The Steenrod Algebra and Its Dual
John Milnor
The Annals of Mathematics, 2nd Ser., Vol. 67, No. 1. (Jan., 1958), pp. 150-171.
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Sun Jul 1 19:31:29 2007
ANNALS
OF MATHEMATICS
Vol. 67, No. 1, January, 1958
Printed in Japan
T H E STEENROD ALGEBRA AND ITS DUAL'
BY JOHN MILNOR
(Received May 15, 1957)
1. Summary
Let .9* denote the Steenrod algebra corrresponding to an odd prime
p. (See $ 2 for definitions.) Our basic results ($3) is that c'!* is a Hopf
algebra. That is in addition to the product operation
there is a homomorphism
satisfying certain conditions. This homomorphism +* relates the cup product structure in any cohomology ring H*(K, 2,) with the action of 9"
on H*(K, 2,). For example if .Pne 9'"(P-l) denotes a Steenrod reduced
pthpower then
The Hopf algebra
has a dual Hopf algebra
The main tool in the study of this dual algebra is a homomorphism
A* : H*(K, 2,)
-+
H*(K, 2,) @ 9j
which takes the place of the action of .3
" on H*(K, 2,). (See $4.) The dual Hopf algebra turns out to have a comparatively simple structure. In fact as an algebra (ignoring the "diagonal homomorphism" 4,) it has the form where E(ri , 2pL- 1)denotes the Grassmann algebra generated by a certain element r1 e X ? , I - ~and
, P(E,, 2p" 2) denotes the polynomial algebra
generated by 6,e . % , L - ~ .
I
T h e author holds an Alfred P. Sloan fellowship. 150 151
THE STEENROD ALGEBRA AND ITS DUAL
In 9 6 the above information about cir, is used to give a new description
of the Steenrod algebra 9 *. An additive basis is given consisting of elements
Q,,EOQIE1
with
by
E, =
0, 1;
Y, 2
... y'
'172'.'
0. Here the elements Q , can be defined inductively
Q" = 8,Q L +=
l '/j21~Q&
-
QLJ71~L
;
while each /"I""A.
is a certain polynomial in the Steenrod operations,"
of dimension
r1(2p - 2) r,(2p" 2) . . r,(2pk - 2) .
The product operation and the diagonal homomorphism in ,Y * are explicitly computed with respect to this basis.
The Steenrod algebra has a canonical anti-automorphism which was
first studied by R. Thom. This anti-automorphism is computed in 97.
Section 8 is devoted to miscellaneous remarks. The equation 8,-1'' = 0 is
studied; and a proof is given that .'/ * is nil-potent.
A brief appendix is devoted to the case p = 2. Since the sign conventions used in this paper are not the usual ones (see 3 2), a second appendix
is concerned with the changes necessary in order to use standard sign
conventions.
+
+
+
2. Prerequisites: sign conventions, Hopf algebras,
the Steenrod algebra
If a and b are any two objects to which dimensions can be assigned,
then whenever a and b are interchanged the sign (- l)dinl""'"'hwill be
introduced. For example the formula for the relationship between t h e
homology cross product and the cohomology cross product becomes
This contradicts the usual usage in which no sign is introduced. In the
same spirit we will call a graded algebra commutative if
Let A = (. , A _ ,, A,, A,, - ) be a graded vector space over a field F.
The dual A' is defined by Al, = Hom ( A _ ,, F). The value of a homomor, ) . I t is understood that
phism a' on a e A will be denoted by < a f a
<a',a = 0 unless dim a' + dim a = 0. (By an element of A we mean an
element of some A , .) Similarly we can define the dual A" of A'. Identify
>
-
T h l s has no relation to the generalized Steenrod operations
,-;/'I
defined by Adem.
152
JOHN MILNOR
each a e A with the element a" E A" which satisfies
a",a' ) = (- 1)dinla" dirn a'
< a fa
, )
(2)
for each a' E A'. Thus every graded vector space A is contained in its
double dual A". If A is of finite type (that is if each A, is a finite dimensional vector space) then A is equal to A".
Now if f : A -t B is a homomorphism of degree zero then f ': B'
A'
and f ": A" -t B" are defined in the usual way. If A and B are both of
finite type it is clear that f = f ".
The tensor product A @ B is defined by ( A@ B), =
j = n Ai @ B j ,
where " " stands for "direct sum ". If A and B are both of finite type
and if A, = B, = 0 for all sufficiently small i (or for all sufficiently large
i) then the product A @ B is also of finite type. In this case the dual
( A@ B)' can be identified with A' @ B' under the rule
<
-+
xi+
In practice we will use the notation A, for a graded vector space A
satisfying t h e condition Ai = 0 for i < 0 . The dual will then be denoted
by A* where An = A'-,= Hom (A,, F). A similar notation will be used
for homomorphisms.
By a graded algebra (A,, 4,) is meant a graded vector space A, together with a homomorphism
+,:
It is usually required
1 e A,. The algebra is
By a connected Hopf
algebra with unit (A,,
A,@A,-tA,.
that #,: be associative and have a unit element
connected if the vector space A, is generated by 1.
algebra (A,, +,, 4,) is meant a connected graded
+,), together with a homomorphism
satisfying the following two conditions.
2.1. 4, is a homomorphism of algebras with unit. Here we refer to the
product operation 4, in A, and the product
(al@I a,) (a,@ a,) = ( - l)diln
'2
'3 (a1a,) @ (a,.a,)
in A,@ A,.
2.2. For dim a > 0 , the element &,(a) has the form a @ 1 1 @I a
b, @ ci with dim b , , dim c, > 0.
A%propriate concepts of associativity and commutativity are defined,
not only for the product operation #, , but also for the diagonal homomorphisms 4,. (See Milnor and Moore [3]).
To every connected Hopf algebra (A,, #, , 4,) of finite type there is as-
+
+
T H E STEENROD ALGEBRA AND I T S DUAL
153
sociated the dual Hopf algebra (A*,$*, +*), where the homomorphisms
are the duals in the sense explained above. For t h e proof that the dual is
again a Hopf algebra see [3].
d
GxG
(As an example, for any connected Lie group G t h e maps G
P
G give rise to a Hopf algebra (H,(G), p , , d,). The dual algebra
(H*(G), v,p*) is essentially the example which was originally studied
by Hopf .)
For any complex K the Steenrod operation LY'~ is a homomorphism
-
-
The basic properties of these operations are the following. (See Steenrod
L41.1
2.3. Naturality. If f maps K into L then f * S =
LgJYf".
2.4. For a e H3(K,Z,), if i > j / 2 then dJkx = 0. If i = j / 2 then
= an.If i = 0 then ../'la = a .
2.5. ~ ~ " ( a v p )
Piaud3j$.
We will also make use of the coboundary operation 6: H9(K,2,) -+
H9+'(K,2,) associated with the coefficient sequence
=x,+,=,
The most important properties here are
2.6. 66 = 0 and
2.7. 6(a v /3) = (6a) v P (- 1 ) " ' " ' " ~ 6P, as well as the naturality
condition.
Following Adem [I] the Steenrod algebra .Y* is defined as follows.
The free associative graded algebra ./r*generated by the symbols 6,
y>0
, /'I,
acts on any cohomology ring H*(K, 2,) by the rule
(OIOL O,).a = (8,(8, .. (8,a) ... )). (It is understood that 6 has dimension 1 in 3%
and that ,lJi
has dimension 2i(p - I).) Let 3"denote
the ideal consisting of all f e F* such t h a t fa = 0 for all complexes K
and all cohomology classes a e H*(K, 2,). Then 9* is defined as the
quotient algebra 9-*j,F*. It is clear that 9* is a connected graded as'* is not commutative.
sociative algebra of finite type over 2,. However 5
(For an alternative definition of the Steenrod algebra see Cartan [2].
The most important difference is that Cartan adds a sign t o the operation
8.1
The above definition is non-constructive. However it has been shown
+
-
154
JOHN MILNOR
by Adem and Cartan that 9,"
is generated additively by the "basic
monomials ''
where each
E~
is zero or 1 and
Furthermore Cartan has shown that these elements form an additive
basis for 9* .
3. The homomorphism 4"
LEMMA
1 . F o r each element 8 of 9 * there i s a unique element +*(8)=
C O',@ 8;' of 5'""@ 9* such that the identity
x
8(a "3) =
(- 1)d'"Oi' d l l n a 8I(a) O;'(P>
i s satisjied for all complexes K and all elements a , 13 e R " ( K ) . Furtheymore
i s a r i n g homomorphism.
(By an "element" of a graded module we mean a homogeneous element.
The coefficient group 2, is to be understood.)
I t will be convenient to let 9* @ 9"act on H * ( X )@ H * ( X ) by the
rule
(of @ 8")(a @ p) ( - 1)di'n8" d l l n a @ ( a )@ d"(P) .
Let c: H * ( X )@ H * ( X )-. H " ( X ) denote the cup product. The required
identity can now be written as
&(a @ p ) = c+"(O)(a@ 9) .
PROOFO F EXISTENCE. Let 5 2 denote the subset of ,5'* consisting of
the required identity
all 19 such that for some p E Y * @ 9"
O P ) = cp(a O P )
is satisfied. We must show that @
, = .9*.
The identities
6(a v,?)
= 6a wl!
and
/ ' n ( a u p )=
+ (-
l ) d n n aw
a 8p
xi+,=,
clearly show that the operations 6 and
to
then the identity
7 J L Uu
PJi3
lJn
belong
to .P. If (I,, O2 belong
THE STEENROD ALGEBRA AND ITS DUAL
155
show that B,8, belongs to A.Similarly 2 is closed under addition. Thus
R is a subalgebra of .i'* which contains the generators 6, .lJn
of 9*.
This proves that 2 = .Y*.
PROOFO F UNIQUENESS. From the definition of the Steenrod algebra we
see that given an integer n we can choose a complex Y and an element
r e H * ( Y ) so that the correspondence
defines an isomorphism of ,Y into H k f i ( Y )f or i 5 n . (For example take
Y = K(Z,, k ) with k > n.) I t follows that the correspondence
defines an isomorphism j of ( 9* @ 9 *)$ into H2""Y x Y ) for i 5 n .
Now suppose that p l y p, e .9* @ 9* both satisfy the identity Bc(a @ P )
= cp,(a @ /3) for the same element 8 of .Y ". Taking X = Y x Y , a =
r x 1, p = 1 x r , we have cp,(a @ p) = j(p,). But the equality j(pl) = j(p,)
with dim p, = dim p, = n implies that p, = p,. This completes the uniqueness proof. Since the assertion that +* is a ring homomorphism follows
easily from the proof used in the existence argument, this completes the
proof.
As a biproduct of the proof we have the following explicit formulas:
THEOREM
1. The homomorphisms
give 9* the structure of a Hopf algebra. Furthermore the product 4" is
associative and the "diagonal homomorphism" +* i s both associative and
commutative.
PROOF.I t is known that (9*, 4") is a connected algebra with unit ; and
that +* is a ring homomorphism. Hence to show that .'/ * is a Hopf algebra it is only necessary to verify Condition 2.2. But this condition is
clearly satisfied for the generators 8, and ,/'" of ,Y *, which implies that
it is satisfied for all positive dimensional elements of .9*.
I t is also known that the product 4" is associative. The assertions that
4" is associative and commutative are expressed by the identities
156
JOHN MILNOR
(2)
T+*B = +*B
for all 8, where T(8' @ 0") is defined a s (- l)d'"e' 8'' @ 8'. Both
identities are clearly satisfied if 8 is one of the generators 8 or :yJn of
.9". But since each of the homomorphisms in question is a ring homomorphism, this completes the proof.
As an immediate consequence we have: COROLLARY
1. There is a dual Hopf algebra with associative, commutative product operation.
4. The homomorphism A*
Let H, , H * denote the homology and cohomology, with coefficients 2, ,
of a finite complex. The action of 9* on H * gives rise to a n action of
9* on H, which is defined by the rule:
for all p E H, , B E 9*, a E H * . This action can be considered as a homomorphism
A*: H , @ 9 * -+H*.
The dual homomorphism
A*: H * -,H * @ .9*
will be the subject of this section.
@ 9' H, has a
Alternatively, the restricted homomorphism Hnti
dual which we will denote by
-+
In this terminology we have
carrying Hn into C,Hn @ .%'. The condition that H * be the cohomology of a finite complex is essential here, since otherwise A* would be a n
infinite sum.
The identity
~ ( 4 8 2=
) (~8,)8,
can easily be derived from the identity (B,8,)a = B,(B,a) which is used to
define the product operation in 9 *. In other words the diagram
THE STEENROD ALGEBRA AND ITS DUAL
--
[A* 0 1
A*
H*@ 9"
is commutative. Therefore the dual diagram
Id*
H,
Ti* @ 1
Ti*
is also commutative. Thus we have proved :
LEMMA2. T h e identity
(A* 0l)A*(a)= (1 @ $*)A*(a)
holds for every a e H*.
The cup product in H* and the
product in .%induce a product operation in H* @
LEMMA3. T h e homomorphism A* : H* -+ H*@ 9
*
i s a ring homomorphism.
PROOF.Let K and L be finite complexes, let B be a n element of 9*,
and let +*(B)=
8; @ 8;'. Then for any a E H * ( K ) , P E H*(L) we have
d . ( a x p) =
(- 1)dllne;'dlln a B',a x B;',l. Using the rule
+,
s.
z
we easily arive a t the identity
In other words the diagram
H,(K) @ H,(L) @ 9%
@ 9%
4'
@
'
@
*'
H,(K) @ H,(L) @ 9*
= H,(K x L)@ 9*
11@~@1
H*(K)€3 9%
@ H*(L) @ 9%
11.
'* @ * *
H J K ) €3 H J L )
= H*(K x L)
is commutative (where T interchanges two factors as in 53). Therefore
the dual diagram is also commutative. Setting K = L , and letting d : K -,
K x K be the diagonal homomorphism we obtain a larger commutative
diagram
158
JOHN MILNOR
Now starting with a @ ,3 e H * @ H * and proceeding to the right and
up in this diagram, we obtain ]"(a w p). Proceeding to the left and up,
and then to the right, we obtain I*(a).I*(P). Therefore
which proves Lemma 3.
The following lemma shows how the action of .Y*on H * ( K ) can be
reconstructed from the homomorphism I*.
we have
LEMMA4. If A*(a) = C ai @ w, then for any 8 e 9'"
ea = C ( - 1)dimaidim w . (8,
.
PROOF.By definition
<P, d a > = <pO,a) = < A * ( P O ~ ) , ~ >
= < P Od , ]*a) = C
< P , a , > < d ,w , )
*
.
Since this holds for each p e H, , the above equality holds.
@ H * -+ H *
REMARK.To complete the picture, the operation 7": 9"
has a dual 7,: H, -. ,S/j @ H, . Analogues of Lemmas 2 and 4 are easily
obtained for 7,. If a product operation K x K -t K is given, so that H,,
@ H,, have product operations; then a straightforward
and hence 9;
proof shows that 7, is a ring homomorphism. (As an example let K denote the loop space of an (n 1)-sphere, or an equivalent CW-complex.
Then H,(K) is known to be a polynomial ring on one generator p E Hn(K).
The element
+
is evidently equal to 1 @ p. Therefore 7,(pk) = 1@ p" for all k. Passing
to the dual, this proves that the action of .ir * on H ' Y K ) is trivial.)
5. The structure of the dual algebra 9
;
As an example to illustrate this operation A* consider the Lens space
X = SLJtl/Z,where N is a large integer, and where the cyclic group Z,
acts freely on the sphere SZS'l.T hus X c a n be considered as the ( 2 N 1)skeleton of the Eilenberg-MacLane space K(Z,, 1). The cohomology ring
H * ( X ) is known to have the following form. There is a generator a E
H1(X) and H ( X ) is generated by 8 = 8a. For 0 2 i _I N, the group
H2i(X)is generated by Pi and HLi+l(X)is generated by a p t .
The action of the Steenrod algebra on H*(X) is described a s follows.
I t will be convenient to introduce the abbreviations
+
159
THE STEENROD ALGEBRA AND ITS DUAL
satisfies M,i3 =
However i f 8
LEMMA5. T h e element M, E .9'pk-3
i s a n y monomial in the operations 6, ,P1, P " .. which i s not of the
VJpY31
t hen 03 = 0. S i m i l a r l y (M,6)a = ppk but Ba = 0
f o r m ,'/'"'-l
i f 0 i s a n y monomial in the operations 6, q5I , q5"
which does not have
_Y16o r 0 = 1.
the f o r m 0 = ";7yk-1
PROOF.It is convenient to introduce the formal operation @
, =1
/'I
9 j 2
. It follows from 2.4 that 9 j p = 3 + 3". Since q5is a
ring homomorphism according to 2.5, it follows that Bp" ((B P")". In
particular if i = p' this gives /lijlir = (3 + / ~ P ) P ' = ,3~' i? p r + l . In other
words
3'.
if j = 0
qjjpuT =
3 n T + l if j = pr
..
..
+
+
+ ..
+
+
1
otherwise .
Since 6p = ip-lap = ipz-'Ma = 0 it follows that the only nontrivial operation 6 or 9 j j which can act on pnr is P p ' . Using induction, this proves
the first assertion of Lemma 5. To prove the second it is only necessary
to add that ?'a = 0 for all j > 0, according to 2.4.
Now consider the operation A*: H * ( X ) -+ H * ( X ) @ % .
LEMMA6. T h e element R*a has t h e f o r m a @ 1 3 @ r, 3'' @ r,
... +Ppr @ rTwhere each r, i s a well defined element of 9 & ~ -and
~ , where
pJ i s the largest pmoer of p w i t h p' 5 N. Similarly R*P has the f o r m
0
+
P @ Eo + P" @ E l
+
+/3"
+
+
@ Er ,
where E, = 1, and zc~hereeach E, i s a well defined element of x p k - , .
PROOF.For any element 8 of 9;
Lemma 5 implies that BP = 0 unless
i is the dimension of one of the monomials M o ,M I ,
: that is unless i
has the form 2pk - 2. Therefore, according to Lemma 4, we see that 2'3
= O unless i has the form 2pk - 2. Thus
...
A*? = Jo(p) + A2p-Yt3) +
... +
Rzpr-2
(8) .
Since Rmk-"p) belongs to H*,(X) @ .9&r-2, it must have the form t3*k @ E,
for some uniquely defined element E, . This proves the second assertion
of Lemma 6. The first assertion is proved by a similar argument.
REMARK.T hese elements E, and r, have been defined only for k S r =
[log, N ] . However the integer N can be chosen arbitrarily large, so we
have actually defined E, and T , for all k 2 0.
Our main theorem can now be stated as follows.
THEOREM
2. T h e algebra 9*i s the tensor product o f the Grassmann algebra generated by T ~T ,~ , and the polynomial algebra generated by El,
E,,
.-..
..
160
JOHN MILNOR
The proof will be based on a computation of the inner products of moand 6. The folnomials in ri and t, with monomials in the operations .7"
lowing lemma is an immediate consequence of Lemmas 4, 5 and 6.
LEMMA7. The inner product
equals one, but (8, 6,) = 0 if B is any other monomial. Similarly
>
but (8, r , = 0 if 8 is any other monomial.
Consider the set of all finite sequences I = (E,, r , , El, r,,
E~ = 0 , l and rt = 0, 1, 2,
. For each such I define
..
.
- ) where
Then we must prove that the collection (w(I)) forms a n additive basis
for
For each such I define
c
~
*
.
where
S,
=
(El
+ ri)pi-l, .,
''
S,
=
CT=,
(Ei + r i ) ~ ' - ~
I t is not hard to verify that these elements B(I) are exactly the "basic
monomials" of Adem or Cartan. Furthermore B(I) has the same dimension as w(I). Order the collection { I ) lexicographically from the right.
(For example (1, 2, 0, - - ) < (0, 0, 1, ...).)
LEMMA8. The inner product 8(1), w(J) i s equal to zero ij- I < J and
*lifI=J.
Assuming this lemma for the moment, the proof of Theorem 2 can be
completed as follows. If we restrict attention to sequences I such that
.
<
>
dim o ( I ) = dim 8(I) = n ,
<
>
then Lemma 8 asserts that the resulting matrix B(I), w(J) is a nonsingular triangular matrix. But according to Adem or Cartan the elements B(I) generate .ir ". Therefore the elements o ( J ) must form a basis
for .i/,;which proves Theorem 2. (Incidentally this gives a new proof of
Cartan's assertion that the @(I)are linearly independent.)
PROOF
O F LEMMA
8. We will prove the assertion (B(I), w(I) ) = + 1
by induction on the dimension. I t is certainly true in dimension zero.
Case 1 . The last non-zero element of the sequence 1 = (E, , r, ,
, E,-, ,
r , , 0, . - - ) i s r , . Set I t =(E,,Y,,. - . , ~ , - , , r , - 1, 0,
so that o ( I ) =
to(I')E,. Then
.
-
0
.
)
THE STEENROD ALGEBRA AND ITS DUAL
...
.
where the summation extends over all sequences ( E : ,
, s;) and ( E ; ,
,
s z ) with EL
&I' = E , and s: sl' = s,. Substituting this in the previous
expression we have
+
+
( B(I),o ( I )) = C
+ < 8'6
. 9'' i , o ( l f>) < Oe6' ... .-Y
Ek
'?';',
>.
But according to Lemma 7 the right hand factor is zero except for the
special case
8';' ... /' = _(3 8 - I . p P . 3 ) I ,
"
..
in which case the inner product is one. Inspection shows that the corresponding expression 6% . " on the left is equal to B(If) and hence
<
. /'
<
;
>
that B(I),o ( I )) = f B ( I ' ) , o(I') = -+ 1.
Case 2. The last non-zero element of I = (E,, Y,, . , r, , E , , 0 ,
&l,
= 1. Define I' = ( E , , r , ,
, r, , 0 , .) so that
..
..
.
...) is
Carrying out the same construction as before we find that the only nonvanishing right hand term is .Ppk-' . y l O , T, = 1. The corresponding left hand term is again < 6 ( I f )o, ( I f ) ) ; so that < 6 ( I ) ,w ( I ) >= +
( @ ( I r )w,( I ' ) ) = -+ 1, with completes the induction.
The proof that B(I),o ( J ) = 0 for I < J is carried out by a similar
induction on the dimension.
Case l a . The sequence J ends with the element rk and the sequence I
ends a t the corresponding place. Then the argument used above shows
that
<
>
..
>
<
Case l b . The sequence J ends with the elements r, , but I ends earlier.
Then in the expansion used above, every right hand factor
,yEb',</~
8%'-I,Ek
<
8;'
>
...
is zero. Therefore ( @ ( I )o, ( J ) ) = 0.
Similarly Case 2 splits up into two subcases which are proved in an
analogous way. This completes the proof of Lemma 8 and Theorem 2.
To complete the description of .9*as a Hopf algebra it is necessary to
compute the homomorphism
. But since is a ring homomorphism it
+,
+,
162
J O H N MILNOR
is only necessary to evaluate it on the generators of S , .
THEOREM3. T h e following formulas hold.
+*(Ed
=
z:=o
Eplt
+*(rk)= Cf=oELi
0rt
O Et
+
~k
@1.
The proof will be based on Lemmas 2 and 3. Raising both sides of the
equation
=
C B"' O E,
to the power p%e obtain
A*(#")
=
pp"'
0E?;I .
Now
(A*
l ) A * ( P ) = (A* @ 1)
= Ci,j8""'
z8""
0 EfY( 0Et .
Comparing this with
(1 0+*)A*(P) = C ppk 63 +*(Ed
We obtain the required expression for +,([,).
Similarly the identity
(A* @ l)A*(a)= (1 0+*),?*(a)
can be used to obtain the required formula for +,(re).
6. A basis for 9"
Let R = (r,,r,, ..) range over all sequences of non-negative integers
which are almost all zero, and define E(R)= E:lEITa . . Let E =
(Eo , E l ,
.) range over all sequences of zeros and ones which are almost all
zero, and define T ( E )= r,%~r,'l
. Then Theorem 2 asserts that the
elements
.
...
Hence there is a dual basis {p(E,R ) }for
form a n additive basis for 9,.
9".
That is we define p(E,R) e .9
by
*
1 if E = E ' , R = R '
0 otherwise.
Using Lemma 8 it is easily seen that p(0, ( r ,0 , 0 , .)) is equal to the
Steenrod power ..Pr. This suggests that we define?-YR as the basis
element p(0, R) dual to E(R). (Abbreviations such as .GJolin place of
6
'"'1 will be frequently be used.)
..
c 0 3
O j h
THE STEENROD ALGEBRA AND ITS DUAL
163
Let Q, denote the basis element dual to r,. For example Qo =
p(1, 0 , .), 0 ) is equal to the operation 6. It will turn out that any basis
,P R .
element p(E, R) is equal to the product & Q,'oQIE1
THEOREM
4a. The elements
.
..
form a n additive basis for the Steenrod algebra .i/ * zchich is, u p to sign,
The elements Q, e .Y 'P'-'
dual to the known basis {r(E):(E)} for
generate a Grassmann algebra: that is they satisfy
z.
They permute with the e l ~ m e n t s3' according to the rule
(By the difference (r, , r 2 , - ) - (s, , s,, - ) of two sequences we mean
the sequence (r, - s,, r, - s,, .). I t is understood, for example, t h a t
. /'R-(~"O,...)is zero in case r, < pk.)
As a n example we have the following where [a, b] denote the " commutator " ab - ( - 1)"'" a c"" 0 ba .
COROLLARY
2. The elements Q, e 9i/-2pk-'
can be defined inductively by
the rule
Qd= 8 , Qk+l= [ 9 p k , Q t I .
..
To complete the description of 9 * a s a n algebra it is necessary to find
the product l J K
/'&.
Let X range over all infinite matrices
of non-negative integers, almost all zero, with leading entry ommitted.
), S ( X ) = (s,, s,,
-), and
For each such X define R ( X ) = (r, , r , ,
T(X) = ( t , , t,, ...), by
(weighted row sum),
rl = C, p'x,,
(column sum),
sj =
xLj
t n = CL+l=n
xlj (diagonal sum).
..
EL
Define the coefficient b(X) =
THEOREM
4b. The product
TI t,! / nx,,!.
. L ~ ' ~ . is
L /equal
'~
to
164
J O H N MILNOR
where the sum extends over all matrices X satisfying the conditions R ( X )
= R , S ( X ) = S.
As a n example consider the case R = ( r ,0 , -), S = (s,0 , -). Then
the equations R ( X ) = R, S ( X ) = S become
...
...=
x l O f pxI1+
= r , x t j = O for i > l ,
x ~ ~ + x ~ ~ S+,
x i j = O for j > 1 , respectively.
Thus, letting x = x,,, the only suitable matrices a r e those of the form
with 0 5 x 5 Min (s,[rip]). The corresponding coefficients b ( X ) are the
binomial coefficients ( r - px, s - x). Therefore we have
3. The product 9'99 s equal to
COROLLARY
The simplest case of this product operation is the following
4. If r , < p, r , < p,
COROLLARY
then 9 R9= ( r , , sl)(r2,s ,)
.
g
j
K+S
As a final illustration we have :
COROLLARY
5. The elements 9( 0 " ' 0 ' 0 "'I can be defined inductively by
The proofs are left to the reader.
4b. Given any Hopf algebra A, with basis {a,)
PROOFOF THEOREM
the diagonal homomorphism can be written as
$*(ad = C j , k cika, O a,
The product operation in the dual algebra is then given by
a ~ a k = $,*(cJJ
g a*) =
(- l ) d i m a ' d i n l a k cjkai ,
xi
where {ai} is the dual basis. I n carrying out this program for the algebra
.'/; we will first use Theorem 3 to compute &(E(T))for any sequence
T = ( t , , t,, --.).
Let [i,, i,,
, i,] denote the generalized binomial coefficient
(i,+ i, +
. - -+ ik)!,lil!i2!
so that the following identity holds
- . a i l , !
;
THE STEENROD ALGEBRA AND ITS DUAL
+ .. + ~
(~1
xil+...
k =) ~
+tk=n
[il,
, i,Iy:l
. Y$
Applying this to the expression
we obtain
+
+
, x,, satisfying xi , - I 2 0, x,,
x,,
summed over all integers x,, ,
= t , . Now multiply the corresponding expressions for k = 1 , 2, 3,
.
Since the product [x,, , xO1][x,, , xll, x,,] [x,, ,
, xO3] is equal to b(X),
we obtain
+*(E(T)) = CTL=,==
b(X)E(R(X)) @ E(S(X)) ,
summed over all matrices X satisfying the condition T ( X ) = X.
In order to pass to the dual +* we must look for all basis elements
r(E)E(T) such that +,(r(E)E(T)) contains a term of the form
(non-zero constant)-E(R) @ E(S) .
However inspection shows that the only such basis elements are the ones
((T) which we have just studied. Hence we can write down the dual formula
+*(PR
@ 9")
= CR(x)-E, S ( x ) - S b ( X ) 9 T ( X .)
This completes the proof of Theorem 4b.
PROOF
O F THEOREM
4a. We will first compute the products of the basis
. . The dual problem is to study the
elements p(E, 0 ) dual to r;or:l
: % -+ 9
;
@ S$ ignoring all terms in .Y* @ .9*which
homomorphism
involve any factor E, . The elements 1 @ El, 1 @ E,,
, El @ 1,
of
genemte a n ideal 3 Furthermore according to Theorem 3:
@
.
+,
+
(mod Y )
(
) = r @ 1 1@ r
(mod 3) .
$*(Ek) = 0
Therefore &(r(E)E(R) 0 if R f 0 and $,(r(E)) =
r(E,) (mod 2).
T he dual statement is that
-
xal+o,-a
* r(El) @
where it is understood that the right side is zero if the sequences El and
E, both have a " 1" in the same place. Thus the basis elements p(E, 0 )
multiply as a Grassmann algebra.
Similar arguments show that the product p(E, O)p(O, R ) is equal to
166
JOHN MILNOR
p(E, R). From this the first assertion of 4a follows immediately.
Computation of dln&,:
We must look for basis elements r(E)[(R') such
that +,(r(E)E(R1)) contains a term
(non-zero constant).E(R) 0r, .
Inspection shows that the only such basis elements are r,E(R), r, +,E(R (pk,0,
)), T*+~E(R
- (0, pk, 0,
)),
etc. Furthermore the corre1 . This proves that
sponding constants are all
+
and completes the proof of Theorem 4.
To complete the description of V *as a Hopf algebra we must compute
the homomorphism +* .
LEMMA9. The following formulas hold
+*(&,I = Q,
0 1 + 10Qk
0, l J.R ~
01 + 10 _j'Uil+
09
+*(PR)
=CRI+Ra=R
+
(For example +*(dJU1l)= gJ
J'O'
OJ1
/3u01
@
J O1.)
REMARK.An operation 0 e ,Y* is called a derivation if it satisfies
B(a v ,3) = (Ba) v 3
+ ( - l)"n'edim a - 8 3 .
a
This is clearly equivalent to the assertion that B is primitive. It can be
, 3' ',
shown that the only derivations in .Y * are the elements Q,, Q, ,
gJ0,1, 2 ° * 0 z 1 , and their multiples.
7. The canonical anti-automorphism
As a n illustration consider the Hopf algebra H,(G) associated with a
Lie group G. The map g -+ g-l of G into itself induces a homomorphism
c: H,(G) -+ H,(G) which satisfies the following two identities:
(1) c(1) = 1
(2) if +,(a) = C a: 0a;', where dim a > 0, then C a;c(al') = 0.
More generally, for any connected Hopf algebra A,, there exists a unique
homomorphism c: A, -+ A, satisfying (1) and (2). We will call c(a) the
conjugate of a. Conjugation is a n anti-automorphism in the sense that
al dim '2 c(a2)c(aL)
.
c(a1a,) = ( - l)dim
;
The conjugation operations in a Hopf algebra and its dual are dual homomorphisms. For details we refer the reader to [3].
For the Steenrod algebra .9
this
*operation was first used by Thom.
(See [5] p. 60). More precisely the operation used by Thom is B
(~(6).
-+
THE STEENROD ALGEBRA AND ITS DUAL
167
If 8 is a primitive element of 9"
then the defining relation becomes
8.1 l.c(B) = 0 so that c(8) = - 8. This shows that c(Q,) = - Q,, ~ ( 9 ~ ' )
- J'~. The elements c(,Y2;'"),n > 0, could be computed from Thom's
identity
+
however it is easier to first compute the operation in the dual algebra and
then carry it back.
By an ordered partition a of the integer n with length l(a) will be
meant a n ordered sequence
,
of positive integers whose sum is n. The set of all ordered partitions of
?z will be denoted by P a r t (n). (For example P a r t (3) has four elements:
(3), ( 2 , l ) (1,2), and ( l , l , l ) . In general P a r t (n) has 2"-l elements.) Given
an ordered partition a E P a r t (n), let a(i) denote the partial sum Cj:: a ( j ) .
the conjzbgate c(E,,) is equal to
LEMMA
10. In, the dual algebra
+
+
EaE? - E ~ E ~ E ? . )
(For example c(E,) = - E3 EIE:
PROOF.Since +,(En) = C E Oti-,@ i t , the defining identity becomes
This can be written as
The required formula now follows by induction.
) an anti-automorphism, we can use LemSince the operation w -+ ~ ( w is
ma 10 to determine the conjugate of an arbitrary basis element i(R).
Passing to the dual algebra p* we obtain t h e following formula. (The
details of the computation are somewhat involved, and will not be given.)
consider the equations
Given a sequence R = (r, , .. , r, , 0,
a )
; where the symbol 6,,,,, denotes a Kronecker delta; and
for i = 1 , 2 , 3 ,
where the unknowns y, are to be non-negative integers. For each solution Y to this set of equations define S ( Y ) = (s, , s,, - ) by
Sn
(Thus s1 = YI,8 2 = Ya
=
C
&Part Cn)
Ya;
.
+ % , I ,etc.) Define the coefficient b(Y) by
T H E STEENROD ALGEBRA A N D ITS DUAL
of the ideal L ~ ~ - ' Ywill
' * also be a solution. Now the identity
P-1,
SISz"'
= (p - 1, S,)JJ T ~ + P - F~2 , ...
JJ
c/j
=
1
0 if s, S 0 (mod p)
-/slsn-l,
sz ... if s, = 0 (mod p)
shows that every element ~7"1'%"'Q,'o . with r, -- -- 1(mod p) actually
belongs to the ideal. Applying the conjugation operation, this proves the
following :
PROPOSITION
1. The equation O,/jl = 0 has as solutions the elements o j
the ideal p * J'"-'. An additive basis i s given by the elements
Next we will study certain subalgebras of the Steenrod algebra. Adem
shown that 9 * is generated by the elements Q,, J", d'", . Let
.Yd*(n)denote the subalgebra generated by Q,, J",
, J, ""-I
PROPOSITION
2 . The algebra .i/ *(n) isfi~zitedimensional, having as basis
the collection of all elements
.
r, < pn, r, < pi'-',
., r, < p .
Thus .'/." is a union of finite dimensional subalgebras .Y" ( n ) . This
clearly implies the following.
COROLLARY
7. Every positive dimensional element of '/ * is nil-potent.
It would be interesting to discover a complete set of relations between
the given generators of .ir *(n). For n = 0 there is the single relation
[QolQu] = 0, where [a, b] stands for ab - (- l)d""""'"h ba. For n = 1 there
are three new relations
[Q, , [./'I1
I, [
Qo]] = 0 , [S
Y",
Qo]] = 0 and (./jl)" = 0
.
For n = 2 there are the relations
as well as several new relations involving Q,. (The relations ( 7")"
=0
/"In = 0 can be derived from the relations above.) The author
and [
has been unable to go further with this.
2. Let -/(n) denote the subspace of .9*
PROOFOF PROPOSITION
spanned by the elements Qo'o ... QnEnC
/Jrl."r, which satisfy the specified
restrictions. We will first show that ,^/(n) is a subalgebra. Consider the
/ / I ) ,
170
JOHN M.LNOR
product
where both factors belong to 2 ( n ) . Suppose that some term b(X)Ptltz"'
on the right does not belong to ,W(n). Then t, must be 2 pn+l-zfor some
I . If xZ0,x ~ - ~...,
, ,x,,
, were all < pn+'-5 then the factor
t, !
'
" ' X0 1 .
X,,!
would be congruent to zero modulo p. Therefore x,, 2 pn+'-zfor some
i j = I. If i > 0 this implies that
+
which contradicts the hypothesis that
0, j = I , then
9"l""n
E ,?/(n).
Similarly if i =
which is also a contradiction.
Since it is easily verified that ,W(n)Qk c ,W(n) for k 5 n, this proves
Since ,-/(n) contains the generators
that ,Y(n) is a subalgebra of 9".
of P * ( n ) , this implies that ~ / ( n )19 " ( n ) .
To complete the proof we must show that every element of &(n) beis the union of the 9 * ( n )
longs to 9 " ( n ) . Adem's assertion that 9"
with k < dim ( 9 g " ) automatically beimplies that every element of 9'"
longs to 9 * ( n ) . In particular we have:
~ ( n ) belongs to 9 * ( n ) .
Case 1. Every element P O . . . O ~ ? S
,r,) lexicographically from the right, t h e
Ordering the indices (r,,
product formulas can be written as
9r l . . T n psl..'Sn = ( r , ,s,)
(r,, s , ) 9 ri+S1," ' 2 r n + s n
(higher terms) .
..
+
Given 9 ' t i " ' t n E -'J(n) assume by induction that
(1) every 9 J r 1 " " m e a/(%) of smaller dimension belongs to 9 * ( n ) , and
(2) every "higher " . 9 ' l " " n e s / ( n ) in the same dimension belongs to
9"(n). We will prove that 9'
ti"'tn E 9"*(n).
t,) = (0
Ot,O
0) where t, is not a power of p.
Case 2. (t,
Choose r , , s, > 0 with r, + s, = t,, ( r , , s,) g 0. Then 9 ' 0 " " i 9 0 " ' ~ =
(r, , s , ) ~ O " ' ~ L(higher terms).
Case 3. Both t, and t, are positive, i < j. Then
9 3 tl.-tLcLy~o...ot
~ + l " ' ~ n= 9tl"'tn + (higher terms) .
.
..
.
+
In either case the inductive hypothesis shows that gJtl"'tn belongs to
Q, belong to 9 * ( n ) by Corollary 3, this completes
9 * ( n ) . Since Q,,
.-a,
T H E S T E E N R O D ALGEBRA A N D I T S D U A L
171
the proof of Proposition 2.
Appendix 1. The case p = 2
All the results in this paper apply to the case p = 2 after some minor
changes. The cohomology ring of the projective space . J " 'is~ a truncated
polynomial ring with one generator a of dimension 1. I t turns out that
P(a)e H"(P"', 2%)
@ .Y" has the form
where C,, = 1 and where each Ci is a well defined element of 9 , 6 - , . The
algebra 9%
is a polynomial algebra generated by the elements TI, C,, . .
Corresponding to the basis {5:1C;2
there is a dual basis
.) for
tiply
to the same for{SqR)for Y " .These elements S ~ ~ 1 ~ ~ . . . m u laccording
The other results of this paper generalize in an obvious
mula as the yJH.
way.
Appendix 2. Sign conventions
The standard convention seems to be that no signs are inserted in formulas 1 , 2, 3 of $ 2 . If this usage is followed then the definition of (1"
becomes more difficult. However Lemmas 2 and 3 still hold as stated, and
Lemma 4 holds in the following modified form.
LEMMA4'. If i"(a)=
a , @ w i then for any 0 E 9":
where d = dim 0.
It is now necessary to define r, E 92,t-,
by the equation
Otherwise there are no changes in the results stated.
1. J. A D E MT, h e relations on Steenrod powers of cohomology classes, Algebraic geometry
and topology, Princeton University Press, 1957, 191-238.
2. H. CARTAN,S u r l'ithration des opdratwns d e s t e e n r o d , Comment. Math. Helv., 29 (1955),
40-58.
3. J . M I L N O Rand J . MOORE, On the structure of Hopf algebras, to appear.
4. N. S T E N R O D ,Cyclic reduced pozc9ersof cohomolog~classes, Proc. Nat. Acad. Sci. U.S.A.,
39 (1953), 217-223.
5. R. T H O M ,Quelques proprihte's globales des varihte's difl&erztinhles, Comment. Math.
Helv., 28 (1954), 17-86.