Operations and poly-operations in Algebraic Cobordism

TL;DR

This paper characterizes all operations from algebraic cobordism to any oriented cohomology theory over characteristic zero fields, introducing poly-operations and a discrete Taylor expansion to handle non-additive cases, and constructs the 0-th symmetric operation.

Contribution

It provides a complete description of operations from algebraic cobordism to other theories, including non-additive ones, using new algebraic tools like poly-operations and discrete Taylor expansion.

Findings

All operations are reconstructible from their action on products of projective spaces.

Constructed the 0-th symmetric operation for arbitrary p.

Proved a general Riemann-Roch theorem for non-additive operations.

Abstract

We describe all operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^* (in the case of a field of characteristic zero). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done earlier, as well as the topological one obtained by T.Kashiwabara. The key new ingredients which permit us to treat the non-additive operations are: the use of "poly-operations" and the "Discrete Taylor expansion". As an application we construct the only missing, the 0-th (non-additive) Symmetric operation, for arbitrary p, which permits to sharpen results on the structure of Algebraic Cobordism. We also prove the general Riemann-Roch theorem for arbitrary (even non-additive) operations (over an arbitrary field). This extends the multiplicative case proved by I.Panin.