Postnikov Invariants and Higher Order Cohomology Operations

Annals of Mathematics

Postnikov Invariants and Higher Order Cohomology OperationsAuthor(s): Emery ThomasSource: Annals of Mathematics, Second Series, Vol. 85, No. 2 (Mar., 1967), pp. 184-217Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970439 .

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Postnikov invariants and higher order cohomology operations*

By EMERY THOMAS

1. Introduction

A classical problem in algebraic topology is to determine whether a given fibre space has a cross-section. Two different methods have been developed to handle this problem. In the first method one assumes that the base of the fibration is a complex. Steenrod [35], generalizing the work of Eilenberg, obtained an obstruction theory for cross-sections using the filtration of the base by skeletons. (Steenrod developed his theory for fibre bundles; Barcus [2] generalized the method to handle (Serre) fibrations.) Subsequent work on this method, especially the problem of the second obstruction, has been done by Liao [18], Boltyanskii [3], Kundert [17], and Hopf [14].

A second approach stems from the work of Postnikov [33] on the homotopy theory of complexes. Moore [30] used the method of Postnikov to develop an obstruction theory for cross-sections based on a filtration (i.e., factorization) of the fibre map itself, rather than of the base. (Moore worked in the category of semi-simplicial fibre spaces. R. Hermann [11], [12] carried out the Moore- Postnikov factorization for a Serre fibration, and gave conditions which sometimes characterize the higher obstructions). In the Moore-Postnikov approach the fibre map is factored into a sequence of principal fibrations such that at each stage the obstruction to lifting a map to the next stage is given by one or more cohomology classes, the Postnikov invariants of the fibration. Recently Mahowald 119], using the point of view of Hermann, has given a detailed description of how one decides precisely which cohomology classes at each stage can be taken as the Postnikov invariants for the next stage.

Both of these methods leave open the problem of computing explicitly the higher order obstructions that occur for a cross-section, though some special results have been obtained. For example Massey [21], [23] used the work of Liao to compute the second obstruction in some cases, while Mahowald and Peterson [20] applied the Adams theory of secondary cohomology operations [1]

* The research for this paper has been supported by the Air Force Office of Scientific Research and by the National Science Foundation. Part of the research for the paper was done while the author was a guest of the Mathematics Research Institute, E. T. H., Zurich.



POSTNIKOV INVARIANTS 185

to calculate obstructions to cross-sections in the (stable) normal bundle of a manifold.

The purpose of this paper is to describe a method (see ? 5) whereby, for certain fibre spaces, one or more of the Postnikov invariants can be expressed in terms of higher order cohomology operations applied to classes coming from the base of the fibration. The operations needed are the higher order twisted operations which generalize the Adams-Maunder operations [1], [26]. (Second- ary twisted operations have been defined by Meyer [28] and by Gitler-Stasheff [10]. McClendon [27] gives an axiomatic characterization of twisted operations of all orders.) When the base space of the fibration is a manifold, one has several techniques for computing these higher order operations, and so the calculation of the Postnikov invariants is sometimes possible. In this paper we give two examples of such computations (see ?? 6-7) which yield the following results.

Let M be a closed connected smooth manifold. We define the span of M to be the maximal number of linearly independent vector fields on M. By the theorem of H. Hopf [13], span M is positive if and only if the Euler charac- teristic of M, X(M), is zero. In particular, every odd-dimensional manifold has positive span, and this result is best possible since the manifold S1 x RP2k, k 0 O. has span 1. (Here Rpq = real projective space of dim q, and Sq = sphere of dimq, for q _ 1.) Even if we consider only orientable odd-dimensional manifolds, the result is still best possible in dimension 4k + 1, k > 0 for span S4k+1 =1 [61, [43]. We shall prove

THEOREM 1.1. Let M be an orientable manifold, and let k be a non- negative integer. If dim M = 4k + 3, or if dim M = 4k + 2 and X(M) = 0, then span M _ 2.

The case dim M = 3 is given by Stiefel [36]. The result for dim M -

4k + 2 is best possible, since the manifold S1 x RP4k+l has span 2. Theorem 1.1 is proved in ? 7. A review of principal fiber spaces and of

Postnikov resolutions is given in ?? 2, 3. In ? 4 we sketch McClendon's theory of higher order twisted cohomology operations. The main theoretical result is given in ? 5, and this is then applied to two fibrations in ? 6.

Further applications of the theory developed here will be found in [39], [40], and [41].

2. Principal fibre spaces

In this section, we review some homotopy theory. We work in the category of spaces with basepoint; all maps and homotopies will respect



186 EMERY THOMAS

basepoints. Given a pair (X, X0), we assume that * G X0, where * denotes the basepoint for any space. We denote by [(X, Xj), (Y Y.0)] the set of homotopy classes of maps from a pair (X, X0) to a pair ( Y. Y0).

For each space Y we denote by P Y the function space ( Y, *)(IO), where I = [0, 1]. Define wr: PY-E Y by

11(X) = x(1), for x c PY.

Then wr is an Hurewicz fibration [15] with fibre the loop space & Y. Now let B and C be spaces, and w: B ). C a map. Denote by p: EP- B

the fibration induced by w from the fibration wr: PC C. Thus EPJ is the subspace of B x PC consisting of those pairs (b, X) such that w(b) = x(1). More generally, if w is a map of pairs, w: (B, B0) (C, CO), then w induces a relative fibration p: (Em, E0) - (B, B0), with w, = w I Bo We call p the principal fibration with classifying map w. (To emphasize the dependence on w we some times write p as pro.)

Define an action map

m: nC x Ew > Ew

by the rule

(c), (b, X)) - (b, w V X)

where co V x denotes the path whose value on t is 'io(2t) for 0 ? t ? 1/2 and is X(2t - 1) for 1/2 ? t _ 1. Notice that m restricts to &2CO x Ewo to give an action map flC0 x Ewo E).O

If (X, X0) is a pair the set [(X, X0), (TIC, flCO)] has a natural group structure using the loop multiplication on 2C, ?2CO. The map m then gives an action of the group [(X, XO), (&2C, D2CO)I on the set [(X, X0), (E, E0)] where we write E = Ew, E0 = EWo. We need the following property of this action (see [32], [38]).

(2.1). Let C1, 2 G [(X, X0), (E, E0)]. Then p*,1 = P*72 in [(X, X0), (B, Bo)1 if and only if there is a class v G [(X, X0), (DC, &2CO)] such that 72 = V

where the dot indicates the action given by the map m. Suppose now that CO = *, and hence &CO = *. (The spaces Bo and E0 are

then homeomorphic by the map b - (b, *), b E Bo.) Identify E and E0 respectively with * x E, * x E0 in UC x E. Let

(EEO)-(& C x E,EO)-) (S7C x E,E)

denote inclusions and

1: (&7C x E, Eo) (E, Eo)




the projection. Notice that m also is a map (fC x E, E0) > (E, EP3). Take cohomology with mod p coefficients. We prove

(2.2). There is a morphism

H H*(Eq EO) H*(fC x E, E)

such that

iJ A - I*

To see this, notice that 1 o i = identity. Moreover, one easily shows that m o X j identity, and hence

i*l* = 1 i*m* = 1

as cohomology morphisms. Since we always have an exact sequence

O >H*(f2C x E, E) j

H*(fC x E, EO) H*(Eq EO) >

and since i*o(m*-l*)= O there is a morphism ,:H*(EEO)H*(f2CxEE) such that jy = m*- 1*. This completes the proof.

Let p be a fixed prime number. Suppose that the space C is a product of Eilenberg-MacLane spaces, say C = K1 x ... x Ks where K, = K(J%, qi + 1), Ji = Z or Zp. Take mod p coefficients and let 0 E Ht(E, EO), where t is less than twice the connectivity of C. Let ei G fqi(7C x E) denote the fundamental class of the factor &2K, in SC. Then,

M(0) =yj(c) 0 Vii where a jG Ap (= mod p Steenrod algebra) and v, G H*(E), deg vi> 0. Combining (2.1) and (2.2) we obtain the following result.

(2.3). Let C, [27 [(X, X0), (E, E0)] be maps such that p*,1 = p*"2 in

[(X, X0), (B, Bo)]. Then there are classes ui e Hi(X, X0; J.), 1 _ i _ s. such that (U2 (u1, us) u8).' and hence

2 0 = aj(ui)-_l*vij .

In subsequent sections we will need the following exact sequence in- volving a.

(2.4). Let q = connectivity of C. For each integer j < 2q there is a morphism

a: Hi(72C x E, E) - Hj+'(B, BO)

such that the following sequence is exact:

*.. * > Hi(&2C x E, E) > Hj+'(B, BO) - Hj+'(E, EO)

--> Hj+'(fC x E, E) > *> H2 (E, EO) .



188 EMERY THOMAS

Moreover the morphism z- has the following properties. (a) Let j: (E, SC) (B, *) denote the map of pairs, let b: (B, *)(B, B0)

denote the inclusion, and let :: Hi(f2C) Hi+'(B, *)/Ker j* denote the transgression. Then, for u E Hi(7CQ), 0 < i < 2q,

b*z(u 1) mod Ker -* = :(u) .

(b) Let v E Hi(2C), w E Hi(B), 0 < i, i + j < 2q. Then

Z-(V X p*w) Z-7(V ( 1)- w .

(c) Let a be an element of the mod p Steenrod algebra, and let u E H*(&7C x E, E) be a class such that deg au < 2q. Then

z(au) = az(u) .

The definition of z- is similar to that of the relative transgression operator defined in [38]. See [42] for details.

In addition, we will need the following naturality properties of e and z. Suppose we have a commutative diagram of principal fibrations, as shown below:

nC E P B { C

f2C f' )E' B ') C'

Here p and p' are the principal fibrations with w, respectively w', as classifying maps. Let u, [e' denote the operator 2.2 associated with the respective fibrations p, p', and let z, z' denote the morphisms given in 2.4. By [42] we then have

(2.5). The following diagram is commutative:

** >Hi(B) ) Hi(E) ) Hi(nC x E, E) , Hi +'(B) ,...

Tg* p,* If* {(hxf)* g* ... --X Hi(B') , Hi(E') - Hi(&2C' x E', E') , Hi+'(B') , * .

where i is less than twice the connectivity of either C or C'.

3. Postnikov invariants

Let X, T. B be complexes with maps as shown below:

X e B < T.

A basic problem in topology is to determine whether factors through T, up to homotopy; i.e., whether there is a map C: X T such that wr C . We




review in this section some elementary facts about Postnikov resolutions that are relevant to this problem. (See [11], [19], and [38], for details.)

Recall that any such map wr can be regarded, up to homotopy, as a fiber map [7]. The fiber of wr is the space E, defined in ? 2, and the natural projection p,: Eu > T defined there plays the role of the fiber inclusion. If 7w is already a fibration then its fiber has the homotopy type of E,. (See [7] for details.)

Thus with no loss of generality we may assume that wr is a fiber map. Let F denote its fiber. We suppose that F is 1-connected and that w1,B acts trivially on F. Suppose that n and s are integers such that

,rF = O. O <i < n w 7nF = G # O. wrF = O. n<j<s, 7rF=A#O.

Let yn H"(F; G) denote the fundamental class, and set w - -(yn) c H7+1(B; G), where z denotes the transgression operator for the fibration wr [34]. Let p: E - B be the principal fibration with w as classifying map (i,e., we regard w as a map B-e K(G, n + 1)). The fiber of p is (2K(G, n + 1) = K(G, n). Since r*w = 0 there is a map q: T E such that pq = 7w. Thus we obtain a commutative diagram

F T

jr {q

(3.1) K(G, n) E

1P B,

where i and j denote inclusions and where r = q I F. It is easily shown (e.g., see [38, Th. 1]) that q can be chosen so that r - an.

Let Er = fibre of r, as above, and let Pr: Er - F denote the natural map. One can show that Er has'the homotopy type of the fibre of q, Eq, and that the composite map

Pr Er * F T

is homotopic to the natural map Pq: Eq - T [7]. Since r Y ran it follows that Eq (_ Er) is (s - 1)-connected and that rsEq A. Let s, E Hs(Eq; A) denote the fundamental class and set k =-'(y8) e Hs l(E; A), where z' denotes the

transgression in the fibration Pq T q E.

Eq I T E.



190 EMERY THOMAS

Notice that

(3.2) q*k = 0 j*k = k-invariant of FG HS+?(K(G, n); A) (see [8], [11]). If A is the integers mod p (p a prime), these properties characterize k, as remarked in [11].

Regard k as a map E -K(A, s + 1), and let p': E' E denote the fibration with k as classifying map. Since q*k = 0, there is a map q': T - E' such that p'q' = q. By an argument analogous to that given above, we see that the fibre of q' is at least s-connected. Thus the morphism

q' :7rT 7riE'

is infective for i ? s and is surjective for i ? s + 1. Therefore, if X is a complex of dim ? s + 1, the set map

(3.3) q* : LX, T] , X, E'] is surjective.

Let d be a map X B as given above. Since w~w = 0 a necessary con- dition for d to lift to T is that d * 0. Suppose this is the case. Then there is a map ,: X E such that p =. Set

k(s)= U2 r*k c HS+l(X; A),

where the union is over all such liftings (. By (3.3) we have:

(3.4). Let X be a complex of dimension ? s + 1 and let i: X B be a map. Then d lifts to T if and only if

(a) d*w = 0, (b) 0 G k(d). In practice the class w will often be some well-known primary invariant

such as a Stiefel-Whitney class or a Chern class. Thus we can assume that (a) poses no problem. Our concern in this paper is (b), the computation of k(s).

We now consider two examples of fibrations with associated invariants "k". In the first example the class k will also be simple to compute, while in the second no simple method seems to exist.

For any topological group G, we denote by BG the classifying space defined by Milnor [29]. Let U(n), n > 1, denote the unitary group of rank n. Recall that H*(BU(n); Z) is a polynomial algebra on the universal Chern classes c1, ** , cn, where deg ct = 2i. The inclusion U(n -k) c U(n) induces a map wr: BU(n - k) - BU(n). Regarded as a fibre map, wr has as fibre the complex Stiefel manifold Wnk = U(n)/U(n - k), 1 ? k < n - 1. (See Borel [4].)




Example I. The fibration

W BU(2s -1) BU(2s + 1) , s > O.

where W = W21,2. If X is a complex, a map co: X BU(2s + 1) determines a complex (2s + 1)-plane bundle over X. The bundle has two (complex) linearly independent cross-sections if and only if the map 0) lifts to BU(2s - 1). Recall [16] that W is (4s - 2)-connected and that

r4s81(W) w 4s+l(W) Z, w 4s(W) = 0

A Postnikov resolution for wr (through dim 4s + 2) is given below:

E -*K(Z, 4s+2)

/ 1~~P BU(2s - 1)- BU(2s + 1) - K(Z, 4s)

7t C2s

Here p denotes the principal fibre map with c28 as classifying map. Let j: K(Z, 4s - 1) E denote the inclusion of the fiber. By using the exact sequence (integer and mod 2 coefficients) given in [38, ? III], one can show that there is a unique class k e H4S+2(E; Z) with the following properties:

q*k = O. j*k = aSq2e4s-1, 2k= P*C2s+1,

where c41_j denotes the fundamental class of K(Z, 4s - 1) and 6 denotes the Bockstein coboundary from mod 2 to integer coefficients. Since &Sq2c41_j is the k-invariant for the space W, it follows from 3.2 that the class k can be taken as the Postnikov invariant for the map w. If ': X - E is a map, where dim X ? 4s + 2, this means that S lifts to BU(2s - 1) if and only if ~*k = 0. Now

2S*(k) = *(2k) = *p*c,+-

and so if H4s+2(X; Z) has no element of order two, then s lifts to BU(2s - 1) if and only if '*p*c21 = 0. Thus we have proved

THEOREM 3.5. Let X. be a complex of dim ? 4s + 2, and let Go be a complex (2s + 1)-plane bundle on X, s > 0. Suppose that H4S+2(X; Z) has no elements of order two. Then Go has two linearly independent cross-sections if and only if c,((a) = c2,1() = 0.

In other words, if c28(o)a=O, then c2,+1(ao))O mod2 and (1/2)c2?+1(a) e k(l(). Let M be a closed connected smooth manifold of dim 2q, q > 0. We say

that M has an almost-complex structure if there is a complex q-plane bundle (o over M whose underlying real bundle is isomorphic to the tangent bundle of M. In this case we define the complex span of M to be the maximal number of (complex) linearly independent cross-sections in ao. From 3.5 we obtain at once



192 EMERY THOMAS

COROLLARY 3.6. Let M be a manifold of dim 4s + 2, s > 0, with an almost-complex structure o. Then complex span M ? 2 if and only if cost = c28+1Co - 0.

Additional results along this line have been obtained by M. Gilmore [9]. For our second example, we consider the various rotation groups SO(n),

n _ 2. Recall that H*(BSO(n); Z2) is a polynomial algebra on the universal Stiefel-Whitney classes w2, *.., we, where wi has degree i. The inclusion SO(n - k) c SO(n) induces a map w: BSO(n - k) - BSO(n). The fiber of w is the real Stiefel manifold Vn,k, 1 ? k n - 1. (See [4].)

Example II. The fibration

V- BSO(4s + 1) >LBSO(4s + 3), s > 0,

where V = V3 2. A map i: X BSO(4s + 3) determines an orientable (4s + 3)-plane bundle over X, which has 2 linearly independent cross-sections if and only if the map d lifts to BSO(4s + 1). To study this lifting problem we construct a Postnikov resolution of w as shown below:

E ->K(Z2,4s+3)

(3.7) P

BSO(4s + 1) - BSO(4s + 3) > K(Z2, 4s + 2) . 77 W4s+2

Here p is the principal fibration with w4,+2 as its classifying map. Since *W4s+2 =0 0 1 factors into pq. Now V is 4s-connected while Z4,-1 V w 4s+2 Van

Z2. (See [31].) Also Sq2H4s+l( V; Z.>) = 0 ([5]). Thus the k-invariant for V is given by Sq2c4s+l E H4?+3(K(Z2, 4s+ 1); Z2), where t4?+l denotes the fundamental class. Therefore by (3.2), the class k is characterized by

q*ko= 0 i*k= Sq2e4s+1I

where i: K(Z2, 4s + 1) - E denotes the fibre inclusion. Notice here that, compared with Example I, there is no simple way to relate k to classes in H*(BSO(4s + 3)).

Consider the exact sequence (mod 2 coefficients) given in ? 2, as applied to this fibration:

P* h H4s -3(E) M H4s+3(K(Z2, 4s + 1) x E, E)

- H48?4(BSO(4s + 3)) P2 ...

(We take B0 = E0 = pt., in the sequence in ? 2.) Since p is induced by w48+2, the fundamental class c48+1 transgresses to w4,+2. Therefore by (2.4a),




Z(4s+1 1) W4s+2

Recall that by Wu [45],

Sq2w4s+2 + w2w4s+2 = 0

Therefore by (2.4b) and (2.4c),

z(Sq2c4s+1 0 1 + c4s+1 0 P*W2) = 0

and so by exactness there is a class k E H4S+3(E) such that

3.8) 1a(k) = Sq2c4s+1 0 1 + c4s+1 (0 P*W2 .

Of course k can be varied by image p*, but by using the exact sequence given in [38, ? III], one can show that k can be chosen uniquely so that (3.8) holds and q*k = 0. This choice of k then coincides with the class k given above. In other words we can think of k as arising because of the relation

Sq2w4s+2 + w2w4s+2 = 0?

which takes place in H*(BSO(4s + 3)). In the following three sections we show how this relation can be used to compute the class k, and hence the set k($). In particular, in the next section we sketch the theory of twisted cohomology operations.

4. Twisted cohomology operations

Let p be a prime, let A be a Hopf algebra over the field Zp, and let M be an algebra over A. Following Massey-Peterson [25] we define a new algebra, A(M), as follows. As a vector space A(M) = M 0 A. To define the multiplication in A(M) we use the diagonal map * in A. Given (m a), (n lb) e A(M), define

(m 0 a) * (n 0 b) =(-1)IaiI.InI J. (me ain) 0 (a, Qb),

where +(a) = 0iai D a'. If N is a left M-module over A, we make N into an A(M)-module by the

rule

(m (D a) * n = m * a(n) .

For each space Y, we denote by H*( Y) the mod p cohomology ring of Y. Let A, (or just A) denote the mod p Steenrod algebra. Then H*( Y) is an algebra over A, and so we may form A(H*( Y)), which we denote simply by A(Y). (See [10].)

Let (X, B) be a pair and i: X Y a map. Then H*(X, B) is an H*( Y)- module over A, by the rule



194 EMERY THOMAS

U.V = -*U V,

where u E H*( Y), v E H* (X, B). Consequently, H* (X, B) becomes an A( Y)- module. We emphasize that this module structure depends upon the map a. Let (X', B') be a second pair and f: (X', B') - (X, B) a map. Then the map e of (= f *e) induces an A( Y)-module structure on H*(X', B') so that f * is a morphism of A( Y)-modules.

Let Y' be a space, and g: Y - Y' a map. Then the morphism

g* 0 1: H*(Y') 0 A ) H*(Y) 0 A

is easily seen to be an algebra morphism from A(Y') to A(Y). To simplify the notation, we will write this morphism also as g*: A(Y') - A(Y). If t: X - Y is a map as above, then the map go t defines an A(Y')-module structure on H*(X, B). Given a' E A(Y'), v E H*(X, B) we have

(4.1) *af v = or'V ,

where the module action on the left (respectively, the right) is given by e (respectively, by go i).

For the rest of the paper we will use the convention that J (occasionally J', J., etc.) will denote either Z or Zp, p a prime; and K(J, s) will denote an Eilenberg-MacLane space of type (J, s), s a positive integer. Now let s and J be fixed, and set K = K(J, s). Identify Y with Y x * in Y x K, where * denotes the basepoint of any space. Let 1: Y x K Y denote the projection. By means of the cohomology morphism 1*, the algebra A(Y) acts on H*(Y x K, Y; J). In particular, if c E H3(K, *; J) denotes the fundamental class, and if v 0 a E A(Y), then

(4.2) (v D a) (1 0Dc) = v D a(c) . We regard a class u E H-(X, B; J) as a map (X, B) (K, *). As above,

given a fixed map I: X Y, we define a map

(4.3) g.: (X, B) - Y x (K, *)

by the composition

(X, B) X x (X, B) x

Y x (K, *),

where d is the diagonal map. Notice that

(4.4) g*(1 (Dc) = u . We now consider higher order operations associated with the algebra

A(Y), following the method of J.F.McClendon. (See [27] for details.) For simplicity, we do only the case p = 2. Let a1, ..., ak, 1, . f 1k be elements in A(Y). Assume that deg ao + deg,8 = q + 1, for some integer q and all




1 i < k. Write

a, * ai ?i in A(Y)

Suppose that aore = 0. With such a relation in A( Y) we associate a secondary operation 4) as follows. To define 4) on classes of dim s, we set C K1 x .. x Kk, where Ki =K(Z2, s+deg/8j). Let id: (Y x K, Y)*(Kj,*) denote a map corresponding to the cohomology class f8 * (1 0 c), and set

A8= (18, Y .. Y 1k) (Y X K, Y) ) (CY *) .

Let p: (E, Y) (Y x K, Y) be the principal fibration with classifying map,8. (See ? 2.) Since 8 restricted to Y(= Y x *) is the constant map, Y is homeomorphic to Y. The fibre of p is saC. Let A(Y) act on H*(&2C x E, E) by means of the composite map

proj p proj &2Cx E - E---Y x K - Y.

Suppose now that the integer s + q is less than twice the connectivity of C. (Recall that the relation a ,8 = 0 has degree q + 1.) Consider the exact sequence given in ? 2:

** ,H-f( x KY Y) H Ha(E Y) HS(C x E, E)

,HS~ ( x K, Y)- .

Let ct. 1 < i < k, denote the fundamental class of i2Kj in N?C. By 2.4a, z(ei 0 1) =8 * (1 0 c). It follows from 2.4b, c, that z- is a morphism of A(Y)- modules, and so

-(ai * (e XD 1)) = (ajh8) * (1 X c)

Consequently, z-(a) = 0, where

a - i- e

Thus by exactness, there is a class p E Hs+q(E, Y) such that 4u(p) a a. We define p to be a representative of the secondary operation 1 associated with the relation af,8 = 0.

Let (X, B) be a pair, let I: X Y be a map, and let u E Hs(X, B; J). Let A(Y) act on H*(X, B; J) by means of t. Suppose that f8i u = 0 for 1 ? i _ k. Then there is a map f: (X, B) (E, Y) such that

of gu Y

where gu is given in 4.3. We then say that 1 is defined on the pair (u, i) and set

4(, i) U f *9 C Hs+q(X, B),



196 EMERY THOMAS

where the union is taken over all maps f such that pf = g,. We define

Indet*(X, B; 4, t) = EaiH*(X, B) .

From 2.3, it is easily seen that 4(u, i) is a coset of the subgroup Indets-q(X, B; J), i).

From the exact sequence given above it follows that

Kernel pa f Hs+q(E, Y) = p*Hs+q(y x K, Y),

and so we can think of 4) as the coset of cp with respect to the subgroup p*Hs+q( Y x K, Y). Thus 4) represents a family of operations associated with the relation ao.s= 0, the members of the family differing by elements of Hs+q(Y x K, Y).

We will say in general that the secondary operations are of order 2, and we write E = E2, Y = Y2. The pair (E2, Y2) will be called a universal example of order 2. As part of the definition, we include the principal fibre map p = P2: (E2, Y2) -( Y x K, Y) with the property that p2 maps Y2 homeomorphically onto Y. To define a universal example of order n, for n > 2, suppose inductively that we have defined a universal example of order n - 1, (E.1, Y,-,), with a fibre map pn-1: (En-1, Y,-,) (Y x K, Y) such that pn-1 maps Y,-1 homeomorphically onto Y. We make H*(En,,l Y,,-) into an A(Y)- module by using the composite map

En-1 ? Y x ) Y. Let 9i, *.,k E H*(E._1, Yn1), and let a1, . . *,ak C A(Y) be such that degaj+ deg (p = t + 1, say, for 1 ? i ? k. Suppose that

Lir at,(p = ?

We now simply repeat the construction given above for the case n= 2, replacing (h8, . * *, Sk) by (gb, * * *, (Pk)(= cp). Thus using qp as a map, we induce a principal fibration qn: (Ens Yn) (En-19 Yn-1) such that qn maps Yn homeomorphically onto Yn-j Then (En r Y) is a universal example of order n, where P = Pn-loqn: (Ens YJ) (Y x K, Y). It will simplify the notation (especially in ? 5) if we agree to identify Yn and Y by Pn. Thus we will write our universal example simply (Ens Y).

The set of cohomology classes = (p , * *p , (k) can be regarded as a map : (En1, Y) - (D, *), where D denotes the appropriate product of Eilenberg-

MacLane spaces. We now assume that the integer t is less than twice the connectivity of D, where t + 1 is the degree of the relation E a- = 0. Again, by using 2.4 for the fibration qn, it follows that there is a class

c C Ht(En, Y) such that




(4.5) -n(*) = Ad a 0i (ii D 1) where Son is the operator (2.2) for the fibration qn, and where ?i denotes the fundamental class of the jth factor in SD. The class * is called a representative for the operation T of order n associated with the relation F ai 0.i = ?.

We say that the operation T is defined on (u, i) if there is a map h: (X, B) (En, Y) such that pnh = gu. If so we set

If(Ug O = Uh h** c: HI(X, B), where the union is taken over all such maps h. The existence of a map h will often be given in an inductive fashion, as will now be shown. Suppose there is a map h,-: (X, B) -E (E1,a Y) such that Pn- o hn-1 = gu. We regard the cohomology classes (qp, , qk) as a multi-valued operation and set

((p1, * * *Pk)(U i) - Uhn-1 (h-l(p1 ly, ...* hn>lnpk)

where the union is taken over all such maps hl1. Then, in this notation, P(u, i) is defined if and only if 0 E Q1, *(**, Pk)(U, I). We define

Indett (X, B; T, y) T(O, t)

where 0 e Hs(X, B). It is shown in [27, ?2.1] that Indett (X, B; T, t) is a subgroup of Ht(X, B), and that T(u, t) is a coset of this subgroup. Notice that, for secondary operations, we have computed the indeterminacy explicitly by means of primary operations.

Let (Emn, Y) be a universal example of order m > 2, let q E H*(Em", Y), and let ID be the operation of order m defined by qp as above. We will need the following properties of the operation (. (As usual, we let (X, B) be a pair, :X - Y a map, and u E Hs(X, B; J).)

(4.6). Naturality. Let (X', B') be a pair, and f: (X', B') (X, B) a map. If (D is defined on (u, i), then (D is defined on (f *u, f *), and

f*41(u, i) c 4D(f*u, f*Th The proof is immediate.

(4.7). Linearity. Let u, W' E Hs(X, B; J), and suppose that ( is defined on (u, I) and (u', i). Then ( is defined on (u + u', i), and

1(u + u', i) = 4(u, i) + 4(u', i),

as cosets of Indets+q (X, B; JD, i). The proof is given in [27]. We will need the following simple computation

of the operations. Suppose that the cohomology class u 0 O. Then the map g.: X - Y x K, given in 4.3, can be factored up to homotopy into i o i, where i: Ye) Y x K is the inclusion. Let ( be an operation of order m(_ 2), with



198 EMERY THOMAS

universal example pm: (Emn, Y) (Y x K, Y). Define h: X Em by the

composition X Y c Emn, where j denotes the inclusion. Then

Pmh= i$ -go

and so h*9p E 4(O, i). But j*q = 0 since q E H*(Em, Y). Thus we have shown:

(4.8) 0 E (0, i)

Remark 1. Gitler and Stasheff define secondary operations associated with relations in A(Y). Since they use the method of functional operations, their operations have a larger indeterminacy than the operations defined in [27].

Remark 2. Let pm: (Em, Y) - (Y x K, Y) denote a universal example of

order m for some m > 2. Let (X, B) be a pair, and h: (X, B) (Em, Y) a map. Suppose that q E H*(Em, Y) represents an operation (. Then h*ap e 4(u, y) for some pair (u, t). How do we find (u, d)? Let

1r YxK- Y, YxK -)K

denote the projections. Then,

(4.9) =lopoh, u = ropmoh,

where h: X Em is the map determined by h.

Remark 3. Suppose that J = Z, so that K = K(Z, n). Let a1, ..., (k,

, / ...k E A(Y). Even though ark - 0 in A(Y) it still may happen that

Lk ak k (1 (D 0 0 in H*(Y x K, Y). In this case we say that we have a relation a /,8 0 over Z. (The simplest example is Sq2 Sq2 = 0 over Z.)

5. Genera ting classes

As in ? 3, we consider the following diagram of complexes and maps:

X t B ( 7T .

Rather than consider a specific Postnikov resolution for w (as given in ? 3), we consider here a more general situation. Namely, suppose that w = (w1, ***, wa) is any vector of cohomology classes of B such that w*w = 0. Then w is a primary obstruction to factoring i, in the sense that if e factors, then d*w = 0. Let p: E - B be the principal fibration over B with w as classifying map. (See ? 2.) Since w*w = 0, there is a map q: TV E such that pq = w. Let k be a mod p cohomology class of E such that q*k = 0. Set

k(s) = Uf! f*k ,

where the union is taken over all maps f: X - E such that pf = d. (For k(s)




to be non-empty we must have $jw = 0.) If k has degree t, say, then k( )Qc Ht(X) (mod p coefficients). Each such set k(s) is a secondary obstruction, for if d factors through T, then 0 E k(s). Thus, regard k(s) as a generalized Postnikov invariant. In this section we consider the problem of computing the set k(s).

Our main result, roughly speaking, is as follows: given certain hypotheses, there will exist classes v, m E H*(B) and a higher order operation &2 such that

( * ) k + p*me 0(p*v) .

By naturality this implies

k(s) + d*m cQ (*v),

which gives us some hold on the coset k(s). In specific examples (see ? 6), we can use (*) to compute k(s) precisely. The exact wording of the main result (Theorem 5.9) is rather complicated, since in its most general form the operation &2 will be given as a twisted higher order operation (as defined in ? 4). We proceed with the details.

Let w be the vector of cohomology classes of B given above; say, w (w1, *..., wa) where wi E H*(B; Jt), each Ji = Z or Z,. Set C = K1 x ... x Ka, where Ki = K(Ji, deg wi). Then w is a map By C, and we obtain the following diagram:

i k U2C E K(Zvy t)

(5.1) / 1P

T ,B ,C 7C W

(Here &2C is the fibre of p and i is the inclusion.) For the remainder of this section we make the following hypothesis:

(5.2). The degree of the cohomology class k(= t) is less than twice the connectivity of C.

Because of 5.2 any class in Ht(&2C x E, E) can be written as a sum of terms of the form

Ej ai P * Vai where et denotes the fundamental class of the factor i2Kj in i2C, where ai E A, and where vi E H*(B). But

ai(ci) (0 p*vi ?(vi (0 a) i (i1)

where (vi 0 aj) denotes an element in A(B), and where A(B) acts on H*(f2C x E, E) by the composite map



200 EMERY THOMAS

proj P

Let A: H*(E) H*(72C x E, E) denote the operator defined in 2.2. By the above remarks we see that there are classes ai E A(B), i = 1, ** , a, such that

U(k) =E a'i (ei @1

Letting "" = (a",, * *, a.), C ( 0 1, **a 0 1), we then have

p(k) =ac .

It follows from 2.3 that if t: X B is a map such that t*w = 0, then k(s) is a coset of the subgroup

(&H*(x)) n Ht(X) , where A(B) acts on H*(X) by the map I.

If the class k is in fact a Postnikov invariant as in ? 3, then its charac- terizing properties are given in (3.2), namely,

(i) q*k 0. (ii) i*k k-invariant of fibre of w.

For the general theory described here it is convenient to let k vary, provided that q*k and i*k remain fixed. In fact we restrict k to belong to a certain coset K, as follows. As above, assume that k E Kernel q*. Set K = coset of k in Ht(E) with respect to the subgroup Ker q* n Ker , n Ht(E). If kl, k2 E K,

then 4a(kl) = (k2), and if we set

Indet* (X; K) = a.H*(X) ,

we then can say that k(s) is a coset of Indett (X; K) for any k E K. Let Y be a space, and A: B-> Y a map. We will say that the vector a',

defined over A(B), is induced by (Y, I) if there are elements (a1, * , a,) in A(Y) and (pl, , Pa) in A such that

(5.3) 1^:, = * (ai pi), 1 _ i _ a,

where A*: A(Y) A(B) is defined in 4.1. (Recall [25] that A is embedded in A(Y) by al 0a, aeA.)

To illustrate these ideas, consider the fibration given in Example II, ? 3. The diagram given there (3.7) is a special case of diagram 5.1, with w = W4"12

C = K(Z2, 4s + 2), t = 4s + 3. Let k E H4s?3(E; Z2) be the the class given in 3.7. Then q*k = 0 and by 3.8, a(k) = Sq2 c4t+l (01 + c48e1 0 p*W2. Thus, for this example,

w =W2 0 1 + 1 0 Sq2 E A(BSO(4s + 3)).




Now take Y K(Z2, 2), a = W2: BSO(4s + 3) Y, a =t2 (9 1 +1 (? Sq2 E A(K(Z2, 2)), p identity. Then,

a= a,

and so a' is induced by (K(Z2, 2), w2). We now come to the main definition of the paper. Suppose that a^ is

induced from a by (Y, ,), as defined in 5.3. Let w be the cohomology vector of BA given at the beginning of the section, and let K c Ht(E) be the coset defined above. For simplicity we now assume that p = 2. We will say that a class v E H*(B) is a generating class for K if the following three conditions, (5.4)-(5.6), are fulfilled.

(5.4). There exist operations pi, ..., q'a, 1 .. * *b (possibly of order > 1) defined over A( Y) such that

(pw, 0) = (cg, Ojv,

where cp ((pig , ). ik= (*i, ., b)

(Here the element p is the element referred to in (5.3). Also, we use the following notational convention: if u is any integral cohomology class, and if 1 e A denotes the identity, then 1 u denotes the mod 2 reduction of u.)

The remaining two conditions in our definition are as follows:

(5.5). There are elements ,81, . . . fib A A(Y), such that

oath + Age = 0

where ,- (ft, . 13b).

(5.6). There is an operation 52 associated with relation (5.5) such that

i2(z*v, A*ge) = w*M,

where M is a coset of Indett (B; 52, A). Notice that the class v occurs in the definition in 5.4 and 5.6, but that

the coset x occurs only through the appearance of the class a in 5.5. As an example of the definition, we will show in ? 6 that the class

W4s E H4s(BSO(4s + 3)) (see Example II, ? 3) is a generating class for the coset determined by the class k in Example II.

We consider in more detail the meaning of (5.4)-(5.6). Suppose that the class v has degree s, and let K = K(Z2, s). If q, r are vectors consisting of primary operations over A(Y), then they can be represented as cohomology vectors of (Y x K, Y), as shown in 4.2. Suppose they are higher order operations, say of order n > 1. Then there is a fibration



202 EMERY THOMAS

built up inductively as described in ? 4, such that q, * are cohomology vectors of (U, Y). (U En, in the notation of ?4.) Thus (5.4) means that there is a map g: (B, (U, Y) such that

*(Pi = piwi 1 _ i_ a, g*j = ,1 j< b l oPnag =' C r oPino?g = V

(See Remark 2 in ?4.) In (5.5) the algebra A(Y) acts on H*(U, Y) by the map 1 op P and the relation is given in terms of cohomology classes. To define the operation D in 5.6 we construct the principal fibration qn11: (W, Y) (U, Y), with classifying map given by the cohomology vector (qa, *). (W En+1, in the notation of ? 4.) According to ? 4 there is a class a e Ht( W, Y) such that

(5.8) Pn+l(0) = (a, ,&) .1 e Ht(F x W, W) . Here F is the fiber of qn+9 1un+1 is the operator given in 2.2, and c denotes the cohomology vector consisting of the fundamental class of each of the factors of F. The operation a2 corresponds then to the choice of one such class w. (Notice that if the cohomology morphism r*: H*(B) H*(T) is surjective in degrees <t, then (5.6) is automatically satisfied for any choice of a2.)

Let L be any space that has the homotopy type of a complex and let f: L - B be a map such that f*w 0 O. Then the operation a is defined on the pair (f*v, f *). For if g: (B, *) (U, Y) is the map given in 5.7, then

(gf)*.i = f*g*qP. = Pjf*W. = 0, (gf)**j = f*0**j = | ? pn ? (gf) = of , r o Pn ? (gf) v of.

Hence by Remark 2 in ?4, 42 is defined on (f*v, f*y). In particular, &2 is defined on the pairs

(pad, p*'O), (7r*vg 7r*'O (*V, An*)

The operation &2 will play a key role in what follows. To emphasize this we will say that v is a generating class for /c, relative to D2.

We now can state the main theoretical result of the paper.

THEOREM 5.9. Let v be a generating class for X, as defined in (5.4)-(5.6). Let D be the operation, and M the coset, given in 5.6. Then there is a class ke X / such that

k, X Q0(p*v, p* )-p*M.

The proof is given in ? 8. Let X be a complex and d: X -B a map as above. Let A(Y) act on H*(X)

by the composite map A*t: X - Y. Suppose that d*w - 0, so that there is a




map f: X E with pf = i. Then by definition, f *k1 X k1(j), where k1 X Ht(E) is the class given in 5.9. But by naturality and 5.9,

f *k1 E 0(e*v, *) -*M

since f *p* = i*. Thus we obtain

COROLLARY 5.10. Let X be a complex, and let d: X B be a map such that d*w 0 O. Suppose that

Indett (X; a?, 0*2)= Indett (X; Kc) . Then,

k1(d) - S0(e*v, W*r)-{J*M}. Here {f*M} denotes the coset of Indett (X; ? *,2) determined by the

subset H*M. In our applications given in ? 6 the vector w and the class k will be chosen

so that for a suitable category of complexes (e.g., those with appropriate dimension, as in 3.4) the coset k1(d) will be the final obstruction to factoring d through T. In other words, in these cases e factors through T if and only if d*w - 0 and E2(d*v, d*O) = {M}.

Recall that the coset K has been defined with respect to a subgroup of the form Kernel q* n Kernel ,u. In our applications (? 6) this subgroup will be zero in the appropriate dimensions, a fact which is implied by the following result. (We retain the notation of diagram 5.1.)

PROPOSITION 5.11. Let d be an integer that is less than twice the connectivity of C. Then

Ker q* n Kera = 0 in dim d if and only if

Ker p* Ker w* in dim d . PROOF. Since ST = pq, one always has Ker p* cz Ker ST*. Notice also that

(**) p*(Ker wr*) = Ker q* n Image p* . By exact sequence 2.4 (taking B. = * in that sequence), it follows that

Ker p = Image p* in dim d . Thus by (**),

p*(Ker wr*) = Ker q* n Ker , in dim d, which implies 5.11.

Remark. If Ker q* n Ker a = 0 in dim t, then the coset K consists of the single element k. In this case we will say that v is a generating class for k, relative to D.



204 EMERY THOMAS

6. Bundles over complexes

In this section we give two examples of the theory developed in the preceding sections.

All cohomology will be taken with mod 2 coefficients, unless otherwise noted.

We consider first Example II, given in ? 3, referring back to diagram 3.7 for our notation. Recall that K(Z2, 4s + 1) is the fibre of the fibration p: E d BSO(4s + 3). Let A(BSO(4s + 3)) act on H*(K(Z2, 4s + 1) x E) by the composite map

K(Z2, 4s + 1) x E-3 E - BSO(4s + 3). Set

a= w2 ( 1 + 1 Sq2 in A(BSO(4s + 3)). It follows from 3.8 that

M(k) = af .(e4.+1 0 1)

Let a = t2 0 1 + 1 (0 Sq2 in A(K(Z2, 2)). As observed following 5.3, or is induced from a by (K(Z2, 2), w2).

A simple calculation shows that

(6.1) a-a + Sql.(a.Sql) = 0

in A(K(Z2, 2)). (Recall that A is embedded in A(K(Z2, 2)) by the map a 1 0 a, a e A.) Let 43 denote a twisted secondary operation (of degree 3) associated with this relation, as described in ? 4. We prove

PROPOSITION 6.2. The class w48 e H 4(BSO(4s + 3)) is a generating class for k, relative to the operation 4D3.

PROOF. In terms of the notation used in 5.4, 5.5, let

(pi = a , *1 = aSq' , 11 = Sq',

all in A(K(Z2, 2)). We show that these operations satisfy 5.4-5.6. By the Wu formulas [44],

aw48 = W s+2 a aSqlw48 = 0 y

and so 5.4 is satisfied. Relation 6.1 gives 5.5. Finally, wr* is surjective (mod 2) in all dimensions, and so 5.6 is trivially fulfilled. Thus w48 is a generating class for the coset / determined by k. But one sees easily that

Ker wr* = Ker p* in dim 4s + 3,

and so by 5.11, k is the sole element in i. Thus w,8 is a generating class for k, and the proof is complete.




For applications, we need to know more about the coset M given in 5.6.

PROPOSITION 6.3. The operation 1%3 can be chosen so that

0 E 4D(w4s, w2) c H4s+3(BSO(4s + 1)) .

PROOF. Let 43 denote a specific choice of the operation. Let

j: BSO(4s - 1) cz BSO(4s + 1) denote the map induced by the inclusion SO(4s - 1) cz SO(4s + 1). Since j*w48 = 0, it follows from 4.6 and 4.8 that

j*4N(W48, W2) cz 4(j*w48, j*w2) = 4(0, j*w2)

= Indet4s+3(BSO(4s - 1); 't3, j*W2)

Since j* is surjective (mod 2) in all dimensions, it follows that 0 e j*D(w48, W2) -

Now Kernel j* in dim 4s + 3 is generated by W4, * W3 and wd8?1, w2. Thus there are mod 2 integers, a and b, such that

aW48*W3 + bw4.,+l w2 e 4D(w48, w2)

Let p: E2- K(Z2, 2) x K(Z2, 4s) denote the universal example for the operation 4?3 (see ? 4). Let p e H4s+3(E2) represent the specific choice of 43 given above. Set

' = (q + p*(aSqlt2 0 t4s + bt2 0D Sq't48), and let V denote the operation determined by q'. Since W3 =Sqlw2, w4s+1

Sq'w48, it follows that 0 e D3(w48, w2), and so J3 is the desired choice of the operation.

As a consequence of 6.3, we can choose the coset M in 5.6 to be the subgroup Indet4s+3(BSO(4s + 3); Iq, W2). Now let e be a bundle over a complex X as given in Example II, ? 3. Recall from ? 5 that

Indet4s+3(X; IC) = a.H4s+l(X)

where A(K(Z2, 2)) acts on H*(X) by the map w2d: X K(Z2, 2). On the other hand, by 6.1 we see that

Indet4s+3(X; 4'3, wl2) = aH4s+l(X) + SqlH4s+2(X)

Therefore by 3.4 and 5.10 we have the following result.

THEOREM 6.4. Let X be a complex of dim ? 4s + 3, and let d be an orientable (4s + 3)-plane bundle over X, s > 0. Suppose that

SqlH4s+2(X) cz a.H4s+l(X) .

Then e has two linearly independent cross-sections if and only if

w4s+2() = 0 and 0 e 43(W4C(), W2())



206 EMERY THOMAS

Here 4)3 is the operation given in 6.3.

Remark. Since w2(SO(3)) = 0, it follows that if d is an orientable 3-plane bundle on a complex of dim ? 3, then e is trivial if and only if w2() = 0.

For our second application of 5.10 we consider the fibration

S4s > BSO(4s) 7 BSO(4s + 1), s > 0 .

A map d: X > BSO(4s + 1) represents an orientable 4s-sphere bundle over X, and a lifting of e corresponds to a cross-section in this bundle. A resolution of wr (through dim 4s + 2) is given below:

E K(Z2, 4s + 2)

BSO(4s) -> BSO(4s + 1) -.

K(Z, 4s + 1). 7t e3~~~W4s

Here a denotes the Bockstein coboundary going from mod 2 coefficients to integer coefficients. The map p is the principal fibre map with classifying map aw 8o The class k e H4s+2(E) is characterized by

q*k = 0 p(k) = a' (g4 )

where a' w 01 + 1 0 Sq2 e A(BSO(4s + 1)), and where A(BSO(4s + 1)) acts on H*(K(Z, 4s) x E; Z) by the composite map

K(Z, 4s) x E P-* E P BSO(4s + 1).

Moreover, if X is a complex of dim ? 4s + 2, and if A: X - E is a map, then , lifts to BSO(4s + 1) if and only if 72*k = 0. The proof of these statements is similar to that given in Example II, ? 3. For details see [19] and [38].

Notice that the class a' is again induced from a in A(K(Z2, 2)) by the map W2: BSO(4s) K(Z2, 2). (Here a = 2 (D1 + 1 0 Sq2.) A simple calculation shows that

(6.5) aoa = 0 over Z, (see Remark 3 in ?4).

Let D* be a twisted secondary operation associated with this relation. We prove:

PROPOSITION 6.6. The class (w482 in H4s-1(BSO(4s + 1); Z) is a generating class for k, relative to 4?.

PROOF. Recall the notational convention used in 5.4, that is, if 1 e A denotes the identity, and if u is an integral cohomology class, then Iu = mod 2 reduction of u. In particular, 1 (3w4, = mod 2 reduction of =w48 = w4s+




But by Wu [44],

a, (3W'4s2) w4s+1

Thus a (3w48_2) =1 .w,8 and so taking qi, = a, we satisfy 5.4. Relation 6.5 satisfies 5.5, and since z* is surjective mod 2, 5.6 is trivially fulfilled. Finally, Kerz* = Ker p* in dim 4s + 3, and so by 5.11, k is the sole element in /. Thus aw48_2 is a generating class for k, as claimed.

Again we seek more information about the coset M given in 5.6. Let j: BSO(4s - 2) ci BSO(4s) denote the map induced by the inclusion. Since j*(3w4.-2) = 0, since j* is surjective, and since Kernel j* in dim 4s + 2 is generated by,

aW4.-2 * Sq'W2, w48 W2

we have:

PROPOSITION 6.7. The operation ID* can be chosen so that

XW48 *W2 E 3*(aw48_2, w2) ci H's+ (BSO(4s))

where X e Z2.

The proof is similar to that given for 6.3, and so is left to the reader. Thus we can take the coset M, given in 5.6, to be the coset of Indet4s+2(BSO(4s + 1); *3, w2) containing the class XW48*W2.

Now let X be a complex, and d: X BSO(4s + 1) a bundle as above. Recall that

Indet4s+2 (X; IC) = ao H4s(X; Z)

Jndet4+2 (X; t*, w24) =.H48(X; Z2)

Therefore, from 3.4 and 5.10 we have:

THEOREM 6.8. Let X be a complex, and let d be an orientable 4s-sphere bundle over X, s > 0. Suppose that a.H4s(X) = a.H4s(X; Z), where A(K(Z2, 2)) acts by the map W2i: X K(Z2, 2). Then d has a cross-section over X4.+2 if and only if

aw ,(d) = 0 and XW4.(,)*w2(i) e 43*(aW4.-2(i), W2())

where X is an integer mod 2 independent of X. Here ID* is the operation given in (6.7).

7. Bundles over manifolds

In this section we specialize the theorems of ? 6 by taking the complex X to be a closed, connected, smooth manifold M. The proof of Theorem 1.1 follows as a consequence. Throughout the section we assume that M is



208 EMERY THOMAS

orientable. In order to apply Theorems 6.4 and 6.8 we must calculate the operations

4)3 and (1g. This is done by means of the following Cartan formula. Recall that 4)3, (13 are given by the following relations in A(K(Z2, 2)):

4()3: a-a + Sql.(a.Sql) = 0 4(*: a0a = 0 over Z,

where a = L2 0 1 + 1 0 Sq2. Let X be a complex, and let u e H*(X; Z), v E H*(X), w e H2(X) be cohomology classes such that 41* is defined on (u, w) and 4(3 on (v, w). Then an easy calculation shows that

Sq2(u v) = 0, Sq 2Sq'(u -v) = 0 and so u - v is in the domain of the (unique) Adams secondary operation P3 [1] associated with the relation

Sq2Sq2 + Sq'(Sq2Sq') 0 .

From the general theory of twisted secondary operations [27], it follows that

4)3(U - V9 0) = @3NU - V)

where the common indeterminacy subgroup is that of the right hand side. The following theorem is proved by analogy with (3, 9.4) in [1]. We

omit the details (cf. [40, 5.3]).

THEOREM 7.1. Let X be a complex, and u, v, w cohomology classes as above. Then there is a mod 2 integer b (independent of X) such that

P3(u >-v)D'4*(u,w) --v + u-A43(V, W) + bSq2u'-Sq'v . We use 7.1 to prove the following result.

THEOREM 7.2. Let M be a manifold as above, and let A(K(Z2, 2)) act on H*(M; Z2) by the map w2M: Ma K(Z2, 2). Suppose that dim M = n > 3; let v e H 3(M; Z2) Then I?3 is defined on (v, w2M), and there is a mod 2 integer X, which depends only upon the choice of the operation 4)3, such that

413(V, w2M) = XSq2Sq'v,

with zero indeterminacy. Similarly, if v e Hn-3(M; Z), then

41*(v, w2M) = 0

with zero indeterminacy.

The proof is given at the end of the section. We consider now the theorems of ? 6, taking the complex X to be a

manifold M, as above.




THEOREM 7.3. Let n be an integer >2, let M be an n-manifold, and let e be an oriented n-plane bundle over M with Euler class x(e). Suppose that

2W() = W2(M) W.-2(0)w2(0) = ?

If either (a) n 3 mod 4 and wn_1(e) 0 ,.

or (b) n =2 mod 4, and X(e) = 3w.-2() O,

then e has two linearly independent cross-sections.

PROOF. Assume (a) and set n = 4s + 3, s > 0. If s = 0 the result is given in the Remark in ? 6, so we suppose that s > 0. We will apply 6.4 to obtain the proof, and so check the hypotheses of that theorem. Since M is orientable, Sq'H--'(M) = 0. Let 4)3 be the specific operation given in 6.3. By 7.2

XSq2Sq'w48() G 4)3(W48(0), w2(M)),

where 0 or 1 depending on the choice of (1)3. But

Sq2Sq'w48() = W48+1(0) W2() = 0

by hypothesis, and soO e (D3(W4J(O) w2(M)). Therefore 7.3a follows from 6.4. To prove part (b), notice that since X(d) = 0, we can write e = e 1,

where 1 denotes the trivial line bundle. Thus d is an orientable (4s + 1)-plane bundle (writing n = 4s + 2) and wj(?2) = wj(i) for i > 0. To prove 7.3b, we apply 6.8 to r. Since w2(?) = w2(M) it follows by Wu that a.H4s(M; Z) = a.H4s(M; Z2) = 0, where a= t2 ? 1 ? 1 0D Sq2 in A(K(Z2, 2)), and where A(K(Z2, 2)) acts on H*(M) by the map w2M: M-a K(Z2, 2). By 7.2, 0 E 3w48W2, w2M). Now (W48C?). (W2?2) = 0 by hypothesis, and so by 6.8, if aw,.8 = 0 then , has a non-zero cross-section. That is if aw4.' = 0, then e has two independent cross-sections, as asserted.

PROOF OF 1.1. In 7.3 we take e to be the tangent bundle of M, As. Suppose first that n = 4s +. 3. By Massey [22], w48+1M = W0,+2M =O and so the first part of 1.1 follows from 7.3a. Suppose then that n = 4s + 2. If we show that

(* ) w48M'w2M = 0. and 3w48M = 0. then the second part of 1.1 will follow from 7.3b.

To prove (*), recall that by Wu [45] w,8M = V2, where Ve H2s(M) is the unique class such that for all u E H2s+2(M),

Sq (U) U p V.

Thus,



210 EMERY THOMAS

w48M-w2M= Sq2(w48M) =Sq2(V2) = (Sq1V)2 = Sq'(V Sq'V) O0

since M is orientable. This proves the first half of (*); the second half is proved by Massey [24].

PROOF OF 7.2. By Wu [45], for any class x e Hn-2(M) (mod 2 coefficients), aox = 0, and thus if v e H -3(M), aoSq'v = 0. Similarly, using Poincare duality, one shows that ao v = 0. Thus the operation 1D3 is defined on (v,w2M), and since a.Hn-2(M) 0 0 and Sq'Hn-'(M) = 0, it follows that 1D3 is defined with zero indeterminacy.

Take an embedding of M in Rn+q for some large q. Let v be the normal bundle of this embedding and let T denote the Thom complex of v. We regard T as the pair (D, S), where D denotes the disk bundle and S the sphere bundle of v. The zero cross-section, M D gives a homotopy equivalence. Let A(K(Z, 2)) operate on H*(D, S) by the class w w Me H2(D) (H2(M)), regarded as a map D K(Z2, 2). Let U e Hq(D, S; Z) denote the Thom class. By Whitney duality w2, = w2M and so by Thom [37], a U = 0. Thus the operation 4D* is defined on U. In a moment we show:

(7.4). There is a mod 2 integer c such that

c(U'w3M) e 4D(U) Assuming this, we complete the proof of 7.2. Consider the class U .*v e H-3(D, S). An easy calculation shows that

Sq2(U. V) 0, Sq2Sq'( U * v) = 0

and so the Adams operation 'I3 is defined on U * v. Let T D/S. The collapsing map -y induces an isomorphism

"/*: H*(T. *) H*(D, S) .

Since v is the normal bundle of an embedding, the top cohomology class of T is spherical, and so 1D3(?*-((U v)) = 0, with zero indeterminacy. (See [25, 4.1.1].) Thus by naturality,

'A3( U * v) = 0 with zero indeterminacy.

On the other hand, by 7.1,

(**) 0 = J3( U*v) D 413*(U, w2M) - v + U - 13(v, w2M) + bSq2U * Sq'v .

By 7.4 it follows that 4W( U, w2M) - V is the set of elements of the form

(c(U w3M) + a. (U y)) -v , where y runs over all classes in H'(M). But one easily shows that a* (U . y)= 0, since Sq2y -0. Thus 4W(U, w2M) - v = (c(Uw3M)) - v, and so by (**),




0 = U - (cw3M v + bw2M.Sq'v + 13(v, w>M)).

Therefore by the Thom isomorphism,

13(v, w2M) = cw3M'v + bwAMSqlv.

But by Wu, w M * Sq'v = Sq2Sq'v, and so w3M*v = Sq w M * v = w M * Sq'v Sq2Sq'v. Thus,

13(v, w-M) = (b + c)Sq2Sq'v .

Since 4'3 was an arbitrary choice of the operation, this completes the proof of 7.2, for the case v is a mod 2 class. The details for v an integral class are left to the reader.

We are left with proving 7.4. But this follows at once from the fact that the bundle v is induced from the canonical bundle Yq over BSO(q) by some map of M into BSO(q). Moreover, the class U is thus induced from the Thom class Uq of Yq. But if Tq denotes the Thom complex of 7q, then Hq+3(Tq) Z, generated by Uq*w3, and so 7.4 follows by naturality.

8. Proof of Theorem 5.9

For the proof of 5.9 we need the following simple fact about principal fibre spaces. Let A, B, C be spaces with maps as shown below:

c b C A B.

Construct the following principal fibrations with classifying maps as given below:

p: E >A, Y pl: El A A, P2: E2 El Here p has (b, c): A B x C as classifying map, p1 has classifying map b, and p2 has c - P.

LEMMA 8.1. There is a homeomorphism h: E E2 such that p1ph= p: E-+A.

The proof is immediate, and is left to the reader. We will identify E and E, (as fibre spaces over A) by h.

The constructions made above give the following commutative diagram:

SDC SDC x f2B E2

jr 1P2

(8.2) QB-> E1 - c AP B A bB,



212 EMERY THOMAS

where i and r denote respectively the inclusion and projection, and j is the inclusion of the fibre &7C x &OB into the total space E2 (= E).

PROOF OF THEOREM 5.9. We divide the proof into two cases, depending on the nature of the vectors w and p.

Case 1. Assume that: (a) each class wi in the cohomology vector w has mod 2 coefficients, and (b) each pi in p is the identity operation, p = (1, 1, *-*, 1). Notice that, with these assumptions, 5.3 and 5.4 become:

(8.3) *r= a , (8.4) (W, 0) = (Pq, 0V.

Now the operation q' is given as a cohomology vector over H*(U, Y). (See discussion following (5.4)-(5.6), for notation.) Since 9v= w, this means we can regard q as a map q: (U, Y) - (C, *), where as before, C = K1 x ... x Ka, Ki = K(Z2, deg wi). Similarly, the operation * can be regarded as a map *: (U, Y) (D, *), where D is an analogous product of K(Z2, q)'s b in number. By definition qua1: (W, Y) - ( U, Y) is the fibration with classifying map (A, p): (U, Y) - (D x C, *). We now apply 8.1, obtaining the following diagram analogous to 8.2:

(SAC,* (SOC x SLD,*) (.Y r jP2 *

(SD, *) > (V, Y) 9(C, *)

{Pi

(U, Y) +(D, *)

where q+1 = p1 P2, and Pi is the principal fibration with + as classifying map.

Let w e Ht( W, Y) be a class such that

p,+#() = (a, j3ic- e HI(nC x nD x W. W) ,

as in 5.8. (Here &?C x SD = F, in 5.8.) Let ,u2 be the operator 2.2 for the fibration P2. By 2.5,

pC^k) = (i x 1)*p+?1(co) = (a, f).((i x 1)*C) = a * C

where, as before, c denotes the cohomology vector (cl 0 1, *.*. , Ca 1) in H*( (C x W, W).

Let g: (B, *) (U, Y) be the map given in 5.7. Since g** = 0, g lifts to a map h: (B, *) (V, Y) such that p1 oh = g. Therefore,




*P,*i (p 9g(Pi = PiWi = Wi, < ? a,

by 5.7 and 8.4. Since p: E- B was defined as the fibration with w as classifying map, this means that we can regard p as the fibration induced by h from the fibre map p2: (W. Y) - (V, Y). Thus we have a commutative diagram, as shown below, where h is the canonical map for an induced fibration:

(E, *) +( W, Y) / 1P jP2 *

(T. ) (B. ) (V, Y) P9(C, ) ST h

Set k, = h *w e Ht(E), where c is the representative for 2 as given in 5.6. Let ,c be the operator 2.2 for the fibration p. Then, by 2.5,

a(k2) = jh*(a) = (1 x h)*f2(w) = (1 x h)*a-C = aC,

since (1 x h)*(ci ( 1) = ei 0& 1, 1 ? i ? a. In the above equation the action

of A(Y) on H*(f7C x E, E) is given by the composite map

gLC E J P B 7

Y.

We also can regard H*(S2C x E, E) as module over A(B), using just the

composite map p o (proj). By (4.1) and (8.3),

aEri = Iaac = Sace,

and so we have

a(k2) - (k),

since by hypothesis, a(k) = &ac. Let pa: (U, Y) - (Y x K, Y) denote the fibre map referred to above 5.7;

let r: Y x Ken K, 1: Y x K - Y denote the two projections. Then by 5.7,

rop 1po h - r op og V

lop~oploh - lop7og 1].

Since p2h hp, it follows by Remark 2 in ? 4 that

k2 E 7(p*V, p*y)0

By naturality (see 4.6), and since 7w = pq,

q *k, C (*v, Ao

and so by 5.6 there is a class u e M such that

q*k2 = r*U .

Set k = k2- p*u e Ht(E). Then,

q*k, = q*k2 - 7r*u 0 .



214 EMERY THOMAS

Also,

p(kl) = p(ks) =M(k), since by the exactness of 2.4, pap* = 0. Therefore, k, e Kernel q*, k, -k E Kernel pa, and so k, e K. But by definition,

k, e 7(p*v, p*) - P*M,

which completes the proof of 5.9 for Case I.

Case II. w and p arbitrary. Let p = (pj, . , Pa) 9 pi e A, and let K,'

K(Z., deg wi + deg pi). Set C' - K' x ... x Ka. We regard p as a map Coo C' and so obtain the commutative diagram below:

DC -> 2C'

E ->E'

(8.5) / P P

T ,B = B

'w pow

C >C'. p

Here p' is the principal fibration with p o w as classifying map, f is the natural map, P - np, and i, i' are fibre inclusions. Recall the class k e Ht(E). In a moment we show:

(8.6). There is a class k' e Ht(E') such that

f*k' = k, a'(k') = e' .

Here pa' is the operator 2.2 for the fibration p', and c' denotes the vector (cl 0 1, * , c' 0 1), where c' is the fundamental class of the factor 7K' in SC'. Assuming (8.6) for the moment, we complete the proof of Theorem 5.9. Set

q' - f oq: T - E'.

Since f *k' = k, we have q'*k' = 0. Let K' denote the coset of k' in Ht(E') with respect to the subgroup

Kernel pa' n Kernel qf* n Ht(E')

Thus, f *K' c K. Now, let w' p o w. All the components of w' have mod 2 coefficients and, by 5.4, (w', 0) e (p, s)(v). By 8.6, la'(k') = a c'. Since 5.5 and 5.6 are already assumed to be true, we see that v is a generating class for K' (still relative to f2) and that the hypotheses of Case I are now satisfied.




Thus, by what we have already proved, there is a class k' e a' such that

k e 7(pP*v, p*y) - p'*M

Setting k= f*k' and recalling that p = p'f, it follows that klutz, kie D(p*v, p`r)-p*M,

as desired, thus completing the proof of the theorem. PROOF OF 8.6. According to 2.5, the commutative diagram 8.5 gives rise

to the following commutative diagram of cohomology groups (mod 2 coefficients):

.. . -, Ht(B) - Ht(E') ) Ht(nC' x E', E') ) Ht+'(B) , ..

11 If IXf)* 11 .. . -, Ht(B) - Ht(E) - Ht(&2C x E, E) - Ht+'(B) *..e.

(Here z-, a' denote the operators given in 2.4.) Let A( Y) act on the cohomology of &?C' x E' and SC x E by the following composite maps:

SfC' x E' j) E' Pt B Ad-- Y.

proj p f2C x E JE ) B I, Y. By hypothesis on the class k and by 5.4,

a(k) = . (aipi)(ci 0 1) ai(pi0i 1) . But piCi = P*e' and so

(a) M(k) = Ei ai(p x f)*(c' 0 1). Consider the class EL ai(ce 0 1) e H*(Q2C' x E', E'). By repeated use of 2.4b, c, and by the commutativity of the above diagram, we have:

Tf(. aai (ce0)) = z* i'(c0l)

ri*(zx(p X f)*(cP O 1)) =Z' ai X f)*(cP0 1) Z= e(k) by (a) above.

But by exactness, za(k) = 0, and so, again by exactness, there is a class kf e Ht(E') such that

p'(kf) = ai(fc 0 1) . Thus a(k - f*k) = 0, and so there is a class u e Ht(B) such that p*u = k - f *k'. Setting

k-= kf + pl*u, we obtain the desired class in 8.6.

UNIVERSITY OF CALIFORNIA, BERKELEY



216 EMERY THOMAS

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Postnikov Invariants and Higher Order Cohomology Operations