Complex Cobordism Modulo c1-Spherical Cobordism and Related Genera

We prove that the ideal in the complex cobordism ring MU* generated by the polynomial generators S = (x1, xk, k = 3) of the c1-spherical cobordism ring W *, viewed as elements in MU* by the 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 c1-spherical cobordism, with the coefficient ring MU* /S. Then any S S is also regular in MU* and therefore gives a multiplicative complex oriented cohomology theory MU*S(-). The generators of W *[ 1/2] can be specified in such a way that for S = (xk, k = 3) the corresponding cohomology is identical to the Abel cohomology previously constructed by Ph. Busato. Another example corresponding to S = (xk, k = 5) gives the coefficient ring of the universal Buchstaber formal group law after being tensored by Z[1/2], i.e., is identical to the scalar ring of the Krichever-Hohn complex elliptic genus.