Complex Cobordism Modulo $c_1$-Spherical Cobordism and Related Genera

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