Stoker, again

Karim Adiprasito

I wanted to quickly mention an “almost” LLM result. I wrote previously about a marvelous new approach of Yuchen Bi to Stoker’s conjecture using differential geometry, but that still critically used quite extensively non-polyhedral techniques. Yesterday, Arseniy Akopyan wrote to me that he chatted with LLM and found a polyhedral only proof.

Not essentially new, Arseniy and I then noticed that the LLM had “stolen” a result of Martin Winter, who in 2023 found the proof without noticing, and wrote it in a very nice paper expanding our knowledge of Wachspress coordinates and Izmestiev matrices. Congratulations, Martin!

Martin Winter

The initial manuscript of Arseniy+LLM was longer, but spent most of its time reproducing a Lemma Martin had in his paper (giving the same proof as him, so I assume it “stole” without being fully aware). So, with that Lemma in mind, the proof is actually trivial. Martin has prepared a short note and allowed me to share it here.

Stoker_s_conjecture-1Download



An icosahedral rational homology 3-sphere.

I want to at this stage remind everyone of one my favorite open problems: Is there a 4-polytope all whose facets are icosahedra. It is still open. For all other regular polytopes, we know whether such a polytope having it as its only facet exists.

The problem, I realized early on, is that the techniques cannot generalize. They all rely on homological properties only, and I realized early on that there are homology 3-spheres every facet of which is an icosahedron. The construction is standard, but I wanted to record it here.


Let

\Delta=[3,5,3]^+ =\langle a,b,c\mid a^3=b^5=c^3=(ab)^2=(bc)^2=(abc)^2=1\rangle .

Define an epimorphism \varphi:\Delta\longrightarrow {\rm PSL}_2({\bf F}_{29}) by sending a,b,c to the projective classes of

A=\begin{pmatrix}14&24\\19&14\end{pmatrix},\qquad B=\begin{pmatrix}5&15\\14&19\end{pmatrix},\qquad C=\begin{pmatrix}9&24\\24&19\end{pmatrix}.

Put K=\ker\varphi. Then K is torsion-free, and the quotient of the regular hyperbolic honeycomb \{3,5,3\} by K is a closed polyhedral $3$-manifold X whose facets are icosahedra. Since

|{\rm PSL}_2({\bf F}_{29})|=12180,\qquad f(X)=(203,2030,2030,203),

the complex has 203 icosahedral facets. Easy coset computation shows that two distinct facets meet only in the empty set, one vertex, or one triangular face, so the complex is strongly regular. Moreover, over {\bf F}_2 the cellular boundary maps have ranks

{\rm rank}\,\partial_1=202,\qquad {\rm rank}\,\partial_2=1828,\qquad {\rm rank}\,\partial_3=202.

Consequently

H_*(X;{\bf F}_2)\cong H_*(S^3;{\bf F}_2),

and hence X is a rational homology $3$-sphere.

The group-theoretic quotient is from G. A. Jones, C. D. Long and A. D. Mednykh, Hyperbolic manifolds and tessellations of type {3,5,3} associated with L2(q), arXiv:1106.0867.

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