Fields of Tangent 2-Planes on Even-Dimensional Manifolds

Annals of Mathematics

Fields of Tangent 2-Planes on Even-Dimensional ManifoldsAuthor(s): Emery ThomasSource: Annals of Mathematics, Second Series, Vol. 86, No. 2 (Sep., 1967), pp. 349-361Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970692 .

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Fields of tangent 2-planes on even-dimensional manifolds

By EMERY THOMAS*

1. Introduction

We consider in this paper the problem of whether a smooth manifold M admits a field of tangent 2-planes. If M has two linearly independent tangent vector fields, these span a field of 2-planes; but we will see that, even if M has no non-singular vector field, it still may have a field of 2-planes. We consider only the case of even-dimensional manifolds. The odd-dimensional case will be treated subsequently [18]. Of course, if dim M = 2, a field of 2-planes exists trivially. Furthermore, Hirzebruch and Hopf [7] have solved completely the problem of fields of 2-planes on 4-dimensional manifolds; and so, for the rest of the paper, we assume that dim M > 4.

All manifolds considered will be smooth, closed, connected, and orientable. Let M be such a manifold, and let k be a positive integer less than dim M.

We will say that M has a k-distribution C, if C is an oriented k-plane sub- bundle of the tangent bundle of M. (This corresponds to a field of oriented tangent k-planes on M; cf. Chevalley [3].) We say that Y is a k-distribution with finite singularities if C is defined over all of M, except at a finite number of points.

Suppose now that k = 2. Recall that oriented 2-plane bundles C are clas- sified by the Euler class x(r)) E H2(M; Z) (because the classifying space for these bundles is an Eilenberg-MacLane space K(Z, 2)). The purpose of this paper is to answer partially the following question.

Question 1. Let M be a smooth manifold as above. For which classes u E H2(M; Z) does there exist a 2-distribution on M with Euler class u?

As usual, we consider first the existence of 2-distributions with finite singularities (and with a prescribed Euler class), and then define an invariant (the index) which measures whether or not one can remove the singularities. We will prove

THEOREM 1.1. Let M be an even-dimensional orientable manifold with dimension greater than four. Then, for each class u E H2(M; Z), there ex-

Research supported by the National Science Foundation. The author is a Professor in the Miller Institute for Basic Research, Berkeley.



350 EMERY THOMAS

ists a 2-distribution on M with finite singularities, whose Euler class is u.

Let C be such a 2-distribution with finite singularities. Set m = dim M. From the point of view of obstruction theory, Y can be regarded as a cross- section over the (m - 1)-skeleton of M of a bundle with fiber Gm,,2. (Here Gm,,2 denotes the Grassmann manifold of oriented 2-planes in Rm). Thus the obstruction to extending C to all of M lies in

Hm(M; 7rmil(Gm,2)) PZ7nml(Gn 2) 1

assuming now that M is oriented. Now for m even and >6,

7Wm1(Gm,2) - Z ?D Z2 .

Thus the obstruction, which classically is called the index of Y [8], [15], [7], has two components, the Z-index and the Z2-index. Our next result charac- terizes those integers that can arise as the Z-index of a 2-distribution with finite singularities.

Denote the Euler characteristic of M by ,. Suppose that M is oriented. For classes u e HP(M; Z) and v E Hq(M; Z), p + q dim M, define an integer P(u, v) by

F(u, v) = (u U v) [M],

where [M] denotes the orientation class of M in Hm(M; Z). For each class u E H2(M; Z), define a class 0(u) E Hm-2(M; Z2) by

0(u) = Ei+J=q-1 Ui U w2j(M)

where m = 2q and where w,(M) denotes the ith Stiefel-Whitney class of M, i > 0. In ? 4, we show that 0(u) is the mod 2 reduction of a class in Hm-2(M; Z). Our result is

THEOREM 1.2. Let M be an m-dimensional oriented manifold, m even, and let u -be a class in H2(M; Z). Then the following integers, and only these, occur as the Z-index of 2-distributions on M with finite singularities, and with Euler class u:

X - F(u, v), where v runs over all classes in H --2(M; Z) such that v mod 2 0 0(u).

Notice that F(u, v) -F(-u, v), and so the set of integers occurring in 1.2 is independent of the choice of orientation given M.

In general we are unable to compute the Z2-index of a 2-distribution r with finite singularities. However, we will show in ?3 that, if dim M -2 mod 4, and if x(r) _ 0 mod 2, then the Z2-index of C vanishes, and thus we will prove

THEOREM 1.3. Let M be an oriented m-manifold, where m 2 mod 4,



FIELDS OF TANGENT 2-PLANES 351

and let u E H2(M; Z). Then there exists a 2-distribution (without singularities) on M, whose Euler class is 2u, if and only if there is a class v E Hm-2(M; Z) such that

v mod 2 = Wm-2(M) and 2F(u, v) =XX

We use here the fact that O(2u) =Wm_2(M).

Suppose that XM = 0 in 1.3. Then we can take u = 0 and so obtain a 2- distribution with zero Euler class; i.e., a 2-plane bundle spanned by 2 linearly independent vector fields, (cf. [17]).

More generally, suppose that M is an even dimensional manifold with

XM # 0. By 1.2 we have: if M has a 2-distribution without singularities, then the second Betti number of M, b2, is positive. Using 1.3 we prove a partial converse.

COROLLARY 1.4. Let M be an orientable m-manifold, with m _ 2 mod 4, such that XM / 0. If wm_2(M) / 0, assume that Hm-2(M; Z) has no 2- torsion; while if wm.2(M) = 0, assume that X _ 0 mod 4. Then M has a 2- distribution without singularities provided that b2 is positive.

PROOF. Choose an orientation for M. Since b2 > 0, it follows by Poincare duality that there are classes u' E H2(M; Z), v' E Hm-2(M; Z), such that F(u', v') = 1. Suppose now that Wm-2(M) / 0. Then the class v' can be chosen so that v' mod 2 = wm-2(M) (see ? 5), since we are assuming that Hm-2(M; Z) has no 2-torsion. Now by Wu [20], since m _ 2 mod 4, wm(M) = 0, and so

X, _ 0 mod 2, say xX = 2s. Set u = su', v = v'; and then by 1.3, M has a 2- distribution with Euler class 2u. A similar proof obtains if wm2(M) = 0. We leave the details to the reader.

For a concrete illustration of 1.3, consider complex projective space CPm, of real dimension 2m. Let x e H2(CPm; Z) denote the canonical generator, so that the orientation of CPm is given by xm. Now the Euler characteristic of CPm is m + 1, so by 1.3 we obtain

EXAMPLE 1.5. Let q and s be integers, with q greater than one. Then there is a 2-distribution on CP2q-1 with Euler class 2sx, if and only if there is an integer t such that

t-qmod2 and st=q.

Remark 1. One can show that for q > 2, CPlq-1 has a 2-distribution that

does not come from a field of complex lines in the (usual) complex tangent bundle of CP2q-1. However T. Heaps [5] has shown that CPlq-1 has an infinite

number of distinct almost-complex structures, and so it might be that every 2-distribution on Cp2q-1 comes from a field of complex lines in some almost-



352 EMERY THOMAS

complex structure on CP

Remark 2. Foliations. Let M be an m-manifold, and let k be a positive integer less than m. The existence of a foliation on M of codimension m - k is equivalent to the existence of a completely integrable (or involutive) k- distribution. (See [141 and 131). Thus Question 1 above suggests

Question 2. Let M be a smooth manifold. For which classes u C H2(M; Z) does there exist an involutive 2-distribution with Euler class u? In particular, which of the 2-distributions on CP'q-1, given in 1.5, can be taken to be involutive?

2. The Postnikov resolution

In this section we study the fibration that gives rise to the results of ? 1. Let Bq, q > 2, denote the classifying space for oriented q-plane bundles, and let Yq denote the canonical bundle over Bq. Choose m to be a fixed even integer greater than four. Let w: Bm_2 X B2 Bm denote the map corresponding to the bundle xy2 X -'2 over Bm_2 x B2. Up to homotopy, the fiber of w is G., 2 (see [11), and the fibration

G Bm-2 XB2 >Bm (G = Gm,2)

is the universal fibration for studying the existence of oriented 2-plane sub- bundles in oriented m-plane bundles.

In order to obtain a Postnikov resolution of w, we record here some facts about the cohomology of G (see 12]). There are classes u e H2(G; Z) and v e Hm-2(G; Z) such that

(2.1 a) H2i(G; Z) = Z generated by u", 0 < i q - 2, where m = 2q; b) Hm-2(G; Z) Z E Z generated by u-1 and v; c) Hm(G; Z) = Z, generated by uv; d) H X+'(G; Z) = Oy 0 < i < q -1.

Let a2 and Jm-2 denote the canonical vector bundles over G with respective dimensions 2 and m - 2. The generators u and v can be chosen so that

(2.2) X(a2) =u, X(fm-2) - Uq - 2v . Since a,2 E fm2 is trivial, we have

(2.3) Uq = 2uv,

and, furthermore, one can show that

(2.4) Sq2v = q(uv) mod 2 .

The groups given in 2.1 can be deduced from the Gysin sequence for the fibration




Si- > V P yG, where V denotes the Stiefel manifold Vm,2. Also from this fibration, one sees that G is 1-connected, that

112(G) W&mn2(G) z Z

with the groups in between all zero, and that

Wmi(G) Z ?9 Z2 .

Since Hm-l(K(Z, 2)) = 0, the Postnikov resolution for G begins

(u, v): G > K(Z, 2) x K(Z,m -2) .

By (2.3), (2.4), one sees that the first k-invariant for G is the pair of classes

(*) (-C~q t$ 1 + 2e2 (0 Cm-2 q 2 (& t_ 1 (D Sq 2m_2)

where c; denotes the fundamental class for K(Z, j), and where for any integral class w, i = w mod 2.

We proceed to factor the map w. Notice that

(2.5) a2 = i*(1 X /2)i flm-2 = j*( ym_2 x 1) X

We agree to identify B2 with K(Z, 2), taking e2 = x('Y2), and so obtain the following homotopy commutative diagram (the maps are explained below):

j G - Bm-2 x B2

IU P

K(Z, 2) > Bm x B2

~1 Bm

Here j, is the inclusion (choosing a basepoint in Bn), while p is given by the pair of maps (w, r), where r is projection on the right component. The map 1 denotes projection on the left component. Then Ip = 1 and, by 2.5, the diagram homotopy commutes.

For any vector bundle $, let (Q) denote the stable equivalence class of d.

Notice that

P*((ym X 1) (1 X 2)) (Yin-2 X 1), and so, if we set

tta Wi((^y / 1) - (1 /e2)) i - 0

we have p*aO = Wi(am2 x 1). In particular,

(6 = t mOm-2) Y



354 EMERY THOMAS

where 3 denotes the Bockstein coboundary operator associated with the exact sequence Z-o Z-o Z2.

Now one can show that the fiber [4] of the map p is the Stiefel manifold V. (See [181.) Moreover, Hm-2( V; Z) Z generated, say, by a. Let

k E Hm-'(Bm x B2; Z)

denote the transgression of a. Then k is the next Postnikov invariant for the map z [6]. Since p*30 = 0, it follows from the Serre exact sequence (for the fibration p) that 30 is a multiple of k. At the end of the section, we show that

(**) 2k= 0,

which means that 30 = k, since 30 #t 0. (Hsiang showed that 0tm- = k mod 2, and conjectured that 30 I k. See [9, p. 410].)

Denote by s: E - Bm x B2 the principal fibration with classifying map 30: Bm x B2 K(Z, m - 1). Now j*60 = 0, and by [16, Th. 1], we can choose a map q: Bm2 x B2 E such that

s o q = p, q I G = (u, v): G - K(Z, 2) x K(Z, m -2).

Thus we obtain the following diagram:

G - Bin2 x B2

(U v) {q

(2.6) K(Zy 2) x K(Z, m -2) j22) E

pI {s

K(Z,2) - BmxB2 0K(Zm-1),

where 12 denotes the inclusion. Now the fiber of q, call it W, is (m - 2)-connected, and w1t-,( W) = Z iD Z2. Thus Hm-'( W; Z) Z, and the transgression of the generator gives a class p in Hm(E; Z). Further, Hm-1( W; Z2) Z2 i3 Z2.

Let bm 1 denote the generator corresponding to the projection Z( Z2 -Z2.

Define A in Hm(E; Z2) to be the transgression of the class bo1. The pair of classes (a, *) is then the Postnikov invariant in dimension m (see Hermann [6]). In particular, suppose that X is a complex of dim < m, and that g: X - E is a map. Then the pair of classes (g*p, g*y) is the obstruction to lifting g to Bm-2 x B2.

To prove Theorem 1.2, we focus on the class Ap. Let d'and C be oriented vector bundles over X, with respective dimensions m and 2. These give a map

($, ): X - Bm x B2,

which lifts to a map g: X > E, if and only if (d, r2)*60 = 0. Suppose that




Hm(X; Z) P Z, with a prescribed generator. Then g*p determines an integer, which we call the index of g. If X is an m-manifold M, and e the tangent bundle, then g corresponds to a choice of 2-distribution C with finite singularities on M.

We proceed to describe Ap. Since W is (m -2)-connected,

q *: H - -2(E; Z) Hm-2(Bm2 x B2; Z)

and so there is a unique class s) E Hm-2(E; Z) such that

q*0) Xrn-2 0 1 (Xm-2 = X(Ym-2))

Since p*O= Wm_2 0 1 and Xm-2 mod 2 = wm_2, we have

(2.7) o) mod 2 s*O .

By 2.2 and 2.5, j*(Xm-2 0) 1) = Uq-2v, and so by the commutativity of (2.6), we have

(2.8) j*a) = ?1 A) 1 - 2(1 0 td m_2)

We now can determine Ap.

LEMMA 2.9. P = S*(Xm 0) 1) - S*(1 0) X2)V(0t

Since q*s* = pl, we have at once that

q*(S*(X. (g) 1) - S(* (l X2)* )) 0 ?

On the other hand,

j2*(S*(X0 (9 1) S*(1 ( X2)) 2-C2?1 + 2(2 (0 Cm-2 Y

by 2.8 and the commutativity of 2.6, and so 2.9 follows from (*). (See Hermann

[6] and [16, ? III].) Now s is a principal fibration with fiber K(Z, m -2). Let

: K(Z, m -2) x E > E

denote the action map for this fibration. (See [16, ? II].) By 2.8 we have

(2.10) ,*o) = -2(1 (C cm_2) + 1 (0 a) O

We use these facts in the next section to prove 1.2 and 1.3. Proof of (**). Consider the following sequence of spaces and maps:

V P. G 2 B.-2 x B2 >Bm x B2X

As remarked above, if p is made into a fiber map, then j o p corresponds to the fiber inclusion. Now V itself fibers over Sm-1 with fiber Stm-2 [15]. Let i: Stm-2 - V denote the fiber inclusion, and set f = p 0 i: Sm-2 G. One can readily show that

f *v generates Hm-2(Sm-2; Z), and



356 EMERY THOMAS

f*3 = tangent bundle of Sm2. Since Xsm-2 = 2, it follows by (2.5) that

(?+)f *(2v) f f *j*(Xm-2 X 1).

But i* is an isomorphism in dimension m - 2 and so

(?+-)2a = p*(2v) = (j ? p)*(X.-2 X 1).

Thus, by exactness, 2a transgresses to zero, and so 2k = 0 as claimed.

3. Proof of Theorems 1.2 and 1.3

We continue with the notation of ? 2. Let X be a complex with bundles (d, (): X Bm x B,. Suppose that (d, Y)*60 = 0, and let g: X E be a lifting. By 2.9

9* ( = X(e)- u.g*ss, and by 2.7,

geco mod 2 =(,)*8,

where u = x(r). Moreover, if g, is any other lifting of (d, A) to E, then there is a class y C Hm-2(X; Z), i.e., a map y: X-) K(Z, m - 2), such that g, = ylg, where the dot indicates the action induced by ,". By 2.10,

g9 a -2y + g*ao,

and so the set of cohomology classes {g*qp}, as g runs over all liftings of (d, (), is simply the set of classes {x(e) - u nix} where x runs over all classes in Hm-2(X; Z), such that x mod 2 = (d, Y)*0. Rewording this for the special case X = M, e = z, we obtain 1.2, since 8(u) = (z, Y)*8, where u = x(r).

To prove 1.3 we study the mod 2 Postnikov invariant * e Hm(E; Z2).

Recall the classes Si C Hi(Bm x B2; Z2), i> 0. By the formula of Wu [19],

Sq2 8 (2 1)8+2 + 0i82

Suppose now that m = 4k + 2, k > 0; set 8 =8 4k as before. Then

Sq2(a8) 8 02.(68) .

Using this fact, together with the exact sequence given in [16, ? III] (relative to the map p), one can easily show that the class * is characterized by the properties

(3.1) q = 0,1 , = Cm-2 ? 8802 + Sq Cm-2 ? 1 + 1 (9

Now let (d, A): X Bm x B2 be bundles as above, such that (d, Y)*60 9 0. We will think of e as fixed, but will allow Y to vary, identifying Y? as usual with its Euler class u. Define




+(u) = Ug g** E Hm(X; Z2),

where g runs over all maps X-) E such that sg = (d, )). We now assume that the reader is familiar with [17, ?? 4-6]. Let A denote

the mod 2 Steenrod algebra, and A(K(Z2, 2)) the semi-tensor algebra

H*(Z2, 2; Z2) 0 A .

(See [12] for details.) We regard a class v C H2(X; Z2) as a map X- K(Z2, 2), and so define a linear action of A(K(Z2, 2)) on H*(X) (mod 2 or integer coef- ficients). In particular, set

a = t2?1 + 10&Sq2 in A(K(Z2,2)).

Then, for x c H*(X),

a(x) = Sq2x + x.v .

To emphasize the dependence of a on v, we write a as a,. One can easily check that there is a relation

(*) rav o a?v = O,

which holds as an operation on integral cohomology classes of X. Let 4? be a twisted secondary operation (of degree 3) associated with this relation. (See [17, ? 4] and [13] for details.) We define 11 on those pairs of classes (x, v), with x C Hi(X; Z) and v C H2(X; Z2), such that a,(x) = 0. And lD(x, v) is then a coset in Hi+3(X; Z2) of the subgroup aHi+'(X; Z2).

Now let d be a fixed bundle over X, as given above. For each u C H2(X; Z), set

oi(d, u) = (d, r)*Oi, i > 0? (u = X(r)

In particular, it follows from 3.1 that +(u) is a coset of aHm-2(X; Z), where we set v= 02(~, i).

Theorem 1.3 will follow at once from

THEOREM 3.2. Let X be a complex of dim < 4k + 2(k > 0), and let d be an oriented (4k + 2)-plane bundle over X. Let u c H2(X; Z) be a class such that 604k($, U) = 0. Suppose that

avH4k(X; Z2) = a.VH4k(X; Z),

and that 04k($, u)v = 0, where v = 02($, u). Then

O(U) = 4(864k-2(l, i), v) I

as cosets of aovH4k(X; Z2).

The proof is very similar to that given for Theorem 6.8 in [17], so we only give a sketch. Set m = 4k + 2. In Hm-1(Bm x B2; Z2), we have



358 EMERY THOMAS

Sq2 a(384 + a(3m_4 02 =mOI

and Om.- = 60 mod 2. Thus by 3.1, we see that a3m-4 is a generating class for A, in the language of [17, ? 5]. (Relation (*) satisfies [17, 5.5].) Now p*6f3m_4 =

awm_4 ? 1 in Hm-3(Bm_2 x B2; Z). Thus by naturality and by [17, 6.7], the operation b can be chosen so that, for some X C Z2,

\p*(0~m_2.02) C b(WWm-4 0 1, W2 0 1), in Hm(Bm-2 x B2; Z2). Theorem 3.2 now follows from [17, Cor. 5.10]. We refer the reader to [17, ?? 5, 6] for details.

Proof of Theorem 1.3. Suppose that X = M, d =, and u = 2x, for some class x C H2(M; Z). Since 8j(2x) = w,(M), it follows by Wu [19] that

a{Hm -2(M; Z2) aHm-2(X; Z) = 0 (v = w2M).

Also, as remarked in [17, Eq. (*) in ? 7], w4k(M).w2(M) 0. Thus the hypoth- eses of 3.2 are satisfied for the pair (z-, A), where x(r) 2x, and so the con- clusion reads:

+(2x) = q(Qw k-2(M), w2(M)),

with zero indeterminacy. Thus by [17, Th. 7.2], +(2x) 0. In other words, in the language of ? 1, the Z2-index of the 2-distribution Y with finite singularities, and with Euler class 2x, vanishes. Since Theorem 1.2 gives the pos- sible Z-indexes for (, the proof of 1.3 is complete.

4. Proof of Theorem 1.1

We revert to the situation in ? 1, where M is an orientable m-manifold, and Y is an oriented 2-plane bundle over M, with Euler class u C H2(M; Z). From the remarks in ? 2, it follows that Y is a 2-distribution with finite singularities if and only if 60(u) = 0, where 0(u) = (z, Y)*O. So the proof of 1.1 consists in showing that 60(u) = 0, for all u C H2(M; Z). This is a consequence of the following result.

PROPOSITION 4.1. Let M be an orientable manifold of dim 2s + 2, s ? 1. Let Q C H2i(M; Z), i ? 0, be a polynomial in 2-dimensional integral classes. Then, 6(Q'w2s-2i(M)) = O.

Since 8(u) is a sum of terms of the form given in 4.1, this shows that 60(u) = 0, hence proving 1.1.

The proof of 4.1 is preceded by two lemmas.

LEMMA 4.2. Let Q C H2i(M; Z), i > 0, be a polynomial in 2-dimensional integral classes, and let j be a positive integer <i. Then there is a class

Q2jC H2i+2j(M; Z), which is again a polynomial in integral 2-dimensional




classes, such that

Q2j mod 2 = Sq2i Q.

PROOF. This follows from the Cartan formula and the fact that, if u is a 2-dimensional integral class, then

Squ u= 0_ Sq2 u = u2 mod 2 .

Let Vj e Hj(JM; Z2), j > 0, denote the Wu class [20].

LEMMA 4.3. Let M be a manifold (of dim 2s + 2), and Q a class (of deg 2i), as in 4.1. Let j and k be non-negative integers such that i + j + k = s. Then

3(Q Sq2 VI,2k) 0 .

PROOF. By Poincare duality [11], it suffices to show that, if t C H2(M; Z) is any torsion class, then

(*) Q*QSq 2j Vk V t = 0

We will show this by an inductive argument on j and k. We note first the following facts.

(a) Let u and v be integral classes such that deg u + deg v = dim M. If one of u, v is a torsion class, then u v = 0.

This is immediate, since M is orientable. (b) aV22k = 0 k > 1. By Wu [20], V2k is a polynomial in the Stiefel-Whitney classes of M. But

aw22i"(M) = 0 and aw2i(M) = 0, since w'i(M)- P(M) mod 2. Thus a V2k =0 as claimed.

We now regard equation (*) as a function of the pair of integers (j, k). For any pair of non-negative integers (j, k), set i s - i- k. If i < 0 we will say that (*) is automatically valid for the pair (j, k). We note the following special case of (*).

(c) If j > k > 0 then (*) is true for the pair (j, k). PROOF. If j > k, then Sq25 V2k 0. If j =c, then 5q2k V2k V2k, and so

(*) follows from (a) and (b) (or from the above convention, if i < 0). Given pairs of non-negative integers (j', k'), (j, k), we say that

(j', k') < (j, k) if k' < k or k' =I, j' < j .

Now let (j, k) be a pair, and assume inductively that (*) is valid for all pairs (j', k') < (j, k). We show that (*) is then true for the pair (j, k), which will complete the proof of 4.3. By (c), we may assume that j < k; also we assume that i > 0 where i = s- - Ik.



360 EMERY THOMAS

In (*), set Q' = Q t. For each integer p > 0 let Q', be a class as given in 4.2 (of degree 2i + 2 + 2p) such that

Q'P mod 2 = Sq P (Q), P > ? .

Notice that

Sq2p+lQ' =Sq1 Sq2pQ' = Sq1 Q'p 0 ? . Suppose first that j > 0. By the Cartan formula,

Q-Sq I V2k-t Q' Sq 2j V2k

= Sq'i (Q'* V20) + Ej=1 Sq Q * Sq 2- 1i V2k

Sq2j(Q' 1V2k) + E Q2. *qi S s V2k.

Since Q' - Q. t, there is a polynomial Q", of degree 2i + 2p, such that

Q11.t ~ ~ ~ p10 Q2 P = 2p- p > 0

Consequently, by the inductive hypothesis,

Q2s Sq V2k = Q q2iSq V2k -t = 0

since s > 0. On the other hand

Sq 2j(Q' - V20) = Q' - V2j V2k ,

since 2j + (2i + 2) + 2k = dim M. And so again

Q. V2j* V2k Sq2k (Q'* V2j) = E Sq2r Q . Sq2k-2r V2j

=r Q~r~ 5q2k

2r V2

Q~ *q2k-2r

V2j . t

0

again by the inductive hypothesis, since j < k. Therefore, Q- Sq2 V2k -t =0. If j = 0, the second half of the above argument shows that Q V2k * t = 0. Thus we have completed the inductive step, and hence the proof of Lemma 4.3.

Proof of 4.1. By Wu [20],

W2S5-2i E qi 17V2s-2i-2j,

and so by 4.3,

(Q'w2s-20) = a(Q.Sq2i V2s-2i-2j) 0

Remark. If we take Q = 1 c H?(M; Z), then 4.1 reduces to a result of Massey [10].

UNIVERSITY OF CALIFORNIA, BERKELEY

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(Received February 14, 1967)



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Fields of Tangent 2-Planes on Even-Dimensional Manifolds