On vertex-minimal simplicial maps to the sphere

Abstract: For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that $\lim_{d\to\infty}\frac{λ(n,d)}d=0$ for any $n\ge3$, disproving O. Musin's conjecture. Using similar idea, for any $C$ we construct a triangulation of $\mathbb{S}<sup>n$, $n\ge3$, for which $\frac{f_j}{f_i}&gt;C$, for any $0\le i&lt;j\le n$ such that $i&lt;\lfloor\frac{n-1}2\rfloor$. All triangulations we obtain are isomorphic to boundaries of convex polytopes in $\mathbb{R}<sup>{n+1}$.

• The paper disproves a longstanding conjecture by proving that for n ≥ 3, the vertex-to-degree ratio in sphere maps tends to zero as degree increases.
• It introduces a join-based construction and central subdivision process that incrementally increases the vertex count by three per unit degree, achieving O(√d) bounds.
• The work establishes flexible control over f-vector ratios in sphere triangulations, opening new directions in computational topology and mesh optimization.

Vertex-Minimal Triangulations and Degree-Bounded Simplicial Maps to the Sphere

Problem Statement and Main Results

This paper investigates the minimal vertex count, $\lambda(n,d)$, in triangulations of $n$-spheres that admit simplicial maps of prescribed degree $d$ to the boundary of an $(n+1)$-simplex (the standard sphere triangulation). This function $\lambda(n, d)$ formalizes a quantitative aspect of combinatorial topology: "How few vertices are required to realize a nontrivial map of controlled degree?" The central result is the disproval of an established conjecture by Apolonskaya and Musin, which posited a positive lower bound on the asymptotic ratio $\frac{\lambda(n,d)}{d}$ for all $n > 0$.

The rigorous result established here is that for all $n \ge 3$, $\lim_{d\to\infty}\frac{\lambda(n,d)}{d}=0$, indicating that for higher dimensions, large-degree maps can be accommodated with remarkably few vertices, contradicting previously held beliefs about this ratio’s lower bound. Additionally, the paper constructs families of triangulations of spheres in which the ratios of $f$-vector entries (number of $j$-simplices to $i$-simplices, for $i<\lfloor\frac{n-1}{2}\rfloor$) are arbitrarily large.

Definitions, Techniques, and Constructions

The study operates in the context of finite simplicial complexes and simplicial maps, with degree defined via homology. A key technical tool is the construction of joins of lower-dimensional spheres, allowing the authors to generate $n$-spheres with controlled combinatorics and map degrees. The paper leverages the multiplicativity property of the degree under the join operation: if $f:K\to K'$ and $g:L\to L'$ are simplicial maps of spheres with degrees $\deg f$ and $\deg g$, their join $f\star g: K\star L\to K'\star L'$ has degree equal to the product $\deg f\cdot \deg g$.

The iterative process for increasing the map degree is facilitated by the central subdivision procedure, which augments the complex by three vertices each time, while simultaneously adjusting the degree by $\pm1$. This construction underpins the bound $\lambda(n, d+1) \leq \lambda(n, d) + 3$ for all $n>0,d>0$.

Figure 1: The subdivisions from proposition~\ref{pr:d+1} illustrating central subdivision and its impact on vertices and degree.

Strong Claims and Numerical Bounds

The results yield strong bounds for $\lambda(n,d)$. Specifically, for $n\geq 3$, $\lambda(n, d)$ can be made sublinear in $d$: for quadratic degree growth $d=k^2$, $\lambda(n, k^2)$ is $O(k)$, so $\lambda(n, d) = O(\sqrt{d})$ as $d \to \infty$. This is sharply at odds with the conjectured fixed-ratio lower bound. For $n=3$, explicit constructions yield $\lambda(3, d) \leq O(\sqrt{d})$, while for general $n$, the bounds are similar with additive constants dependent on $n$.

The $f$-vector constructions, leveraging repeated joins of cyclic $1$-sphere triangulations, demonstrate controllable blow-up of the higher-dimensional face count ($f_j$) relative to lower-dimensional faces ($f_i$), for those $i<\lfloor\frac{n-1}{2}\rfloor$, and this holds for arbitrary constants $C$ given sufficiently large parameter values. These triangulations are isomorphic to the boundaries of convex polytopes in $\mathbb{R}^{n+1}$, preserving their combinatorial regularity.

Theoretical Implications

The disproval of Musin's conjecture realigns our understanding of the combinatorial bottlenecks in sphere mapping. For $n \geq 3$, combinatorial constructions surpass prior limitations, showing that sphere maps of large degree can be realized with vanishingly small vertex-to-degree ratios. This result suggests finer-grained connections between combinatorial topology and piecewise-linear geometry; specifically, that complexity can be traded off in higher-dimensional polytopal boundaries through join operations, yielding high-degree maps with modest vertex counts.

Additionally, the $f$-vector estimates open new directions on the combinatorial flexibility of triangulated spheres. By embedding high-dimensional joins as boundaries of convex polytopes, the constructed complexes are within reach of both algebraic and discrete geometric investigations.

Practical Perspectives and Future Directions

The immediate practical applications of such vertex-minimal constructions lie in computational topology and geometric modeling, where minimizing combinatorial complexity is desirable for error control, efficient algorithms, and mesh optimization. These bounds may also affect lower bounds in algorithmic complexity in topological data analysis, as the results provide new minimal examples for testing homological and mapping algorithms.

On the theoretical front, the paper raises open questions about the behavior of $\lambda(n,d)$ under stricter constraints (such as non-degenerate or orientation-preserving maps), and whether sharper, possibly polynomial bounds in higher dimensions exist. The possibility that the bound $\lambda(n,d)^{\lfloor\frac{n+1}{2}\rfloor}/d$ remains nonzero is specifically noted as an unresolved question.

Further investigation may clarify the landscape of optimal triangulations for prescribed mapping degrees, both in simplicial and PL categories. The combinatorial constructions suggest possible extensions to other manifolds and to the study of secondary invariants in topology, such as mapping degree spectra and face vector optimizations.

Conclusion

This paper establishes that for spheres of dimension $n \geq 3$, high-degree simplicial maps to standard triangulations can be realized with sublinear growth in the required number of vertices, refuting the established conjecture. The join-based constructive approach yields not only tight bounds on $\lambda(n,d)$ but also flexible control over $f$-vector ratios, achieved by polytopal triangulations. These results recalibrate both theoretical and practical expectations in combinatorial topology and present open problems for future research regarding the minimal triangulation requirements for prescribed mapping degrees.