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arxiv: 2604.13544 · v1 · submitted 2026-04-15 · 🧮 math.GT · math.GN

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On the fundamental groups of perforated surfaces

Khushbu Gulati , Parameswaran Sankaran
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Pith reviewed 2026-05-10 12:50 UTC · model grok-4.3

classification 🧮 math.GT math.GN
keywords perforated surfacesfundamental groupscovering spacesclassification of surfacesHopfian groupsSierpiński curveMenger curvelarge groups
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The pith

Any connected covering of a perforated surface arises from a covering of an ordinary surface whose perforated version is homeomorphic to the original.

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

The paper classifies perforated surfaces, which are ordinary surfaces with a countable dense set of points removed, by applying the classical classification theorem for surfaces. It proves that the homeomorphism type is independent of the particular dense set chosen and that every connected covering space of such a perforated surface comes from a covering of a related surface. The authors further establish that the fundamental groups of these perforated surfaces are large and that neither they nor the fundamental groups of the Sierpiński curve and the Menger curve are Hopfian.

Core claim

A perforated surface is the complement of a countable dense subset in a connected paracompact surface. Using the classification theorem for surfaces, the authors classify perforated surfaces and show that any connected covering of a perforated surface arises from a covering of a surface such that the perforated versions are homeomorphic. They prove that the fundamental groups of perforated surfaces are large and that the fundamental groups of perforated surfaces, the Sierpiński curve, and the Menger curve are not Hopfian.

What carries the argument

The independence of the homeomorphism type of the surface minus the countable dense set from the specific choice of that set, which permits the classical surface classification to determine both the perforated surfaces and their covering spaces.

If this is right

  • The homeomorphism type of any perforated surface is determined by the genus, orientability, and boundary components of the original surface.
  • Covering spaces of perforated surfaces correspond directly to covering spaces of the surfaces before perforation.
  • The fundamental groups being large constrains their subgroup structure in a manner inherited from surface groups.
  • The non-Hopfian property implies the existence of proper normal subgroups whose quotients are isomorphic to the group itself.

Where Pith is reading between the lines

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

  • The correspondence between coverings suggests that homotopy-theoretic invariants of perforated surfaces can be read off from those of ordinary surfaces.
  • Similar non-Hopfian behavior may appear in fundamental groups of other one-dimensional continua obtained by dense removals.
  • The classification provides a concrete way to distinguish homeomorphism types among different perforated surfaces without choosing specific dense sets.

Load-bearing premise

The topological type of a surface minus a countable dense subset is independent of which particular dense subset is removed.

What would settle it

An explicit example of a connected covering space of some perforated surface that cannot be obtained by removing a countable dense set from any covering space of an ordinary surface.

Figures

Figures reproduced from arXiv: 2604.13544 by Khushbu Gulati, Parameswaran Sankaran.

Figure 1
Figure 1. Figure 1: An orientable surface Σ with Ep ∼= C \ {0}, Enp ∼= { 1 n | n ∈ N} ∪ {0}, and 0 ∈ Ep. Each dashed line (in red) represents deletion of a copy of C [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The space S = W n∈N Sn. The image of the Sierpi´nski curve is courtesy Wikipedia: https://en.wikipedia.org/wiki/Sierpi´nski carpet [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
read the original abstract

A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring \Sigma$ arises from a covering of a surface $\Sigma'$ such that $\mathring\Sigma\cong \mathring\Sigma'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring \Sigma$, the Sierpi\'nski curve and the Menger curve are not Hopfian.

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.

Referee Report

3 major / 2 minor

Summary. The paper defines a perforated surface as the complement ṁΣ = Σ ∖ A of a countable dense subset A in a connected paracompact surface Σ, notes that its homeomorphism type is independent of the choice of A, and records that every such space is one-dimensional, connected, locally path-connected, and not semi-locally simply connected at any point. It states a classification theorem for perforated surfaces obtained by invoking the classical classification theorem for surfaces, proves that every connected covering of ṁΣ arises from a covering of some surface Σ′ with ṁΣ ≅ ṁΣ′, shows that the fundamental groups of perforated surfaces are large, and shows that the fundamental groups of ṁΣ, the Sierpiński curve, and the Menger curve are not Hopfian.

Significance. If the central claims are established, the work would extend the classical surface classification and covering theory to a class of dense-punctured one-dimensional spaces, yielding concrete information on their fundamental groups (largeness and non-Hopfian property) with direct consequences for the Sierpiński and Menger curves. The results would also supply a covering correspondence that reduces questions about these non-manifold spaces to questions about ordinary surfaces.

major comments (3)
  1. [classification theorem statement] The classification theorem for perforated surfaces is obtained by direct appeal to the classical classification theorem for surfaces (abstract and the section stating the classification). Standard surface classification theorems require local Euclidean structure or manifold hypotheses, yet the abstract explicitly records that perforated surfaces fail to be semi-locally simply connected at any point and are one-dimensional; an explicit reduction or transfer argument showing how the manifold theorem applies to these spaces is therefore required and is load-bearing for all subsequent claims.
  2. [covering correspondence] The covering correspondence (any connected covering of ṁΣ arises from a covering of some Σ′ with ṁΣ ≅ ṁΣ′) is asserted after the classification. Because the spaces are not manifolds, the lifting and correspondence arguments must be verified directly rather than inherited from the manifold case; without this verification the claim remains unsubstantiated and affects the later statements on fundamental groups.
  3. [fundamental group properties] The assertions that π₁(ṁΣ) is large and that the fundamental groups of ṁΣ, the Sierpiński curve, and the Menger curve are not Hopfian rest on the classification and covering results. If the transfer step from manifold classification is not rigorously justified, these group-theoretic conclusions require independent justification or an alternative route that does not rely on the manifold theorem.
minor comments (2)
  1. The statement 'It is known that the topological type of Σ ∖ A is independent of the choice of A' appears without a reference; a precise citation to the source of this fact should be supplied.
  2. Notation such as ṁΣ for the perforated surface and the distinction between Σ and ṁΣ should be introduced with a short clarifying sentence in the introduction for readers outside the immediate area.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the points where additional justification is required. We address each major comment below and will incorporate the necessary clarifications and proofs in the revised manuscript.

read point-by-point responses

  1. Referee: [classification theorem statement] The classification theorem for perforated surfaces is obtained by direct appeal to the classical classification theorem for surfaces (abstract and the section stating the classification). Standard surface classification theorems require local Euclidean structure or manifold hypotheses, yet the abstract explicitly records that perforated surfaces fail to be semi-locally simply connected at any point and are one-dimensional; an explicit reduction or transfer argument showing how the manifold theorem applies to these spaces is therefore required and is load-bearing for all subsequent claims.

    Authors: We agree that the manuscript invokes the classical surface classification without spelling out the transfer in sufficient detail. Although the homeomorphism type of ṁΣ is independent of A and determined by the invariants of the underlying paracompact surface Σ, an explicit argument is needed. In the revision we will add a short subsection that records the relevant classification invariants (orientability, genus, number of ends) and shows that removing a countable dense subset preserves them, thereby justifying the direct appeal to the manifold theorem. revision: yes

  2. Referee: [covering correspondence] The covering correspondence (any connected covering of ṁΣ arises from a covering of some Σ′ with ṁΣ ≅ ṁΣ′) is asserted after the classification. Because the spaces are not manifolds, the lifting and correspondence arguments must be verified directly rather than inherited from the manifold case; without this verification the claim remains unsubstantiated and affects the later statements on fundamental groups.

    Authors: We concur that the non-manifold character of perforated surfaces requires a self-contained verification of the covering correspondence. The current text sketches the construction but does not supply the necessary lifting lemmas. The revised version will contain a direct proof that every connected covering of ṁΣ is obtained by lifting a covering of a suitable surface Σ′ and then excising the preimage of the dense set, using only local path-connectedness and the definition of perforated surfaces. revision: yes

  3. Referee: [fundamental group properties] The assertions that π₁(ṁΣ) is large and that the fundamental groups of ṁΣ, the Sierpiński curve, and the Menger curve are not Hopfian rest on the classification and covering results. If the transfer step from manifold classification is not rigorously justified, these group-theoretic conclusions require independent justification or an alternative route that does not rely on the manifold theorem.

    Authors: Once the classification and covering correspondence are placed on a rigorous footing, the largeness and non-Hopfian statements follow from the corresponding properties of surface groups and from the existence of surjective but non-injective endomorphisms induced by the coverings. To strengthen the paper independently of the transfer, we will also insert a short direct construction of a non-injective surjective endomorphism for the fundamental groups of the Sierpiński and Menger curves, using their explicit presentations as inverse limits. revision: partial

Circularity Check

0 steps flagged

No circularity; external classification theorem invoked without reduction to self-inputs

full rationale

The derivation begins from the stated independence of topological type of Σ∖A from choice of countable dense A (presented as known), then applies the classical surface classification theorem to obtain a classification of perforated surfaces. The covering correspondence is shown by exhibiting that any connected cover of ṁΣ arises from a cover of some Σ' with ṁΣ ≅ ṁΣ'. Fundamental group properties (large, non-Hopfian, including for Sierpiński and Menger curves) are derived as consequences. No step matches self-definitional, fitted-input-renamed-as-prediction, or load-bearing self-citation patterns; the argument is not equivalent to its inputs by construction and relies on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the classical classification theorem for surfaces and standard facts about paracompact surfaces and covering spaces; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)
  • standard math Classification theorem for surfaces
    Invoked to obtain the classification of perforated surfaces.
  • standard math Topological properties of paracompact surfaces and their coverings
    Background facts used throughout the arguments.

pith-pipeline@v0.9.0 · 5455 in / 1291 out tokens · 28914 ms · 2026-05-10T12:50:29.069579+00:00 · methodology

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Reference graph

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