)
0
951*
?
9;*
?1
0
H
14(K)
1H*(L)0 y* = H*(K X L)?
I11?T01
H*(K)
?$
*
?
9
|A*
H*(K) ?
H*(L)
H*(L)
= H*(K x L)
is commutative
twofactorsas in ? 3). Therefore
(whereT interchanges
thedualdiagramis also commutative.
SettingK = L, andlettingd: K -*
we obtaina largercommutative
K x K be thediagonalhomomorphism
diagram
H*
l&H~ 19 s
l @+H* (&H*R
- (1
11?T
H*(
9-( H
9
H*
H
=91*
=H*(K x K) (9,*
-H
*
H*(K xK)
H* (9 9
IH*
H*
158
JOHN MILNOR
Now startingwith a ? ife H* ? H* and proceedingto the rightand
up in this diagram,we obtain d*(a ). Proceedingto the left and up,
and thento the right,we obtain2*(aQ)
-*(d). Therefore
2*(aP)= 2*(002*0
whichprovesLemma 3.
The followinglemma shows how the actionof SY* on H*(K) can be
A*.
fromthe homomorphism
reconstructed
?
=
for any 0 e a* we have
then
wa
LEMMA4. If A*(a)
a,
dimco~idim
woi <O
Oa =E-1
cojaj)>0
PROOF. By definition
Oa,
0x> = <Pa, a>
= <( p
< MP 00), a >
, ,A*a> = E i <P, ai><O,
a)>,
Since this holdsforeach pue H*, the above equalityholds.
H*
REMARK. To complete the picture, the operation rf*: 9* 0 H*
2
and
4
of
are
Lemmas
easily
has a dual (*: H* -A*9? HH. Analogues
obtainedfor7*. If a productoperationK x K -- K is given,so that H.,
and hence 9* 0 H*, have productoperations;then a straightforward
(As an example let K deproofshows that 7* is a ringhomomorphism.
note the loop space of an (n + 1)-sphere,or an equivalent CW-complex.
ringon one generatorp e Hn(K).
Then H*(K) is knownto be a polynomial
The element
7*(p) e (,o O Hn) () ((
(D)(X~' ( Ho)
Hn-) (D) *.*.*
r*(pk)
equal to 1 0 u. Therefore
is evidently
-
1
0
forall k. Passing
pk
to the dual, this provesthat the actionof t9* on H*(K) is trivial.)
5. The structure of the dual algebra M*
As an example to illustratethis operationA*considerthe Lens space
X - S2N+1/Zp where N is a large integer,and where the cyclic group Z,
acts freelyon the sphereS2N+1. ThusX can be consideredas the (2N + 1)skeletonof the Eilenberg-MacLanespace K(Z,, 1). The cohomologyring
H*(X) is known to have the followingform.There is a generatora e
For 0 < i < N. the group
H1(X) and H2(X) is generated by PB=a.
H20(X) is generatedby dBand H20+'(X) is generatedby ad'.
The actionof the Steenrodalgebra on H*(X) is describedas follows.
It will be convenientto introducethe abbreviations
MO-
1
,
M
-
$,1
2 =
9'P9
**1
.M
=
.
.
5
,p
159
THE STEENROD ALGEBRA AND ITS DUAL
LEMMA 5. The elementM, e
C 2"P-2
satisfies M9
- ,2X. However if 0
is any monomialin the operations3, 9?1, ? 2,**
whichis not of the
1
'
p29D then d: = 0. Similarly(M,3)a = dP but Oa = 0
form P1-*
_d
3, 91
if 0 is anymonomial
in theoperations
theform0 = ?7p -P ... 913 or 0 1.
... whichdoesnothave
92,
PROOF. It is convenientto introducethe formaloperation v = 1 +
,91 + t2 + -- . It followsfrom2.4 that ,BP= P + dP. Since P is a
(b9+ dP). In
ringhomomorphism
accordingto 2.5, it followsthat 9Aji
pT + [BpT+1
= (p +
IIn ther
particularif X - pr this gives
o9p)i =
words
if j =0
Adr
,9i13fT
{
AP~T1if j
0
p
otherwise
Since 3~i = ij3-13j = id-1e3a = 0 it follows that the only nontrivialoperation 3 or 9J whichcan act on pr is 2 r. Using induction,this proves
the firstassertionof Lemma 5. To provethe secondit is only necessary
to add that i9Aa = 0 forall j > 0, accordingto 2.4.
Now considerthe operationA*:H*(X) - H*(X) ? 4.
+
LEMMA 6. The element 2Aa has the form a ? 1 + P 0To + dP mrT
... +/P" ? r whereeachr, is a well definedelementof _Vpkl , and where
has theform
pr is thelargestpowerofp withpr < N. SimilarlyAdd
d0$0)to+d/0(9
1 + * **+0
$g
r
Y
where$%= 1, and whereeach k is a well definedelementof _9fpk 2.
PROOF. For any element0 of 9", Lemma 5 impliesthat Op = 0 unless
i is the dimensionof one of the monomialsMo, M1, *--: that is unless i
has the form2pk - 2. Therefore,accordingto Lemma 4, we see that Aid
-0 unless i has the form2pk - 2. Thus
A =0(p)
+ 21p-2(,)+ *- + 22P-2(0)
0 9,k-2, itmusthavetheformgpk (? $k
forsome uniquelydefinedelement$k. This proves the second assertion
of Lemma 6. The firstassertionis provedby a similarargument.
REMARK.These elements$k and zrkhave been definedonlyfork < r =
[logpN]. However the integerN can be chosenarbitrarilylarge, so we
have actuallydefined k and zrkforall k > 0.
Our main theoremcan now be stated as follows.
THEOREM2. The algebra 9 is thetensorproductof theGrassmannal-
SinceA2Pk-2) belongstoH2pk(X)
gebra generated by roz-,
Al
$2
.
...
and the polynomial algebra generated by $,
160
JOHN MILNOR
The proofwill be based on a computationof the innerproductsof monomialsin ri and If withmonomialsin the operations "Iand a. The followinglemmais an immediateconsequenceof Lemmas 4, 5 and 6.
LEMMA 7. The inner product
<
equals one,but< 6,
>
k >
Klf,
i 6 is any othermonomial.Similarly
if
=
<Mk,
1
-r>
i 6 is any other monomial.
but < 6, -r > =
if
Consider the set of all finite sequences I =
r1, &,, r2, ***) where
&j = 0, 1 and ri = 0, 1, 2, - - - . For each such I define
cw(l) =
An}0e$'rIrzl2r2 . .
Then we must prove that the collection{co(I)} formsan additivebasis
for 4.
For each such I define
6(I)
S2 . . .
813
=
where
+ ri)p'
It is not hard to verifythat these elements6(I) are exactly the "basic
monomials"of Adem or Cartan. Furthermore6(I) has the same dimension as co(I). Orderthe collection{I} lexicographicallyfromthe right.
(For example (1, 2, 0, *-.) < (0, O 1, *..).)
LEMMA 8. The innerproduct< 6(I), w(J)> is equal to zeroif I < J and
SI
? 1 if I=
I (8- + ri)p,* ,
,
sk
=
c (8i
J.
Assumingthis lemma forthe moment,the proofof Theorem2 can be
completedas follows. If we restrictattentionto sequences I such that
dim co(I) = dim 6(I) = n,
then Lemma 8 asserts that the resulting matrix < d(I), w(J) > is a non-
singular triangularmatrix. But accordingto Adem or Cartan the elementsO(I) generate 7n . Thereforethe elementsw(J) mustforma basis
for >X; whichprovesTheorem2. (Incidentallythis gives a new proofof
Cartan's assertionthat the 6(I) are linearlyindependent.)
PROOF OF LEMMA 8. We will prove the assertion < d(I), co(I) > = ? 1
by inductionon the dimension.It is certainlytrue in dimensionzero.
Case 1. The last non-zero element of the sequence 1
.rk,O, ...) is rk. Set ' = (80, rl, .., k, rk-l,0
co(I')k . Then
)
(&O,r1, . . ., 8k -1
so that co(1)=
161
THE STEENROD ALGEBRA AND ITS DUAL
< 6(I),
(I) > = < d(I), v%(w(I')
= <0*O(I),
Since 6(1) -
...
*-
asoi
E
d()=
?
)>
$t >
kwe have
8
k-1?
0~~~~
@(I')
?
k
iao...a
X
a
g0
k'
where the summation extends over all sequences (8K, **, Sk) and (80', ***,
with 6 + j' =
and st + s' = si. Substitutingthis in the previous
Sk)
expressionwe have
K()
w(I)>
<a6
:E
S
D?
C
(1T) > <
0
* **
9 k,
k>
But accordingto Lemma 7 the righthand factor is zero except for the
special case
S
8-S . . .
=
_5
65p
in whichcase the innerproductis one. Inspectionshows that the correspondingexpression 'O'*--- k on the left is equal to 6(1'); and hence
that <8(I), w(I)> = + <(I'), co(I')>= ? 1.
Case 2. The last non-zero element of I = (o, r,
8k = 1. Define '= (0,rl, ... rk, . .. ) so that
*
Er
r,,
8
0,
O.
*.. )
is
Cw(I) = Cl(F)7k .
as beforewe findthat the onlynonCarryingout the same construction
vanishing right hand term is
< As
p .-
*
, -rk>
=
1. The correspond-
?
ing left hand term is again <6(1'), co(I')>; so that <8 (I), wj(I)>
1, withcompletesthe induction.
< 6(1'), CO(')>
The proofthat < 6(I), co(J)> = 0 for I < J is carried out by a similar
inductionon the dimension.
Case la. The sequence J ends withthe elementrkand the sequence I
place. Then the argumentused above shows
ends at the corresponding
that
< (1'), '0(J')>= 0 .
Case lb. The sequence J ends withthe elementsrk, butI ends earlier.
Then in the expansionused above, everyrighthand factor
<K(I), @(J)>=
<
0@
9
+
1
.**8k-1k
is zero. Therefore < 6(I), co(J)> = 0.
>
SimilarlyCase 2 splits up into two subcases which are provedin an
analogousway. This completesthe proofof Lemma 8 and Theorem2.
To completethe descriptionof A'*-as a Hopf algebra it is necessaryto
it
p*. But since O* is a ringhomomorphism
computethe homomorphism
162
JOHN MILNOR
is onlynecessaryto evaluate it on the generatorsof S*.
THEOREM3. The followingformulas hold.
Ei)= if
=
0*(7k)
0
=0 $kp-i (
Tif
e
?k-i 9
-r
+
_rk
The proofwill be based on Lemmas 2 and 3. Raising bothsides of the
equation
to the powerpi we obtain
Ep ?d(j
2*(0=
2*(9Pi)= E p+i (? ePi
Now
(A*(
1)2*(W)
=
(A* ? 1) I
=
E
a ?
e
?m+SXe~
;i
?&
.
Comparingthis with
= E p (?
(1 (? b*)2*(p)
for b*(k).
We obtaintherequiredexpression
0b*($k)
Similarlythe identity
= (1 0I sb*)2*Qx)
(2* (8 1)2*Qxt)
can be used to obtainthe requiredformulaforq*(rk).
6. A basis for j7*
range over all sequences of non-negative integers
which are almost all zero, and define $(R) = $1r1$22r ... . Let E =
(80, 6, **.. ) rangeoverall sequencesof zerosand ones whichare almostall
zero, and definezr(E)= -ro.r1.* . Then Theorem 2 asserts that the
Let R
=
(r1, r2,
**
*)
elements
{z(E)$(R) }
Hence thereis a dual basis {p(E, R)} for
forman additivebasis for S
j7** That is we definep(E, R) e 59* by
1 'if E = E', R = OR
< p(E, R), -r(E')$(R)> =
{
o
Using Lemma 8 it is easily seen that p(O,(r, 0, 0, *..)) is equal to the
2 9 R as the basis
Steenrodpower _ r. This suggests that we define
as
such
9"' in place of
element p(O, R) dual to $(R). (Abbreviations
0
1,
?.) will be frequentlybe used.)
e (0
THE
STEENROD
ALGEBRA
AND ITS
163
DUAL
Let Qk denote the basis element dual to rk. For example Q0 =
p(1, 0, *-- ), 0) is equal to the operation8. It will turnout that any basis
--element o(E, R) is equal to the product + QOsoQ1,si
THEOREM4a. The elements
R
.. *7
Q oeQlel
forman additivebasisfor theSteenrodalgebra 9C* whichis, up to sign,
dual to the knownbasis {-r(E)e(E) } for -5. The elementsQk, e 9
generatea Grassmannalgebra: thatis theysatisfy
QJQQk +
Theypermutewiththeelements
Q_
QkRv
=
R-
QC+
(By the difference(r1, r2,
accordingto therule
R
(p
0 .
=
QkQJ
)
?'
++
- (si,
)
Q
R-(O,
0 **-) +
**.*) of two sequences we mean
the sequence (r1- s1,r2 - S,* *.* )- It is understood,for example, that
(P'0
RXI---) is zero in case
r, < pa.)
As an example we have the followingwhere [a, b] denotethe " commutator"ab - (- 1)dimadinb ba.
COROLLARY2. The elementsQ, e 972 -1 can be definedinductivelyby
therule
=
[QU
Q=
t9,
Qk]
To completethe descriptionof 7-*as an algebra it is necessaryto find
matrices
. Let X range over all infinite
the product
*
Xo1 X02 .
X
, 1 1?
0
0
11.
.
.
.
.
.
.
.
.
.
.
.
.
.
of non-negativeintegers,almostall zero, with leading entryommitted.
For each such X define R(X) = (r1, r2, ...),
T(X)
S(X) = (s,, so, ...),
-
(t1, t2, *), by
ri = EJ pjxi
> = Ad xij
tn = Ei+j=n
Xi}
(weightedrow sum),
(columnsum),
(diagonal sum).
Definethe coefficient
b(X) = II tj! /fIxij!.
THEOREM4b. The product
A:
RnX
iJR?7S
=
ySX)
is equal to
bX
9
T(X)
and
164
JOHN MILNOR
wherethesum extendsoverall matricesX satisfyingthe conditionsR(X)
= R, S(X) = S.
Then
As an example consider the case R = (r, 0, ... ), S = (s, 0, ..*).
the equations R(X) = R, S(X) = S become
x10+ px + *--- r,
x0i+ x1 + S.. = s,
xi=
xf
O for i > 1,
0 for j > 1, respectively.
Thus, lettingx = x1,,the onlysuitablematricesare those of the form
Ss-x O
x O*
r-px
o
0
0-
b(X) are the
with0 < x < Min (s, [r/p]). The correspondingcoefficients
binomial coefficients(r
px, s
-
EMi i(s,
r/Ip])(r
(For example Z95~l? l =
x). Therefore we have
-
3. The product
COROLLARY
p r 9Ps
sis
px, s
-
2??PP+2
+
equal to
x)
-
v 1I1,)
The simplestcase of this productoperationis the following
COROLLARY4. If ri < p, r, < p, *-- then
ARIS
= (r1, s1)(r2,sH) ...
95R+S
As a finalillustrationwe have:
COROLLARY5. The elements 9 (0..010 ) can be defined inductivelyby
P
[0,1
ta]
p0,0,1
=
[is
etc.
p?Z0,1]
The proofsare leftto the reader.
PROOF OF THEOREM4b. Given any Hopf algebra A* with basis {ad
can be writtenas
the diagonalhomomorphism
P*(a0)= ok c 6a, a.
The productoperationin the dual algebra is thengiven by
ada
-
I*(ai
0
ak) =
(_
V
l)dimran dim ak
i
where {a'} is the dual basis. In carryingoutthisprogramforthealgebra
M* we will firstuse Theorem 3 to compute )*(s(T)) for any sequence
T= (t1, t2,.) .
Let [U1i,2,
ik] denotethe generalizedbinomialcoefficient
(so
t he flo!
w! i+ i2 +i
hkl!+
so that the followingidentityholds
k)!/
165
THE STEENROD ALGEBRA AND ITS DUAL
(Y1 +
+ y")f =
+ik=n [i1t
i
..
* 9 ik]Yl
k
Applyingthis to the expression
)* (ok)
=
$k
$1 +
***+
xok]('ko~kfkPlk-11
...
+
$Pk-l 0
$P -1@ &
k-l
+
$
k
we obtain
c+
((kk)
-
-
E
[xo,
*
**,
Ad [Xk9, *...,
S
XOk] (pk
*k
X1k-l,
kf
*
,
Xlk-1)
XkO) &
X
$(Xk
...
(ark-ll
1 1,*9xo.
,
$ Ok)
)
summedoverall integersxk, **, x~k satisfying Xik > 0, Xk3 +
+ Xok
Now multiplythe correspondingexpressionsfor k = 1, 2, 3,9tk.
Since the product[x10,X01]
[x20,x11,x02][x30,*.. , x03] ... is equal to b(X),
we obtain
-..
-
q)*(s(T))_=
T(X)=T
b(X)s(R(X)) & $(S(X)) ,
summedover all matricesX satisfyingthe conditionT(X) = X.
In order to pass to the dual qp*we must look for all basis elements
r(E)$(T) such that k*(zr(E)$(T))containsa termof the form
(non-zeroconstant).-(R) ? a(S) .
Howeverinspectionshows that the onlysuch basis elementsare the ones
i(T) whichwe have just studied. Hence we can writedownthe dual formula
cp*(9
R
=
?9)
1R(X)-R
S(X)=S
b(X)_9T(X)
This completesthe proofof Theorem4b.
PROOF OF THEOREM4a. We will firstcompute the products of the basis
elements p(E, 0) dual to r-o-rill -.- . The dual problem is to study the
S A*
? * ignoringall termsin A~*?
which
homomorphism
0*:
*..
involveany factor $k. The elements 1 ? $1 1 ? S, ... sl,
of
?
an
ideal
Furthermore
to
3:
&* Jg generate
X
according Theorem
k*(rk)rk
1+
1?
rk
(mod
J)
(mod Y) .
0 if R # 0 and P*(r(E)) =
?,1+E2R
0
Therefore0*(r(E)$(R)
r(E.9)(mod 93).
The dual statement is that
+
z(E1)
J
p(El, O)p(E2,0) = + p(El + E2, 0),
whereit is understoodthat the rightside is zero if the sequences E1 and
E, bothhave a " 1 " in the same place. Thus the basis elements p(E, 0)
multiplyas a Grassmannalgebra.
Similar argumentsshow that the productp(E, 0) p(0, R) is equal to
166
JOHN MILNOR
p(E, R). Fromthis the firstassertionof 4a followsimmediately.
Computationof ' RQk: We mustlookforbasis elementsr(E)$(R') such
that k*(z-(E)$(R'))containsa term
(non-zeroconstant).-(R) ? zInspection shows that the only such basis elements are red(R), rT+l,(R
(pk.
0,
...
)),
-
rk+2$(R
(0,
...
0,
pk.
)),
-
the correetc. Furthermore
...
spondingconstantsare all + 1. This provesthat
Q=
9AQk
-
+
Qk+llI)R-(up,
.*-)
+
.
and completesthe proofof Theorem4.
To completethe descriptionof A* as a Hopfalgebra we mustcompute
sb*.
the homomorphism
LEMMA9. The followingformulas hold
=
0*(Qk)
t
(gR)
(For examples(91Q11) =
Qk
= ,R1+R2=R
_9O11
1+ 1
Qk
yR1
01 + 1 0
?
SOR2
011+
&J 01
o + vJ001
'
pul
,z9j01.)
REMARK.An operation9 e A* is called a derivationif it satisfies
sa- _E
j) = (dao) x,
P+
(_ 1)dimodimaat as-.
This is clearly equivalent to the assertionthat 9 is primitive.It can be
shown that the only derivations in !S7* are the elements Q , Q1, **., SD',
99 91, !?7001
...
*
and their multiples.
7. The canonical anti-automorphism
As an illustrationconsiderthe Hopf algebra H*(G) associated with a
Lie group G. The map g -- g- of G intoitselfinducesa homomorphism
c: H*(G)
H*(G) which satisfies the following two identities:
(1) c(l) =1
aa&0 a", wheredima > 0, then ? ac(al') = 0.
(2) if 0b*(a)
Moregenerally,foranyconnectedHopfalgebra A*,,thereexistsa unique
c: A* - A* satisfying(1) and (2). We will call c(a) the
homomorphism
in the sense that
conjugateof a. Conjugationis an anti-automorphism
c(a,aCt)
=(
)dim a dima2
c(a,)(a,)
The conjugationoperationsin a Hopf algebra and its dual are dual homomorphisms.For details we referthe reader to [3].
For the Steenrodalgebra 9?* this operationwas firstused by Thom.
(See [5] p. 60). More precisely the operation used by Thom is 9
(
_ Jmdii
{ Cblah
167
THE STEENROD ALGEBRA AND ITS DUAL
If 0 is a primitiveelementof R9* thenthe definingrelationbecomes
0-1 + 1-c(8) = Oso thatc(8) = - 6. Thisshowsthatc(Qk) =-QkQ,
c(g1)
The elementsC(d ), n > 0, could be computedfromThom's
=- Jl.
identity
i,
a'-
)
tC(9
=;0
howeverit is easier to firstcomputethe operationin the dual algebraand
thencarryit back.
By an orderedpartitiona of the integern with lengthl(a) will be
meantan orderedsequence
(a(l), a(2),
*,
a(l(a)))
of positiveintegerswhose sum is n. The set of all ordered partitionsof
n will be denotedby Part (n). (For examplePart (3) has fourelements:
(3), (2,1) (1,2), and (1,1,1). In generalPart (n) has 2n-1 elements.) Given
an orderedpartitiona e Part (n), let o-(i) denotethe partialsum fa a(j).
LEMMA 10. In thedual algebra 9, theconjugatec(en) is equal to
(-)Wi
cEmePart~n
1_ ira5
(pTMf
n1(a)
;i
e-l + ES + $2i -e1$1P$*
becomes
identity
thedefining
$,
0pn-i(
,(Xn) == En=o
(For examplec(t3)
PROOF. Since
,=
nLjc($J)
=
0
This can be writtenas
C(n)
=
-n
-
-
C(t1)PnP
* -(n-1)
The requiredformulanow followsby induction.
we can use LemSince the operationw -- c(@) is an anti-automorphism,
ma 10 to determinethe conjugate of an arbitrarybasis element d(R).
Passing to the dual algebra 9* we obtainthe followingformula.(The
are somewhatinvolved,and will not be given.)
detailsofthecomputation
Givena sequence R = (r1, ..., rk .0 ...) considerthe equations
r1 =
( * )
for i = 1, 2, 3,
...
Eno=
(n)
EwEPart
P
a)
El
uy0
,
; where the symbol 3i, (J)denotes a Kronecker delta; and
wherethe unknownsy, are to be non-negativeintegers. For each solution Y to this set of equations define S(Y)
Sn
=
E
aEPart
=
W Ya
(s1, s8,
...
) by
e
(Thus s, = yi, s = y2+ y1,1,etc.) Define the coefficientb(Y) by
168
JOHN MILNOR
ii]
b(Y) = [y2,Y11][Y3,
Y21,Y12,Y111i
=
JnSn!
Ya s
la,
.
THEOREM5. The conjugatec(?A9R)is equal to
b(YOf)
99 s(y)
wherethesummationextendsoverall solutionsY to theequations(*).
To interpretthese equations(*) note that the coefficient
( _
]
E
)rl+***+rk
1(^) a. (j) p010)
of yp,in the ith equation is positiveif the sequence
a
= (Ml(),
a(l(a)))
*..,
contains the integeri, and zero otherwise. In case the lefthand side r,
is zero,thenforeverysequence a containingthe integeri it followsthat
ya = 0. In particularthis is true forall i > k.
As an example, suppose that k = 1 so that R = (r, 0, 0, ***). Then the
integersye,mustbe zero whenevera containsan integerlargerthanone.
Thus the only partitionsa which are left are: (1), (1,1), (1,1,1),
Therefore we have si = y1, s = y11,S3 = y111,
etc. The equations (*) now
reduceto the singleequation
r = si +
(1 + p)S2 +
(1 +
p +
p2)s3 +
*..
But this is just the dimensional restriction that dim JS = (2p - 2)s, +
(2p2 - 2)s8 + *-- be equal to dim '9ar = (2p - 2)r. Thus we obtain:
S where the
COROLLARY 6. The conjugate c( sfi r) is equal to (- 1)r E
having the correctdimension.(For example
sum extendsover all JS
2P+3
+
p+2,1
p+3
-
9
1,2)
8. Miscellaneous remarks
The followingquestion, which is of interestin the study of second
ordercohomologyoperations,was suggested to the author by A. Dold:
Whatis theset of all solutions6 e jte* to theequation6SY1 = 0 ? In view
of the results of ? 7 we can equally well studythe equation ja) 16 = 0.
The formula
'~la 1,r-l
implies that this equation
r
916
spannedby the elements
a)
withr, _-1
+
=r(1
r1r2
=
rj)?A)l+rir2.
0 has as solution the vector space
Q0'oQ1%l
. . .
(modp). The firstsuchelementis SD p-, and everyelement
169
THE STEENROD ALGEBRA AND ITS DUAL
will also be a solution.Now the identity
of the ideal ~93 P-Ijj7*
?)
p1*
a9)
1, S1)?3
(p-
S1S2.=
sl+p -1,S2.
*
if s, E 0 (modp)
S1+p-1, S2
if s,
0
_1
0 (modp)
shows that everyelement r)r2r. Q0S0 ... with r, = -- 1 (modp) actually
belongsto the ideal. Applyingthe conjugationoperation,thisprovesthe
following:
0 has as solutions the elementsof
PROPOSITION1. The equation 6591
-
the ideal
* J7
-i
Q oSQl
An additive basis is given by the elements
...
1 (modp)
with r --
C(*S*/)9 rlr2 .)
Next we will studycertainsubalgebrasof the Steenrodalgebra. Adem
Let
shown that J/* is generated by the elements Q0, P91, ?>P, *-.
/P,
9*(n) denote the subalgebra generated by Q0, 5)1, **
PROPOSITION2. The algebra5*(n) is finitedimensional,havingas basis
thecollectionof all elements
QOo
...
Qn en
rl ,n
pn, r2
<
Pn-1
.
whichsatisfy
r, <
rn < P
Thus a/* is a union of finitedimensionalsubalgebras S a(n). This
clearlyimpliesthe following.
COROLLARY7. Everypositivedimensionalelementof 5s * is nil-potent.
It would be interestingto discovera completeset of relationsbetween
the given generatorsof 9) *(n). For n 0 there is the singlerelation
1 there
1)dim-adimbba. For n
[Q0,Q0]= 0, where[a, b] standsforab are threenew relations
[Q0,[y91,
Qo]]
0
Q]]
=
0
and
(a)
1)
=
0
.
For n = 2 thereare the relations
[i1,
[tP,
7
1]]=
0,
and
[J/)7,
(,LI
I')
[ -
'?P9
L) I[1P
1]]
Iy
0
ll}P-l
= 0
as well as several new relationsinvolvingQo. (The relations()p)2p
and [ 9 )ip = 0 can be derivedfromtherelationsabove.) The author
has been unable to go furtherwiththis.
PROOF OF PROPOSITION 2. Let 7v(n) denote the subspace of 9'*
spannedby the elementsQSo ...**Qnen~r,. rn whichsatisfythe specified
restrictions.We willfirstshow that 1V(n) is a subalgebra.Considerthe
170
JOHN M LNOR
product
?r.
sn =
rnS
b(X)iiP
(X)-(s1,
jR(X)=(rs,
T(X)
wherebothfactorsbelongto ??(n). Supposethatsometermb(X)?9 tlt2
on therightdoes not belongto J%(n). Then t, mustbe > pn+l-l forsome
1. If x1o,xl,1 ..., xo,were all < pn+,-z thenthe factor
to!
..
x10! **x01!
would be congruentto zero modulop. Thereforex}, > pn+1-lfor some
i + j = 1. If i > 0 this impliesthat
ri =
pjxij >
Aj
pjpn+l-z = pn+l-i
whichcontradictsthe hypothesisthat ?9dri .
0, j = 1, then
Si
=
En
Xij >
pkl+l-
=
e .7(n).
Similarlyif i
pk+1-j
whichis also a contradiction.
Since it is easily verifiedthat JV(n)Qk C V(n) fork < n, thisproves
that 37(n) is a subalgebraof A*. Since _V(n) contains the generators
of 97*(n), this implies that 33(n) D 9*(n).
To completethe proofwe mustshow that everyelementof s7(n) belongs to 9?*(n). Adem's assertion that ?0-* is the unionof the 9;'*(n)
impliesthat everyelementof ?0k withk < dim(9X ) automaticallybelongs to ?9*(n). In particularwe have:
Case 1. Every element ?0 ..Pi in ?7(n) belongs to 9*(n).
Ordering the indices (r1, . . ., rn) lexicographically from the right, the
productformulascan be writtenas
X7l
i..rn
?
S1...
=
(r1, Si) * -.(rn,
sn
)&9
rl+S1
rn+Sn+
(higherterms)
Given 0 t1"fl e 37(n) assume by inductionthat
(1) every ?0 l1 rn e, 3(n) of smaller dimensionbelongsto S0*(n), and
belongsto
(2) every"higher" q rl rne (n) in thesamedimension
597*(n). We will provethat ?0 11"' e .9*(n).
Case 2. (t1 ... tn) = (0 ... 0ti0 ... 0) where tj is not a power of p.
Si
Choose ri, si > 0 with r, + s, = ti, (ri, si) t 0. Then ?0.. ri0..
(ri, sj)?9?0t- + (higher terms).
Case 3. Both tj and t, are positive,i < j. Then
...
9t
_)-ti+1'tn
=
?0t1...
tn +
(higherterms).
In either case the inductivehypothesisshows that 0" tn belongsto
97*(n). Since Q0, **, Qnbelong to 9*(n) by Corollary 3, this completes
.
171
THE STEENROD ALGEBRA AND ITS DUAL
the proofof Proposition2.
Appendix 1. The case p= 2
All the results in thispaper apply to the case p = 2 aftersome minor
changes. The cohomologyringof the projectivespace &' N is a truncated
polynomialring with one generatorca of dimension1. It turnsout that
2*(a) e H*(PN, Z2) ? A& has the form
at
4Co +
g
a2 (8) C1 +***+
3
where o = 1 and where each Ci is a well defined element of 9.2 -1. The
algebra 54 is a polynomialalgebra generatedby the elementsC C2 *
Corresponding to the basis
{C171C2T2 ...
} for 94 there is a dual basis
for * These elementsSqrlr2 multiplyaccordingto the same for{SqR}
mulaas the d9)
X. The otherresultsof this papergeneralizein an obvious
way.
Appendix 2. Sign conventions
The standardconventionseems to be that no signs are insertedin formulas 1, 2, 3 of ? 2. If this usage is followedthen the definitionof A*
However Lemmas 2 and 3 stillholdas stated,and
becomesmoredifficult.
Lemma 4 holdsin the followingmodifiedform.
LEMMA 4'. If 2*(Qa)=
ai (? w thenfor any 0 e C*:
(
)'d(d-1)+ddima;
<0
>
whered = dimd.
i by the equation
It is now necessaryto definerie -41
A*(a) = a (& 1-
-z P(9 r0
(0- i-
- --
Otherwisethereare no changes in the resultsstated.
PRINCETON UNIVERSITY
RE FERENCES
1. J. ADEM, The relations on Steenrod powers of cohomology classes, Algebraic geometry
and topology, Princeton University Press, 1957, 191-238.
des operationsde Steenrod,Comment. Math. Helv., 29(1955),
2. H. CARTAN, Sur l'iteiration
40-58.
3. J. MILNOR and J. MOORE, On the structure of Hopf algebras, to appear.
classes, Proc. Nat. Acad. Sci. U.S.A.,
4. N. STEENROD, Cyclicreducedpowersofcohomology
39 (1953), 217-223.
Comment. Math.
5. R. THOM, Quelques propri~tesglobales des varietes di flerentiables,
Helv., 28 (1954), 17-86.