Lifting Cohomology Operations to Smooth Extensions

Definitions and mathematical statement

Let $E,F$ be generalized cohomology theories and let $\widehat E,\widehat F$ be chosen smooth (differential) extensions on smooth manifolds. In a standard model, $\widehat E^n(M)$ has an underlying-class map $I_E:\widehat E^n(M)\to E^n(M)$, curvature $R_E$ valued in closed differential forms with real graded coefficients, and a map $a_E$ from forms modulo exact forms, satisfying the usual exactness and Chern-character compatibility axioms. These choices are part of the extension, not merely a topology on $E^n(M)$.

For a natural cohomology operation $\theta:E^n(-)\to F^m(-)$, the minimum lifting requirement is a natural transformation \[ \widehat\theta:\widehat E^n(-)\longrightarrow\widehat F^m(-), \qquad I_F\circ\widehat\theta=\theta\circ I_E. \] Naturality here concerns pullback along smooth maps. A differential refinement normally also specifies a compatible operation $\theta_\Omega$ on curvature forms and requires \[ R_F\circ\widehat\theta=\theta_\Omega\circ R_E, \] with appropriate compatibility on form classes and real cohomology. Such compatibility data are not automatic from the underlying set-map equation and must be fixed. Stable additive operations, unstable or nonlinear operations, products, and secondary operations may require different formulations.

The source asks to determine when cohomology operations admit these lifts. It does not specify which operations, which theories or smooth extensions, whether only the underlying-class square or the full differential structure must be respected, or whether the desired statement is existence for each compatible operation rather than an unrestricted universal theorem. No uniqueness or multiplicativity requirement is added unless it belongs to the chosen operation.

The supplied literature triage checked 21 August 2026 reports differential/smooth refinements of many particular operations, but says the cited literature does not yield a lifting theorem for all operations and all smooth extensions. This preserves partial progress and does not assert either universal liftability or a general nonexistence theorem.

Short English statement

When does a natural operation on generalized cohomology have a natural differential refinement respecting underlying classes and the required curvature data? The operation class and smooth-extension axioms must be specified. The supplied assessment reports many special refinements, without a universal lifting theorem.

Sources

• Local UnsolvedMath registry, permanent ID 30000248, exact source code catalogue entry; Lifting Cohomology Operations to Smooth Extensions. Proposed year 2005, status partially_solved; exact supplied question.
• Supplied report citation: Geometric Topology and Connections with Quantum Field Theory (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/27, pp. 1505--1546, https://doi.org/10.4171/owr/2005/27. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "Bunke smooth extensions", https://doi.org/10.1112/jtopol/jtq002; "Differential cohomology axioms", https://dspace.sunyconnect.suny.edu/items/529173c7-634f-4f32-b0df-26ef0017a804; "OWR report", https://doi.org/10.4171/owr/2005/27. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.