Abstract
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.
