On functional cup-products and the transgression operator

Vol. XII, 1961 435

On Funct ional Cup-Products and the Transgression Operator

By

EMERY TtIO~AS 1)

1. Introduction. Le t g be a m a p f rom a topological space X to a topo lo~ea l space Y. Take cohomology groups with coefficients in a fixed ring 2 and denote by g* the homomorphism from H * ( Y ) to H * (X) induced b y g. Suppose tha t u e H r + I ( Y ) and v e Hq+I(Y) are elements such t h a t u v v = 0 and either g*u = 0 or g*v = 0. Then STEENROD [6] has defined a funct ional cup-product

u v v e Hr+q +1 (X) rood L (g, u, v), g

where 2) L (g, u, v) = g* H~+q+1 ( y ) + g* u ~ Hq (X) § H~ (X) ~ g* v.

Suppose now t h a t X and Y are the total spaces of respective proper tr iads and tha t g is a t r iad map. The purpose of this paper is to show tha t it is then possible, in certain cases, to express the functional cup-product in terms of an ordinary cup-product . We apply this to obtain a classical result about the t I o p f invar iant and to obta in some new informat ion about the cohomology rings of certain classifying spaces.

Assume then t h a t (X; X0, X1) and (Y; 170, 171) are proper triads [3; Chap. I , w 14] and t h a t g is a t r iad map. Denote by

gj : (X, X j ) -+ ( Y, Yj) (j = 0,1)

the map determined by g and set

A - - - - - X o n X 1 , B - = Y 0 n Y1, / = g [ A .

For the remainder of this section assume tha t X = X0 w X1 and Y = Y0 ~9 Y1- One then has the (absolute) Mayer-Vietoris cohomology sequence [3; Chap. I. , w 15] for each triad, which contains the coboundary

A : H r (A) --> H r+l (X) (r ~ 0),

with a similar coboundary for Y. For elements u i e H * ( X i ) ( j ~ 0,1) we set

u0 * u l = A (~r~ u0 ~ ~ * u l ) e H* (X),

where ~* denotes the homomorphism induced by the inclusion A c Xj (j = 0,1).

1) Fellow of the John Simon Guggenheim Foundation. This research was partially supported by the Air Force Office of Scientific Research.

�9 2) I f A1, A.2 . . . . , An are sub,cups of an abe]Jan group B. we denote by AI ~ "'" -b An the smallest subgroup of B containing all the Ai.

28*

436 E. THOMAS ARCH. MATH.

We will say that an element x e H p ( X i ) (i = 0 or 1) is transgressive (with respect to g) if there is an element y ~ HP +1 ( Y, Y~) such that

5~x = g ' y ,

where 6i denotes the relative coboundary Hp (Xi) --> H p*I (X, X~). We will say that the element n* y ~ HP +1 (Y) represents the transgression o/ x, where n* is induced by the inclusion Y r Y~-). Of course the representative is determined only up to the image by n* of the kernel of g*.

We shall prove

Theorem 1.1. Let x, ~ H p ( X i ) (i = 0 or 1) be an element that is transgressive and let wi e H ~+1 (Y) represent its transgression. Let v ~ Hq (B) be an element such that

Vo * " ____. v ~"ll V a ~ a

where ca ~ H * (X0), v~ ~ H * (X1) a n d / * is induced b y / . Then,

and g*w~ = O, w~ ~ Av = 0 = Av .-~ w~,

wo ~ zJv - ~ ( - -1)p+l(xo ,~ va) * vj m o d L ( g , wo, ,Av) (i] i = O), g a

z J V v W l -~ Z ( - - 1 ) q v a * ( V a V x l ) m o d L ( g , Av, wl) (i/ i = 1). r a

We will say that an element v e/-/q (B) is primi t ive (with respect to / ) if

/* v = v0 + v l ,

where vj ~ H q ( X j ) ( j ---- 0,1). We term the element v ja projection of v on Xj. In the Mayer-Vietoris sequence for X the kernel of/1 is the subgroup spanned by the image of ~* and ~*. Since g is a triad map we have the commutativity relation g*A ---- , j]*, and therefore: an element v ~ Hq ( B) is pr imi t ive i/, and only i], g* ~Jv = O. Thus as a special case of Theorem 1.1 we obtain

Corollary 1.2. Let xl and wi (i -~ 0 or 1) be the elements given in Theorem 1.1. Let v ~ Hq (B) be a pr imi t ive element with projection vj on X j ( j = 0,1). Then

g * w ~ = O , g * A v = O , w ~ A v = O = A v , ~ w i ,

and w o v A v - ( - - 1 ) p + l x o * v l m o d g * H p + q + l ( Y ) (i/ i = O ) ,

g

A v ' ~ w l - ( - - 1 ) q v o * x l m o d g * H P + q + l ( Y ) (i] i = 1). g

Suppose now that u e H P ( B ) is a primitive element with projection uj on Xj (j = 0.1). One may then show that uj is transgTessive and that s j A u represents its transgTession, where e0 ---- -- 1, el -~ 1. Thus from Corollary 1.2 we obtain

Corollary 1.3. Let u ~ H P ( B ) and v ~ H q ( B ) be pr imi t ive elements with respective projections u i and vj on X j ( j = 0,1). Then

Vol. XII, 1961 Functional Cup-Products 437

g* A u ~- O, g* A v ----- O, A u ,~ A v ---- O,

and A.u -.~ A v ~ ( - - 1)1~ u0 , Vl r o o d g * H p + q + l ( y ) .

g

Denote by ~- the cone on X attached to Y by the map g. One then has an exact sequence (see [1 ; w 3])

g* �9 " > Hq ( Y ) ~* > Hq (:Y) - - > Hq ( X ) ~ >. Hq +1 ( Y ) > . . . ,

where j * is induced by the inclusion Y r and/~ is a Mayer-Vietoris coboundary. Let x~, wi (i -~ 0 or 1) and v be the elements W e n in Corollary 1.2, and let u be the element W e n in Corollary 1.3. Since g * w i = g* A u ---- g* Av =. O, by exactness there are elements

such that j*~,, = w , , j * ~ = / l u , j * ~ = Av.

Corollary 1.4.

(a) ~(uo*vl)= ( - 1 ) ~ . (b) #(x0 * vl) = (-- 1)~+1 w0 ~ v

~a (v0 * Xl) = (-- 1)q v -J Wl

( q i = 0 ) ,

(if i = 1) .

The proof follows from Corollaries 1.2, 1.3 and the formulation of the functional cup-product due to A D ~ [1 ; w 4].

In the following section we give two applications of these results, while the re- maining sections of the paper are devoted to the proof of Theorem 1.1.

2. Applications. Our examples will use the join construction. Let A0 and A1 be topological spaces. We define their join, A o * A 1 , to be the set of points toao + t l a l ,

where aj e A j, tj ~ 0, and to ~ tl = 1. This set is topologized by the strong topology, defined by M ~ O R in [4]. Thus we obtain the proper triad (A0 * A 1 , A o , A1) , where A0 is the set of points (1 - - t) ao (~ ta l with t ~ 1/2 and -41 is the similar set of points with t ~ 1/2. Identify the spaces A0, -41, and -4o • -41 with the subspaees of -40 * A1 consisting of all points (1 - - t) ao 0 t a l with t = 0, t = 1, and t = 1/2 respectively. Then Aj is a deformation retract of .4~- and A0 • -41 ~- A0 c~ A1. I f we compose the inclusion -4o • -dj- with the deformation retract Aj -~ -dj, we obtain the natural projection ~j : A0 • A1 -~ Aj. (j = 0,1).

For our first application of the results of section 1 denote by S~ and S~ two copies of the n-sphere S ~ (n ~ 1). We identify S 2n+1 with its homeomorph S~ * S~ and obtain the triad (S 2~+1, ~ , S~), where S~ • S~ S~ ~ -~ ---- S I. Pick generators uj

H n (S~) (integer coefficients) and denote by ~j the corresponding element in H n (S~) (j ---- 0,1). Then uo * ul is a generator for H2n+l(S2n+l) .

Consider now the suspension triad (~n Co, C1), where ~n is obtained from S n • [0,1] by identifying S" • {]} to a point x j ( j = 0,1), and where Cj is the cone S n • [j, 1/2].

Identify S n with Co c~ C1, and S n+l with ~n.


Let m be a map from S~) • S~ to S n. Then m determines a triad map

: (82~+1, ~ , p~) ~ (S~+1, Co, C1), by

~'n(toso ~ tl sl) = (re(so, 81), tl)

for sj e S~, tj >_ 0, and to ,'-- tl ----= 1. Let ej e S i (3 ---- 0,1) be a basepoint and define a map ]j : S} ~ ~ S~ • S 1 by

/0(s0) = (so, e l ) , / 1 ( s l ) = (e0, s~) ,

for sj e S~. Choose a generator u for H n (Sn). We ,sTill say tha t the map m has type (p, q) if

( m o / o ) * u ~ - p u o , ( m o / 1 ) * u = g u l ,

where (mo/3")* denotes the cohomology homomorphism induced by m o/1(J = 0,1). Clearly the element u is primitive (with respect to m) ; and m has type (p, q) if, and only if,

m*u =puo Q 1 & 1 ~ ) q u l .

Now the Mayer-Vietoris coboundary in the exact sequence for the tr iad (S n+l, Co, C1) becomes simply the suspension isomorphism a : H r ( S n) ~---H r+l(Sn+l). Thus au is a generator for Hn+l(Sn+l). Applying Corollary 1.3 we obtain

au ~ au : (-- 1)npq(uo *uz),

which proves the following classical result, using STEP,ROD'S definition [6; w 17] of the Hopf invariant.

Theorem 2.1. Let m be a map ]rom S~ • S~ to S n and let ~n : S 2n+1 --> S n+l be the map obtained ]rom m by the Hop] construction. Suppose that m has typ, e (p, q). Then the HoT/invariant o/r~ is -4-p q.

For our second example denote by En(n ~ 1) the (n q- 1)-fold join of a topological gToup G ~dth itself. Define X , to be the orbit space of En with respect to the action of G on E~ by right translation (for details, see MIL~ro~ [4]). MrLNOR shows that En is the total space of a principal G-bundle (En, G, X , , pn), whose projection Pn is simply the identification map En ~ Xn. Since the join construction is associative we may regard En as En-1 * G (n _>_ 1, E0 = G), and hence obtain a tr iad (En, En-z, "G) where En-1 n G ---- En-1 • G and where En-1 and G are deformation retracts respectively of ~'n-1 and G.

Now MIL~o~ [4] shows tha t Xn is in fact homeomorphic to the cone on En-1 at tached to Xn-z by the map Pn-1- That is, we have a tr iad (Xn, Mn-1, Cn-z) (n _>-- 1), where Cn-1 is the cone on En-1, M , - I is the mapping cylinder ofpn-1 and

X n = Mn-1 u Cn-1, .En-1 = Mn-1 ~ Cn-1.

Furthermore, Xn-1 is a deformation retract of M . - I and Pn is in fact a tr iad map

(En, E , -~ , G) --> (Xn, Mn-~, Cn-~)

such tha t Pn [ En-~ = Pn-~. We set ran-1 ~- Pn] En-1 X G : En-1 x G --> En-1. For


example, ff n = 1 then m0 is the m a p from G • G to G given by

mo (x, y) = x y -1 (x. y e G).

Take cohomologT groups with coefficients in some fixed ring R and let u l , u2, . . . . un e H*(G) . Then by i teration of the coboundary A we obtain an element u l . . . . * u~ e H * ( E ~ - 0 .

Lemma 2.2. Suppose that u l , u2 . . . . . un are pr imi t ive elements (with respect to the product in G). Then Ul * "'" * un is 19rimitive w~th respect to the map ran-1.

The proof is not difficult bu t involves some computat ion. We omit the details. Since the ident i ty element, e, of G acts as the ident i ty t ransformat ion on En-1 ,

we see tha t the composition i

En-1 - - > En-1 X G mn-~ En-1

is Vhe ident i ty map, where i (x) = (x, e). Thus if u E H * (En-1) is primitive with respect to mn-1 , its projection on En-1 is s imply S) u.

We will say t h a t an element x e H * (G) is n-transgre8sive (n ~ 1) if it is transgressive with respect to the map Pn - - t ha t is, in the usual fibre bundle sense, x is transgressive in the bundle (En, X u , G).

Denote b y # , - 1 the Mayer-Vietoris coboundary for the t r iad (Xn , M n - 1 , Cn-1). Thus, #n-z : H r (En-1) --> H r+l(Xn) (r ~ 0). Applying Corollary 1.2 and L e m m a 2.2 we obtain

Theorem 2.3. Let u l , u2 . . . . . un e H*(G) be pr imi t ive elements (n ~ 1). Let x ~ HP(G) be an n-transgressive element and let w ~ HP+I(Xn) represent its transgression. Then,

p ~ / . ~ n - - t (U l * " ' " * U n ) = 0 , p ~ W = 0 , ~ B - - I ( U l * " ' " * U . ) ~-~ W = 0 ,

and

where

/~n-1 (ul * " " * un) ~ w = ( - - 1)q ul * " " * un * x modp~* HP+q +1 (Xn) ,

q = (n - 1) + Z d i m u ~ .

Using Corollary 1.4 we show

Theorem 2.4. Let u l , u2 . . . . . u ,+ l e H*(G) be elements which are (n + 1)-transgressive (n ~ 1) and let w l , w2 . . . . . wn+l ~ H * (Xn+l) represent their respective transgres- sions. Then,

~ (ul * " " * ~n+l) = • w l v w2 . . . . . w ~ + l .

Consider first the case n = 1. Then X1 is homeomorphic to the suspension of G and #0 is s imply the suspension isomorphism g. Moreover one m a y show t h a t 4)

k* wz = / ~ 0 (Ul),

s) For the remainder of this section we make the identifications t t * (En-1 )~ H*(E,-z) , H* (G) ~ H* (G), where the isomorphisms are induced by the canonical deformation retracts.

4) The proof is essentially that given by J~_~Es-T~oMAs for Equation 1.2, On homotopy- commutativity, :4an. of Math., II. Ser. (1962).

440 E. THOI~IAS ARCH. MATH.

where k* is induced by the inclusion X1 c X2. B y exactness ~o*k*wl = 0 and therefore (see section 1) u l is primitive with respect to Pl . On the other hand by na tura l i ty u2 is 1-transgressive and k ' w 2 represents its transgression. Since X2 is homeomorphic to the cone on E1 a t tached to X1 b y P l , we see t h a t Theorem 2.4 (for the case n----- 1) follows a t once from Corollary 1.4 (b). The proof of the Theorem for an a rb i t ra ry integer is now a simple mat te r of induction. We leave the details to the reader.

Recall tha t the limit of the spaces Xn, X ~ , is a classifying space for G. ROTHE~- BERG [5] has used Theorem 2.4 to describe the cohomology ring H * (Xn), in the ease tha t the coefficient domain is a field and H* (G) is an exterior algebra. His result, roughly speaking, is t h a t H * (Xn) is then the direct sum of a t runca ted polynomial ring ~dth an ideal which has trivial multiplication. I n part icular he obtains a new proof of Theorem (19.1) of [2], by BOR~.L.

3. A relative product. Let (X; X0, X1) and (Y; Y0, Y1) be arb i t rary proper tr iads and g a t r iad map. Take coefficients in some fixed ring R and suppose t h a t

Y0 e HP +1 (Y, Y0) and Yl e H q+l ( Y, I71)

are elements such tha t y0 ~ Y l = 0 (in HP+q+2(Y, Y0 (9 Y1)) and g*y~ = 0, for i = 0 or 1. We then have a relative functional cup-product

yo ~ y l e H~+q +t (X; X0 u X1) rood L (g, Y0, Yl), g

where

L(g, Yo, Yl) = g*H~+q+l(Y; Yo w Yl) + g~Yo ~ H a ( X , X1) + H P ( X , Xo) "-~ g ' Y 1 .

Taking X0 = X1 = Y0 ---- }71 = Z , the e m p t y set, we obta in the (absolute) func- t ional cup-product described in section 1. STEX~ROI)'S original definition of these operations was done in an invar iant manner, using the mapping cylinder technique. For our purpose it is more convenient to give cochain definitions, based on the singular cohomology theory. Let {~ be a coeycle representat ion for yj (j = 0,1). Since * g~ yt ---- 0, for i = 0 or 1, there is a cochain a~ �9 C * ( X , X i ) such tha t 6a~ = = g#i-Yi, where 6 denotes the cochain coboundary and g~ is the eoehain homomorphism

determined by gi. Let C*(Y; Y0, Yl) denote the subgroup of C * ( Y ) consisting o f those cochains which vanish both on C . ( Y o ) and C. (Y1). Since Y0 ~ Yl = 0, there is a cochain b �9 C~+a+I(Y; Yo, Yl) such tha t 6b = Y0 v Yl. Define

w0 = g # b - - a0 ~ g~ .~1 (if i ---- 0), (3.1)

wl ---- g # b + ( - -1 )~g~y0 ~ a l (if i---- 1),

where p ----- dim a0. One readily verifies that. w~ is a cocycle, and STEENROD shows [6 ; w 16] t h a t the eohomology class of w~ is a representat ive for Y0 v Yl- Notice tha t if

g

go Yo = g~yl = 0, then w0 and wl are cohomologous so tha t the symmetr ic nota t ion Y0 v Yl is well defined.

g

Suppose now tha t (X; X0, X1) and ( Y; Y0, ]71) are the proper triads considered in section 1. Thus, X = Xo w X�92 Y ---- Yo u Y1. Denote by I the inclusion (X, Z , Z ) c


c (X; Xo, X1) and by l j ( j = 0,1) the inclusion (X, ~ ) c (X, Xj). We then have the composite maps h ---- g o l, hj -~ gl o lj ( j ---- 0,1):

l (X, ~ , ~ ) c (X, Xo , X~) ~ ( Y; 7o, Y1),

(x, z)~(x, xj) ~(r , 7j). Suppose tha t zo �9 H p+I ( Y, Yo) and zl �9 H q*l ( Y, }71) are elements such tha t h~z~ -~ 0

for i -~ 0 or 1. Since Yo u Y1 ~- Y we have zo v zl -~ 0, and therefore the functional cup-product

zo v zl �9 H ~+q+l (X) mod L (h, zo, zl) h

is defined. Let z1 be a eocycle representing zj (j ---- 0,1). Since h~z~ ---- 0 for i -~ 0 or 1,

there is a eochain c~ �9 C*(X) such tha t 6c~ ---- h/#zi. Since zo v zl -~ 0 there is a cochain b �9 CP+q +1 (Y; Y0, Yl) such tha t 6 b : ~o v ~1. Now b m a y be regarded as a cochain in C * ( Y , t~.) for j ~ 0,1. Fur thermore, the cup-product of the absolute cochain ci with

the relative cochaia g~_,.Zl-~ is again a relative cochain - - in fact an element of CP+q +1 (X, XI-~). Thus if we define

= b-co g z (ifi=ol, (3.2)

vo = g#o b ~- (-- 1)~go~Zo v cl (if i : 1),

we obtain a cochain vl_~ �9 C~+q +~ (X, X~-~) (i ---- 0 or 1). Since hi# = g~ o l~ (j = 0,1)

it is clear t ha t l~_ivl-i is a cocycle which represents zo ~ Zl. Thus h

0 = ~ 11~ v~-t = l~_i ~v~-~.

But l~_s is a monomorphism, since C* (X, XI-~) is s imply a subgroup of C* (X). There- fore, vl-~ is a cocycle. We define

(3.3) zo o zl = {vl-~} �9 H p+q+l (X, X l - i ) rood L (g, zo, Zl) , g

where /~(g, zo zl) denotes the s u b ~ o u p H* (X) * , ~ gl zl, ff i = 0, and the subgToup g~zo ~ H * (X) if i = 1. Thus we have proved

Lemma 3.4. Let zo �9 H ~+1 ( Y, Yo), Zl �9 Ha+~ ( Y, Y1) be elements such that h~z~ = 0 /or i -- 0 or 1. Then there is a cohomology class

zoo Zl �9 HP+q+I(X, X~_~) mod/~(g, zo, Zl) such that

l*_i (zoo Zl) ~ zo ,~z~ �9 Hp+q+~ (X) rood L (h, zo, z l ) . g h

B y the na tura l i ty of the functional cup-product we obtain

Corollary 3.5.

l~* (zoo z~) - n~ zo ~ n ~ z~ �9 Hp+q+~ (X) rood L (g, n~ zo, n* Zl), g g

where n* is induced by the inclusion Y c ( Y, Y~) ( j -~ 0,1).

4 4 2 E . THOMAS &RCH. MATH.

Let xi �9 H~ (Xi) (i = 0 or 1) be a transgTessive element and choose zi �9 H~ +1 ( Y, Yi) such tha t g*zt = ~ xi �9 H ~+1 (X, Xi). Let zl- i be an arb i t rary element in Hq+I(Y, Yl-i). Then zo v Zl = 0, since Yo w Yl = Y- Furthermore, h*zi = 0, by natural i ty. Thus

A

we can define the class zoo zl e H p+a+l (X, X l - t ) rood L(g, zo, zl). We show g

Lemma 3.6. k~" (zo o zl) - -- (xo v k* g~z*) rood m~ H~ (X) ~ Ic~ g*zl (i/ i = 0),

g

k~ (zo o Zl) -- ( - - 1)P (k~ g~zo ~ zl) rood k~g~zo ~ m~Hq (X) (i/ i = 1). g

Here k* is induced by the inclusion (XI-s , A) c (X, Xj) and m* b y the inclusion X $ c X (j = 0,1). Consider first the case i----0. Let xoeZ2~ be a cocycle representat ive for xo. Ex t end xo to a cochain xo defind on all of X ; ~o �9 C~ (X). Then 8 xo is a coeycle representing 6oxo. Since g~' zo ---- ~o xo, there is a eochain eo �9 C~ (X, Xo) such tha t

go # zo - - ~ ~o ---- ~ co,

where ~j. is a cocycle representing zj (j = 0,1). Thus Xo + eo is a cochain in C~ (X) such

t h a t 6(xo + co) = ho~zZo. Choose b �9 C~+q+I(Y;Yo,Y1) such tha t ($b = z0 -~ zl. Then by (3.2), we can define a representat ive eocyele for zo o zl by

r

Vl = gl # b - - (xo + co) v gl # zl-

Denote by kl# the cochain m a p induced by kl. Then,

since g~b and eo ~ gl#zl vanish on C. (Xo). Fur thermore

since xo was defined to be the extension of ~o to all of X. Therefore,

-

and passing to cohomology classes we obtain

as was asserted. The indeterminancy arises f rom the equat ion k ~ ( H P ( X ) ~ g*z~)= --~' / /~(X) k* *z - - ~ 1 gl 1, by the natura l l ty of the cup-product.

The proof for the case i = 1 is just the same and is left to the reader.

4. Proof of Theorem 1.1. We continue with the nota t ion of section 1 ; tha t is, with proper tr iads (X; Xo, X]), (Y; Yo, Y1), a tr iad map g, and

A = X o ~ X ~ , B = Y o ~ Y l , ] = g ] A .

Consider the following sequence of homomorphisms (j = 0,1) :

E - l* (4.1) H r (A) ~ H "+1 (XI-~, A) 2"~ H r+~ (X, X~) - + H r+l (X) .


Here 6s" is the coboundary and k*, l* are induced by inclusions. Since (X; X0, Z l ) is a proper triad, k~" is an isomorphism for all r _> 0 and j ---- 0,1. Define ~j (j ---- 0,1) to be

the composition k*- lo 6i. Thus, ~ : H r ( A ) - - + H r + I ( X , X~). Then the composition As"----l* o ~s" is simply a Mayer-Vietoris coboundary [3], with A0 q-A~ =-0. The operator A, used in sections 1 and 2, we define to be A1.

Now let vs" be the counterpart, for the triad (Y; Yo, Yz), of the map ~s'. Since g is a tr iad map we obtain a commutat ive diagram (j ~ 0,1):

H r (A) ---> H r+l (X, X~)

(4.2) ~* ~ ~ ~- H~(B) , ~ H~+~(: [ ; :Ys").

We use this to show

Theorem 4.3. Let x~ eHI~ (i ~ 0 or 1) be a transgressive element and choose z~ e Hp+I ( y , y~) such that g~ ~i --- ~ix~. Let v e Hq (B) be an element such that

a

where va e H* (Xo), v~ e H* (X1) and ~* is induced by the ir~clusion A c X j ( j = 0,1). Then

h[~z~ = 0, z~ ~ ~l-~v---- 0, and

z o o v l v - (--1)P+l ~ ~ l ( z r ~ ( x o v v a ) ~ : z*v~)modL(g , zo,~,lv) (i/ i = O ) , g a

~ 0 v o ~ l - - - ( - 1 ) q ~ 0 ( ~ $ v ~ * ' = "-'~1 ( v ~ v x l ) ) m o d L ( g , vov, zl) ( i l l 1). g a

Again we do only the case i : 0. By Lemma 3.6 and 4.2 and the definition of 21 we have

~ (~0 o ~ v) - - (~0 ~ ~ g~ ~,) - - (~0 ~ k? ~1 /* v) - g

= - (~0 ~ g~ /* ~) rood ~ H ~ (X) ~ k * gl ~ ~ .

By (3.3) of [6],

Therefore, �9 o ~ (~11" ~) = ( - 1)~ ~ (:~'2 xo ,~ I* v ) .

kt (zo o lv) - ( - 1) +1 E (x0 v v;). g a

Applying k~ -1 to both sides of this and recalling tha t 21 ---- k~ -1 o 51, we obtain the desired result. The proof for the case i = I is entirely similar, except tha t we now use (3.2) of [6].

To prove Theorem 1.1 define w~ = n~ z. Then w~ is a representative for ~he transgression of xl. Furthermore, n~ ~,~v ---- A~v. Thus, by Corollary 3.5 we obtain

l* (zo o , , l v) - w o v A l v ( i f i=0) , g r

l~ (~ovoz l ) -- A o v ' ~ w l (ff i = 1). g g

Applying l~_(to both sides of (4.3) we then obtain the desired result.


5. Appendix. Bo th the examples given in section 2 m a y be subsumed under a more general construction. Let E, F , B and X be topological spaces and m and p be maps as below:

E • F--> B--> X .

We will say tha t p is stable with respect to m, if, for all e e E,

p(m(e,/1)) =p(m(e,/2)) (II,/2eF).

We consider the join t r iad (E * iv, E , 2") and the t r iad (X1, Mp, C B ) where X1 is the cone on B a t tached to X b y p , C B is the cone on B and M~ is the mapping cylinder o f p (see w 2 of [lJ). One has

X, = M ~ u C B , B = M ~ n C B .

I f p is stable, then m extends to a t r iad map

~n : (E * 2", E , 2") -+ ( X 1 , M p , C B ) .

I n part icular if X is a point, then X , is the suspension of B and ~ is the map obtained f rom m by the H o p f construction. This was the case in our first example in section 2, with E ~ S~, 2' ~-- S~, and B ~ S n. I n the second example E ---- B -~ En -1 , F ~ G,

X ~-- X n - 1 , X 1 -~ X n , m ~ ran - l , p ~ P , - 1 , and ~ ~ Pn. I t is clear t ha t the results of section 1 m a y be applied to the cohomology groups of the spaces in this more general construction. I n a for thcoming note with W. BROWDEI~ Corollary 1.4 is applied to compute the rood 2 cohomology ring for the "project ive p lane" associated wi th an H-space.

Bibliography

[1] J. ADEM, Un criterio cohomologico para determina y composiciones esenciales de transfor- maciones. Bol. Mat. Mex. 1, 38--48 (1956).

[2] A. BOWEL, Sur la cohomologie des espaces fibres principaux et des espaees homogenes de groupes de Lie compacts..4zm, of Math., II. Ser. 57, 115--206 (1953).

[3] S. E*LENBERG and N. STEEN~OD, Foundations of algebraic topology. Princeton 1952. [4] J. MXL~COZ, Construction of universal bundles. Ann. of l~Iath,, II. Ser. 63, 430--436 (1956). [5] M. ROTHENBERG, Thesis. Univ. of Calif. (Berkeley) 1961. [6] ~q. STEENROD, Cohomology invariants of mappings. Ann. of Math., II. Ser. 50, 954--988

(1949).

Anschrift des Au~ors: Emery Thomas Department of Mathematics University of California Berkeley (Cal.), USA

Eingegangen am 28. 8/1961

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On functional cup-products and the transgression operator