Locally Combinatorially Defined Manifolds

• LCD is a framework where compact PL n-manifolds are defined by finitely many local triangulation types, establishing a clear local specification.
• The approach demonstrates an equivalence between finite local combinatorial models and proper PL immersions into a compact branched manifold, unifying local and global perspectives.
• Applications include classifying torus bundles over S¹ and closed 3-manifolds with Thurston geometries, illustrating LCD's broad impact in topological manifold studies.

Searching arXiv for papers on “Locally Combinatorially Defined” and related terminology. Locally combinatorially defined (LCD) denotes, in the PL-topological sense, a family of compact $n$-manifolds specified by finitely many local triangulation types: a set $T$ of PL $n$-manifolds is LCD if there exists a finite set of local models $M$ such that $T=PL(M)$ (Cooper et al., 7 Oct 2025). In the 2025 formulation, this notion is equivalent to the existence of a compact branched $n$-manifold $W$ for which the family consists exactly of the compact PL $n$-manifolds that properly PL immerse into $W$, thereby identifying a finite local combinatorial presentation with a universal immersion target (Cooper et al., 7 Oct 2025).

1. Formal definition

A local model of dimension $n$ is a pair $T$0, where $T$1 is a simplicial complex and $T$2 is a vertex, such that $T$3 is PL homeomorphic to the cube $T$4 (Cooper et al., 7 Oct 2025). A family $T$5 of compact PL $T$6-manifolds is locally combinatorially defined if there exists a finite set $T$7 of such local models and

$T$8

where $T$9 denotes the set of PL $n$0-manifolds admitting a triangulation modeled on $n$1 (Cooper et al., 7 Oct 2025).

This definition isolates a strictly local specification principle. Membership in the family is determined by whether the stars appearing in a triangulation can be assembled from finitely many prescribed local configurations. A plausible implication is that LCD families are designed to capture global manifold classes through bounded local data rather than through a priori global geometric structures.

2. Branched manifolds as universal targets

A branched $n$2-manifold in this setting is a compact polyhedron $n$3 equipped with a finite system of PL maps, the local projections, from subpolyhedra $n$4, so that locally $n$5 is a union of parameterized sheets (Cooper et al., 7 Oct 2025). Given such a branched manifold $n$6, one writes $n$7 for the family of compact PL $n$8-manifolds that properly PL immerse into $n$9 (Cooper et al., 7 Oct 2025).

The central theorem is the equivalence

$M$0

that is, a family of compact PL $M$1-manifolds is LCD if and only if it is BM (Cooper et al., 7 Oct 2025). Here BM abbreviates the class of families arising from proper PL immersions into a compact branched $M$2-manifold.

The significance of this equivalence is structural. It identifies two ostensibly different modes of classification: finite local combinatorial rules on triangulations, and immersion into a single compact branched object. In this sense, LCD is not merely a condition on triangulations; it is also an immersion-theoretic characterization.

3. How the equivalence is realized

For the implication $M$3, a finite set of local models $M$4 is used to construct a universal branched manifold $M$5 encoding all local combinatorial types in the family (Cooper et al., 7 Oct 2025). The local neighborhoods in $M$6 correspond to the local models via a refined system of coloring and “geography” labels, and any manifold modeled on $M$7 can be immersed into $M$8 by mapping simplices according to their labels (Cooper et al., 7 Oct 2025).

For the implication $M$9, one begins with a compact branched $T=PL(M)$0-manifold $T=PL(M)$1, triangulates $T=PL(M)$2 nicely so that the branch locus is a simplicial subcomplex and the projections are well behaved, and then observes that the family of compact $T=PL(M)$3-manifolds immersing into $T=PL(M)$4 is determined by finitely many star types occurring in that triangulation (Cooper et al., 7 Oct 2025). Those finitely many local stars provide the required local models.

An additional point in the 2025 treatment is that the necessary labelings, including colors and “geography” data for neighborhoods, can be encoded combinatorially, so labelled LCD is equivalent to LCD (Cooper et al., 7 Oct 2025). This removes any essential dependence on auxiliary decorations external to the triangulation itself.

4. Examples and stated applications

One explicit example is the family of torus bundles over $T=PL(M)$5. The construction described in the 2025 account forms a branched 3-manifold $T=PL(M)$6 by gluing together torus bundles over the circle, one for each generator of $T=PL(M)$7; every torus bundle over the circle immerses into this $T=PL(M)$8, and conversely any manifold immersing into $T=PL(M)$9 is a torus bundle over $n$0 (Cooper et al., 7 Oct 2025). By the LCD–BM equivalence, the family of torus bundles over $n$1 is LCD.

The same source states that, in subsequent papers, the equivalence will be used to show that for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD (Cooper et al., 7 Oct 2025). It also records that, by earlier work of Cooper and Thurston, all closed orientable 3-manifolds can be modeled on a finite set of local models; thus that family is LCD and there is a universal branched 3-manifold that all such manifolds immerse into (Cooper et al., 7 Oct 2025).

These examples indicate the intended range of the concept: LCD is aimed at families naturally described by local geometric-topological structure but recoverable through finite combinatorial data.

5. Related local notions on simplicial complexes

A related but distinct notion appears in the study of numerical invariants of finite simplicial complexes. There, a function

$n$2

is called combinatorially locally determined if there exists a function $n$3, invariant under combinatorial equivalence of links, such that

$n$4

for every complex $n$5 in the class under consideration (Bloch, 2014).

Within that framework, the Euler characteristic is locally determined: $n$6 whereas not every linear combination of simplex counts is local in this sense (Bloch, 2014). In particular, the Charney–Davis quantity

$n$7

is not locally determined in either the combinatorial or geometric sense on any class containing all flag spheres of a fixed odd dimension at least $n$8 (Bloch, 2014).

This earlier literature is conceptually adjacent to LCD families of manifolds because both theories use finite local combinatorial data to recover global objects. The two notions are nevertheless different: one concerns families of manifolds specified by local triangulations, while the other concerns numerical functions reconstructed from vertex links.

6. Terminological ambiguity and contrast with coding theory

The acronym “LCD” is heavily overloaded. In coding theory it usually means linear complementary dual, namely a linear code $n$9 with

$W$0

together with variants such as Euclidean LCD, Hermitian LCD, Galois LCD, and $W$1-LCD codes (Carlet et al., 2017). That usage is standard across work on bounds, constructions, masking schemes, and adder-channel coding, including mixed-alphabet settings and linear complementary pairs (LCPs) (Jose et al., 2024).

The topological usage “locally combinatorially defined” is therefore unrelated to the coding-theoretic LCD literature. This suggests that acronym disambiguation is essential in cross-disciplinary settings, particularly on repositories where both subjects coexist.

In the topological sense, LCD has a precise role: it identifies manifold families describable by finitely many local triangulation types and, equivalently, by proper PL immersions into a compact branched $W$2-manifold (Cooper et al., 7 Oct 2025). That equivalence places local combinatorial specification and universal branched targets within a single classification framework.