Exotic invariant for 6-manifolds: a direct construction

Some of the author's previous works, dealing with the classification problem for simply connected closed 6-manifolds, contain a construction of a certain “exotic” invariant $\Gamma$. This construction is substantially indirect and based on nontrivial calculations. In the present paper, a direct construction is suggested, which does not depend on the calculations mentioned and involves only some simple surgery, plus some well-known identities for Stiefel-Whitney and Pontryagin classes, namely, “modulo 2” and “modulo 4” Wu formulas.