Title:

Divisibility of normal chern numbers

Following the work of Rees, Thomas and Barton on the divisibility properties of certain normal chern numbers some chem numbers of the MilnorNovikov generators of the cobordism ring are examined. The divisibility properties, at least up to 2torsion, of these chem numbers are computed and these properties are then used to construct the manifolds whose chern numbers realize the minimum divisibility. As an example of how these direct methods can be employed an observation of Libgobcr and Wood is verified and improved upon. Odd torsion is also examined. It is observed that a proof from the work of Barton and Rees is incomplete and that proof is duly completed. Symmetric functions are introduced to form natural coefficients for a formal sum of the chem numbers of a manifold. Using this construction a bound on the primes contained in the hcf of any chem number is obtained, where this bound is dependent upon the length of the partition which indexes the chern number. A systematic method for lowering this bound (often eliminating odd torsion completely) for particular examples is demonstrated. As a digression the link between chem numbers and symmetric functions is examined in its own right. In particular the combinatorial side is addressed through the generalization of a partition of a number to a partition of a set. The general case of an arbitrary chem number of an arbitrary cross product of projective spaces is considered in detail and a general formula is obtained using the language of the lattice of partitions of a set. Examples to demonstrate the viability of this approach are presented.
