174 ARCa. MATH.

On Tensor Products of n-plane Bundles

By Eg~RY THOMAS*) in Berkeley (Cal.)

1. Introduction. We consider in this note fibre bundles whose fibre is Bn (R = real numbers) and whose structural group is the general linear group L n - i.e., the group of all real, non-singular n X n matrices. As usual we term such a bundle an n-plane bundle, or often simply a vector bundle. In [4] Mn~OR gives an intrinsic definition of such bundles, and then axiomatizes the Stiefel-Whitney characteristic classes associated with these bundles: The purpose of this note is to define the tensor product of two vector bundles, and to give a new proof (see [1]) of the formula for the Stiefel-Whitney classes of this tensor product bundle.

Let 7] and ~ denote respectively r and s-plane bundles over the same base space X. By taking intersections of open sets if necessary, we may assume that ~ and ~ have the same coordinate neighborhoods {V~} covering X (see [5] for definitions used).

Let

be the coordinate transformations for the respective bundles ~ and C. Define

hj, : V~ c~ r i l L , , by

(1.1) %~(x) = ej~(x) | a~(~) (z e V~ n D ) ,

where the | sign indicates the tensor product Of the matrices gj~ (x) and gji(x)- .By (16) in chapter VII of [3], one sees tha t

h~j (x) hj, (x) = hk, (z),

for x e V, C~ V~ c~ Vk. Thus, by 3.1 of [5], the covering {V,} together with the functions hij constitute a system of coordinate transformations for X, relative to the group Lrs. We define the tensor product bundle, ~ / | r to be the bundle given by theorem 3.2 of [5] - - with fibre/~rs, group ]~rs and coordinate transformations h,jl).

Recall that given an n-plane bundle ~ over X, we associate with ~ a set of coho- mology classes (the Stiefel-Whitney classes, see [4]):

w(~) = I + w~(~) + ... +-w.(~)

*) This research has been partly supported by U.S. Air 1%ree Contract _A_F 49(638)--79. 1) Alternatively, we can define r/(~ r intrinsically, using the definition of n-plane bundles

given in [4]. This is, in fact, the procedure followed in [2; w 3.6b)]. However, for the lemmas stated in section 2, the definition given here is more convenient.

Vol. X, 1959 On Tensor Products of n-plane Bundles 175

where 2) Wi(~)s l i t (X) . The problem is to determine W(~ Q ~) in terms of W(~) and W(~). For this, let Pr,~(r, s ~ 1) denote the polynomial ring (over the integers mod. 2) in variables xl . . . . . xr and yl . . . . . Ys. Let al . . . . , ~r (resp. vl . . . . . Ts) denote the elementary symmetric functions in x l . . . . , xr (resp. Yl . . . . . Ys). The following lemma is easily proved, using the elementary properties of the symmetric functions.

Lemma 1.1. There is a unique polynomial qSr, s in Pr, ~ such that

Cr,~(al . . . . . ~r, n . . . . . ~ ) - 1- / (1 + x~ § yj ) ,

where the product is taken over all integers i, ] such that 1 ~ i ~-- r, 1 ~ ] <= s. Suppose now that ~ and ~ are respectively r and s-plane bundles over X. Set

w (~) --- 1 + w ; + . . . + w : , w (o = 1~ + w;" + . . . + w;" .

We then have, I t t t

Theorem I. W (7 @ ~) = Cr, , (W; . . . . , W;, W~ . . . . , W, ). This theorem is due to BOREL and H~ZV.B~VCH, and occurs as a consequence of

their general theory of characteristic classes (see [1 ; w 11]). The proof given below is quite elementary and serf-contained; and so is presented for these reasons.

2. Lemmas on tensor bundles. We state in this section the geometric properties of tensor bundles needed for the proof of Theorem I. We omit the proofs as they are straightforward applications of Lemma 2.10 in [5].

Lemma 2.1. Let ~7 and ~ be two vector bundles over X . Then, the bundles ~ ~ ~ and ~ ~? are equivalent3). For any two vector bundles ~7, $ over X, we denote by ~ @ ~ the Whitney-sum

of these bundles (see [4; w I]).

Lemma 2.2. Let ~, ~7, and ~ be vector bundles over a space X . Then, the bundles

~ "(~l ~ ~) and (~ ~ ~?) ~ (~ Q ~) are equivalent.

Lemma 2.3. Let ~ be an n-plane bundle over X , and let ~1 denote the trivial (product) 1-plane bundle over X . Then, the bundles

@ v l , ~, and z I Q

are all equivalent. Let ~ be any bundle over a space Y, and suppose we are given a map / from a

space X to Y. We denote b y / * ~ the bundle over X induced by ] (see [5; w 10.1]).

Lemma 2.4. Let $~ (i = 1, 2) be vector bundles over a space Y, and let / be a m a t X --> Y. Then,

(/*~1) @ (/*~9.) is equivalent to /*($1 Q ~2), as bundles over X."

This follows from Lemma 2.10 and w 10.1 of [5].

Corollary 2.5. Let ~, ~ be vector bundles over respective spaces Y1, Ye. Denote by ~1, ~2 the bundles over the cartesian product Y1 • Y2 induced by the respective pro-

u) All cohomology groups considered will be singular cohomology groups with the integers rood. 2 as coefficients. ~) See either [5; w 2.7] or [4; w 1] for the definition of equivalence.

176 E. THo~s ' ~em ~ATH.

jections Y l • Y2 ---> YI (4 .~ 1, 2). Let fl be a map X ---> Yt, and define a map / from X to Y1 • Y~ by / (x) = (/1 (x), /2 (x)). Then,

(/1"~7) @ (/2"~) is equivalent to ]*(~1 @ ~2) as bundles over X .

3. Reduction to a special ease. We will prove theorem I by showing tha t the general result follows from a special ease. For each positive integer n, let Gn denote the set of n-dimensional subspaces of/~oo. In [4] it is shown tha t Gn may be topo- logized as a countable CW-eomplex, the infinite Grassmann complex. In addition, an n-plane bundle, 7n, over each Gn is defined such that 7n is a classifying bundle for n-plane bundles over paracompact spaces (see [5; w 19.3] and [4; Theorem 7]). Let ~7 and ~ be respectively r and s-plane bundles over such a space X. Then, there are maps

/ :X--->Gr, g:X--->Gs such that

V - f * 7 r , ~ - g * y s ,

where the symbol -- denotes equivalence of fibre bundles. Let 7~ and y~ be the bundles over Gr • Gs defined in Corollary 2.5. Define a map-

ping h f rom X to Gr • Gs by h (x) ---- (/(x), g(x)). Then, by Corollary 2.5, one has

(3.1) v | ~ -= h * ( ~ | ~ ) .

For each n recall that H*(Gn) is a polynomial algebra (over the integers rood.2) in W1, W2 . . . . . Wn, where We = Wl (yn). Therefore, by the Kfinneth theorem, H*(Gr • Gs) is a polynomial algebra in WI* . . . . . Wr*, WI**, . . . , Ws**. Here

Wi* = 7~l*Wi(Tr), Wy** ~--- ~2* Wj(ys) ,

where ~k = k TM projection map (k ~ 1, 2). Consequently, there is a polynomial ~r, s ~ Pr, s such that

(3.2) w(~I | ~ ) = 7 " ~ , ~ ( w ~ * . . . . , w ~ * , w ~ * * . . . . , w ~ * * ) .

Using (3.1), (3.2), and the notation of section 1, one easily obtains :

Lemma 3.1. W07 @ ~) = Tr, s(W~ . . . . , W:, W;', . . . . W:'). Therefore, the proof of Theorem I consists in showing that the polynomials ~Sr. s

(see section 1) and ~1r, s are identical. We do this by computing two special cases. As usual, let poo denote the real, infinite dimensional projective space; i.e.,

poo = G1. Denote by ~i the ith projection map from (poo)r+s to poo (4 ----- 1, 2 . . . . , r + 8). Let a be the generator of Hl(poo), and set

t

a i -= 7~l*a, ,ui = 7gl*~/I (3.3)

ai ~ x~r+ja*, vl _-- ~zr+iy* 1 Of course,

Denote by a~ . . . . , ~ �9 �9 t l I t

al . . . . . ar (resp. a 1 . . . . . a8 ). One then has, by w I I of [4],

(3.4) w~ (~) -- ~', w~ (f19 = ~/ '

( 4 = 1 , 2 . . . . , r ) ;

�9 (i = 1 ,2 . . . . . s ) .

W (/~) = 1 --~ ai' , W (vj) --~ 1 -~ ai" .

(resp. a ~ ' , . . . , (r's') the elementary symmetric functions in

Vol. X, 1959 On Tensor Products of n-plane Bundles 177

Here, ~ = ~ I | ~ - ~ i | 1 7 4

and 1 ~ i ~ r, l _ ] ~ 8. Therefore, by Lemma 3.1 we obta in t t t r t t

L e m m a 3.2. W (:cr Q ~s) ___ ~ r , s ( a l . . . . . a~, a l , . . . , as ). �9 P I s t t

Now H * ( ( P ~ ) r+s) is a polynomial algebra in the classes al . . . . . a r, a 1 . . . . . a~. Thus, the subalgebra generated b y the e lementary symmetr ic functions al . . . . , at,

I t H al . . . . . as is itseff a polynomial algebra in these r -~ 8 variables. Consequently, to show t h a t two polynomials ~ and ~v in Pr, ~ are identical, i t suffices to show t h a t

. . . . . . ) ( . . . . . . ) ( ~ l , - - . , f i r , ~ z . . . . , a s : ~ qz . . . . , ~ r , a l , . . . , a ~ .

I n particular, then, Theorem I will follow from Lemma 3.1 when we show t h a t Cr, s and Tr,8 agree when evaluated on the classes az . . . . , a;, al . . . . . a~'. Thus, b y L e m m a 3.2, Theorem I is proved when we show

�9 / J I t /

Lemma 3.3. W (otr Q fls) = qSr, s((y 1 . . . . . a t , a l . . . . , as ).

We prove L e m m a 3.3 by using a special case: namely, when r ---- s ---- 1. Then,

Also, r ~1) = 1 -{- Xl q- Yl,

f rom the definition ~ v e n in section 1 (since a l ---- x l , ~1 ~ Y l , when r ---- s ---- 1). Thus, as a special case of Lemma 3.3, we have:

t � 9 1 4 9

L e m m a 3.4. W(y~ | y~) ---- 1 q- a~ q- a~.

We ~ v e the proof of Lemma 3.4 in the next section. Let us use this special case to prove Lemma 3.3.

P r o o f o f L e m m a 3.3. Notice f rom (3.2) and L e m m a 3.4 t h a t

~rrl, l(X , y) ---- 1 -~- x ~- y .

Therefore, b y L e m m a 3.1 we have : I I r

(3.5) W (/~ | ~i) = 1 + a~ + a i ,

where /~, v~ are the bundles over (pcc)r+s defined in (3.3). Now by Lemmas 2.1 and 2.2 we have

- |

where the summat ion indicates W h i t n e y bundle sum and is taken over all integers i, ] such tha t 1 --< i --< r, 1 ~ ?" ~ s. Then, by w I I of [4],

w (:~ | ~ ) = H w (~ | ~) . i,1

Hence, b y (3.5) and L e m m a 1.1 we have

/ 7 ( . . . . . . . . . W(ocr | - - - 1 - ~ a i + a i ) = ~br, s (a l . . . . . ar, a z . . . . . , a s ) .

This completes the proof of Lemma 3.3 and hence of Theorem I. ~krchiv der l~athematik X 12

178 E. TI~OI~AS ARCS. Y.ATH.

4. Proof of Lemma 3.4. As remarked above, we m a y write ffz for ~ and vl for y~. s t t

We also set a ' ---- al, a " ---- a 1 . Thus, in this nota t ion we arc to show

W ( f f l @ v l ) = l + a ' + a " .

Now/z l @ vl is a 1-plane bundle over po~ • and hence is induced by some map h f rom PC~ • poo to Pr162 Since W0 (if1 | vx) = 1, L e m m a 3.4 is proved when we show (4.1) h*a = a' + a" ,

where a is the non-zero class of H 1 (P:r (i. e., W (?!) ---- 1 + a). Recall two things about the space P ~ :

i) P ~ is an Eilenberg-MacLane space of type (Z2, 1); and hence, ii) Pr is an H-space ; i.e., there is a map m from P ~ • Pc~ to Pr such tha t

m ( x , e) --~ x = m(e , x) (x e poo) , where e is a basepoint in P ~ .

L e t / , (i = 1, 2) be the maps f rom poo into pr x poo given b y

/ l ( z ) = (~, e ) , /2(z) = (e, z ) .

Then, m oft -----identity; hence m * a = a' + a" .

Therefore, (4.1) ~ l l follow ff we show tha t the map h is homotopic to m. Bu t by i) above, maps f rom pr x poo in to poo are characterized by their behaviour on the class a e H 1 (poo). This in t u rn shows :

Lemma 4.1. Let k be a m a p / r o m poo • pco to poo. Then, lc is homotopic to m i],

and only i/, k o/~ "" identity (i = 1 , 2 ) .

We use Lemma 4.1 to show t h a t h is homotopic to m; and hence prove (4.1). Define ~ : PC~ --> Pr162 b y z (x) = e (x e PC~ Then,

/ l = l x z , 1 2 = ~ x l , where 1 = identi ty. B y 10.5 in [5], we have

vl - z*(~, i) ,

where zl is the trivial 1-plane bundle o v e r P ~. Clearly, 1"71 - 71. Thus, by Corol- la ry 2.5,

/1"(ff l | ~1) -= ~1 | v l , /2"(ff l | ~1) - vl |

B u t b y definition of the map h, h * y I -- /,el | v l .

Thus, by 10.5 of [5] and Lemma 2.3, we have

(4.2) (h/1)*yl - 71 , (hl2)*yl = 71 .

Now, the equivalence classes of 1-plane bundles over PC~ are in 1 - - 1 correspondence with the h o m o t o p y classes of maps P ~ -~ P ~ . Therefore, by (4.2) we have :

hl1~-- 1 , h12"~ 1 .

Vol. X, 1959 On Tensor Products of n-plane Bundles 179

Hence, by Lemma 4.1, the map h is homotopic to m. Thus, h*a = a' + a"; which proves (4.1) and hence Lemma 3.4.

5. Appendix: the function Cr, s. The application of Theorem I is limited by our knowledge of the function ~5r, s- We state two simple results to show the complexity of the function. The proofs are a simple mechanical verification and are omitted.

L e m m a 5.1. qSn, l (a l . . . . . (;n, z) = ~. (1 § T ) n - ~ . iffi0

Corollary 5.2. Let ~n and (~ be respectively n-plane and 1-plane bundles over a space X . .Let

W(~ n) = 1 + WI + "" + W n , W(co 1 ) = l § Then,

In particular,

W($ n (~)a) 1) = ~ (1 § a)n-IW~. i = 0

Wk(~n | o91) = ~ , (n - - k , k - - i)ak-J W~ . ] = 0

Here ]~ = 1, 2 . . . . . n and (r, s) denotes in general the binomial coefficient

(r § s)!/r! s ! .

Lemma 5.3. For any polynomial 79 in Pr, s, let q~(Ic ~ 1) denote the term8 in 7~ o/ degree < lc. Then,

r . . . . �9 . . , ( ~ r , T l , ,Ts) -~-

----~ 1 § r~l § s (~1 § (r - - 2, 2) ~12 § (s -- 2, 2) (;12 § rT2 § s q2 § (rs -- 1) Zl a l .

Corollary 5.4. Let ~ and ~ be respectively r and s-plane bundles over a space X . Set i p t p r l

W (v) - - l + W~ + . . . + w ~ , w (~) = l + W~ + . . . + W~ .

Then, W l (v | ~) = 1 + s w l + r Wi" ;

�9 t] l tl

W2 (~ | r --- (s -- 2, 2) W~2 + (r -- 2,2) W~ '~' 4- s W~ + r W~ 4- (rs - I) WI W~ .

In particular, if ~ and ~ are even dimensional, then ~ ~ ~ is orientable.

Bibliography

[1] A. BOREL and F. HIRZEBRUCH, Characteristic classes and homogeneous spaces. Amer. J. Math. 80, 458--538 (1958).

[2] F. HIRZEBRUCH, Neue topologische Methoden in der algebraischen Geometric. Springer 1956. [3] N. JAeOBSON, Lectures in abstract algebra, Vol. II. Van Nostrand 1953. [4] J. ~ILNOI% Lectures on characteristic classes. Mimeographed lecture notes, Princeton Univ-

ersity 1957. [5] N. STEENROD, The topology of fibre bundles. Princeton University 1951.

Eingegangen am 5.2.1959

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