Topological applications of the theory of two-valued formal groups

V. M. Buchstaber

Izvestiya. Mathematics , Tome 12 (1978) no. 1, pp. 125-177

Résumé

The Pontryagin classes of real vector bundles are constructed in selfconjugate cobordism theory. In terms of these classes a construction of the two-valued formal group in cobordism is carried out and a topological interpretation of the main results of the algebraic theory of two-valued formal groups is given. The ring $Л^{\,*}(X)=\operatorname{Hom}_{A^U}(U^*(M\operatorname{Sp}),U_*(X))$ is computed for the basic nontrivial cases where $X=(\text{point})$ and $X=\mathbf CP(\infty)$. An effective description of the cobordism classes of Stong manifolds is given, and a number of problems on the cobordism classes associated with the universal Pontryagin classes are solved. The principal characteristics of the cobordism ring of selfconjugate manifolds are calculated, including the image of this ring in the cobordism ring of nonorientable manifolds. Bibliography: 29 titles.