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 L2k(p) the component of L(p) in H2k(M,Z) and by w2k Î H2k(M,Z2) the 2k-th Stiefel-Whitney class of M. I have strong reasons to believe that the reduction of L2k(p) modulo 2 is equal to to w2k. . Is it really so? And, if it is, how to prove it?
Remark: if k is odd, so that 2k is not divisible by 4, then, of course, L2k(p) = 0 by dimensional reason. In this case, Massey [Amer. Jornal of Math. 82, 92-102] showed that w2k=0. Thus, if k is odd, the answer to my question is positive.