Mathematics Colloquia and Seminars

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In some earlier papers by the speaker, for the class of simply connected closed 6-manifolds M was constructed an invariant Gamma, which can be described as a family of (non-linear) functions Gamma_a, a = 0, 1, 2 ..., defined on the sets of cohomology classes omega in H^2(M; Z_{2a}) satisfying the "spin condition" \bar{omega} = w_2(M) (bar denotes reduction modulo 2), and having their values in the groups Z_a=Z/aZ. Invariant Gamma is not, in general, defined by the tangential homotopy type (homotopy type plus tangent bundle) of the manifold in question. Along with the other known invariants, Gamma forms a complete set and allows to give a classification of the manifolds of the type indicated (St. Petersburg Math. J. 12(2001), 605-680).

The (original) construction of invariant Gamma is essentially indirect, and based on non-trivial (therefore difficult-to-verify) calculations. It would be good, therefore, to have a more "straightforward" construction, independent of the above-mentioned calculations. In this talk we consider such a construction, using only some simple surgery and some well-known relations for Pontryagin and Stiefel-Whitney classes ("mod 2" and "mod 4" Wu formulas).