Higher Associativity Structures on Mapping Cones

Definitions and mathematical statement

The supplied record concerns multiplicative structures on mapping cones in the context of Rabinowitz Floer homology. It uses the notation $A_3^+$ for structures encoding associativity and $A_\infty^+$ for structures encoding associativity up to homotopy.

The requested research task is to develop the theory of $A_3^+$-structures for these multiplicative structures on mapping cones, and, more generally, the theory of $A_\infty^+$-structures encoding associativity up to homotopy.

The superscript $+$ is source-specific notation whose definition is not supplied. The record also does not specify the ambient category, coefficient ring, grading and differential conventions, maps whose cones are considered, multiplication data, or higher operations and their compatibility identities. Consequently, this is a research-program statement: the supplied evidence does not determine a precise system of axioms or a universally quantified existence assertion. In particular, the notation must not be silently identified with ordinary $A_3$- or $A_\infty$-algebra structures, nor does the record assert that every multiplication on every mapping cone admits such structures.

Short English statement

Develop the source-specific $A_3^+$ theory of associativity for multiplicative structures on mapping cones, and its $A_\infty^+$ extension describing associativity up to homotopy. The supplied record omits the defining data and identities, so a precise axiomatic formulation requires further source clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30005709$, title "Higher Associativity Structures on Mapping Cones," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied source citation: Mini-Workshop: Flavors of Rabinowitz Floer and Tate Homology (2024), Oberwolfach Reports, pp. 3139--3171, DOI 10.4171/owr/2023/56. Supplied source URL and literature-source link, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2023/56.
• The original statement is attributed within the record to the contribution "Cone perspective on Rabinowitz Floer homology," Alexandru Oancea, joint work with Kai Cieliebak and Nancy Hingston. The registry gives proposed year 2023 and no separately identified proposer; the contribution attribution does not establish an individual proposer.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes development of the requested higher associativity structures and their $A_\infty$ theory as an open problem, and reports that no complete resolution was located. This is supplied dated curation, not independently verified literature assessment. The listed source checked is the Oberwolfach report linked above.
• The supplied registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims, not independent verification performed for this draft. The background explicitly requests independent verification of time-sensitive status and cited claims.