On the secondary classes of foliations 321

generates R^^" when n = 0. Now induct on n: assume this holds for some n >0; then we show it holds for n + 1.

Let Sen -^п+1~Уп and assume Э„ is not empty. First, note that

Zi^«n : H * ( M ; Z ) ( 8 ) H „ , i ( G ; Z ) ^R^^ " (3.10)

is the zero map. For any yjCj € T* we can use Theorem 3.1 to obtain the expansion

ài { yiCj ) = 1 ±^ИУгС/)®Тг. (3.11)

i=rùi "

Given any y^Cj e У^ such that in the sum (3.11) there is a term with deg yj- = n -f 1, then by the definition of D(yjCj) either Tj« = 0 or A^^iyi'Cj) = 0. Therefore, A^^iy^Cj) vanishes on the domain of (3.10).

To complete the inductive step, it suffices to show the union over aesd of the images of

Zi^n : H , c ( M ; Z ) ( 8 ) H , ^i ( G ; Z ) ^R^^ " (3.12)

generates R^^". The assumption УяУо implies that for any yjCjeS„, there is a unique decomposition I = IoÜIS such that Tj^'^O, r>(yrCj) = deg yjr^'= n + 1 and yii,Cjey. So Formula (3.11) implies

when restricted to the domain of the map (3.12). Since the classes Tj^g H^+iiG; Z) are as independent as the yi- and it is given that У is independent and variable for the foliations on M, it follows that the images of (3.12) generate the range. ■

§4 . Proofs of the main theorems

In this section we combine some known properties of FFg with the machinery and results of sections 1 to 3 to prove Theorems 1 through 5.

Proof of Theorem 1. Our construction of foliated manifold with non-zero rigid classes begins with a theorem of Mather and Thurston [27], [22], [25]:

THEOREM 4.1. Forq^l the space РГ^ is {q^^iyconnected.

For q = 2 this implies the map р:ВГ2 -> BO2 is 4-connected. Let giS^xS^-^ BO2 be a map such that g^ : H4(S^ x S^) -^ H4(B02) is an isomorphism. Since p is