bravchick (bravchick) wrote in geometry_qa,

Relationship between L-polynomial and the Stiefel-Whitney class

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.

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