ИСТИНА

The development of algebraic topology in the 1960s culminated in the description of the special unitary bordism ring. Most leading topologists of the time contributed to this result, which combined the classical geometric methods of Conner-Floyd, Wall and Stong with the Adams-Novikov spectral sequence and formal group law techniques that emerged after the fundamental 1967 work of Novikov. Thanks to toric topology, a new geometric approach to calculations with SU-bordism has emerged, which is based on representing generators of the SU-bordism ring and other important SU-bordism classes by quasitoric manifolds and Calabi-Yau hypersurfaces in toric varieties. Both geometric and algebraic approaches to describing the structure of SU-bordism are based on the study of an intermediate theory between U- and SU-bordism, the c1-spherical bordism W. We describe the algebra of SU-linear operations in the theory MU of complex bordism and prove that it is generated by the well-known geometric operations~$\partial_i$. For the theory W of c1-spherical bordism, we describe all SU-linear multiplications on W and projections MU to W. We also analyse complex orientations on W and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the W-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on W, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism. The talk is based on joint work with Zhi Lu, Ivan Limonchenko and Georgy Chernykh.