Overview
The study investigates families of compact $n$-manifolds, focusing on those characterized by local combinatorial properties. It introduces the concept of a family being 'locally combinatorially defined' (LCD) and establishes an equivalence between this property and the existence of a specific topological structure, a compact branched $n$-manifold, into which members of the family can immerse.
Research Context
A family of compact $n$-manifolds is defined as locally combinatorially defined (LCD) if its specification can be achieved through a finite number of local triangulations. This definition provides a combinatorial framework for classifying and understanding such families. The research aims to provide a topological characterization that is equivalent to this combinatorial definition.
Findings
The core finding of the research is the demonstration that if a family of compact $n$-manifolds is locally combinatorially defined (LCD), then there exists a compact branched $n$-manifold, denoted as $W$, such that the entire family consists precisely of those manifolds that immerse into $W$. Conversely, the paper states that if such a compact branched $n$-manifold $W$ exists, then the family of all compact $n$-manifolds that immerse into $W$ is locally combinatorially defined. This establishes a bidirectional equivalence between the combinatorial characterization (LCD) and the immersibility into a specific compact branched manifold.
Why This Matters
The established equivalence between locally combinatorially defined families of compact $n$-manifolds and their immersion into a compact branched $n$-manifold will be utilized in subsequent research. Specifically, this equivalence is intended to demonstrate that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that particular geometry is LCD. This suggests a systematic approach to characterizing geometric 3-manifolds using combinatorial and immersion properties.