Manifolds from Partitions

TL;DR

The work introduces a discrete inverse-function framework whereby a $d$-manifold $G$ mapped onto a partition complex $P$ via a $P$-onto map $f$ yields a pullback $H(G,f,P)$ that is a $(d-k)$-manifold or empty, with a proof structure that mirrors Sard-type results in a discrete setting. It develops partition complexes $P=K_{\{(n_0,\dots,n_k)\}}$ and their algebraic structure (as a submonoid under join, with addition and multiplication), connects them to Alexandrov topology and irreducible representations, and provides a Mathematica-based toolkit to construct and analyze $H(G,f,P)$ for arbitrary dimensions. The paper combines a theoretical discrete Sard-type theorem with extensive computational statistics and concrete gallery examples, illustrating how finite combinatorial maps generate a rich variety of sub-manifolds, including cobordism-like decompositions, within finite simplicial complexes. This bridges combinatorial topology, representation-theoretic perspectives on partitions, and practical computation of sub-manifolds in finite settings, with potential applications to discrete geometry and visualization of manifold-like structures.

Abstract

If f maps a discrete d-manifold G onto a (k+1)-partite complex P then H(G,f,P),the set of simplices x in G such that f(x) contains at least one facet in P defines a (d-k)-manifold.

Figures (8)

• Figure 1: For Sard $G \to \mathbb{R}$, where studying $\{ f =c \}$ is determined by the map ${\rm sign}(f-c)$ onto $\{(-1,1)\}$, the complex $K_2=K_{(1,1)}$ is the only relevant choice KnillSard. For Sard $G \to \mathbb{R}^2$, where ${\rm sign}(f-c)$ map into $\{(-1,1),(1,1),(-1,-1),(-1,-1)\}$, we needed to pick the unique 2-dim partition complex $P=K_{(1,1,2)}$ that exists for $n=4$DiscreteAlgebraicSets. For $G \to \mathbb{R}^3$, the map ${\rm sign}(f-c)$ targets $\{-1,1\}^3$ which has 8 points. The 3-dimensional partitions of $n=8$ are $(5,1,1,1), (4,2,1,1),(3,3,1,1),(3,2,2,1), (2,2,2,2)$. In DiscreteAlgebraicSets, we picked $p=(1,1,1,5)$ by forced the target signs to be $(-1,1,1),(1,-1,1),(1,1,-1)$ and requiring to reach a 4th point. This complex $P$ is displayed to the right. For maps into $\mathbb{R}^m$ we chose $p=(1,1,\dots, 1,2^m-m)$ in DiscreteAlgebraicSets.
• Figure 2: As a 4-manifold $G=K_{(2,2,2)} \oplus C_{10}$ we took the join of the Octahedron complex with a circular complex. The 2-dimensional partition complex $P$ is by $p=(1,2,3)$ and a random function from $G$ to $\{1,\dots,6\}$. Now we can look at the $6$ facets $y$ of $P$ and color each of these patches $f^{-1}(y)$ differently. The 6 patches cover the manifold. The number of patches for $n=(n_0, \dots, n_k)$ is $\prod_{j=0}^k n_j$.
• Figure 3: We see the 5 integer partitions for $n=4$ and the corresponding 4 positive dimensional complexes. We left out $K_{\{5\}} = \overline{K_5}$. There is only one 2-dimensional complex $P=K_{(1,1,2)}$, the kite complex. This complex was involved when looking at maps $G \to \mathbb{R}^2$, where $\{ f = 0, g=0\}$ involved the 4 cases $\{ 1,1\},\{1,-1\},\{-1,1\},\{-1,-1\} \}$ and where we had imposed that 2 points $(1,-1),(-1,1)$ and at least one third point are in the image $f(G)$. We rephrase this now as that the image has to contain at least one maximal simplex in $P$.
• Figure 4: All 15 integer partitions for $n=7$ and the corresponding 15 complexes. We used Cuisenaire tools in first grade CuisenaireGattegno were partitions were studied for the competition KnillSchweizerJugendForscht. As far as I know, this was the first use of such a pedagogical tool for research rather than for teaching. Beside known results like the Euler pentagonal theorem, unusual results appeared like like $p(n)=\sum_{k=1}^n \sigma(k) p(n-k)/n$, where $\sigma(k)$ is the total number of prime factors of $k$. This can be seen by looking at $n p(n)$ as the area of the Cuisenaire rectangle listing all the partitions.
• Figure 5: A 2-manifold obtained as a co-dimension-3 manifold in the 5-manifold $S^2 \oplus S^2$, where $S$ is an icosahedron complex. The target space was the 3-dimensional $K_4$. In this case, we got a genus $2$ surface $H(G,f,P)$.

Theorems & Definitions (8)

• Theorem 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof
• Lemma 1
• proof