Soft Barycentric Refinement

TL;DR

The paper introduces soft Barycentric refinement, a refinement that preserves manifold structure and controls vertex-degree growth, and establishes a universal spectral limit for the associated Kirchhoff Laplacians that depends only on the maximal dimension $q$. It proves a dichotomy for the chromatic number under soft refinement: for a $q$-manifold $G$, $c(\psi(G))=q+1$ while $c(\phi(G))=(q-1)+c(\hat{G})\le q+2$, with $c(\hat{G})\le 3$ via a dual acyclic 3-coloring principle. The Fisk complex is introduced to analyze odd-degree dual structures and its refinement under soft Barycentric rules, elucidating how colorability and Euler characteristic behave in higher dimensions, with explicit 2D results tying to Eulerian conditions. In 2D specifically, the approach yields a concrete coloring strategy for $G'$ that preserves $\chi(G)$ and shows $c(G')\le 4$ (with $c(G')=3$ exactly when $G$ is Eulerian). Overall, the work links spectral, topological, and combinatorial properties of manifolds under refinement, with implications for universal limits, dual-graph colorability, and 2D special cases.

Abstract

The soft Barycentric refinement preserves manifolds with or without boundary. In every dimension larger than one, there is a universal spectral central limiting measure that has affinities with the Barycentric limiting measure one dimension lower. Ricci type quantities like the length of the dual sphere of co-dimension-2 simplex stay invariant under soft refinements. We prove that the dual graphs of any manifold can be colored with 3 colors, which is in the 2-dimensional case a special case of the Groetzsch theorem. It follows that the vertices of a soft Barycentric refined q-manifold G' can be colored by q+1 or q+2 colors.

Figures (8)

• Figure 1: We see to the left the 4'th soft Barycentric refinement of a wheel graph with chromatic number $c(G)=c(\phi^4(G))=4$ and the 3'rd softly refined projective plane $G$ with $c(\phi^3(G))=5$. No 2-manifold with $c(\phi(G))<c(G)$ is known. Giving such an example would settle a conjecture of Albertson and Stromquist AlbertsonStromquist which says $c(G) \leq 5$ for 2-manifolds. The smallest manifold with $c(\phi(G))<c(G)$ we know of is the $q=5$-sphere $G=C_5 \oplus C_5 \oplus C_5$ (the graph join of three $1$-spheres) with $c(G)=3 c(C_5)=3*3=9$ for which our result shows $c(\phi(G))=q+2=7$.
• Figure 2: We see approximations of the limiting measure for Barycentric refinement in dimension $q=1$ and $q=2$. These measures are supported on the real half line $[0,\infty) \subset \mathbb{R} \subset \mathbb{C}$. The existence of the limiting measures in any dimension $q$ is an older story KnillBarycentricKnillBarycentric2.
• Figure 3: We see approximations of the limiting measure for the soft Barycentric refinement limit in dimension $q=2$ and $q=3$. This is new. For $q=2$, we have still an absolutely continuous measure of compact support. A Fourier transform allows to express the integrated density of states. It shows a van Hove singularityVanHove1953, which is related to critical points of the Fourier transform $\hat{L}(x,y)=6-2\cos(x)-2\cos(y)-2\cos(x+y)$ on $L^2(\mathbb{T}^2)$, the Laplacian $L$ of the hex lattice.
• Figure 4: The figure shows a 3-manifold with boundary and its first soft Barycentric refinement. Our definition of soft Barycentric refinement features that the boundary undergoes the usual Barycentric refinement.
• Figure 5: A 3-connected planar triangle-free graph that is not the dual graph of a 2-manifold. The reason is that it is not 3-regular. It is a subgraph of the prism graph $C_8 \oplus \overline{K_2}$. Grötsch's theorem applies to this graph, but not to the dual 3 color theorem.

Theorems & Definitions (10)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2: Dual 2 forest and 3 color theorem
• Corollary 1
• proof
• Theorem 3: Acyclic 3-color theorem
• proof
• Lemma 2