Degrees of Asymptotically Conical Expanders Under Connected Sum

Definitions and mathematical statement

The supplied record uses $\deg_{\exp}(X)$ for a degree associated with asymptotically conical expanders on $X$. It does not define this degree, the geometric equation defining the expanders, the relevant moduli space, or the hypotheses under which the degree is defined. Accordingly, $\deg_{\exp}$ is retained as source notation rather than assigned an invented definition.

Write $X_1\#_{\partial}X_2$ for the boundary connected sum indicated by the source. The intended class of manifolds, their dimensions, any orientation or boundary requirements, and the gluing conventions are not specified in the supplied passage. Define the standard closed ball and circle by \[ D^3=\{x\in{\mathbb{R}}^3:\|x\|\leq 1\},\qquad S^1=\{y\in{\mathbb{R}}^2:\|y\|=1\}. \] The concrete manifold in the second question is their product $D^3\times S^1$.

Requested mathematical statement, subject to the missing source hypotheses. For pairs $X_1,X_2$ in the intended domain for which the boundary connected sum and all three degrees are defined, determine a relation, if any, between \[ \deg_{\exp}(X_1),\qquad \deg_{\exp}(X_2),\qquad \deg_{\exp}(X_1\#_{\partial}X_2). \] Also determine \[ \deg_{\exp}(D^3\times S^1). \] No particular connected-sum formula is asserted. A fully specified mathematical formulation requires the missing definition of the degree and its admissibility hypotheses, including those needed for the displayed product.

Short English statement

The problem asks how the degree associated with asymptotically conical expanders behaves under boundary connected sum, and asks for its value on $D^3\times S^1$. The supplied extract omits the definition of this degree and the required manifold hypotheses, so both questions remain conditional on recovering that context.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005584$, source problem code catalogue entry; supplied collection: Oberwolfach Reports Open Problems. No individual proposer is supplied.
• Supplied source citation: Differentialgeometrie im Grossen (2024), Oberwolfach Reports, pp. 1617--1670, DOI 10.4171/owr/2023/29. This is also the literature source labeled "original OWR report": https://doi.org/10.4171/owr/2023/29. The record gives proposed year 2023; the relationship between that year and the citation's 2024 label is not explained.
• Supplied literature source labeled "later OWR report": https://oa.tib.eu/renate/bitstreams/0157a756-a5e7-48ad-abe4-dcc06bc0796f/download. No further bibliographic identification or passage location is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that the connected-sum question remains posed in later reports, and states that no computation of $\deg_{\exp}(D^3\times S^1)$ or general connected-sum law was identified. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and that definitions were restored. These are supplied provenance claims; no independent source visit or verification is asserted here, and the degree's definition is absent from the supplied record.