# Comptes Rendus
Mathématique

Sumanta Das and Siddhartha Gadgil

## Surfaces of infinite-type are non-Hopfian

Volume 361 (2023), p. 1349-1356

Published online: 31 October 2023

https://doi.org/10.5802/crmath.504

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Geometry and Topology / Géométrie et Topologie

# Surfaces of infinite-type are non-Hopfian

*Les surfaces de type infini sont non-Hopfian*

Sumanta Das*, a and Siddhartha Gadgil a

**Abstract.** We show that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an oriented surface Σ is of finite-type if and only if every proper map $f: Σ \to Σ$ of degree one is homotopic to a homeomorphism.

**Résumé.** Nous montrons que les surfaces de type fini sont caractérisées par un analogue topologique de la propriété de Hopf. A savoir, une surface orientée Σ est de type fini si et seulement si toute application propre $f: Σ \to Σ$ de degré 1 est homotope à un homéomorphisme.

**2020 Mathematics Subject Classification.** 57K20, 55S37.

Manuscript received 3 January 2023, revised 17 March 2023, accepted 6 April 2023.

## 1. Introduction

All surfaces will be assumed to be connected and orientable throughout this note. We will say a surface is of *finite-type* if its fundamental group is finitely generated; otherwise, we will say it is of *infinite-type*.

Recall that a group $G$ is said to be *Hopfian* if every surjective homomorphism $\varphi: G \to G$ is an isomorphism. It is well known that a finitely generated free group is Hopfian, for instance, as a consequence of Grushko's theorem. On the other hand, a free group generated by an infinite set $S$ is not Hopfian as a surjective function $f: S \to S$ that is not injective extends to a surjective homeomorphism on the free group generated by $S$ which is not injective.

In this note, we show that there is an analogous characterization for orientable surfaces of *finite-type*. The natural topological analog of a surjective homomorphism is a proper map of degree one, and that of an isomorphism is a homotopy equivalence.

One-half of this characterization is classical, namely that any proper map of degree one from a surface of finite-type to itself is a homotopy equivalence. For instance, a theorem of Olum (see [2, Corollary 3.4]) says that every proper map of degree one between two oriented manifolds of the same dimension is $\pi_1$-surjective. Now, the fundamental group of any surface is residually finite (see [4]). Also, any finitely generated residually finite group is Hopfian. Thus, every degree one

<footnote><sup>a</sup> Department of Mathematics, Indian Institute of Science, Bangalore 560012, India</footnote>

<footnote>E-mails: sumantadas@iisc.ac.in, gadgil@iisc.ac.in</footnote>

<footnote>* Corresponding author.</footnote>
self map of a finite-type surface is a weak homotopy equivalence, hence a homotopy equivalence by Whitehead's theorem.

Our main result is that infinite-type surfaces are not Hopfian.

**Theorem 1.** Let $\Sigma$ be any infinite-type surface. Then there exists a proper map $f: \Sigma \to \Sigma$ of degree one such that $\pi_1(f): \pi_1(\Sigma) \to \pi_1(\Sigma)$ is not injective. In particular, $f$ is not a homotopy equivalence.

## 2. Background

A surface is a connected, orientable two-dimensional manifold without boundary and a *bordered surface* is a connected, orientable two-dimensional manifold wit non-empty boundary. A (possibly bordered) subsurface $\Sigma'$ of a surface $\Sigma$ is an embedded submanifold of codimension zero.

Let $\Sigma$ be a non-compact surface. A *boundary component* of $\Sigma$ is a nested sequence $P_1 \supseteq P_2 \supseteq \dots$ of open, connected subsets of $\Sigma$ such that the followings hold:

* the closure (in $\Sigma$) of each $P_n$ is non-compact,

* the boundary of each $P_n$ is compact, and

* for any subset $A$ with compact closure (in $\Sigma$), we have $P_n \cap A = \emptyset$ for all large $n$.

We say that two boundary components $P_1 \supseteq P_2 \supseteq \dots$ and $P'_1 \supseteq P'_2 \supseteq \dots$ of $\Sigma$ are *equivalent* if for any positive integer $n$ there are positive integers $k_n, \ell_n$ such that $P_{k_n} \subseteq P'_n$ and $P'_{\ell_n} \subseteq P_n$. For a boundary component $\mathcal{P} = P_1 \supseteq P_2 \supseteq \dots$, we let $[\mathcal{P}]$ to denote the equivalence class of $\mathcal{P}$.

The *space of ends* $\text{Ends}(\Sigma)$ of $\Sigma$ is the topological space having equivalence class of boundary components of $\Sigma$ as elements, i.e., as a set $\text{Ends}(\Sigma) := \{|\mathcal{P}| \mid \mathcal{P}$ is a boundary component}\}$; with the following topology: For any set $X$ with compact boundary, at first, define

$$X^{\dagger} := \{[\mathcal{P} = P_1 \supseteq P_2 \supseteq \dots] \mid X \supseteq P_n \supseteq P_{n+1} \supseteq \dots \text{ for some large } n\}.$$

Now, take the set of all such $X^\dagger$ as a basis for the topology of $\text{Ends}(\Sigma)$. The topological space $\text{Ends}(\Sigma)$ is compact, separable, totally disconnected, and metrizable, i.e., homeomorphic to a non-empty closed subset of the Cantor set.

For a boundary component $[\mathcal{P}]$ with $\mathcal{P} = P_1 \supseteq P_2 \supseteq \dots$, we say $[\mathcal{P}]$ is *planar* if $P_n$ are homeomorphic to open subsets $\mathbb{R}^2$ for all large $n$. Define $\text{Ends}_{\text{np}}(\Sigma) := \{[\mathcal{P}] : [\mathcal{P}] \text{ is not planar}\}$. Thus, $\text{Ends}_{\text{np}}(\Sigma)$ is a closed subset of $\text{Ends}(\Sigma)$. Also, define the *genus* of $\Sigma$ as $g(\Sigma) := \sup g(\mathcal{S})$, where $\mathcal{S}$ is a compact bordered subsurface of $\Sigma$.

**Theorem 2 (Kerékjártó's classification theorem [7, Theorem 1]).** Let $\Sigma_1, \Sigma_2$ be two non-compact surfaces. Then $\Sigma_1$ is homeomorphic to $\Sigma_2$ if and only if $g(\Sigma_1) = g(\Sigma_2)$, and there is a homeomorphism $\Phi: \text{Ends}(\Sigma_1) \to \text{Ends}(\Sigma_2)$ with $\Phi(\text{Ends}_{\text{np}}(\Sigma_1)) = \text{Ends}_{\text{np}}(\Sigma_2)$.

Let $\Sigma$ be a non-compact surface, and let $\mathcal{E}_{\text{np}}(\Sigma) \subseteq \mathcal{E}(\Sigma)$ be two closed, totally-disconnected subsets of $\mathbb{S}^2$ such that the pair $\text{Ends}_{\text{np}}(\Sigma) \subseteq \text{Ends}(\Sigma)$ is homeomorphic to the pair $\mathcal{E}_{\text{np}}(\Sigma) \subseteq \mathcal{E}(\Sigma)$. Consider a pairwise disjoint collection $\{D_i \subseteq \mathbb{S}^2 \setminus \mathcal{E}(\Sigma) : i \in \mathcal{A}\}$ of closed disks, where $0 \le |\mathcal{A}| \le g(\Sigma)$, such that the following holds: For $p \in \mathbb{S}^2$, any open neighborhood (in $\mathbb{S}^2$) of $p$ contains infinitely many $D_i$ if and only if $p \in \mathcal{E}_{\text{np}}(\Sigma)$. [7, Theorem 2] describes constructing such a collection of disks.

Now, let $M := (\mathbb{S}^2 \setminus \mathcal{E}(\Sigma)) \setminus \bigsqcup_{i \in \mathcal{A}} \text{int}(D_i)$ and $N := \bigsqcup_{i \in \mathcal{A}} S_{1,1}$, where $S_{1,1}$ is the genus one compact bordered surface with one boundary component. Define a non-compact surface $\Sigma_{\text{handle}}$ as follows: $\Sigma_{\text{handle}} := M \bigsqcup_{\partial M \equiv \partial N} N$. Then we have the following theorem.

**Theorem 3 (Richards' representation theorem [7, Theorems 2 and 3]).** The surface $\Sigma_{\text{handle}}$ is homeomorphic to $\Sigma$.
**3. Proof of Theorem 1**

Let $M$ and $N$ be two non-compact, oriented, connected, boundaryless smooth $n$-manifolds.
Then the singular cohomology groups with compact support $H_c^n(M; \mathbb{Z})$ and $H_c^n(N; \mathbb{Z})$ are infinite
cyclic with preferred generators $[M]$ and $[N]$. If $f: M \to N$ is a proper map then the degree of $f$ is
the unique integer $\deg(f)$ defined as follows: $H_c^n(f)([N]) = \deg(f) \cdot [M]$. Note that $\deg$ is proper-
homotopy invariant and multiplicative. See [2, Section 1] for more details.

We will use the following well-known characterization of degree.

**Lemma 4 ([2, Lemma 2.1b].).** Let $f: M \to N$ be a proper map between two non-compact, oriented, connected, boundaryless smooth $n$-manifolds. Let $D$ be a smoothly embedded closed disk in $N$ and suppose $f^{-1}(D)$ is a smoothly embedded closed disk in $M$ such that $f$ maps $f^{-1}(D)$ homeomorphically onto $D$. Then $\deg(f) = +1$ or $-1$ according as $f|f^{-1}(D) \to D$ is orientation-preserving or orientation-reversing.

We will prove Theorem 1 by considering the following three cases:

(1) $\Sigma$ has infinite genus.

(2) $\Sigma$ has finite genus and the set of isolated points $\mathcal{I}(\Sigma)$ of $\mathcal{C}(\Sigma)$ is finite.

(3) $\Sigma$ has finite genus and the set of isolated points $\mathcal{J}(\Sigma)$ of $\mathcal{C}(\Sigma)$ is infinite.

**Remark 5.** If $\Sigma$ is an infinite-type surface of a finite genus, then $\mathcal{C}(\Sigma)$ is an infinite set.

Our first result proves Theorem 1 in the case with infinite genus.

**Theorem 6.** Let $\Sigma$ be a surface of the infinite genus. Then there exists a degree one map $f: \Sigma \to \Sigma$ which is not $\pi_1$-injective.

**Proof.** Since $\Sigma$ has infinite genus, there exists a compact bordered subsurface $\mathcal{S} \subset \Sigma$ such that $\mathcal{S}$ has genus one and one boundary component. Define $\Sigma' := \Sigma/\mathcal{S}$ be the quotient of $\Sigma$ with $\mathcal{S}$ pinched to a point and let $q: \Sigma \to \Sigma'$ be the quotient map. Thus, $\Sigma'$ is also an infinite genus surface. Further, there are compact sets in $K \subset \Sigma$ and $K' \subset \Sigma'$ whose complements are homeomorphic, so the pair $(\mathcal{C}(\Sigma), \mathcal{E}_{np}(\Sigma))$ is homeomorphic to the pair $(\mathcal{C}(\Sigma'), \mathcal{E}_{np}(\Sigma'))$. Hence, by Theorem 2, there is a homeomorphism $\varphi: \Sigma' \to \Sigma$.

Let $f: \Sigma \to \Sigma$ be the composition $f = \varphi \circ q$. By Lemma 4, the quotient map $q: \Sigma \to \Sigma'$ is of degree $\pm 1$. Thus, $\deg(f) = \pm 1$ as homeomorphisms have degree $\pm 1$. Notice that $f$ sends $\partial\mathcal{S}$ to a point. But $\partial\mathcal{S}$ does not bound any disk in $\Sigma$, i.e., $\partial\mathcal{S}$ represents a primitive element of $\pi_1(\Sigma)$, see [1, Theorem 1.7. and Theorem 4.2.]. Hence, $f$ is not $\pi_1$-injective. If $\deg(f) = 1$, then we are done. Otherwise, we replace $f$ by $f \circ f$ to get a map that has degree one and is not injective on $\pi_1$. $\square$

For the remaining two cases, we use a map from the sphere to the sphere, which has degree $\pm 1$ but with some disks identified. We will replace these disks with appropriate surfaces to get $\Sigma$.

**Lemma 7.** There exist pairwise disjoint closed disks $\mathcal{D}_0, \mathcal{D}_1 \subseteq \mathbb{S}^2$ and a map $f: \mathbb{S}^2 \to \mathbb{S}^2$ such that the following hold:

• $f^{-1}(\mathcal{D}_0) = \mathcal{D}_0$ and $f|_{\mathcal{D}_0}: \mathcal{D}_0 \to \mathcal{D}_0$ is the identity map.

• $f^{-1}(\mathcal{D}_1)$ is the union of pairwise-disjoint closed disks $\mathcal{D}_{1,1}, \mathcal{D}_{1,2}$, and $\mathcal{D}_{1,3}$ in $\mathbb{S}^2$; and
$f|_{\mathcal{D}_{1k}}: \mathcal{D}_{1k} \to \mathcal{D}_1$ is a homeomorphism for each $k \in \{1, 2, 3\}$.

Further, there is a loop $\gamma$ in $\mathbb{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3})$ which is not homotopically trivial in $\mathbb{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3})$ but such that $f(\gamma)$ is null-homotopic in $\mathbb{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_1)$.

**Proof.** For each $k \in \{0, 1, 2, 3\}$, choose $(a_k, b_k) \in \mathbb{R}^2$ such that if we define $\mathcal{B}_k := \{(x, y) \in \mathbb{R}^2 : (x - a_k)^2 + (y - b_k)^2 \le 1\}$, then $\{\mathcal{B}_0, \mathcal{B}_1, \mathcal{B}_2, \mathcal{B}_3\}$ is a pairwise-disjoint collection of closed disks.
Define $X := \mathbb{S}^2 \setminus \bigcup_{i=0}^3 \text{int}(\mathcal{B}_i)$ and $Y := \mathbb{S}^2 \setminus \bigcup_{i=0}^1 \text{int}(\mathcal{B}_i)$. Next, define a map $f: \partial X \to Y$ as follows:

* $f|_{\partial \mathcal{B}_k}: \partial \mathcal{B}_k \to \partial \mathcal{B}_k$ is the identity map for each $k \in \{0, 1\}$;

* $f|_{\partial \mathcal{B}_2}: \partial \mathcal{B}_2 \to \partial \mathcal{B}_1$ is defined as $f(x, y) := (-x + a_2 + a_1, y - b_2 + b_1)$ for all $(x, y) \in \partial \mathcal{B}_2$.

* $f|_{\partial \mathcal{B}_3}: \partial \mathcal{B}_3 \to \partial \mathcal{B}_1$ is defined as $f(x, y) := (x - a_3 + a_1, y - b_3 + b_1)$ for all $(x, y) \in \partial \mathcal{B}_3$.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** The four-holed sphere X by attaching a 2-cell.</figcaption></figure>

For each $k \in \{0, 1, 2\}$, let $\gamma_k: [0, 1] \to X$ be an embedding such that $\text{im}(\gamma_k) \cap \partial X$ consists of $\gamma_k(0) = (a_k + 1, b_k) \in \partial \mathcal{B}_k$ and $\gamma_k(1) = (a_{k+1} - 1, b_{k+1}) \in \partial \mathcal{B}_{k+1}$.

Define $\Gamma_0: [0, 1] \to Y$ as $\Gamma_0(t) := \gamma_0(t)$ for all $t \in [0, 1]$. Let $\Gamma_1, \Gamma_2: [0, 1] \to Y$ be the constant loops based at the points $(a_1 + 1, b_1) \in \partial Y$ and $(a_1 - 1, b_1) \in \partial Y$, respectively.

Next, define $X^{(1)} := \partial X \cup \text{im}(\gamma_0) \cup \text{im}(\gamma_1) \cup \text{im}(\gamma_2)$. Extend $f: \partial X \to Y$ to a map $X^{(1)} \to Y$, which we again denote by $f: X^{(1)} \to Y$, by mapping $\gamma_0$ onto $\Gamma_0$ by the identity, and, for each $k = 1, 2$, mapping $\gamma_k$ to the constant loop $\Gamma_k$.

Let $\theta_0$ (resp. $\theta_3$) be the simple loop that traverses $\partial \mathcal{B}_0$ (resp. $\partial \mathcal{B}_3$) in the counter-clockwise direction starting from $(a_0 + 1, b_0)$ (resp. $(a_3 - 1, b_3)$).

Let $\theta_{1,l}$ (resp. $\theta_{1,u}$) be the simple arc that traverses $\partial \mathcal{B}_1 \cap \{y \le b_1\}$ (resp. $\partial \mathcal{B}_1 \cap \{y \ge b_1\}$) counterclockwise direction. Similarly, define $\theta_{2,l}$ and $\theta_{2,u}$.

Now, $X \cong X^{(1)} \cup_{\varphi} \mathbb{D}^2$, (see Figure 1) where the attaching map $\varphi: \mathbb{S}^1 \to X^{(1)}$ can be described as
$$ \varphi := \theta_0 * \gamma_0 * \theta_{1,l} * \gamma_1 * \theta_{2,l} * \gamma_2 * \theta_3 * \bar{\gamma}_2 * \theta_{2,u} * \bar{\gamma}_1 * \theta_{1,u} * \bar{\gamma}_0. $$

Notice that $f(\gamma_1) = \Gamma_1$ and $f(\gamma_2) = \Gamma_2$ are constant loops. Also, as in Figure 2, $\overline{f \circ \theta_{1,l}} = f \circ \theta_{2,l}$ and $\overline{f \circ \theta_{1,u}} = f \circ \theta_{2,u}$. Thus, $f \circ \varphi$ is homotopic to $(f \circ \theta_0) * \Gamma_0 * (f \circ \theta_3) * \bar{\Gamma}_0$.

If $r: Y \cong \mathbb{S}^1 \times [0, 1] \to \mathbb{S}^1$ is the projection then $r \circ f \circ \theta_0$ and $r \circ f \circ \theta_3$ traverse $\mathbb{S}^1$ in opposite directions. Since $r$ is a strong deformation retract, $(f \circ \theta_0) * \Gamma_0 * (f \circ \theta_3) * \bar{\Gamma}_0$, and hence $f \circ \varphi$ is null-homotopic. Now, the null-homotopic map $f \circ \varphi: \mathbb{S}^1 \to Y$ can be extended to a map $\mathbb{D}^2 \to Y$. Thus $f: X^{(1)} \to Y$ can be extended to a map $X \cong X^{(1)} \cup_{\varphi} \mathbb{D}^2 \to Y$, which will be again denoted by $f: X \to Y$.

Note that every homeomorphism $\mathbb{S}^1 \to \mathbb{S}^1$ can be extended to a homeomorphism $\mathbb{D}^2 \to \mathbb{D}^2$ naturally. Thus, we can extend $f: X \to Y$ to a map $\mathbb{S}^2 \to \mathbb{S}^2$, which will be again denoted by $f: \mathbb{S}^2 \to \mathbb{S}^2$. Let $\mathcal{D}_0$ (resp. $\mathcal{D}_1$) be any closed disk, which is contained in $\text{int}(\mathcal{B}_0)$ (resp. $\text{int}(\mathcal{B}_1)$).

Finally, observe that if $\gamma = \theta_{1u} * \theta_{1l} * \gamma_1 * \theta_{2l} * \theta_{2u} * \bar{\gamma}_1$, then $\gamma$ is a loop in $\mathbb{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3})$ which is not homotopically trivial in $\mathbb{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3})$, but such that $f(\gamma)$ is null-homotopic in $\mathbb{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_1)$, as claimed. $\square$

We now prove Theorem 1 in the two remaining cases, in both of which we have a finite genus surface. Note that for a finite genus surface, all ends are planar, so in applying Theorem 2, it suffices to consider the genus and the space of ends.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** The map on the $X^{(1)}$</figcaption></figure>

**Theorem 8.** Let $\Sigma$ be a finite genus infinite-type surface such that $\mathcal{E}(\Sigma)$ has finitely many isolated points. Then there is a degree one map $f: \Sigma \to \Sigma$ which is not $\pi_1$-injective.

**Proof.** Let $\mathcal{I}(\Sigma)$ be the set of all isolated points of $\mathcal{E}(\Sigma)$, let $k \in \mathbb{N} \cup \{0\}$ be the cardinality of $\mathcal{I}(\Sigma)$, and let $g$ be the genus of $\Sigma$. Then $\mathcal{C}(\Sigma) := \mathcal{E}(\Sigma) \setminus \mathcal{I}(\Sigma)$ is a non-empty, perfect, compact, totally-disconnected, metrizable space as it is infinite (by Remark 5) and has no isolated points. Thus $\mathcal{C}(\Sigma)$ is a Cantor space (see [5, Theorem 8 of Chapter 12]).

Let $\mathcal{D}_0, \mathcal{D}_1, \mathcal{D}_{1,1}, \mathcal{D}_{1,2}, \mathcal{D}_{1,3} \subseteq \mathcal{S}^2$, $f: \mathcal{S}^2 \to \mathcal{S}^2$, and let $\gamma$ be as in the conclusion of Lemma 7. Let $C_1 \subset \text{int}(\mathcal{D}_1)$ be a subset homeomorphic to the Cantor set and let $\mathcal{I} \subset \text{int}(\mathcal{D}_0)$ be a set consisting of $k$ points (hence homeomorphic to $\mathcal{I}(\Sigma)$). Let $C_{1,j} = f^{-1}(C_1) \cap \mathcal{D}_{1,j}$ for $j = 1,2,3$. Note that each $C_{1,j}$ is homeomorphic to the Cantor set. See Figure 3.

As $f^{-1}(\mathcal{D}_0) = \mathcal{D}_0$ and $f|_{\mathcal{D}_0}: \mathcal{D}_0 \to \mathcal{D}_0$ is the identity map, we can say that $f^{-1}(\mathcal{I}) = \mathcal{I}$. Let $\Sigma_1$ be the surface obtained from $\mathcal{S}^2 \setminus (\mathcal{I} \cup C_1)$ by attaching $g$ handles along disjoint disks $\Delta_k \subset \text{int}(\mathcal{D}_0) \setminus \mathcal{I}$, $1 \le k \le g$ and let $\Sigma_2$ be the surface obtained from $\mathcal{S}^2 \setminus (\mathcal{I} \cup C_{1,1} \cup C_{1,2} \cup C_{1,3})$ by attaching $g$ handles along the (same) disks $\Delta_k$, $1 \le k \le g$. Then $f$ induces a proper map, which we also call $f$, from $\Sigma_2$ to $\Sigma_1$. By Lemma 4, $\deg(f) = \pm 1$.

Further, we claim that $f: \Sigma_2 \to \Sigma_1$ is not injective on $\pi_1$. Namely, the fundamental group of $\Sigma_2$ is the amalgamated free product of four groups, one of which is $\pi_1(\mathcal{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3}))$. As $\gamma$ is not homotopic to the trivial loop in $\mathcal{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3})$, and components of an amalgamated free product inject, $\gamma$ is not homotopic to the trivial loop in $\Sigma_2$. However, $f(\gamma)$ is homotopic to the trivial loop in $\mathcal{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_1)$ and hence in $\Sigma_1$. Hence, $f$ is not injective on $\pi_1$.

Both $\Sigma_1$ and $\Sigma_2$ have genus the same as $\Sigma$, and the space of ends homeomorphic to that of $\Sigma$ (as a finite disjoint union of Cantor spaces is a Cantor space by the universality of the Cantor set) with all ends planar. Hence, by Theorem 2 both $\Sigma_1$ and $\Sigma_2$ are homeomorphic to $\Sigma$.

Identifying $\Sigma_1$ and $\Sigma_2$ with $\Sigma$ by homeomorphisms, we get a proper map $f: \Sigma \to \Sigma$ which is not injective on $\pi_1$. As homeomorphisms have degree $\pm 1$, it follows that $\deg(f) = \pm 1$. Replacing $f$ by $f \circ f$ if necessary, we obtain a proper map of degree one that is not injective on $\pi_1$. $\square$

**Theorem 9.** Let $\Sigma$ be a finite genus surface such that $\mathcal{E}(\Sigma)$ has infinitely many isolated points. Then there is a degree one map $f: \Sigma \to \Sigma$ which is not $\pi_1$-injective.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** A non $\pi_1$-injective degree $\pm 1$ map $f: \Sigma \to \Sigma$, where $g=3$ and $|\mathcal{I}|=4$.</figcaption></figure>

**Proof.** Let $\mathcal{I}(\Sigma)$ be the set of all isolated points of $\mathcal{E}(\Sigma)$ and let $g$ be the genus of $\Sigma$. Let $\mathcal{D}_0, \mathcal{D}_1, \mathcal{D}_{1,1}, \mathcal{D}_{1,2}, \mathcal{D}_{1,3} \subseteq \mathcal{S}^2$, $f: \mathcal{S}^2 \to \mathcal{S}^2$, and let $\gamma$ be as in the conclusion of Lemma 7. Let $\mathcal{E}$ be a subset of $\text{int}(\mathcal{D}_0)$ such that $\mathcal{E}$ is homeomorphic to $\mathcal{E}(\Sigma)$. Also, let $p_1 \in \text{int}(\mathcal{D}_1)$ and $p_{1,i} \in \text{int}(\mathcal{D}_{1,i})$, $i=1,2,3$ be points such that $f(p_{1,i}) = p_1$ for each $i=1,2,3$. See Figure 4.

Recall that $f^{-1}(\mathcal{D}_0) = \mathcal{D}_0$ and $f|_{\mathcal{D}_0}: \mathcal{D}_0 \to \mathcal{D}_0$ is the identity map. Thus $f^{-1}(\mathcal{E}) = \mathcal{E}$. Now, let $\Sigma_1$ be the surface obtained from $\mathcal{S}^2 \setminus (\mathcal{E} \cup \{p_1\})$ by attaching $g$ handles along disjoint disks $\Delta_k \subset \text{int}(\mathcal{D}_0) \setminus \mathcal{I}$, $1 \le k \le g$ and let $\Sigma_2$ be the surface obtained from $\mathcal{S}^2 \setminus (\mathcal{E} \cup \{p_{1,1}, p_{1,2}, p_{1,3}\})$ by attaching $g$ handles along the same disks $\Delta_k$, $1 \le k \le g$. Then $f$ induces a proper map, which we also call $f$, from $\Sigma_2$ to $\Sigma_1$. By Lemma 4, $\text{deg}(f) = \pm 1$.

Further, we claim that $f: \Sigma_2 \to \Sigma_1$ is not injective on $\pi_1$. Namely, the fundamental group of $\Sigma_2$ is the amalgamated free product of four groups, one of which is $\pi_1(\mathcal{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3}))$. As $\gamma$ is not homotopic to the trivial loop in $\mathcal{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_{1,1} \cup \mathcal{D}_{1,2} \cup \mathcal{D}_{1,3})$, and components of an amalgamated free product inject, $\gamma$ is not homotopic to the trivial loop in $\Sigma_2$. However, $f(\gamma)$ is homotopic to the trivial loop in $\mathcal{S}^2 \setminus \text{int}(\mathcal{D}_0 \cup \mathcal{D}_1)$ and hence in $\Sigma_1$. Hence, $f$ is not injective on $\pi_1$.

Both $\Sigma_1$ and $\Sigma_2$ have genus the same as $\Sigma$ and, by Lemma 10 below, $\mathcal{E}(\Sigma_1)$ and $\mathcal{E}(\Sigma_2)$ are homeomorphic to $\mathcal{E}(\Sigma)$. Further, all ends of $\Sigma$, $\Sigma_1$ and $\Sigma_2$ are planar. Hence, by Theorem 2 both $\Sigma_1$ and $\Sigma_2$ are homeomorphic to $\Sigma$.

Identifying $\Sigma_1$ and $\Sigma_2$ with $\Sigma$ by homeomorphisms, we get a proper map $f: \Sigma \to \Sigma$ which is not injective on $\pi_1$. As homeomorphisms have degree $\pm 1$, it follows that $\text{deg}(f) = \pm 1$. Replacing $f$ by $f \circ f$ if necessary, we obtain a proper map of degree one that is not injective on $\pi_1$. $\square$
<figure><img src="image_6.png" /><figcaption>**Figure 4.** A non $\pi_1$-injective degree $\pm 1$ map $f: \Sigma \to \Sigma$, where $g=5$ and $\mathcal{I}$ is an infinite set.</figcaption></figure>

**Lemma 10.** Let $\mathcal{E}$ be a closed totally disconnected subset of $\mathbb{S}^2$. Let $\mathcal{I}$ be the set of all isolated points of $\mathcal{E}$. Assume $\mathcal{I}$ is infinite. If $\mathcal{F}$ is a finite subset of $\mathbb{S}^2 \setminus \mathcal{E}$, then $\mathcal{E} \cup \mathcal{F}$ is homeomorphic to $\mathcal{E}$.

**Proof.** Let $\mathcal{A} := \{a_1, a_2, \dots\}$ be a subset of $\mathcal{I}$ such that $a_n \to \ell \in \mathcal{E}$ ($\mathcal{A}$ exists as $\mathcal{E}$ is compact and infinite). Define $\mathcal{B} := \mathcal{A} \cup \mathcal{F}$. Write $\mathcal{B}$ as $\mathcal{B} = \{b_1, b_2, \dots\}$. Then the map $g: \mathcal{E} \cup \mathcal{F} \to \mathcal{E}$ defined by

$$g(z) := \begin{cases} z & \text{if } z \in (\mathcal{E} \cup \mathcal{F}) \setminus \mathcal{B}, \\ a_n & \text{if } z = b_n \in \mathcal{B}, \end{cases}$$

is a homeomorphism. To prove this, note that $g$ is a bijection from a compact space to a Hausdorff space, so it suffices to show that $g$ is continuous. But observe that $g$ restricted to the closed set $(\mathcal{E} \cup \mathcal{F}) \setminus \mathcal{B}$ is the identity, so $g$ is continuous on $(\mathcal{E} \cup \mathcal{F}) \setminus \mathcal{B}$. Also $g$ restricted to the closed set $\mathcal{B} \cup \{\ell\}$ is continuous as $b_n \to \ell$ and $g(b_n) = a_n \to \ell = g(\ell)$, and all other points of $\mathcal{B} \cup \{\ell\}$ are isolated. Thus $g$ is continuous, as required. $\square$

**Remark 11.** In the paper [3], the authors have proved that for every infinite-type surface $\Sigma$, there exists a subsurface homeomorphic to $\Sigma$ such that the inclusion map is not homotopic to a homeomorphism. As our surfaces are connected, this type of inclusion map can't be proper because of the following two facts:

* Any injective map between two boundaryless topological manifolds of the same dimension is an open map. This follows from the invariance of domain.

* Any proper map between two topological manifolds is a closed map, as manifolds are compactly generated spaces, see [6].
Also, notice that all our results are related to proper maps.

## Acknowledgements

The first author is supported by a fellowship from the National Board for Higher Mathematics. We would like to thank Hugo Parlier for telling us about the excellent result of [3]. We are grateful to the anonymous referee for his careful reading of the paper and his comments and suggestions, which helped considerably in improving the manuscript.

## References

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[2] —, "The degree of a map", *Proc. Lond. Math. Soc.* **16** (1966), p. 369-383.

[3] F. Fanoni, T. Ghaswala, A. McLeay, "Homeomorphic subsurfaces and the omnipresent arcs", *Ann. Henri Lebesgue* **4** (2021), p. 1565-1593.

[4] J. Hempel, "Residual finiteness of surface groups", *Proc. Am. Math. Soc.* **32** (1972), p. 323.

[5] E. E. Moise, *Geometric topology in dimensions 2 and 3*, Graduate Texts in Mathematics, vol. 47, Springer, 1977, x+262 pages.

[6] R. S. Palais, "When proper maps are closed", *Proc. Am. Math. Soc.* **24** (1970), p. 835-836.

[7] I. Richards, "On the classification of noncompact surfaces", *Trans. Am. Math. Soc.* **106** (1963), p. 259-269.