Kirby Problem 2.45

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An end-periodic oriented manifold has finitely many ends, each with a neighborhood modeled on one half of an infinite cyclic cover of a compact manifold. An end-periodic automorphism agrees, sufficiently far out on each end, with a covering translation in that model. Use either the smooth category, with diffeomorphisms, or the topological category, with homeomorphisms.

An oriented end-periodic cobordism from $(S_0,f_0)$ to $(S_1,f_1)$ is an oriented 3-manifold $W$ with $\partial W=S_0\sqcup(-S_1)$, equipped with an end-periodic automorphism restricting to $f_0,f_1$ on the boundary. All manifolds, end models and automorphisms are in the chosen category. The group $\Delta^e_2$ consists of cobordism classes of these surface-automorphism pairs; addition is disjoint union and inversion reverses orientation. The end-periodic restriction is imposed on the extension as well as on the boundary maps.

Statement recorded in the supplied source.

Compute the oriented end-periodic cobordism group $\Delta^e_2$ of surface automorphisms, in the diffeomorphism and homeomorphism categories. Determine its group structure rather than substituting the ordinary compact surface-automorphism cobordism group. [S2]

Short English statement

Compute cobordism classes of end-periodic surface automorphisms, requiring the cobordism extension to remain end-periodic, in both smooth and topological categories.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2793 (KP-2.45). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 2.45 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.