Let * M *be a closed manifold of odd dimension * n=2k+1*. Let * L(p)* denote the Hirzebruch L-polynomial in the Pontrjagin classes of *M*. Denote by * L _{2k}(p) *the component of

*L(p)*in

*H*and by

^{2k}(M,**Z**)*w*the

_{2k}Î H^{2k}(M,**Z**_{2})*2k*-th Stiefel-Whitney class of

*M.*I have strong reasons to believe that the reduction of

*L*modulo 2 is equal to to

_{2k}(p)*w*. Is it really so? And, if it is, how to prove it?

_{2k. }**Remark**: if *k * is odd, so that *2k * is not divisible by 4, then, of course, *L _{2k}(p) = 0 * by dimensional reason. In this case, Massey [Amer. Jornal of Math.

**82**, 92-102] showed that

*w*Thus, if

_{2k}=0.*k*is odd, the answer to my question is positive.