Barycentric Collapse and Simplicial Embeddings

• Barycentric collapse is a process in combinatorial topology where barycentric subdivision converts convex and star-shaped complexes into collapsible simplicial complexes.
• For convex polytopes, one subdivision suffices, while star-shaped complexes require at most d-2 derived subdivisions, with proofs leveraging link and cone lemmas.
• This process also enables linear embeddings in ℝ^(2d) for collapsible complexes with n facets using fewer than n subdivisions, establishing sharp bounds with practical implications.

Barycentric collapse denotes the phenomenon in combinatorial topology whereby the application of barycentric subdivision converts convex or certain structured polytopal complexes into collapsible simplicial complexes. Specifically, any linear subdivision of a convex $d$-polytope becomes simplicially collapsible after one barycentric subdivision; more generally, any star-shaped polyhedron in $\mathbb{R}^d$ becomes collapsible after at most $d-2$ derived subdivisions. For collapsible $d$-complexes with $n$ facets, fewer than $n$ barycentric subdivisions suffice to admit a linear embedding in $\mathbb{R}^{2d}$, a bound that is shown to be sharp in the worst case. The concept and proofs underlying barycentric collapse intricately combine elements of PL topology, combinatorial geometry, and the theory of embeddings and collapses (Adiprasito et al., 2017, Adiprasito et al., 2014).

1. Foundational Definitions and Preliminaries

A simplicial complex $K$ is a finite collection of subsets (faces) of a vertex set $V$ such that every subset of a face is also a face, and every singleton $\{v\}$ is included for each $\mathbb{R}^d$0. Its underlying space $\mathbb{R}^d$1 is the union of the convex hulls of its faces in some $\mathbb{R}^d$2.

A linear subdivision of $\mathbb{R}^d$3 is a simplicial complex $\mathbb{R}^d$4 with $\mathbb{R}^d$5 and with every face of $\mathbb{R}^d$6 lying in some face of $\mathbb{R}^d$7. The barycentric subdivision $\mathbb{R}^d$8 is a refinement where vertices correspond to barycenters of faces, and higher-dimensional simplices correspond to chains of faces $\mathbb{R}^d$9 in $d-2$0. The $d-2$1-fold iterated barycentric subdivision is denoted $d-2$2.

A face $d-2$3 is said to be free if it is properly contained in exactly one other face $d-2$4. An elementary collapse removes the pair $d-2$5. A simplicial complex is collapsible if it can be reduced to a single vertex by a sequence of elementary collapses.

2. Main Results: Theorems Governing Barycentric Collapse

Two central theorems formalize barycentric collapse for polyhedral complexes:

• Convex polytopes: For any convex $d-2$6-polytope $d-2$7 and any linear subdivision $d-2$8 of $d-2$9, the barycentric subdivision $d$0 is simplicially collapsible (Adiprasito et al., 2017).
• Star-shaped complexes: Let $d$1 be any (polyhedral) complex whose underlying space is star-shaped. Then $d$2 is collapsible. This result verifies that up to $d$3 derived subdivisions suffice in this broader class.

Furthermore, collapsible $d$4-complexes with $d$5 facets can be embedded linearly into $d$6 after fewer than $d$7 barycentric subdivisions. This is sharp: for certain 2-complexes (e.g., cones over non-planar graphs), the bound cannot be reduced because topological obstructions prevent embedding in $d$8 regardless of the number of subdivisions (Adiprasito et al., 2014).

3. Proof Strategy: Inductive Scheme and Key Tools

The proof of barycentric collapse for convex polytopes proceeds by induction on dimension $d$9 and is a higher-dimensional extension of Chillingworth’s 3D argument. The inductive step relies crucially on two ingredients:

• Link–collapsibility lemma: If $n$0, then $n$1. If $n$2 is collapsible, then $n$3.
• Cone lemma: Any simplicial cone is collapsible.
• Derived-neighborhood lemma: $n$4 can be collapsed stepwise via vertex deletions in the derived order, leveraging the collapsibility of barycentrically subdivided links.

At each inductive step, a generic linear functional orders the vertices. The highest vertex has a link—a convex spherical complex in the unit sphere—whose barycentric subdivision is collapsible by the inductive hypothesis. This establishes that the vertex is free and enables a full sequence of elementary collapses.

For embedding collapsible $n$5-complexes with $n$6 facets, the construction also proceeds by induction, refining the embedding at each step by carving out new facets and using combinatorial tools that control the embedding's extension over newly subdivided simplices (Adiprasito et al., 2014).

4. Sharpness, Limitations, and Exemplary Cases

The “fewer than $n$7 subdivisions” bound for embedding collapsible $n$8-complexes is optimal. For any non-planar graph $n$9 (e.g., $n$0, $n$1), its cone $n$2 forms a 2-dimensional collapsible simplicial complex. However, $n$3 cannot be embedded in $n$4, even after arbitrarily many subdivisions, since its 1-skeleton $n$5 fails to embed in $n$6.

In low dimensions, explicit examples such as a triangle subdivided by an internal vertex demonstrate the collapse after barycentric subdivision: each barycenter becomes sequentially free, and successive elementary collapses eventually reduce the subdivided complex to a point (Adiprasito et al., 2017).

5. Computational Complexity and Practical Consequences

Determining whether a given simplicial complex is collapsible is NP-complete. Barycentric subdivision entails a combinatorial explosion: the number of faces increases by an exponential factor in $n$7. This complexity affects the potential implementation of barycentric collapse-based algorithms, especially for large or high-dimensional input complexes. However, the theorems guarantee that, for convex or star-shaped inputs, a finite and uniform number of barycentric subdivisions will render any linear subdivision collapsible, facilitating topological simplifications in combinatorial and computational topology.

6. Corollaries, Extensions, and Open Questions

The main results extend to convex polytopal complexes, collapsible cubical complexes, and finite CAT(0) cube complexes: for any collapsible $n$8-cubical complex with $n$9 cubes, fewer than $\mathbb{R}^{2d}$0 cubical barycentric subdivisions suffice for embeddability in $\mathbb{R}^{2d}$1 (Adiprasito et al., 2014). If a 2-complex $\mathbb{R}^{2d}$2 admits a collapsible PL thickening to a 3-manifold $\mathbb{R}^{2d}$3, then $\mathbb{R}^{2d}$4 embeds in $\mathbb{R}^{2d}$5 by the same inductive argument.

Classical conjectures have been resolved up to one subdivision: Lickorish's conjecture (“Every subdivision of the simplex is collapsible”) is true after one barycentric subdivision. Goodrick's conjecture (star-shaped implies collapsible) holds after $\mathbb{R}^{2d}$6 derived subdivisions. Hudson's problem on compatibility of collapses with further subdivisions is similarly resolved.

Several further directions remain open, including the possibility of more efficient subdivision procedures, application to other complex classes, and purely combinatorial approaches avoiding spherical geometry (Adiprasito et al., 2017).