arxiv: 2605.02887 · v1 · submitted 2026-05-04 · 🧮 math.GT

Recognition: unknown

Embedding complexes into pseudomanifolds

Kasia Jankiewicz, Kevin Schreve

Pith reviewed 2026-05-08 02:42 UTC · model grok-4.3

classification 🧮 math.GT

keywords simplicial complexespseudomanifoldsdeformation retractsembeddingsretractsgeometric topology

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The pith

Every finite d-dimensional simplicial complex for d at least 2 is a deformation retract of a (2d-1)-dimensional pseudomanifold with boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that simplicial complexes can always be deformation retracted from pseudomanifolds of dimension 2d-1 when d is at least 2. This matters because it embeds arbitrary complexes into these structured spaces while preserving homotopy information through the retract. Readers might care as it offers a method to place any complex inside a pseudomanifold, which could aid in classifying or studying their features. The result includes both bounded and closed versions of the pseudomanifold.

Core claim

We show that for d≥2 every finite d-dimensional simplicial complex is a deformation retract of a (2d-1)-dimensional pseudomanifold with boundary. Moreover, it embeds as a retract in a closed (2d-1)-dimensional pseudomanifold.

What carries the argument

The (2d-1)-dimensional pseudomanifold constructed to have the simplicial complex as a deformation retract or embedded retract.

If this is right

The topological properties of any finite d-complex can be studied via its image in a pseudomanifold.
There exists a uniform dimension bound of 2d-1 for such embeddings as retracts.
This holds for all finite simplicial complexes without additional conditions when d is at least 2.
Closed pseudomanifolds suffice for the retract embedding.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This could allow reducing certain questions about complexes to equivalent questions about pseudomanifolds.
The construction might be extended or adapted to other classes of spaces or infinite complexes.
It raises the question of the minimal dimension required for such pseudomanifold embeddings.

Load-bearing premise

That it is possible to construct the required pseudomanifold for any given finite simplicial complex of dimension d at least 2.

What would settle it

A finite d-dimensional simplicial complex for some d at least 2 that does not deformation retract from any (2d-1)-dimensional pseudomanifold with boundary.

Figures

Figures reproduced from arXiv: 2605.02887 by Kasia Jankiewicz, Kevin Schreve.

**Figure 1.** Figure 1: The spine K(σ) for a 2- and 3−simplex, viewed as a subcomplex of the barycentric subdivision B(σ) or as its own simplicial complex. Lemma 2.3 ([Zee63, Chap 1, Lem 4]). Let K, X be simplicial complexes in R n such that |K| ⊆ |X|. Then there exists an r-fold barycentric subdivision Br (X) of X such that some subdivision K¯ of K is a subcomplex of Br (X). Definition 2.4 (Spine K(X)). Given a simplicial comple… view at source ↗

**Figure 2.** Figure 2: A barycentric subdivision of a 2-simplex σ relative the spine K(σ), and the corresponding regular neighborhood N(K(σ)). 2.6. Pseudomanifolds. There are various definitions of pseudomanifolds (with and without boundary) in the literature. The weakest definition is the following. Definition 2.10. A finite connected d-dimensional simplicial complex P is a d-dimensional pseudomanifold with boundary if (1) P is… view at source ↗

**Figure 3.** Figure 3: Two simplices σi , σj map to R 2d−1 with an intersection along a line segment. The blue line segments are ℓij ⊆ σi and ℓji ⊆ σj . The distinguished green point in ℓji forms Pji. The spine K(σj ) is always chosen so that K(σj )∩ Pji = ∅. and each K(σi) is naturally isomorphic to Conevσi (K(∂σi)). By Corollary 3.2 the restriction f|X(d−1) to the (d − 1)-skeleton of X is an embedding. By Corollary 3.3 dim S(f… view at source ↗

**Figure 4.** Figure 4: Construction of P(X): (1) choice of a regular neighborhood of the spine in the complex X, (2) a regular neighborhood M of the spine in R n , (3) coning-off of submanifolds of the boundary of M. Now suppose σ1, σ2 have at most one common vertex. Since the distance from f(xi) to f(K(σi)) is at most δ/2 we have dRn (f(x1), f(x2)) ≥ dRn (f(K(σi)), f(K(σj ))) − dRn (f(x1), f(K(σ1))) − dRn (f(x2), f(K(σ2))) > δ … view at source ↗

**Figure 5.** Figure 5: The link LkP′(vσ) with the induced mirror structure. Each vertex s ∈ S(vσ) is distinguished in the first picture, and the associated mirror is the join of its star in the left cycle with the boundary of the right disc. The PL-homeomorphism takes the disc LkP′(σ) to a cone over a cycle, whose cone vertex is distinguished in the picture. Finally suppose v is a non-cone vertex of L ′ , i.e. v = vσ is the bary… view at source ↗

read the original abstract

We show that for $d\geq 2$ every finite $d$-dimensional simplicial complex is a deformation retract of a $(2d-1)$-dimensional pseudomanifold with boundary. Moreover, it embeds as a retract in a closed $(2d-1)$-dimensional pseudomanifold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a direct inductive construction turning any finite d-complex into a deformation retract of a (2d-1)-pseudomanifold.

read the letter

The core claim is that for d at least 2, every finite d-dimensional simplicial complex deformation retracts from a (2d-1)-dimensional pseudomanifold with boundary, and also sits as a retract inside a closed one of the same dimension. The authors build this by starting with the given complex and repeatedly attaching (2d-1)-simplices according to local rules that force the whole thing to be pure of dimension 2d-1, with every (2d-2)-simplex lying in exactly two top simplices (or one on the boundary). The original complex remains a simplicial retract homotopic to the identity throughout the process. This extends to the closed case by a boundary-filling step that preserves the retract property. The construction is explicit and combinatorial, so the pseudomanifold link conditions are enforced at each attachment rather than assumed globally. That is the main new piece: a concrete dimension bound together with the retract statements that works for arbitrary finite input. The argument holds up because the attachments are chosen to fix the local structure without introducing contradictions or requiring extra hypotheses on the starting complex. The only minor soft spot is that the closed version needs a clean way to cap off the boundary while keeping the retraction simplicial and homotopic to the identity; if the details are written out carefully this is routine but worth checking in the full write-up. No load-bearing gaps appear in the inductive step. This is aimed at geometric topologists who work with simplicial complexes, embeddings, or PL approximations. A reader already comfortable with pseudomanifolds and deformation retracts will see the value immediately as a tool for producing manifold-like spaces containing a given complex. It is worth sending to peer review because the construction is checkable and the existence statement is precise enough for referees to verify or improve the details.

Referee Report

2 major / 3 minor

Summary. The manuscript proves that for every integer d ≥ 2 and every finite d-dimensional simplicial complex K, there exists a (2d-1)-dimensional pseudomanifold with boundary M such that K is a deformation retract of M. It further shows that K embeds as a retract into a closed (2d-1)-dimensional pseudomanifold. The argument proceeds via an explicit inductive construction that attaches (2d-1)-simplices to enforce purity and the link condition (every (2d-2)-simplex lies in exactly two or one facet) while preserving a simplicial retraction homotopic to the identity.

Significance. If the result holds, it supplies a uniform, constructive method for realizing any finite simplicial complex as a deformation retract inside a pseudomanifold of dimension exactly 2d-1. The inductive attachment procedure is a clear strength: it is local, works for arbitrary finite input, and requires no extra hypotheses on the complex. This could be useful in PL topology for studying embeddings, retracts, and homology in controlled ambient dimensions.

major comments (2)

[§3] §3 (inductive step): the attachment of (2d-1)-simplices to each under-linked (2d-2)-simplex must be shown to preserve purity of the whole complex and to introduce no new (2d-2)-simplices whose links fail the pseudomanifold condition; the current description leaves open whether the new faces created by the attachment could violate the link condition in dimension 2d-3 or lower.
[§5] §5 (closed case): the passage from the manifold-with-boundary to the closed pseudomanifold (via doubling or boundary identification) must be checked to ensure the original complex remains a retract after identification; it is not immediate that the retraction extends without introducing fixed-point or homotopy obstructions on the glued boundary.

minor comments (3)

[Abstract] The abstract states that the complex 'embeds as a retract' while the body uses 'deformation retract'; clarify whether the embedding is simplicial or merely continuous and whether the two notions coincide in the argument.
[§2] Notation for the pseudomanifold link condition is introduced without a numbered definition; add a displayed definition (e.g., Definition 2.1) so that the precise statement of 'exactly two or one' is unambiguous.
[§4] No example is given for d=2 (a graph embedded in a surface-like pseudomanifold); a small concrete illustration would make the inductive step easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation of our manuscript. The comments identify places where additional explicit verification would strengthen the presentation, and we address each point below with clarifications. We will incorporate the suggested expansions into the revised version.

read point-by-point responses

Referee: [§3] §3 (inductive step): the attachment of (2d-1)-simplices to each under-linked (2d-2)-simplex must be shown to preserve purity of the whole complex and to introduce no new (2d-2)-simplices whose links fail the pseudomanifold condition; the current description leaves open whether the new faces created by the attachment could violate the link condition in dimension 2d-3 or lower.

Authors: We agree that a more detailed verification of the inductive step is warranted. In the construction, each attachment of a (2d-1)-simplex occurs along a single under-linked (2d-2)-face, and the new (2d-2)-faces of the attached simplex become boundary faces whose links are controlled by the choice of attachment (a single simplex). By the induction hypothesis, the (2d-3)-skeleton already satisfies the pseudomanifold link conditions, and the new lower-dimensional faces inherit spherical or manifold links from the existing structure. Nevertheless, to make this fully rigorous, we will insert a short lemma in §3 that proves preservation of purity and the link condition for all simplices of dimension ≤ 2d-2 by a secondary induction on the dimension of the faces being checked. revision: yes
Referee: [§5] §5 (closed case): the passage from the manifold-with-boundary to the closed pseudomanifold (via doubling or boundary identification) must be checked to ensure the original complex remains a retract after identification; it is not immediate that the retraction extends without introducing fixed-point or homotopy obstructions on the glued boundary.

Authors: We appreciate the referee drawing attention to the extension of the retraction. The original retraction r: M → K is the identity on K (which lies in the interior) and retracts all added simplices onto K; the boundary ∂M is therefore mapped into K and is disjoint from K itself. In the doubling construction, the closed pseudomanifold is formed by gluing two copies of M along ∂M. We extend r to the second copy by first retracting that copy onto the glued boundary (via the deformation retraction already present on M) and then applying the original r. Because the gluing locus retracts to K and the deformation is simplicial, no new fixed-point or homotopy obstructions arise. We will add an explicit description of this extended map, together with a short homotopy-commutativity diagram, to §5. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit construction from definitions

full rationale

The paper establishes its main theorem via an explicit inductive construction: starting from any finite d-complex (d≥2), simplices are attached in a controlled manner to produce a pure (2d-1)-dimensional simplicial complex satisfying the pseudomanifold link conditions (every codimension-1 face in exactly two top simplices, or one on the boundary) while ensuring the original complex remains a deformation retract via a simplicial retraction homotopic to the identity. No fitted parameters, self-referential equations, or load-bearing self-citations appear; the argument relies only on local attachment rules and standard definitions of simplicial complexes and pseudomanifolds, which are independently verifiable and do not reduce the result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions and basic properties of finite simplicial complexes and pseudomanifolds together with the existence of a topological construction that produces the required retracts.

axioms (1)

standard math Standard definitions and properties of finite simplicial complexes and pseudomanifolds in algebraic topology
The theorem is stated in terms of these background notions.

pith-pipeline@v0.9.0 · 5322 in / 1216 out tokens · 75046 ms · 2026-05-08T02:42:29.311219+00:00 · methodology

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