On the Mod 2 Cohomology of Certain H-spaces

by EMERY THOM.~S1), Univers i ty of California (Berkeley)

1. Introduction. Let X be a topological space and G an abelian group. We denote by H~(X; G) the i- th (singular) cohomology group of X with coefficients in G. Define an integer-valued funct ion va b y

ca (X) : least positive integer n such t h a t Hn(X; G) ~: 0 .

In case X is G-acyclic we set ca(X) : 0. For simplicity we write the func- tion as vr if G : Z , , the i n t e g e r s m o d r , r ~ 2 .

Suppose now tha t X is an H-space - t ha t is, X has a continuous multipli- cation with unit - and suppose (for the remainder of the paper) t h a t X satis- fies the following conditions.

(1) The integral cohomology groups of X are finitely genera ted in each dimension.

(2) H * ( X ; Z~) is finitely generated as a vector space, and is pr imit ively generated as a HOPF algebra.

In [9] we showed t ha t for such H-spaces, v~(X) = 2 ~ - 1 for some r ~ 0 . The purpose of this note is to prove

Theorem 1. Let X be an H-space satis[ying the above two conditions. Then

v~(X) = O, 1,3,7 or 15.

Moreover i/ X has no 2-torsion, then

v~(X) : vQ(X) --~ O, 1 , 3 or 7,

where Q denotes the rational numbers. Examples of H-spaces having these values of v2 are the various LIE groups

(for v2 = 0, l , 3) and the sphere of dimension seven (for v2 : 7). At present no H-space is known for which ~z : 15, and it seems unlikely t h a t this value can occur.

The method of proof for Theorem 1 is based on the work of ADAMS [1], [2]. For re la ted results in case X is a topological group, see C L ~ K [7].

I would like to t hank W. BROWDER for arousing m y interest in this problem and for his helpful suggestions about its solution.

2. Truncated polynomial algebras. We work in the category of commu- tat ive, associative, graded and connected algebras of finite type over Z~. For

1) This research was supported by the Jo~N SIMO~ GUGO~.~rm~M Foundation and by the Air Force Office of Scientific Research.

E ~ R Y THO~S On the Mod 2 Cohomology of Certain H-spaces 133

such algebras A we identify A0 with Z 2 and denote the complement b y A. Following w 1 we define v (A) to be the least positive integer n such tha t A~ =~ O.

Define

D 1 A = A , D ~ A = D n - I A . A ( n > 2 ) .

We call D 2 A the ideal of decomposable elements, and if an element a ~ D 2 A

we call it indecomposable. We will say tha t A is a truncated polynomial algebra o /he igh t n (for n ~_ 2)

if A ~ A ' /D '~A ' , where A' is a polynominal algebra. We call A finite- dimensional if it is so as a rood 2 vector space.

We now assume tha t A is a finite-dimensional t runca ted polynomial algebra of height three, and fur thermore t ha t A is an algebra over the rood 2 S T ~ - ROD algebra, ~ � 9 Such algebras were studied in [9], and we record here the results needed from tha t paper�9

Le t a e A be an indecomposable element and suppose tha t deg a = 2r+lk + 2 r, where r , k ~ 1. Then (see (2.1) and (3.3) of [9]):

(2.1) There is an indecomposable element a' o[ degree 2r+lk such that

a ~ Sq 2~(a') rood D~A .

(2.2) Sq 2~ (a) = 2: ~=o Sq 2~ (di), where d i ~ D 2 A . I n part icular, i / (D~A): ----

= 0, /or 1---- 2~+~(/~+ 1 ) - - 2'(0 ~ i < r ) , then Sq2' (a) ---- 0.

Here Sq ~ denotes the (mod 2) STEE~ROD operator. I t follows from (2.1) t ha t

(2.3) v ( A ) = 2 r, /or some r ~ O .

Let M = 2' , for some s ~ 2, and let a ~ AM+~r, where 0 < r < s -- 2.

Since A is an .~-algebra, a ~ : S q M + 2 r ( a ) . B y the ADEM relations [3]

SqM+S r = SqS r+l SqM-2 r + Sq M S q 2 ' ,

and since r ~ s - - 2, M - - 2 ~ - - 2 ~ + ~ e + 2 ~ for some e ~_ 1. Therefore b y (4.2) of [9],

Sq M-2" : ~ '=o Sq ~* (x~ ,

where ~ E L~', and consequently

(2.4) a ~ ---- Sq 2'+1 [Z ~=o Sq 21 ~ (a ) ] + Sq M SqZ'(a) .

For the remainder of the section we assume that A~ : 0 i / i is odd. We use (2.4) to prove

134 EM~Y THOMAS

($. 5) Lemma . Suppose that v (A) = 2q, where q > 3. Then A s = 0, /or

2 q <: 8 ~ 2 q ~- 2 q-1. M o r e o v e r i / S q 9q-1 A q =- O, then A~ ~- O, /or 2 q ~ t ~ 2 q+l. Notice t h a t the second assert ion follows a t once f rom 2.1 and the first asser-

t ion. B y 2. I (and the above hypothesis) i t suffices to prove the first assert ion jus t for the case s ----- 2 q -~- 2 r , where 1 ~ r ~ q -- 2. B y the definition o f ~(A) we have (D~A)~ = 0 for ?" < 2 TM . Le t a e A be on e lement of degree 2 q ~ - 2 r . Thus a is indeeomposable and hence if a 2 ----- 0, t hen a = 0, since A is a t runca ted polynomial algebra of height three. Set M = 2 q in 2.4. Then

deg ~ ( a ) = 2 q + l - 2 ~ ( 0 ~ i < r ~ q - - 2)

and hence each e lement c~ (a) is also indecomposable . Therefore b y 2.2,

Sq ~'(a) = 0, Sq ~' a , (a ) = 0 (0 < i < r)

and hence b y (2.4) a 2 = 0, showing t h a t A2q+~ , = 0 as asserted.

I n w 4 we use (2.5) to show tha t v~(X) _~ 15, i f X is an H-space sat isfying the two conditions in w 1.

Fo r the remainder of the section let A denote a fixed f ini te-dimensional t r unca t ed po lynomia l a lgebra of height 3, which is an algebra over ~ , such t h a t v ( A ) = 16. Then b y (2 .5) , Ale+~ ~ ----- 0 for 1 ~ i ~ 3. We show

( 2 . 6 ) L e m m a . (i) Sq ~ A u ----- 0, /or i = 1 , 3 .

(ii) Sq 2 j A ~ = 0, /or j - - - -2 ,3 ;

(iii) Sq ~ A ~ = 0, /or k = 1 , 2 , 3 .

Set D = D~A. Notice t ha t (2.2) implies a t once t h a t

SqSAu = Sq4A~ ---- Sq~Aso = 0 ,

since D s = 0 for ?" ~ 32. Thus to comple te the p roof of (2.6) (i) we m u s t show t h a t S q ~ ( a ) = 0, for a EA~a. Sett ing b = Sq~'(a), we have b ~ = = Sq ~6 (b). B y the ADaM relat ions [3],

Sq2a = Sq a Sq ~ Sq ~~ § ~ Sq I -~- ~x z Sq z ,

where ~ , o% r ~ ' . Moreover Sq ~ Sq ~ : Sq s Sq ~ and Sq ~_= 0 in A since the odd-dimensional e lements of A are zero. Therefore

b ~ = Sq ~ Sq ~ Sq ~~ (b) .

Since deg Sq~~ = 46, it follows f rom (2.1) t ha t

Sq~(b) = Sq~(c) + d ,

On the Mod 2 Cohomology of Certain H-spaces 135

where c~A44 and d~D~6. Moreover by 2 .5 D 4 , = A 1 8 . A 3 o , and b y the C~TAN product formula, Sq 2 (A16 �9 As0) = 0. Hence

b 2 ~ - - - Sq ' SqS[Sq2(c) § d] = 0 ,

which shows t ha t b = 0, thus proving (2.6) (i). The remaimng cases in (2.6) are proved similarly (using (2.1) and (2.5)) and are left to the reader.

We set

n+ : ~ A~ . A , , D ; = Z A , . A~, D - = X D ; , 1=1 o<i<j n--3

i + j = n

which are well-defined subspaces of A since A is a t runca ted polynomial algebra of height three. Moreover, as a mod 2 vector space

D = D + O D - , a splitting we use to show

(2 .7 ) Lemma. Aa~+s i = 0 , /or 1 < i < 3 .

Since Da2+s ~ = 0 for 1 < ] < 3, it suffices (by (2.1)) to show tha t Au

= A3e = 0, and to do this it suffices to show tha t if a e A82+2~ (r = 1 ,2 ) , then

a s = 0. Now by (2.2), Sq ~' (a) = 0, and therefore by (2.4) (taking M ----- 32

and r = 1 or 2),

a s = ~q~ L~= 1 Sq s~ ~ (a)] .

Now deg c~(a) : 64 ~ 2 i and hence by (2.2), i

Sq "~ ~ ( a ) = 27 SqSt(d~), i - 1

where d t ~D~4_sj (1 < j < i < r < 2). Bu t

D~_~ : Ale �9 A~_2~ + A~ �9 A4o-2~ ~- A28 �9 As6-s~ ~- A3o �9 A~_s~,

and hence by the C~3~TAN produc t formula and (2.6),

Sq s'+~ Sqd(D~4_~) c D - .

Since a s e D + and since D + (3 D - = 0, this shows t h a t a s = 0, completing the proof of the lemma.

3. The operations o f / k D ~ S . The proof of Theorem 1 will require the secon- da ry cohomology operations of J . F. ADAMS, defined in [1]. These are functions r defined for each pair of integers, i , j such t ha t 0 < i < 7" and i # ] -- 1. We will use the following propert ies of these operations (see [1; w167 3, 4.2]).

(3.1) Let X be a space. The operation q~ is de/ined on cohomology classes

136 EMEry THOMAS

u E H * ( X ) (mod 2 coe//icients) such that Sq~r(u) = 0 /or 0 < r < ]. I /

u �9 Hm(X) (m > 0), then q~i~(u) is an element o/ the group Hm+2i+2i-l(X),

modulo an indeterminancy subgroup Qi~ (X) . Moreover, i / i < ~, then

Qi~(X) = Sq 2~ H'~+~-~(X) + Z b~ H'~+~t-~(X) , 0</<~

where b~ �9 ..~'and deg bt -= 2 i + 2~ -- 2 I.

Take an integer k >__ 3 and suppose tha t u E H ~ (X) (m > 0) is a class such

tha t Sq~r(u) ~ O, for 0 < r < k. ADAMS' main result is ([1; 4.6.1]):

(3 .2) Theorem (ADAMS). There is a relation

2: aij q~ij (u) = [Sq ~k+l (u)] ,

(independent o/ X ) which holds modulo the total indeterminaney o/ the le/t-hand side.

Here the summation is taken over all i , ?" such t h a t 0 < i < ~" _< k and i :fi ~" -- 1. We apply this in the next section to prove Theorem 1, using in fact the following simple consequence of (3.2).

(3.3) Let X be a space and let m, k be positive integers with k > 3. Suppose

that H~t-I(X)--~0 /or m < i <_ m + 2~ and that S q 2 ' H 2 m ( X ) = 0 /or

0 < i <_ k. T he n /o r any element u ~ H~r~(X),

Sq~k+l(u) = Z~<~<~ a~ ~0j(u) ,

with zero indeterminancy, where the elements a~ E .)I. This follows at once from 3.1 and 3.2.

4. Proof of Theorem 1. The relation between the work of w 2 and the proof of Theorem 1 is given by the projective plane, P 2 X , for an H-space X . This was defined by STASHrFF [8] and the cohomology of P 2 X is studied in [6], where a group homomorphism t is defined with t: H q + I (P2 X) -+ H g (X) . (For the remainder of the paper we use rood 2 coefficients.) t is an .~-homo- morphism and the image of t is the subspaee of H* (X) spanned by the pri- mitive classes. Moreover if v~(X) = 2d - 1, then (see [6; 3.1]).

(4.1) t is a monomorphism /or q < 4d -- 2 and is an epimorphism /or q < 4 d - - 3 .

Denote by P - the subspace of H * ( X ) spanned by the odd-dimensional primitive classes. Let {ui} be a basis for P - and choose classes {a~} in

H * ( P ~ X ) so tha t t a i -~ u i. Define A to be the subalgebra of H * ( P ~ X ) generated by {ai}. The following result is proved in [6].

On the Mod 2 Cohomology of Certain H-spaces 137

(4.2) The algebra A is a truncated polynomial algebra o/ height three. Moreover there is an ideal N in H* (P~X) such that

H* (P~X) ~-- ~1 Q N , (group direct sum)

and J2"I (N ) c N , where .~/1 denotes the subalgebra of J /generated by the squares o /even degree. Furthermore i/ v2(X) ~ 2d - 1, then

H2i(P2 X ) ----- A2~, /or 1 < i < 2d .

Thus if we set A' : H * ( P 2 X ) / N ,

we obtain a finite~dimensional t runca ted polynomial algebra of height three, which is an L~l-algebra. Moreover

(4.3) A~i ~ H 2i(P2X), ]or 1 < i < 2d ,

and the isomorphism is an ~ - m a p . Le t ~ denote the ideal of ~ generated by Sq 1 . Then .:~ ~- ~ (D 2~1, as a Z2-module. Since the elements of A' all have even degree we can make A' into an :~'-algebra by setting ~::~(A ~) ---- 0. Fur thermore , v(A') ~ 2d.

We use these facts to show

(4. 4) Lemma. Suppose that ~,2(X) ~ 2d - 1, where d is an even integer 4. Then

H 2~§ (1"2 X)

Furthermore if Sq~ H 2d(p2x )

H ~+* (P2X)

The proof uses the following result of W. BICOWDER [5].

= 0 , /or O ~ s ~ d .

0, then

-~ O, ]or O ~ t ~ 2 d .

(4. 5) Theorem (BI~OWDER). Let X be an H-space as in w 1 and let v e H2q(X) (q ~_ i) be a primitive class. Then there is a class u E H~q-I(X)

such that v = Sq 1 (u).

Clearly if q _~ v2(X), then the class u is i tself primitive.

P r o o f o f (4. 4) . I t follows from (4.1) and (4.5) tha t

H 2~+l(P2X) -~ Sq lH2d(p~X) �9

Let u c H2d(P2X ) and set v ---- Sql (u) . Then,

v ~ = Sq ~+1 (v) = Sq 2d+1 Sq 1 (u) .

138 EMERY THOMAS

B y the ADE~ relations [3], since d is even,

Sq~a+l = Sq ~ Sq~a-1 + Sq ~a Sq 1 and therefore,

v ~ = Sq ~ Sq2a-l(v) = Sq 2 Sq 1 Sq~a-~(v) ,

using the fact t h a t Sq 1 S q l = 0 and tha t Sq 1 S q 2 a - ~ = Sq 2a-I. Le t w = = Sq2a-2(v). Then t(w) is a pr imit ive class of degree 4 d - 2. Hence b y (4.5) there is a pr imi t ive class y ~ H 4a-8 (X) such t h a t Sq 1 (y) = t (w), and therefore b y (4.1) there is a class x EH4a-~(P2X ) such t h a t S q l ( x ) = w. Hence

v ~ = Sq 2 Sq 1 Sq l(x) = 0.

Bu t b y Theorem 1.1 of [6], v 2 = 0 implies t h a t v = 0, and hence H 2a+I(P~X) = O.

As r emarked above we h a v e a l ready proved tha t v~(X) = 2q -- 1 for some q > 0 [9]. Thus in the present s i tuat ion we can assume t h a t d = 2~-1 with q _> 3. Therefore by 4 .3 and 2 .5 applied to the algebra A ' ,

H 2a+2~(P2X) = A ' 2d+2i = 0

for O < i < d / 2 , and also for d/2 ~ i < d i f Sq dH2a(P~X) = 0 . Therefore b y 4 .5 and 4.1,

H 2a+t(P2X) = 0

tbr 0 < t < d , and also for d <_ t<2d if SqaH2a(p2x ) = 0, which com- pletes the proof of the l emma.

We now can prove the first pa r t of Theorem 1. Suppose t h a t X is an H-space such t h a t v~(X) = 2q - 1, where q > 5. We show t h a t this leads to a contradict ion.

Le t k denote the least posi t ive integer such t h a t Sq ~k+l H~(P~X) :/: O. B y (4.4), k > _ q - - 2 and hence k > 3 . Le t uEH~ q(P~X) he a class such

t ha t S q ~ + l ( u ) ~ 0. Since

Sq~q(u) = u 2 4= 0 ,

we see t h a t k = q - 1 or q - 2. F r o m (4.4) and (3.3) ( taking 2 m = 2 ~ in (3.3)) i t follows t h a t

S q 2/c+1 (u) = X2_<]< k a~ O0t(u) .

But deg ~0~(u) = 2 q ~- 2t, where 2 < ] _< k, and therefore b y (4.4) and

the definition of the integer k, ~bo~(u ) = 0. Hence Sq 2k+x (u) = 0, which is a contradict ion. Therefore q < 4, which proves the first assert ion in Theorem 1.

Suppose now t h a t X is an H-space with no 2-torsion. We comple te the proof

On the Mod 2 Cohomology of Certain H-spaces 139

of Theorem 1 by showing t ha t q < 3 (i.e., t ha t v2(X) = 0, 1, 3 or 7). Since X has no 2-torsion it is immediate tha t v2(X) = vQ(X).

By BOREL [4; Prop. 7.2] H* (X) is generated by odd-dimensional primit ive generators, since X is wi thout 2-torsion. F rom this it follows (see [9]) t ha t the ideal N given in 4 .2 is an 3,r-submodule of H*(P~X); and moreover, t h a t if v 2 ( X ) = 2 d - - 1, then

(4.6) U2 t - i (paX ) = 0, 1 ~ )" ~ 3d -- 1; Hk(P2X) = "4k, 0 <I t < 6d -- 2,

where A is the algebra given in 4.2.

Suppose now tha t X is an H-space wi thout 2-torsion such tha t v~(X) = 15. We show tha t this leads to a contradict ion. For this X, (4.6) implies

(4.7) H~I-t(P2X ) = 0, 1 ~ j ~ 23; A~ ~ H k ( p ~ x ) , 0 < k < 46,

where, as before, A' = H* (PaX)/N. Since N is an L~'-submodule, the above isomorphism is an ~ - m a p . Applying (2.6) and (2.7) to A' we obtain the following facts.

(4.8) Sq2'Ha4(P2X ) = 0 for i = 0, 1, 3 .

Sq dH~s(P~X) = 0 for j = 0 , 2 , 3 .

Sq 2kHa~ = 0 for k = 0 , 1 , 2 , 3 .

(4.9) Ha~+at(PaX ) -~ 0 for 1 < i ~ 3 .

We now obtain the contradiction, caused by assuming tha t vz(X) = 15. Firs t of all i t follows f rom (3.3), (4.7), (4.8) and (4.9), t h a t Sq ts H 3~ (P2X) = O. But by the AD]~I relations and (4.8) this shows tha t Sq a~ Ha~ = O.

Thus if u E Hs~ u s = Sqa~ = 0 ,

and therefore u = 0, showing t ha t Ha~ = 0. Now

H a~ (PAX) = Sq 2 H ~s (P~X)

(see (2.1) and (4.7)) and hence by (4.8), Sq*JHas(P2X) = 0 for 0 < ~ ~ 3, Le t v ~ H ~s ( P ,X ) . By the ADv,~ relations

3 SqaS = Sqia Sq16 + ~Z' fli Sq ~i,

i=O

where fli E ~7, and therefore

v 2 = Sq as (v) = Sq la Sq 16 (v).

Hence by 3.3, v 2 = Sq 1~ a, (q~0~(v)) ,

since Has(P,X) = 0. Now deg a a = 12 and therefore by the ADEM relations 2

Sq t2 aa = 2: at Sq aj (a~ E ~ r ) .

140 EMERY THOMAS On the Mod 2 Cohomology of Certain H-spaces

Since ~b02(v ) E H 3 2 ( p 2 x ) , this shows by (4 .9) that v 2 ---- 0. Thus v ~- 0 and hence H 2 8 ( P a X ) ---- O.

Since H 2s ( P A X ) ~ Sq4H ~a ( P A X ) we obtain by (4.8) that Sq 2 i l l ~ ( P ~ X ) ~- 0

for 0 < i < 3. The fact that H ~ ( P ~ X ) ~- 0 now follows by a similar ar- gument to that used above. One must use (4.33) of [1], setting ]c ---- 3 in that lemma. We leave the details to the reader.

Finally, since H ~ ( P 2 X ) ---- 0 we see by (2.5) (taking q---- 4) and (4 .7) that H l s + 1 ( P 2 X ) : 0 for 0 < ] < 16. Hence by (3.3), S q 1 6 H I s ( P 2 X ) : O,

which implies that H l e ( P 2 X ) ~ O. But this is a contradiction, since va (X) ----15. Hence, assuming that q----4 has led to a contradiction, and therefore q < 3 -- that is, va(X) ---- 0, 1, 3, or 7, completing the proof of the theorem.

REFERENCES

[I] J.F.ADAMS, On the non-exlstence a] elements o] Hop] Invariant one, Ann. of Math., 72 (1960), 20-104.

[2] J.F.AD~.Ms, H-spaces with #w cells, Topology, I (1962), 67-72. [3] J.AD~.M, The relations on Steenrod powere o] cohomology classes, in Algebraic Geometry and

Topology, Princeton, 1957. [4] A. BOREL, Sur la cohomologie des espaces fibr$s principaux et des espaces homog~nes de groupes

de Lie compact, Ann. of Math., 57 (1953), 115-207 [5] W. BROWDER, Higher torsion in H-spaces, to appear. [6] W. BROWDER and E. T H o ~ s . On the projective plane o/ an H-space, Ill. J . Math., to appear [7] A. CLARK, Some applications o] Hop] algebras and eohomology operations, thesis, Princeton

Universi ty, 1961. [8] J . STASn~FF, On homotopy abelian H.spaces, Proe. Camb. Phil. Soe., 57 (1961), 734-745. [9] E. THO~LS, Steenrod Squares and H-spaces, Ann. of Math. , to appear.

(Received March 19, 1962)