On the fundamental groups of perforated surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911594707943424 |
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| author | Gulati, Khushbu Sankaran, Parameswaran |
| author_facet | Gulati, Khushbu Sankaran, Parameswaran |
| contents | A perforated surface is the complement $\mathringΣ:=Σ\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $Σ$. It is known that the topological type of $Σ\setminus A$ 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 $\mathring Σ$ arises from a covering of a surface $Σ'$ such that $\mathringΣ\cong \mathringΣ'$. We show that the fundamental group of perforated surfaces are large.
We also show that the fundamental groups of $\mathring Σ$, the Sierpiński curve and the Menger curve are not Hopfian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the fundamental groups of perforated surfaces Gulati, Khushbu Sankaran, Parameswaran Geometric Topology General Topology 54F50, 57M07, 20F99, 20E26 A perforated surface is the complement $\mathringΣ:=Σ\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $Σ$. It is known that the topological type of $Σ\setminus A$ 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 $\mathring Σ$ arises from a covering of a surface $Σ'$ such that $\mathringΣ\cong \mathringΣ'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring Σ$, the Sierpiński curve and the Menger curve are not Hopfian. |
| title | On the fundamental groups of perforated surfaces |
| topic | Geometric Topology General Topology 54F50, 57M07, 20F99, 20E26 |
| url | https://arxiv.org/abs/2604.13544 |