Expander Degree Under Boundary Connected Sums

Definitions and mathematical statement

Write \[ D^3=\{x\in{\mathbb{R}}^3:\|x\|\leq 1\},\qquad S^1=\{x\in{\mathbb{R}}^2:\|x\|=1\}. \] Thus $D^3\times S^1$ is a compact four-dimensional manifold with boundary $S^2\times S^1$. The notation $\deg_{\mathrm{exp}}(X)$ denotes the expander degree referred to in the supplied record and its Bamler--Chen reference. The record does not give the definition of this invariant, its normalization, its admissible class of spaces, or any auxiliary data required to define it.

In the usual smooth manifold convention, the boundary connected sum $X_1\mathbin{\#_\partial}X_2$ of compact smooth $n$-manifolds with nonempty boundary is formed by attaching a one-handle $D^1\times D^{n-1}$ to their disjoint union, with one attaching disk in each boundary, and smoothing corners. This explains the notation only: the record does not specify whether this category, additional structures, or restrictions on the attaching data are intended.

Requested relation and value. For every pair $X_1,X_2$ satisfying the hypotheses of the intended expander-degree theory, and for which the boundary connected sum is admissible, determine a general relation among \[ \deg_{\mathrm{exp}}(X_1),\qquad \deg_{\mathrm{exp}}(X_2),\qquad \deg_{\mathrm{exp}}(X_1\mathbin{\#_\partial}X_2). \] In particular, determine \[ \deg_{\mathrm{exp}}(D^3\times S^1). \] The source asks whether there is any relation; it does not propose a specific identity. The admissibility hypotheses, degree definition, and dependence on auxiliary or attaching data are unresolved in the supplied record, so this is a conditional formulation of the requested problem rather than a fully specified standalone statement.

Short English statement

Determine how expander degree behaves under boundary connected sum, and compute its value on $D^3\times S^1$. The supplied record does not define expander degree or specify the hypotheses on the spaces, so these must be recovered from the cited theory before the question has a complete standalone formulation.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30006223$, title Expander Degree Under Boundary Connected Sums, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project registry paths are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; no independent inspection of either file is claimed.
• MATRIX-MFO Tandem Workshop: Nonlinear Geometric Diffusion Equations (2025), Oberwolfach Reports, report 11/2025, pp. 485--530, DOI https://doi.org/10.4171/owr/2025/11. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No individual proposer or precise location within the report is supplied.
• Bamler--Chen, Expander degree theory, https://doi.org/10.1002/cpa.70024. This is a supplied literature reference; its contents have not been independently consulted here.
• Supplied literature assessment, dated exactly 2026-08-22: the problem is labelled open, and the assessment states that the expander-degree paper proves the degree for $D^4$ and several orbifolds while listing the value for $S^1\times D^3$ and boundary-connected-sum behavior as open questions. It consequently describes the requested $D^3\times S^1$ value as unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.