Complex cobordism modulo $c_1$-spherical cobordism and related genera

Abstract

We prove that the ideal in complex cobordism ring $\MU^*$ generated by the polynomial generators $S=(x_1, x_k, k\geq 3)$ of $c_1$-spherical cobordism ring $W^*$, viewed as elements in $\MU^*$ by forgetful map is prime. Using the Baas-Sullivan theory of cobordism with singularities we define a commutative complex oriented cohomology theory $\MU^*_S(-)$, complex cobordism modulo $c_1$-spherical cobordism, with the coefficient ring $\MU^*/S$. Then any $\Sigma\subseteq S$ is also regular in $\MU^*$ and therefore gives a multiplicative complex oriented cohomology theory $\MU^*_{\Sigma}(-)$. The generators of $W^*[1/2]$ can be specified in such a way that for $\Sigma=(x_k, k\geq 3)$ the corresponding cohomology is identical to the Abel cohomology, previously constructed in \cite{BUSATO}. Another example corresponding to $\Sigma=(x_k, k\geq 5)$ gives the coefficient ring of the universal Buchstaber formal group law after tensored by $\mathbb{Z}[1/2]$, i.e., is identical to the scalar ring of the Krichever-Hoehn complex elliptic genus \cite{KR}, \cite{H}.