$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory

TL;DR

This paper investigates $SU$-linear operations in complex cobordism and $c_1$-spherical bordism, revealing their structure and how formal group laws relate to the coefficient rings, extending classical results from Buchstaber.

Contribution

It characterizes $SU$-linear operations in complex cobordism and describes all $SU$-linear multiplications on $W$, extending Buchstaber's analysis of formal group laws.

Findings

$SU$-linear operations are generated by geometric operations $ ext{ extbackslash}partial_i$

All $SU$-linear multiplications on $W$ are described explicitly

Coefficients of the formal group law $F_W$ do not generate $ ext{ extbackslash}Omega^W$

Abstract

We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $\partial_i$. For the theory $W$ of $c_1$-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 $\varOmega^W$ 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 $\varOmega^W$, unlike the situation with complex bordism.