Orientable Cohomology Theories with Universal Chern Operations

Definitions and mathematical statement

Let $K$ be a generalized cohomology theory. The supplied record does not specify the category on which $K$ is defined, its grading, its coefficient ring, or additional hypotheses on $K$.

Source question. For a generalized cohomology theory $K$, can one formulate a notion of $K$-orientability for cohomology theories, together with distinguished operations analogous to Chern classes, such that these operations freely generate all operations from $K$ into an oriented cohomology theory $E$, under appropriate hypotheses to be specified?

This is a request for a framework and a universal generation property. The record does not define $K$-orientability or explain its relationship to the orientation of the target $E$. It also does not specify the domains, codomains, degrees, indexing, or identities of the proposed Chern-class operations. Even the class of operations to be generated is unspecified: additive, multiplicative, stable, and unrestricted natural operations need not give the same question. The algebraic structure and coefficient ring relative to which generation is free, and any completion or finiteness conditions, are likewise absent. Thus the question above preserves the supplied target but cannot be made into a fully quantified universal-property assertion from this record alone.

The supplied dated assessment identifies constructions for Morava $K$-theory with values in Chow groups and in $p^n$-typical oriented theories, including free-generation results under stated hypotheses. Those hypotheses are not reproduced in the record; these reported special cases do not specify a general formulation for arbitrary $K$.

Short English statement

The question asks for a notion of $K$-orientability and analogues of Chern classes that freely describe operations from $K$ to oriented theories. The ambient category, allowed theories and operations, and meaning of free generation remain unspecified. The supplied curation reports partial results for Morava $K$-theory, subject to hypotheses not included here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003046$, title Orientable Cohomology Theories with Universal Chern Operations, exact source problem code catalogue entry. The record attributes the question to the Oberwolfach Reports workshop collection and gives the proposed year as 2016; no individual proposer is identified.
• Algebraic Cobordism and Projective Homogeneous Varieties (2016), Oberwolfach Reports, pp. 215--265, DOI 10.4171/owr/2016/5. Supplied report reference: https://doi.org/10.4171/owr/2016/5. No problem-specific page is supplied.
• Supplied literature reference, labelled "Sechin on Morava K-theory Chern classes": https://arxiv.org/abs/1605.04444.
• Supplied literature reference, labelled "Sechin on p^ n-typical theories": https://arxiv.org/abs/1805.09050.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports Chern operations from Morava $K$-theory to Chow groups and to $p^n$-typical oriented theories, with free-generation results under stated hypotheses. It also states that a general formulation of $K$-orientability is not supplied. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; the status and cited claims require independent verification.