Kirby Problem 4.28

Definitions and mathematical statement

For a closed connected oriented three-manifold $M$, define $s_{\mathrm{sm}}(M)$ as the least $n\ge0$ for which $M$ embeds smoothly in $\#_n(S^2\times S^2)$. Define $s_{\mathrm{top}}$ by locally flat topological embeddings instead. Local flatness means that the pair is locally modeled on $(\mathbb R^4,\mathbb R^3)$. Let $E=S^2\widetilde\times S^2$ be the nontrivial oriented sphere bundle over the sphere, diffeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}}^{2}$. Replacing $S^2\times S^2$ by $E$ defines $\widetilde s_{\mathrm{sm}}$ and $\widetilde s_{\mathrm{top}}$. An empty connected sum means $S^4$.

Compute these functions for substantial classes of three-manifolds, distinguishing the two categories and the two ambient families. Specifically:

(a) Determine their values for lens spaces and Brieskorn homology spheres. Use $L(p,q)=S^3/((z_1,z_2)\sim(\zeta z_1,\zeta^qz_2))$, where $\zeta^p=1$ and $\gcd(p,q)=1$; a Brieskorn sphere is the link of $z_1^a+z_2^b+z_3^c=0$ for pairwise coprime integers $a,b,c\ge2$.

(b) Is there an integral homology sphere $M$ with $s_{\mathrm{sm}}(M)=1$? This requires a smooth embedding in $S^2\times S^2$ and no smooth embedding in $S^4$.

(c) Are the four functions algorithmically computable from a finite triangulation of $M$? A computation must halt and return the minimum, not merely find an embedding providing an upper bound. No particular time bound is required.

Short English statement

Compute the least number of sphere-bundle summands needed to embed a three-manifold, smoothly or locally flatly.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2904 (KP-4.28), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.28. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.