On the failure of the first \v{C}ech homotopy group to register geometrically relevant fundamental group elements

TL;DR

This paper constructs a space where the canonical homomorphism from the fundamental group to the cech homotopy group is not injective, despite the space having many properties that typically relate these groups.

Contribution

It provides a counterexample demonstrating the failure of the first cech homotopy group to detect certain fundamental group elements, challenging assumptions about their relationship.

Findings

The space onstructed has a non-injective canonical homomorphism.

emonstrates the failure of cech homotopy group to register all fundamental group elements.

The space exhibits properties like being homotopically Hausdorff and having a simply connected generalized covering space.

Abstract

We construct a space $\mathbb{P}$ for which the canonical homomorphism $\pi_1(\mathbb{P},p) \rightarrow \check{\pi}_1(\mathbb{P},p)$ from the fundamental group to the first \v{C}ech homotopy group is not injective, although it has all of the following properties: (1) $\mathbb{P}\setminus\{p\}$ is a 2-manifold with connected non-compact boundary; (2) $\mathbb{P}$ is connected and locally path connected; (3) $\mathbb{P}$ is strongly homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5) $\mathbb{P}$ is 1-UV$_0$; (6) $\mathbb{P}$ admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) $\pi_1(\mathbb{P},p)$ naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) $\pi_1(\mathbb{P},p)$ is locally free.

Equations

$${\mathbb{P}}_1^{+}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$d_n\circ {\ell }_n={\rho }_{n,1}\cdot {\ell }_{2n}\cdot {\rho }_{n,1}^{-}\cdot {\rho }_{n,2}\cdot {\ell }_{2n+1}\cdot {\rho }_{n,2}^{-}.$$

$$[{\ell }_n]=[{\rho }_{n,1}][{\ell }_{2n}][{\rho }_{n,1}{]}^{-1}[{\rho }_{n,2}][{\ell }_{2n+1}][{\rho }_{n,2}{]}^{-1}\in {\pi }_1(\mathbb{P},b_0).$$

$$\{[{\rho }_{i,j}]\mid 1⩽i⩽m-1,1⩽j⩽2\}\cup \{[{\ell }_i]\mid m⩽i⩽k\}.$$

$$\mathcal{N}$$

$${\mathbb{H}}_n^{+}=\underset{⟵}{\lim }\left({\mathbb{H}}_{n,n-1}^{+}\stackrel{r_{n,n-1}}{⟵}{\mathbb{H}}_{n,n}^{+}\stackrel{r_{n,n}}{⟵}{\mathbb{H}}_{n,n+1}^{+}\stackrel{r_{n,n+1}}{⟵}\cdots \right)\approx \mathbb{H},$$

$$((r_{n,k}{)}_{\#}{)}_{k⩾n-1}:{\pi }_1({\mathbb{H}}_n^{+},b_0)\to \underset{⟵}{\lim }\left({\pi }_1({\mathbb{H}}_{n,n-1}^{+},b_0)\stackrel{(r_{n,n-1}{)}_{\#}}{⟵}{\pi }_1({\mathbb{H}}_{n,n}^{+},b_0)\stackrel{(r_{n,n}{)}_{\#}}{⟵}\cdots \right)$$

$$\underset{⟶}{\lim }\left({\pi }_1({\mathbb{H}}_1^{+},b_0)\stackrel{d_{1\#}}{⟶}{\pi }_1({\mathbb{H}}_2^{+},b_0)\stackrel{d_{2\#}}{⟶}{\pi }_1({\mathbb{H}}_3^{+},b_0)\stackrel{d_{3\#}}{⟶}\cdots \right)$$

$$\pi (\alpha ,U)=\{[\alpha \cdot \delta \cdot {\alpha }^{-}]\mid \delta \text{mbox}aloopinU\}⩽{\pi }_1(X,x_0).$$

$$\pi (x,U)=⟨\pi (\alpha ,U)\mid \alpha (1)=x⟩⩽{\pi }_1(X,x_0).$$

$${\pi }^s(X,x_0)=\bigcap _{\mathcal{U}\in Cov(X)}\pi (\mathcal{U},x_0)=\bigcap _{\mathcal{U}\in Cov(X)}⟨\pi (x,U)\mid x\in U\in \mathcal{U}⟩$$

$$\bigcap _{U\in {\mathcal{T}}_x}\bigcup _{\alpha (1)\in U}\pi (\alpha ,U)=\{1\}.$$

$$g_n=(g_{n,k}{)}_{k⩾n-1}:{\pi }_1({\mathbb{H}}_n^{+},b_0)\to F_n^{+}=\underset{⟵}{\lim }\left(F_{n,n-1}^{+}\stackrel{T_{n,n-1}}{⟵}{F}_{n,n}^{+}\stackrel{T_{n,n}}{⟵}{F}_{n,n+1}^{+}\stackrel{T_{n,n+1}}{⟵}\cdots \right).$$

$${\omega }_{n,m}(x)=R_{n,m}\circ R_{n,m+1}\circ \cdots \circ R_{n,k-1}(g_{n,k}(x)),$$

$${\mathcal{W}}_n^{+}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\pi }_1({\mathbb{H}}_{n,k+1}^{+},b_0)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{W}}_{1,k}^{+}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\chi =({\chi }_n{)}_{n\in \mathbb{N}}:{\pi }_1(\mathbb{P},b_0)\hookrightarrow \underset{⟵}{\lim }\left({\mathcal{W}}_{2,1}^{+}\stackrel{E_1}{⟵}{\mathcal{W}}_{3,2}^{+}\stackrel{E_2}{⟵}{\mathcal{W}}_{4,3}^{+}\stackrel{E_3}{⟵}\cdots \right)$$

$${\mathcal{W}}_{2,1}^{+}$$

$${\omega }_{n,n-1}([\beta ])=R_{n,n-1}\circ R_{n,n}\circ \cdots \circ R_{n,k-1}\circ g_{n,k}([\beta ])$$

$${\omega }_{n+1,n}([{\beta }^{\prime }])=R_{n+1,n}\circ R_{n+1,n+1}\circ \cdots \circ R_{n+1,k-1}\circ g_{n+1,k}([{\beta }^{\prime }]).$$

$$E_{n-1}\circ R_{n+1,n}\circ R_{n+1,n+1}\circ \cdots \circ R_{n+1,k-1}\circ D_n=R_{n,n-1}\circ R_{n,n}\circ \cdots \circ R_{n,k-1}$$

$$E_{n-1}({\omega }_{n+1,n}([{\beta }^{\prime }]))$$

$$g_{n,k}([{\beta }^{(s)}])\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{F}}_{n,k}^{+}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\overline{D}({\omega }_{n,m-1}([{\beta }^{(s)}]))$$

$${\mathcal{W}}_{2,1}^{+}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\pi }_1(\mathbb{P},b_0)\hookrightarrow \underset{⟵}{\lim }\left(F_{2,1}^{+}\stackrel{J_1}{⟵}{F}_{3,2}^{+}\stackrel{J_2}{⟵}{F}_{4,3}^{+}\stackrel{J_3}{⟵}\cdots \right),$$

$$R_{n,n-1}\circ R_{n,n}\circ \cdots \circ R_{n,k-1}\circ g_{n,k}([\beta ])={\omega }_{n,n-1}([\beta ]).$$

$$R_{n,n-1}\circ R_{n,n}\circ \cdots \circ R_{n,k-1}\circ R_{n,k}\circ \cdots \circ R_{n,m-1}\circ g_{n,m}([{\gamma }^{\prime }])={\omega }_{n,n-1}([{\gamma }^{\prime }]),$$

Full text

1. Introduction

The geometric significance of the elements of the fundamental group $\pi_{1}(X,x)$ of a connected and locally path-connected metric space $X$ is most prominently on display in the context of a simply connected covering space (if it exists) where these elements comprise the group of deck transformations of the covering projection. In this situation, the canonical homomorphism $\pi_{1}(X,x)\rightarrow\check{\pi}_{1}(X,x)$ to the first Čech homotopy group (also called the first shape group [26]) is an isomorphism [22].

In fact, as long as all fundamental group elements are accounted for, that is, if $\pi_{1}(X,x)\hookrightarrow\check{\pi}_{1}(X,x)$ is injective, the standard covering construction yields a generalized covering projection $p:\widetilde{X}\rightarrow X$ with connected, locally path-connected and simply connected $\widetilde{X}$ [22]. It is characterized by the usual unique lifting property and we have $\pi_{1}(X,x)\cong Aut(\widetilde{X}\stackrel{{\scriptstyle p}}{{\rightarrow}}X)$. Examples of spaces for which $\pi_{1}(X,x)\hookrightarrow\check{\pi}_{1}(X,x)$ is injective include all one-dimensional separable metric spaces [11, 17], all planar spaces [21], the Pontryagin sphere, the Pontryagin surface $\Pi_{2}$, and similar inverse limits of higher-dimensional manifolds [19].

Several weaker properties that quantify the geometric relevance of the elements of $\pi_{1}(X,x)$ can be found in the literature. The strongly homotopically Hausdorff property, for example, stipulates that for every essential loop in $X$ there should be a limit to how small it can be made at a particular point by a free homotopy [9]. The homotopically path Hausdorff property, on the other hand, calls for $\pi_{1}(X,x)$ to be T1 in the quotient topology induced by the compact-open topology on the loop space $\Omega(X,x)$ [2]. Both of these properties are implied by $\pi_{1}(X,x)\hookrightarrow\check{\pi}_{1}(X,x)$ being injective [20].

Then there are properties that guarantee, in and of themselves, the existence of a simply connected generalized covering space. These include the homotopically path Hausdorff property above and the 1-UV0 property, which requires small null-homotopic loops to contract via small homotopies [4, 20].

Even if a generalized covering projection with simply connected domain exists, it might not be a fibration and the monodromies between fibers might not be continuous [18]. (Such is the case for the Hawaiian Earring.) However, for all one-dimensional spaces and for all planar spaces, these monodromies have discrete graphs; a fact implicitly used in the work of Eda [14, 15] and Conner-Kent [8]. If any two spaces with this property (cf. Definition 9.2) are homotopy equivalent, then their respective wild sets (points at which they are not semilocally simply connected) are homeomorphic (Theorem 9.15).

As far as the algebraic structure of fundamental groups of low-dimensional spaces is concerned, we recall that the fundamental group of an arbitrary planar Peano continuum (not necessarily homotopy equivalent to a one-dimensional space) is isomorphic to a subgroup of the fundamental group of some one-dimensional planar Peano continuum [6]. In turn, fundamental groups of one-dimensional path-connected separable metric spaces are locally free [11] and, in the compact case, have structures similar to that of the Hawaiian Earring $\mathbb{H}$, where the injective function $\pi_{1}(\mathbb{H})\hookrightarrow\check{\pi}_{1}(\mathbb{H})$ factors through the limit $\mathcal{M}$ of an inverse system ${\mathcal{M}}_{1}\stackrel{{\scriptstyle R_{1}}}{{\longleftarrow}}{\mathcal{M}}_{2}\stackrel{{\scriptstyle R_{2}}}{{\longleftarrow}}{\mathcal{M}}_{3}\stackrel{{\scriptstyle R_{3}}}{{\longleftarrow}}\cdots$ of free monoids ${\mathcal{M}}_{n}$ on $\{\ell_{1}^{\pm 1},\ell_{2}^{\pm 1},\cdots,\ell_{n}^{\pm 1}\}$ with $R_{n-1}$ deleting the letters $\ell_{n}$ and $\ell_{n}^{-1}$ from every word [12, 24].

This raises the following

Question: Is there a space $X$ for which $\pi_{1}(X)\rightarrow\check{\pi}_{1}(X)$ is not injective, but with all of the other properties discussed above: low-dimensional, strongly homotopically Hausdorff, homotopically path Hausdorff, 1-UV0, admits a generalized universal covering $p:\widetilde{X}\rightarrow X$ whose monodromies between fibers have discrete graphs, admits a natural injective function $\pi_{1}(X)\rightarrow{\mathcal{M}}$, and has locally free fundamental group?

We present a relatively simple and prototypical construction of a two-dimensional space that yields a positive answer to this question.

Specifically, in Section 2, we define the space $\mathbb{P}$ mentioned in the abstract, by attaching countably many “pairs of pants” to the Hawaiian Earring $\mathbb{H}$, and identify its fundamental group as a direct limit of groups each isomorphic to $\pi_{1}(\mathbb{H})$ with injective bonding homomorphisms. In particular, $\pi_{1}(\mathbb{P})$ is locally free (Proposition 2.6).

We show that $\pi_{1}(\mathbb{P})\rightarrow\check{\pi}_{1}(\mathbb{P})$ is not injective (Theorem 3.6), but that $\mathbb{P}$ is both strongly homotopically Hausdorff (Theorem 4.5) and homotopically path Hausdorff (Theorem 6.4); as far as the authors know, it is the first such example (Remarks 4.2 and 6.3). The proof of the latter property hinges on the fact that $\pi_{1}(\mathbb{P})$ naturally injects into the inverse limit of monoids associated with the Hawaiian Earring(Theorem 5.3), despite $\pi_{1}(\mathbb{P})$ not being isomorphic to a subgroup of an inverse limit of free groups (Remark 5.4). Moreover, we show that $\mathbb{P}$ is 1-UV0 (Theorem 7.3) and that $\pi_{1}(\mathbb{P})$ is locally free (Proposition 2.6).

After a brief review of generalized covering space theory (Section 8) we show that the monodromies for the simply connected generalized covering space of $\mathbb{P}$ have discrete graphs (Theorem 9.14). We also discuss some general aspects of this property, such as its relationship to the homotopically Hausdorff property relative to a subgroup of the fundamental group (Proposition 9.11) and its impact on the stability of wild subsets under homotopy equivalence (Theorem 9.15).

2. The Hawaiian pants $\mathbb{P}$

Let $C_{n}\subseteq\mathbb{R}^{2}$ be the circle of radius $\frac{1}{n}$ centered at $(\frac{1}{n},0)$ and let $\mathbb{H}=\bigcup_{i=1}^{\infty}C_{i}\subseteq\mathbb{R}^{2}$ be the usual Hawaiian Earring with basepoint $b_{0}=(0,0)$. Define $\ell_{n}:[0,1]\rightarrow C_{n}$ by $\ell_{n}(t)=(\frac{1}{n}(1-\cos 2\pi t),\frac{1}{n}\sin 2\pi t))$.

For $n\in\mathbb{N}$, let $D_{n,1}$ and $D_{n,2}$ be two disjoint disks in the interior $D_{n}^{\circ}$ of a disk $D_{n}\subseteq\mathbb{R}^{2}$ and consider the “pair of pants” $P_{n}=D_{n}\setminus(D^{\circ}_{n,1}\cup D^{\circ}_{n,2})$. Let $\alpha_{n},\beta_{n},\gamma_{n}:[0,1]\rightarrow P_{n}$ be parametrizations of the boundaries $\partial D_{n}$, $\partial D_{n,1}$, and $\partial D_{n,2}$, respectively, with clockwise orientation.

Let $\mathbb{P}$ be the space obtained from $\mathbb{H}$ by attaching all $P_{n}$ via maps $f_{n}:\partial P_{n}\rightarrow\mathbb{H}$ such that $f_{n}\circ\alpha_{n}=\ell_{n}$, $f_{n}\circ\beta_{n}=\ell_{2n}$ and $f_{n}\circ\gamma_{n}=\ell_{2n+1}$. That is, we put $\mathbb{P}=\mathbb{H}\cup_{f}\left(\coprod_{n\in\mathbb{N}}P_{n}\right)$ where $f:\coprod_{n\in\mathbb{N}}\partial P_{n}\rightarrow\mathbb{H}$ and $f|_{\partial P_{n}}=f_{n}$. We refer to $\mathbb{P}$ as the “Hawaiian Pants”. (See Figure 2.)

We observe that exactly one pair of pants is attached to $C_{1}$, namely $P_{1}$ via $f_{1}|_{\partial D_{1}}$, and that for each $n\geqslant 2$, exactly two pairs of pants are attached to $C_{n}$, namely $P_{n}$ via $f_{n}|_{\partial D_{n}}$, and either $P_{n/2}$ via $f_{n/2}|_{\partial D_{n/2,1}}$ (if $n$ is even) or $P_{(n-1)/2}$ via $f_{(n-1)/2}|_{\partial D_{(n-1)/2,2}}$ (if $n$ is odd). In particular:

Proposition 2.1**.**

• (a)

$\mathbb{P}\setminus\{b_{0}\}$* is a 2-manifold with boundary $C_{1}\setminus\{b_{0}\}$.*

• (b)

$\mathbb{P}$* is connected and locally path connected.*

We have $f^{-1}_{n}(b_{0})=\{a_{n},b_{n},c_{n}\}$ with $a_{n}\in\partial D_{n}$, $b_{n}\in\partial D_{n,1}$ and $c_{n}\in\partial D_{n,2}$. Choose arcs $A_{n,1},A_{n,2}\subseteq P_{n}$ such that $A_{n,1}\cap\partial P_{n}=\partial A_{n,1}=\{a_{n},b_{n}\}$, $A_{n,2}\cap\partial P_{n}=\partial A_{n,2}=\{a_{n},c_{n}\}$, and $A_{n,1}\cap A_{n,2}=\{a_{n}\}$, configured as in Figure 2. Let $B_{n,j}$ be the image of $A_{n,j}$ in $\mathbb{P}$ when attaching $P_{n}$. Then $B_{n,j}$ is homeomorphic to a circle. We regard $P^{\circ}_{n}=D^{\circ}_{n}\setminus(D_{n,1}\cup D_{n,2})$ as a subspace of $\mathbb{P}$.

Let $\mathbb{H}_{n}=\bigcup_{i=n}^{\infty}C_{i}$ and let $\mathbb{P}_{n}$ be the space obtained from $\mathbb{H}_{n}$ by attaching $P_{n},P_{n+1},P_{n+2},\dots$. Define $\mathbb{P}_{n}^{+}=\bigcup_{i=1}^{n-1}(B_{i,1}\cup B_{i,2})\cup\mathbb{P}_{n}$ for $n\geqslant 2$ and put $\mathbb{P}_{1}^{+}=\mathbb{P}_{1}=\mathbb{P}$. Then $\mathbb{P}=\mathbb{P}^{+}_{1}\supseteq\mathbb{P}^{+}_{2}\supseteq\mathbb{P}^{+}_{3}\supseteq\cdots$.

Since each $P_{n}$ deformation retracts onto $\partial D_{n,1}\cup A_{n,1}\cup A_{n,2}\cup\partial D_{n,2}$, there are deformation retractions $\phi_{n}:\mathbb{P}^{+}_{n}\times[0,1]\rightarrow\mathbb{P}^{+}_{n}$ such that

• (i)

for all $p\in\mathbb{P}^{+}_{n}$, we have $\phi_{n}(p,0)=p$;

• (ii)

for all $p\in\mathbb{P}^{+}_{n}$, we have $\phi_{n}(p,1)\in\mathbb{P}^{+}_{n+1}$;

• (iii)

for all $p\in\mathbb{P}^{+}_{n+1}$ and all $t\in[0,1]$, we have $\phi_{n}(p,t)=p$;

• (iv)

for all $p\in\mathbb{P}^{+}_{n}\setminus\mathbb{P}^{+}_{n+1}$ and all $t\in[0,1]$, we have that $\phi_{n}(p,t)$ lies in the image of $P_{n}$ in the quotient $\mathbb{P}^{+}_{n}$.

Put $\mathbb{H}_{n,k}=\bigcup_{i=n}^{k}C_{i}$ (where $\mathbb{H}_{n,k}=\emptyset$ for $k<n$) and $\mathbb{H}_{n,k}^{+}=\bigcup_{i=1}^{n-1}(B_{i,1}\cup B_{i,2})\cup\mathbb{H}_{n,k}$. Likewise, put $\mathbb{H}_{n}^{+}=\bigcup_{i=1}^{n-1}(B_{i,1}\cup B_{i,2})\cup\mathbb{H}_{n}$. Note that $\mathbb{H}^{+}_{n,n-1}=\bigcup_{i=1}^{n-1}(B_{i,1}\cup B_{i,2})$ and $\mathbb{H}_{1}^{+}=\mathbb{H}$.

Defining $d_{n}(p)=\phi_{n}(p,1)$ and letting $r_{n,k}:\mathbb{H}^{+}_{n}\rightarrow\mathbb{H}^{+}_{n,k}$ denote the canonical retractions with $r_{n,k}(\bigcup_{i=k+1}^{\infty}C_{i})=\{b_{0}\}$, we have the following commutative diagrams:

[TABLE]

We may assume that there are parametrizations $\rho_{n,j}:([0,1],\{0,1\})\rightarrow(B_{n,j},b_{0})$ such that

[TABLE]

Here, “$\;\cdot\;$” denotes the usual concatenation of paths and $\rho_{n,j}^{-}$ denotes the reverse path of $\rho_{n,j}$, given by $\rho_{n,j}^{-}(t)=\rho_{n,j}(1-t)$. Taking path homotopy classes, and noting that $[\ell_{n}]=[d_{n}\circ\ell_{n}]\in\pi_{1}(\mathbb{P},b_{0})$, we have

[TABLE]

Lemma 2.2**.**

For every $k\geqslant 2n+1$, $d_{n\#}:\pi_{1}(\mathbb{H}_{n,k}^{+},b_{0})\rightarrow\pi_{1}(\mathbb{H}_{n+1,k}^{+},b_{0})$ is injective.

Proof.

For $m\in\{n,n+1\}$, the group $\pi_{1}(\mathbb{H}_{m,k}^{+},b_{0})$ is free on the set

[TABLE]

So, the claim follows from Equation (2.1) and the fact that $d_{n}\circ\rho_{i,j}=\rho_{i,j}$ for $1\leqslant i\leqslant n-1$, $1\leqslant j\leqslant 2$ and $d_{n}\circ\ell_{i}=\ell_{i}$ for $n+1\leqslant i\leqslant k$. ∎

Notation**.**

We will denote functions $\sigma:\mathcal{L}\rightarrow\mathcal{N}$ to inverse limits

[TABLE]

as sequences $\sigma=(\sigma_{n})_{n\geqslant 1}$ with $\sigma_{n}=\mu_{n}\circ\sigma:{\mathcal{L}}\rightarrow{\mathcal{N}}_{n}$, where $\mu_{n}:\mathcal{N}\rightarrow\mathcal{N}_{n}$ are the projections.

Lemma 2.3**.**

For every $n$, $d_{n\#}:\pi_{1}(\mathbb{H}_{n}^{+},b_{0})\rightarrow\pi_{1}(\mathbb{H}_{n+1}^{+},b_{0})$ is injective.

Proof.

Let $1\not=[\alpha]\in\pi_{1}(\mathbb{H}_{n}^{+},b_{0})$. Since

[TABLE]

we have that

[TABLE]

is injective [5]. Therefore, there is a $k\geqslant 2n+1$ such that $(r_{n,k})_{\#}([\alpha])\not=1$. The claim now follows from Lemma 2.2 and applying $\pi_{1}$ to the diagram above. ∎

Lemma 2.4**.**

$\pi_{1}(\mathbb{P},b_{0})$* is isomorphic to the direct limit*

[TABLE]

with canonical homomorphisms $\iota_{n\#}:\pi_{1}(\mathbb{H}_{n}^{+},b_{0})\rightarrow\pi_{1}(\mathbb{P},b_{0})$ induced by inclusion $\iota_{n}:\mathbb{H}_{n}^{+}\hookrightarrow\mathbb{P}$.

Proof.

For $n\in\mathbb{N}$ and $[\alpha]\in\pi_{1}(\mathbb{H}_{n}^{+},b_{0})$, we have $[d_{n}\circ\alpha]=[\alpha]\in\pi_{1}(\mathbb{P},b_{0})$. Hence $\iota_{n+1\#}\circ d_{n\#}=\iota_{n\#}$. In order to verify the universal property, let $h_{n}:\pi_{1}(\mathbb{H}_{n}^{+},b_{0})\rightarrow G$ be a homomorphisms with $h_{n+1}\circ d_{n\#}=h_{n}$. Let $[\alpha]\in\pi_{1}(\mathbb{P},b_{0})$. Since $\alpha([0,1])$ is compact, there is an $n\in\mathbb{N}$ such that $\alpha([0,1])\cap P^{\circ}_{i}=\emptyset$ for all $i\geqslant n$. Put $\beta=d_{n-1}\circ d_{n-2}\circ\cdots\circ d_{1}\circ\alpha$. Then $\beta:([0,1],\{0,1\})\rightarrow(\mathbb{H}_{n}^{+},b_{0})$ and $\iota_{n\#}([\beta])=[\alpha]\in\pi_{1}(\mathbb{P},b_{0})$. Moreover, $h_{n+1}([d_{n}\circ\beta])=h_{n}([\beta])$. Put $h([\alpha])=h_{n}([\beta])$. Once we show that $h:\pi_{1}(\mathbb{P},b_{0})\rightarrow G$ is well-defined, it is clear that $h$ is a homomorphism with $h\circ\iota_{n\#}=h_{n}$ and that $h$ is the unique homomorphism with this property. To this end, suppose that $[\alpha]=[\widetilde{\alpha}]\in\pi_{1}(\mathbb{P},b_{0})$. Let $H:[0,1]\times[0,1]\rightarrow\mathbb{P}$ be a homotopy with $H(t,0)=\alpha(t)$, $H(t,1)=\widetilde{\alpha}(t)$, and $H(0,t)=H(1,t)=b_{0}$ for all $t\in[0,1]$. Since $H([0,1]\times[0,1])$ is compact, for $n\in\mathbb{N}$ sufficiently large, we have $d_{n-1}\circ\cdots\circ d_{2}\circ d_{1}\circ H([0,1]\times[0,1])\subseteq\mathbb{H}_{n}^{+}$ so that $[d_{n-1}\circ\cdots\circ d_{2}\circ d_{1}\circ\alpha]=[d_{n-1}\circ\cdots\circ d_{2}\circ d_{1}\circ\widetilde{\alpha}]\in\pi_{1}(\mathbb{H}_{n}^{+},b_{0})$. ∎

Lemma 2.5**.**

For every $n$, $\iota_{n\#}:\pi_{1}(\mathbb{H}_{n}^{+},b_{0})\rightarrow\pi_{1}(\mathbb{P},b_{0})$ is injective.

Proof.

This follows from Lemmas 2.3 and 2.4. ∎

Recall that a group $G$ is called locally free if every finitely generated subgroup of $G$ is free.

Proposition 2.6**.**

$\pi_{1}(\mathbb{P},b_{0})$* is locally free.*

Proof.

Since $\pi_{1}(\mathbb{H}^{+}_{n},b_{0})$ is isomorphic to a subgroup of an inverse limit of free groups of finite rank, it is locally free [11, Theorem 1]. Then, by Lemma 2.4, $\pi_{1}(\mathbb{P},b_{0})$ is a direct limit of locally free groups, and thus locally free [7, Lemma 24]. ∎

Remark 2.7** (Metric Hawaiian Pants $\mathbb{P}^{\ast}\subseteq\mathbb{R}^{3}$).**

While $\mathbb{P}$ is not metrizable (it is not first countable at $b_{0}$), it is naturally homotopy equivalent to the metrizable space formed by attaching the pants $P_{n}$ to the union $\bigcup_{i=1}^{\infty}Z_{i}\subseteq\mathbb{R}^{3}$ of the cylinders $Z_{n}=C_{n}\times[1-n,n-1]$ via identifying $\partial D_{n}$ with $C_{n}\times\{1-n\}$, $\partial D_{n,1}$ with $C_{2n}\times\{2n-1\}$, and $\partial D_{n,2}$ with $C_{2n+1}\times\{2n\}$.

If we change the embedding $\mathbb{H}\subseteq\mathbb{R}^{2}$, this procedure yields a subspace of $\mathbb{R}^{3}$ that is homotopy equivalent to $\mathbb{P}$: For each $n\in\mathbb{N}$, choose a triangle $C^{\prime}_{n}\subseteq\mathbb{R}^{2}$ of diameter less than $1/n$ such that for all $i\not=j$, $C^{\prime}_{i}\cap C^{\prime}_{j}=\{b_{0}\}$ and the bounded components of $\mathbb{R}^{2}\setminus C^{\prime}_{i}$ and $\mathbb{R}^{2}\setminus C^{\prime}_{j}$ are disjoint. Then $\bigcup_{i=1}^{\infty}C^{\prime}_{i}\subseteq\mathbb{R}^{2}$ is homeomorphic to $\mathbb{H}$. The adjunction space resulting from attaching the pants $P_{n}$ to the union $\bigcup_{i=1}^{\infty}Z_{i}^{\prime}\subseteq\mathbb{R}^{3}$ of the corresponding cylinders $Z_{n}^{\prime}$ can now readily be implemented in $\mathbb{R}^{3}$ by forming the union of $\bigcup_{i=1}^{\infty}Z_{i}^{\prime}$ with appropriate sets $P_{n}^{\prime}\subseteq\mathbb{R}^{3}$ that are homeomorphic to $P_{n}$.

To obtain a subspace $\mathbb{P}^{\ast}\subseteq\mathbb{R}^{3}$ which is semilocally simply connected at all but one point and homotopy equivalent to $\mathbb{P}$, slightly deform the cylinders $Z_{n}^{\prime}$ so that their only common point of contact is the origin. Then there is a bijective homotopy equivalence $h:\mathbb{P}\rightarrow\mathbb{P}^{\ast}$ such that $h(C_{n})=C_{n}^{\prime}\times\{0\}$ for all $n\in\mathbb{N}$ with homotopy inverse $g:\mathbb{P}^{\ast}\rightarrow\mathbb{P}$ collapsing the cylinders such that $g|_{P^{\prime}_{n}}:P_{n}^{\prime}\rightarrow P_{n}$ is a homeomorphism, $g(Z_{n}^{\prime})=C_{n}$, $g|_{C_{n}^{\prime}\times\{0\}}=(h|_{C_{n}})^{-1}$, and $g(h(B_{n,j}))=B_{n,j}$ for all $n\in\mathbb{N}$ and $j\in\{1,2\}$. The resulting deformation retraction of the cylinders allows the proofs in this paper for those properties of $\mathbb{P}$ that are not homotopy invariant (such as the 1-UV0 property and the discrete monodromy property) to go through for $\mathbb{P}^{\ast}$ with only minor changes. However, working with $\mathbb{P}$ is conceptually simpler.

3. Non-injectivity into the first Čech homotopy group

Let $X$ be a path-connected topological space and $x_{0}\in X$. For a point $x\in X$, let ${\mathcal{T}}_{x}$ denote the set of all open neighborhoods of $x$ in $X$. For $U\in{\mathcal{T}}_{x}$ and a path $\alpha$ in $X$ from $x_{0}$ to $x$, consider the subgroup

[TABLE]

Let $\pi(x,U)$ denote the normal closure of $\pi(\alpha,U)$ in $\pi_{1}(X,x_{0})$, i.e., the subgroup generated by all $\pi(\beta,U)$, with $\beta$ a path from $x_{0}$ to $x$:

[TABLE]

Remark 3.1**.**

Suppose $U\in{\mathcal{T}}_{x}$ is path connected and let $[\gamma]\in\pi_{1}(X,x_{0})$. Then $[\gamma]\in\pi(x,U)$ if and only if there is a map $g:D\setminus(D^{\circ}_{1}\cup D^{\circ}_{2}\cup\cdots\cup D^{\circ}_{j})\rightarrow X$ from a “disk with holes” to $X$ with $g|_{\partial D}=\gamma$ and $g(\partial D_{i})\subseteq U$ for all $i\in\{1,2,\dots,j\}$. (Here, $D_{1},D_{2},\dots,D_{j}$ are pairwise disjoint disks in the interior $D^{\circ}$ of the disk $D$.)

Let $Cov(X)$ denote the set of all open covers of $X$. For ${\mathcal{U}}\in Cov(X)$, let $\pi({\mathcal{U}},x_{0})$ denote the subgroup of $\pi_{1}(X,x_{0})$ generated by all $\pi(x,U)$ with $U\in{\mathcal{U}}$ and $x\in U$.

Definition 3.2** (Spanier group [29]).**

The subgroup

[TABLE]

of the fundamental group $\pi_{1}(X,x_{0})$ is called the Spanier group of $X$.

Remark 3.3**.**

If $X$ is locally path connected, then for a given $H\leqslant\pi_{1}(X,x_{0})$, there is a (classical) covering projection $p:(\widetilde{X},\widetilde{x})\rightarrow(X,x_{0})$ with $p_{\#}\pi_{1}(\widetilde{X},\widetilde{x})=H$ if and only if there is a ${\mathcal{U}}\in Cov(X)$ such that $\pi({\mathcal{U}},x_{0})\leqslant H$ [29].

Remark 3.4**.**

The Spanier group $\pi^{s}(X,x_{0})$ is contained in the kernel of the natural homomorphism $\Psi_{X}:\pi_{1}(X,x_{0})\rightarrow\check{\pi}_{1}(X,x_{0})$ to the first Čech homotopy group [22]. If $X$ is locally path connected and metrizable, then $\pi^{s}(X,x_{0})$ equals this kernel [3].

Let $\iota=\iota_{1}:\mathbb{H}\hookrightarrow\mathbb{P}$ denote inclusion.

Lemma 3.5**.**

$\iota_{\#}\pi_{1}(\mathbb{H},b_{0})\leqslant\pi^{s}(\mathbb{P},b_{0})$.

Proof.

Let $[\alpha]\in\pi_{1}(\mathbb{H},b_{0})$ and $\mathcal{U}\in Cov(\mathbb{P})$. Choose $U\in{\mathcal{U}}$ with $b_{0}\in U$ and fix $n\in\mathbb{N}$ with $\iota(\mathbb{H}_{n})\subseteq U$. Express $[\alpha]=[\gamma_{1}][\delta_{1}][\gamma_{2}][\delta_{2}]\cdots[\gamma_{m}][\delta_{m}]$ with loops $\gamma_{j}$ in $C_{1}\cup C_{2}\cup\cdots\cup C_{n-1}$ and loops $\delta_{j}$ in $\mathbb{H}_{n}$. Then $\iota_{\#}([\delta_{j}])\in\pi(b_{0},U)$ for every $j$. Also, for every $j$, we have $\iota_{\#}([\gamma_{j}])\in\left<[\ell_{1}],[\ell_{2}],\dots,[\ell_{n-1}]\right>\leqslant\pi_{1}(\mathbb{P},b_{0})$, so that repeated application of Equation (2.2) yields $\iota_{\#}([\gamma_{j}])\in\pi(b_{0},U)$. Hence, $\iota_{\#}([\alpha])\in\pi^{s}(\mathbb{P},b_{0})$. ∎

Theorem 3.6**.**

$\Psi_{\mathbb{P}}:\pi_{1}(\mathbb{P},b_{0})\rightarrow\check{\pi}_{1}(\mathbb{P},b_{0})$* is not injective.*

Proof.

Combining Lemma 2.5 (with $n$=1), Lemma 3.5, and Remark 3.4, we obtain $1\not=\iota_{\#}\pi_{1}(\mathbb{H},b_{0})\leqslant\pi^{s}(\mathbb{P},b_{0})\leqslant\ker\Psi_{\mathbb{P}}$. ∎

Recall that if the fundamental group of a Peano continuum does not (canonically) inject into the first Čech homotopy group, then it is not residually n-slender [16]. However, since $\mathbb{P}$ is not a Peano continuum, we verify this separately:

Definition 3.7** (Noncommutatively slender [13]).**

A group $G$ is called noncommutatively slender (n-slender for short) if for every homomorphism $h:\pi_{1}(\mathbb{H},b_{0})\rightarrow G$, there is a $k\in\mathbb{N}$ such that $h([\alpha])=1$ for all loops $\alpha:([0,1],\{0,1\})\rightarrow(\mathbb{H}_{k},b_{0})$.

Recall that a group $G$ is called residually n-slender (respectively residually free) if for every $1\not=g\in G$ there is an n-slender (respectively free) group $S$ and a homomorphism $h:G\rightarrow S$, such that $h(g)\not=1$.

Proposition 3.8**.**

$\pi_{1}(\mathbb{P},b_{0})$* is not residually n-slender.*

Proof.

Consider $1\not=[\ell_{1}]\in\pi_{1}(\mathbb{P},b_{0})$. Let $h:\pi_{1}(\mathbb{P},b_{0})\rightarrow S$ be a homomorphism to an n-slender group $S$. It suffices to show that $h([\ell_{1}])=1$. If we precompose $h$ with the homomorphism $\iota_{\#}:\pi_{1}(\mathbb{H},b_{0})\rightarrow\pi_{1}(\mathbb{P},b_{0})$, induced by inclusion $\iota:\mathbb{H}\hookrightarrow\mathbb{P}$, and note that $S$ is n-slender, we see that $h([\ell_{k}])=h\circ\iota_{\#}([\ell_{k}])=1$ for all but finitely many $k$. However, by Equation (2.2), we have $[\ell_{n}]=[\rho_{n,1}][\ell_{2n}][\rho_{n,1}]^{-1}[\rho_{n,2}][\ell_{2n+1}][\rho_{n,2}]^{-1}$ in $\pi_{1}(\mathbb{P},b_{0})$ for all $n$. Hence, $h([\ell_{n}])=1$ for all $n$. ∎

Remark 3.9**.**

Every free group is n-slender [13]. So, $\pi_{1}(\mathbb{P},b_{0})$ is not residually free.

4. The strongly homotopically Hausdorff property

Definition 4.1** (Strongly homotopically Hausdorff [9]).**

A path-connected space $X$ is called strongly homotopically Hausdorff at $x\in X$ if for every essential loop $\gamma$ in $X$, there is an open neighborhood $U$ of $x$ in $X$ such that $\gamma$ cannot be freely homotoped into $U$, that is, if

[TABLE]

(If $U$ is path connected, we may replace “$\alpha(1)\in U$” by “$\alpha(1)=x$”.) The space $X$ is called strongly homotopically Hausdorff if it is strongly homotopically Hausdorff at every point $x\in X$.

Remark 4.2**.**

If the natural homomorphism $\pi_{1}(X,x)\hookrightarrow\check{\pi}_{1}(X,x)$ is injective, then $X$ is strongly homotopically Hausdorff [20]. However, the converse does not hold in general [20, Example $Z^{\prime}$].

Remark 4.3** (The Hawaiian Mapping Torus).**

Let $f:\mathbb{H}\rightarrow\mathbb{H}$ be the map given by $f\circ\ell_{n}=\ell_{n+1}$, $n\in\mathbb{N}$. The Hawaiian Mapping Torus is the space $M_{f}=\mathbb{H}\times[0,1]/\sim$, where $(x,0)\sim(f(x),1)$ for all $x\in\mathbb{H}$. Identifying $\mathbb{H}$ with the image of $\mathbb{H}\times\{0\}$ in $M_{f}$, the inclusion $i:\mathbb{H}\hookrightarrow M_{f}$ induces an injection $i_{\#}:\pi_{1}(\mathbb{H},b_{0})\rightarrow\pi_{1}(M_{f},b_{0})$. Consider the loop $\rho:([0,1],\{0,1\})\rightarrow(M_{f},b_{0})$ where $\rho(s)$ is the image of $(b_{0},s)$ in $M_{f}$ and put $t=[\rho]\in\pi_{1}(M_{f},b_{0})$. From two applications of van Kampen’s Theorem, we get that $\pi_{1}(M_{f},b_{0})$ is isomorphic to the quotient of $\pi_{1}(\mathbb{H},b_{0})\ast\left<\,t\,\right>$ by the relations $g=tf_{\#}(g)t^{-1}$, $g\in\pi_{1}(\mathbb{H},b_{0})$ (see [28]). Iterating these relations, we see that each $[\gamma]\in\pi_{1}(\mathbb{H},b_{0})$ factors in $\pi_{1}(M_{f},b_{0})$, for every $n\in\mathbb{N}$, as a conjugate $[\rho]^{n}[f^{n}\circ\gamma][\rho]^{-n}$ where the diameter of the loop $f^{n}\circ\gamma$ shrinks to [math].

While $M_{f}$ is a Peano continuum that embeds into $\mathbb{R}^{3}$ and has many of the same properties as $\mathbb{P}$, it is not strongly homotopically Hausdorff, since $\ell_{1}$ is freely homotopic to $\ell_{n}$ for all $n\in\mathbb{N}$. Our detailed treatment of $\mathbb{P}$ is motivated by the fact that $\pi_{1}(\mathbb{P},b_{0})$ exhibits a somewhat more intricate algebraic phenomenon: in order to write an element $g\in\iota_{\#}\pi_{1}(\mathbb{H},b_{0})\leqslant\pi_{1}(\mathbb{P},b_{0})$ as a product of conjugates of homotopy classes of arbitrarily small loops (as in the proof of Lemma 3.5), it takes an exponentially growing number of distinct conjugating elements, namely products of $[\rho_{i,j}]$.

A path $\alpha:[a,b]\rightarrow X$ is called reduced if for every $a\leqslant s<t\leqslant b$ with $\alpha(s)=\alpha(t)$, the loop $\alpha|_{[s,t]}$ is not null-homotopic in $X$. For a one-dimensional metric space $X$, every path $\alpha:[a,b]\rightarrow X$ is homotopic (relative to endpoints) within $\alpha([a,b])$ to either a constant path or a reduced path, which is unique up to reparametrization [14]. A path $\alpha:[a,b]\rightarrow X$ is called cyclically reduced if $\alpha\cdot\alpha$ is reduced.

Lemma 4.4** (Lemma 3.11 of [4]).**

Let $\lambda:([0,1],0)\to(X,x_{0})$ be a reduced path in a one-dimensional metric space $X$, $\delta:[0,1]\to X$ be a reduced loop based at $\lambda(1)$, and $\gamma$ be a reduced representative of $[\lambda\cdot\delta\cdot\lambda^{-}]$. Then there exist $s,t\in[0,1]$ such that $\lambda|_{[0,t]}\circ\phi=\gamma|_{[0,s]}$, for some increasing homeomorphism $\phi:[0,s]\rightarrow[0,t]$, and $\lambda([t,1])\subseteq\delta([0,1])$.

Theorem 4.5**.**

$\mathbb{P}$* is strongly homotopically Hausdorff.*

Proof.

For $n\in\mathbb{N}$, we define $U_{n}=\mathbb{H}\cap\{(x,y)\in\mathbb{R}^{2}\mid x<\frac{2n+1}{n(n+1)}\}$. Since $\operatorname{diam}(C_{n+1})=\frac{2}{n+1}<\frac{2n+1}{n(n+1)}<\frac{2}{n}=\operatorname{diam}(C_{n})$, we see that the sequence $U_{1}\supseteq U_{2}\supseteq U_{3}\supseteq\cdots$ forms a neighborhood basis for $\mathbb{H}$ at $b_{0}$. For every pair $n,k\in\mathbb{N}$, $f_{k}^{-1}(U_{n})$ has three components $L^{0}_{k,n}$, $L^{1}_{k,n}$, and $L^{2}_{k,n}$, each of which is an open arc or a circle (there are four cases based on the position of $n$ relative to $k$, $2k$, and $2k+1$), such that $a_{k}\in L^{0}_{k,n}\subseteq\partial D_{k}$, $b_{k}\in L^{1}_{k,n}\subseteq\partial D_{k,1}$ and $c_{k}\in L^{2}_{k,n}\subseteq\partial D_{k,2}$. Since, for a given $k$, $L^{i}_{k,n+1}\subseteq L^{i}_{k,n}$, we may choose three pairwise disjoint open neigborhoods $N^{0}_{k,n}$, $N^{1}_{k,n}$, $N^{2}_{k,n}$ of $L^{0}_{k,n}$, $L^{1}_{k,n}$, $L^{2}_{k,n}$ in $P_{k}$, respectively, such that $N^{i}_{k,n+1}\subseteq N^{i}_{k,n}$, $N^{i}_{k,n}\cap\partial P_{k}=L^{i}_{k,n}$, and $N^{i}_{k,n}$ deformation retracts onto $L^{i}_{k,n}$. Define $V_{k,n}=N^{0}_{k,n}\cup N^{1}_{k,n}\cup N^{2}_{k,n}$. (See Figure 3.) Put $V_{n}=\bigcup_{k\in\mathbb{N}}f_{k}(V_{k,n})$. Then $V_{n}$ is an open neighborhood of $b_{0}$ in $\mathbb{P}$, $V_{n+1}\subseteq V_{n}$, and $V_{n}$ deformation retracts onto $U_{n}$. (Note that $V_{1}\supseteq V_{2}\supseteq V_{3}\supseteq\cdots$ does not form a neighborhood basis for $\mathbb{P}$ at $b_{0}$.)

Now suppose, to the contrary, that $\mathbb{P}$ is not strongly homotopically Hausdorff. Then $\mathbb{P}$ is not strongly homotopically Hausdorff at $b_{0}$, since $\mathbb{P}\setminus\{b_{0}\}$ is locally contractible. Hence, there is a $1\not=[\gamma]\in\pi_{1}(\mathbb{P},b_{0})$ such that for every $n\in\mathbb{N}$, there is a path $\lambda_{n}:[0,1]\rightarrow\mathbb{P}$ from $\lambda_{n}(0)=b_{0}$ to $\lambda_{n}(1)\in V_{n}$ with $[\gamma]\in\pi(\lambda_{n},V_{n})$.

Since each $V_{n}$ is path connected, we may assume that $\lambda_{n}(1)=b_{0}$. Choose loops $\delta_{n}$ in $V_{n}$ with $[\gamma]=[\lambda_{n}][\delta_{n}][\lambda_{n}]^{-1}\in\pi_{1}(\mathbb{P},b_{0})$. Since $V_{n}$ deformation retracts onto $U_{n}$, we may assume that $\delta_{n}$ lies in $U_{n}$. We may also assume that each $\delta_{n}$ is reduced in $\mathbb{H}$. This implies that $\delta_{n}$ lies in $\mathbb{H}_{n+1}$, because $C_{m}$ is not fully contained in $U_{n}$ if $m\geqslant n$. Replacing each $\lambda_{n}$ by $\lambda_{1}^{-}\cdot\lambda_{n}$, we may assume that $\gamma=\delta_{1}$, which lies in $U_{1}$. There is a maximal $s_{0}\in[0,1]$ such that $\gamma|_{[0,s_{0}]}$ is a reparametrization of $(\gamma|_{[t_{0},1]})^{-}$ for some $t_{0}\in(s_{0},1]$. Then $\gamma(s_{0})=\gamma(t_{0})=b_{0}$, $\gamma|_{[s_{0},t_{0}]}$ is cyclically reduced, and $[\gamma]=[\gamma|_{[0,s_{0}]}][\gamma|_{[s_{0},t_{0}]}][\gamma|_{[0,s_{0}]}]^{-1}$. Replacing each $\lambda_{n}$ by $\gamma|_{[0,s_{0}]}^{-}\cdot\lambda_{n}$ and replacing $\gamma$ by $\gamma|_{[s_{0},t_{0}]}$, we may assume that $\gamma$ is cyclically reduced.

Let $m$ be minimal such that $\gamma([0,1])$ intersects $C_{m}\setminus\{b_{0}\}$. Then $\gamma$ fully traverses $C_{m}$, at least once, and $\gamma$ lies in $\mathbb{H}_{m}$. Let $F$ be a homotopy from $\gamma$ to $\lambda_{m}\cdot\delta_{m}\cdot\lambda_{m}^{-}$ (relative to endpoints) within $\mathbb{P}$. Choose $n>m$ such that the image of $F$ misses all $P^{\circ}_{i}$ with $i>n$. Then $F^{\prime}=d_{n}\circ d_{n-1}\circ\cdots\circ d_{1}\circ F$ is a homotopy from $\gamma^{\prime}=d_{n}\circ d_{n-1}\circ\cdots\circ d_{1}\circ\gamma$ to $\lambda\cdot\delta\cdot\lambda^{-}$ (relative to endpoints) within $\mathbb{H}^{+}_{n+1}$, where $\lambda=d_{n}\circ d_{n-1}\circ\cdots\circ d_{1}\circ\lambda_{m}$ and $\delta=d_{n}\circ d_{n-1}\circ\cdots\circ d_{1}\circ\delta_{m}$.

On one hand, $\gamma^{\prime}$ is a cyclically reduced loop in $\mathbb{H}^{+}_{n+1}$ which traverses $B_{m,1}$. On the other hand, the image of $\delta$ is disjoint from $B_{m,1}\setminus\{b_{0}\}$. (In particular, $\lambda$ is not null-homotopic in $\mathbb{H}^{+}_{n+1}$.) Therefore, using reduced representatives $\lambda^{\prime}$ and $\delta^{\prime}$ of $[\lambda]$ and $[\delta]$, respectively, we have $s,t>0$ in Lemma 4.4. Applying Lemma 4.4 to $\gamma^{\prime-}$, as well, we see that $\gamma^{\prime}$ is not cyclically reduced; a contradiction. ∎

5. An inverse limit of finitely generated free monoids

Let ${\mathcal{W}}^{+}_{n,k}$ denote the set of finite words (including the empty word) over the alphabet ${\mathcal{A}}_{n,k}=\{\rho^{\pm 1}_{i,j}\mid 1\leqslant i\leqslant n-1,1\leqslant j\leqslant 2\}\cup\{\ell_{i}^{\pm 1}\mid n\leqslant i\leqslant k\}$. Then ${\mathcal{W}}^{+}_{n,k}$ forms a free monoid on the set ${\mathcal{A}}_{n,k}$ under concatenation.

The deletion of a subword of the form $\rho_{i,j}^{+1}\rho_{i,j}^{-1}$, $\rho_{i,j}^{-1}\rho_{i,j}^{+1}$, $\ell_{i}^{+1}\ell_{i}^{-1}$, or $\ell_{i}^{-1}\ell_{i}^{+1}$ from an element of ${\mathcal{W}}^{+}_{n,k}$ is called a cancellation. A word is called reduced if it does not allow for any cancellation. Recall that, starting with a fixed element of ${\mathcal{W}}^{+}_{n,k}$, every maximal sequence of cancellations results in the same reduced word.

Let $F^{+}_{n,k}\subseteq{\mathcal{W}}^{+}_{n,k}$ be the subset of all reduced words. Then $F^{+}_{n,k}$ forms a free group on the set $\{\rho_{i,j}\mid 1\leqslant i\leqslant n-1,1\leqslant j\leqslant 2\}\cup\{\ell_{i}\mid n\leqslant i\leqslant k\}$ under concatenation, followed by maximal cancellation. Moreover, we have isomorphisms $h_{n,k}:\pi_{1}(\mathbb{H}^{+}_{n,k},b_{0})\rightarrow F^{+}_{n,k}$, mapping $[\rho_{i,j}]\mapsto\rho_{i,j}$ and $[\ell_{i}]\mapsto\ell_{i}$.

Let $R_{n,k}:{\mathcal{W}}^{+}_{n,k+1}\rightarrow{\mathcal{W}}^{+}_{n,k}$ denote the function that deletes every occurrence of the letters $\ell_{k+1}$ and $\ell_{k+1}^{-1}$ from a word. Let $S_{n,k}:{\mathcal{W}}^{+}_{n,k}\rightarrow F^{+}_{n,k}$ be the function that maximally cancels words and put $T_{n,k}=S_{n,k}\circ R_{n,k}|_{F^{+}_{n,k+1}}:F^{+}_{n,k+1}\rightarrow F^{+}_{n,k}$. Then $R_{n,k}$, $S_{n,k}$, and $T_{n,k}$ are monoid/group homomorphisms.

Define $g_{n,k}=h_{n,k}\circ r_{n,k\#}:\pi_{1}(\mathbb{H}^{+}_{n},b_{0})\rightarrow F^{+}_{n,k}\subseteq{\mathcal{W}}^{+}_{n,k}$. Since $T_{n,k}\circ g_{n,k+1}=g_{n,k}:\pi_{1}(\mathbb{H}^{+}_{n},b_{0})\rightarrow F^{+}_{n,k}$ and $T_{n,k}\circ h_{n,k+1}=h_{n,k}\circ r_{n,k\#}:\pi_{1}(\mathbb{H}^{+}_{n,k+1},b_{0})\rightarrow F^{+}_{n,k}$, we have, for each $n$, as in the proof of Lemma 2.3, an injective homomorphism into an inverse limit $F^{+}_{n}$ of free groups:

[TABLE]

(Note that $g_{n}$ is not surjective and that $F^{+}_{n}$ is not free.)

Remark 5.1**.**

The image of $g_{n}$ equals the locally eventually constant sequences, where $(y_{n,k})_{k\geqslant n-1}\in F^{+}_{n}$ is called locally eventually constant if for every $m\geqslant n-1$, the sequence $(R_{n,m}\circ R_{n,m+1}\circ\cdots\circ R_{n,k-1}(y_{n,k}))_{k\geqslant m}$ is eventually constant [27].

For every $n\in\mathbb{N}$, $x\in\pi_{1}(\mathbb{H}^{+}_{n},b_{0})$, and $m\geqslant n-1$, the sequence $(R_{n,m}\circ R_{n,m+1}\circ\cdots\circ R_{n,k-1}(g_{n,k}(x)))_{k\geqslant m}$ of (unreduced) words is eventually constant, so that we may define functions $\omega_{n,m}:\pi_{1}(\mathbb{H}^{+}_{n},b_{0})\rightarrow\mathcal{W}^{+}_{n,m}$ by

[TABLE]

for sufficiently large $k$, and obtain commutative diagrams

[TABLE]

with injective functions $\omega_{n}=(\omega_{n,k})_{k\geqslant n-1}$ that output stabilized word sequences. (See [12] or [24], for example.)

For $k\geqslant 2n+1$, let $D_{n}:{\mathcal{W}}^{+}_{n,k}\rightarrow{\mathcal{W}}^{+}_{n+1,k}$ be the monomorphism that replaces every occurrence of the letter $\ell_{n}$ by $\rho_{n,1}\ell_{2n}\rho_{n,1}^{-1}\rho_{n,2}\ell_{2n+1}\rho_{n,2}^{-1}$ and every occurrence of the letter $\ell_{n}^{-1}$ by $\rho_{n,2}\ell_{2n+1}^{-1}\rho_{n,2}^{-1}\rho_{n,1}\ell_{2n}^{-1}\rho_{n,1}^{-1}$. Then, for every $k\geqslant 2n+1$, we obtain the following commutative diagram:

[TABLE]

Note: For every $\omega\in F^{+}_{n,k}$, the word $D_{n}(\omega)$ is reduced.

Remark 5.2**.**

In view of the above, the direct limit structure of Lemma 2.4 suggests the possibility of labelling the elements of $\pi_{1}(\mathbb{P},b_{0})$ using sequences of finite words over finite alphabets that gradually exclude all letters $\ell_{n}^{\pm 1}$ in favor of only using the conjugating letters $\rho_{n,j}^{\pm 1}$. Since such a shift causes conjugating pairs to become adjacent in words at later levels, we use monoid structures to prevent their cancellation. This, in turn, requires us to work with $D_{n}:{\mathcal{W}}^{+}_{n,k}\rightarrow{\mathcal{W}}^{+}_{n+1,k}$ in what follows, at a level just deep enough to stabilize the appropriate word sequences.

Recall that ${\mathcal{W}}^{+}_{n+1,n}$ is the set of finite words over $\{\rho_{i,j}^{\pm 1}\mid 1\leqslant i\leqslant n,1\leqslant j\leqslant 2\}$. In particular, the set ${\mathcal{W}}^{+}_{1,0}$ contains only one element: the empty word.

Let $E_{n-1}:{\mathcal{W}}^{+}_{n+1,n}\rightarrow{\mathcal{W}}^{+}_{n,n-1}$ denote the epimorphism deleting every occurrence of the letters $\rho^{\pm 1}_{n,1}$ and $\rho^{\pm 1}_{n,2}$. Then, for $k\geqslant 2n+1$, we obtain commutative trapezoids:

[TABLE]

Theorem 5.3**.**

There is a well-defined injective function

[TABLE]

defined as follows: for a given $[\alpha]\in\pi_{1}(\mathbb{P},b_{0})$ and sufficiently large $n\geqslant 2$, choose $[\beta]\in\pi_{1}(\mathbb{H}_{n}^{+},b_{0})$ with $\iota_{n\#}([\beta])=[\alpha]$ and put $\chi_{n-1}([\alpha])=\omega_{n,n-1}([\beta])\in{\mathcal{W}}^{+}_{n,n-1}$.

Proof.

First we show that $\chi$ is well-defined. By Lemmas 2.4 and 2.5 it suffices to show that for any $[\beta]\in\pi_{1}(\mathbb{H}_{n}^{+},b_{0})$, we have $E_{n-1}(\omega_{n+1,n}(d_{n\#}([\beta])))=\omega_{n,n-1}([\beta])$, making the following diagram commute:

[TABLE]

(Recall that the underlying set of a direct limit of groups is the direct limit of the underlying sets.) To this end, put $\beta^{\prime}=d_{n}\circ\beta$. Then $[\beta^{\prime}]\in\pi_{1}(\mathbb{H}_{n+1}^{+},b_{0})$. By definition of $\omega_{n,n-1}$ and $\omega_{n+1,n}$, respectively, for sufficiently large $k\geqslant 2n+1$, we have (cf. Remark 5.1)

[TABLE]

and

[TABLE]

Noting that $g_{n+1,k}([\beta^{\prime}])=h_{n+1,k}\circ r_{n+1,k\#}([\beta^{\prime}])=h_{n+1,k}\circ r_{n+1,k\#}\circ d_{n\#}([\beta])=h_{n+1,k}\circ d_{n\#}\circ r_{n,k\#}([\beta])=D_{n}\circ h_{n,k}\circ r_{n,k\#}([\beta])=D_{n}\circ g_{n,k}([\beta])$ and that

[TABLE]

(see Diagram (5.3)) we obtain the desired equality:

[TABLE]

Now we show that $\chi$ is injective. Suppose $[\alpha^{(1)}]\not=[\alpha^{(2)}]\in\pi_{1}(\mathbb{P},b_{0})$. Choose $n\geqslant 2$ sufficiently large (as in the proof of Lemma 2.4) so that $\beta^{(s)}([0,1])\subseteq\mathbb{H}_{n}^{+}$, where $\beta^{(s)}=d_{n-1}\circ d_{n-2}\circ\cdots\circ d_{1}\circ\alpha^{(s)}$ for $s\in\{1,2\}$. Then $\iota_{n\#}([\beta^{(s)}])=[\alpha^{(s)}]$ and $[\beta^{(1)}]\not=[\beta^{(2)}]\in\pi_{1}(\mathbb{H}^{+}_{n},b_{0})$. Hence, there is an $m\geqslant n$ such that $\omega_{n,m-1}([\beta^{(1)}])\not=\omega_{n,m-1}([\beta^{(2)}])$. We may assume that $m$ is even. Put $\gamma^{(s)}=d_{m-1}\circ d_{m-2}\circ\cdots\circ d_{n}\circ\beta^{(s)}$. Choose $k\geqslant 2(m-1)+1$ sufficiently large, such that for $s\in\{1,2\}$, we have:

[TABLE]

For $n\leqslant j\leqslant m-1$, let $\overline{D}_{j,m-1}:\mathcal{W}_{j,m-1}^{+}\rightarrow\mathcal{W}_{j+1,m-1}^{+}$ be the monomorphism that replaces every occurrence of the letter $\ell_{j}$ (respectively $\ell_{j}^{-1}$) by $\rho_{j,1}\ell_{2j}\rho_{j,1}^{-1}\rho_{j,2}\ell_{2j+1}\rho_{j,2}^{-1}$ (respectively $\rho_{j,2}\ell_{2j+1}^{-1}\rho_{j,2}^{-1}\rho_{j,1}\ell_{2j}^{-1}\rho_{j,1}^{-1}$) if $2j+1\leqslant m-1$, but instead replaces it by $\rho_{j,1}\rho_{j,1}^{-1}\rho_{j,2}\rho_{j,2}^{-1}$ (respectively $\rho_{j,2}\rho_{j,2}^{-1}\rho_{j,1}\rho_{j,1}^{-1}$) if $2j\geqslant m$.

Since each $\overline{D}_{j,m-1}$ with $n\leqslant j\leqslant m-1$ is injective, so is their composition $\overline{D}=\overline{D}_{m-1,m-1}\circ\overline{D}_{m-2,m-1}\circ\cdots\circ\overline{D}_{n,m-1}$. Moreover, the following diagram commutes:

[TABLE]

Hence, for $s\in\{1,2\}$, we have

[TABLE]

Since $\omega_{n,m-1}([\beta^{(1)}])\neq\omega_{n,m-1}([\beta^{(2)}])$ and $\overline{D}$ is injective, we have $\chi_{m-1}([\alpha^{(1)}])\neq\chi_{m-1}([\alpha^{(2)}])$. ∎

Remark 5.4**.**

Although $\bigcup_{i=1}^{\infty}(B_{i,1}\cup B_{i,2})\subseteq\mathbb{P}$ is a bouquet of circles that is not homeomorphic to $\mathbb{H}$, one can algebraically set up a commutative diagram as follows:

[TABLE]

However, there does not exist an injective homomorphism

[TABLE]

because $\pi_{1}(\mathbb{P},b_{0})$ is not residually free (Remark 3.9).

6. The homotopically path Hausdorff property

Definition 6.1** (Homotopically path Hausdorff [20]).**

A path-connected space $X$ is called homotopically path Hausdorff if for every two paths $\alpha,\beta:[0,1]\rightarrow X$ with $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$ such that $\alpha\cdot\beta^{-}$ is not null-homotopic, there is a partition $0=t_{0}<t_{1}<\cdots<t_{n}=1$ of $[0,1]$ and open subsets $U_{1},U_{2},\dots,U_{n}$ of $X$ with $\alpha([t_{i-1},t_{i}])\subseteq U_{i}$ for all $1\leqslant i\leqslant n$ and with the property that if $\gamma:[0,1]\rightarrow X$ is any path with $\gamma([t_{i-1},t_{i}])\subseteq U_{i}$ for all $1\leqslant i\leqslant n$ and with $\gamma(t_{i})=\alpha(t_{i})$ for all $0\leqslant i\leqslant n$, then $\gamma\cdot\beta^{-}$ is not null-homotopic.

Remark 6.2**.**

We recall from [2] that a connected and locally path-connected space $X$ is homotopically path Hausdorff if and only if $\pi_{1}(X,x)$ is T1 in the quotient topology induced by the compact-open topology on the loop space $\Omega(X,x)$.

Remark 6.3**.**

If the natural homomorphism $\pi_{1}(X,x)\hookrightarrow\check{\pi}_{1}(X,x)$ is injective, then $X$ is homotopically path Hausdorff [20]. However, the converse does not hold in general [20, Example $Y^{\prime}$].

Theorem 6.4**.**

$\mathbb{P}$* is homotopically path Hausdorff.*

Proof.

Let $1\not=[\alpha]\in\pi_{1}(\mathbb{P},b_{0})$. We wish to find a partition $0=t_{0}<t_{1}<\cdots<t_{s}=1$ and open subsets $U_{1},U_{2},\dots,U_{s}\subseteq\mathbb{P}$ with $\alpha([t_{i-1},t_{i}])\subseteq U_{i}$ for all $1\leqslant i\leqslant s$ such that the following property holds: if $\gamma:[0,1]\rightarrow\mathbb{P}$ is any loop with $\gamma(t_{i})=\alpha(t_{i})$ for all $0\leqslant i\leqslant s$ and $\gamma([t_{i-1},t_{i}])\subseteq U_{i}$ for all $1\leqslant i\leqslant s$, then $[\gamma]\not=1\in\pi_{1}(\mathbb{P},b_{0})$. (This property is preserved if we add a subdivision point $t^{\prime}\in(t_{i-1},t_{i})$ and choose $U^{\prime}=U_{i}$. Therefore, checking this statement for all essential loops $\alpha$ based at $b_{0}$, validates Definition 6.1 for all essential loops $\alpha\cdot\beta^{-}$.)

By Theorem 5.3, there is an $n\in\mathbb{N}$ such that for $\beta=d_{n-1}\circ d_{n-2}\circ\cdots\circ d_{1}\circ\alpha$, we have $\beta([0,1])\subseteq\mathbb{H}^{+}_{n}$ and $\chi_{n-1}([\alpha])=\omega_{n,n-1}([\beta])\in{\mathcal{W}}^{+}_{n,n-1}$ is not the empty word. Choose $k\in\mathbb{N}$ sufficiently large, so that

[TABLE]

Consider $\beta:[0,1]\rightarrow\mathbb{H}^{+}_{n}\subseteq\mathbb{P}^{+}_{n}$ and $r_{n,k}:\mathbb{H}^{+}_{n}\rightarrow\mathbb{H}^{+}_{n,k}$. Since $\mathbb{H}^{+}_{n,k}$ is a finite bouquet of circles, there is an open cover $\mathcal{B}$ of $\mathbb{H}^{+}_{n,k}$ with the following property: if $\eta,\tau:([0,1],\{0,1\})\rightarrow(\mathbb{H}^{+}_{n,k},b_{0})$ are two loops that are $\mathcal{B}$-close, i.e., if for every $t\in[0,1]$, there is a $B\in{\mathcal{B}}$ with $\{\eta(t),\tau(t)\}\subseteq B$, then $[\eta]=[\tau]\in\pi_{1}(\mathbb{H}^{+}_{n,k},b_{0})$. Choose an open cover $\{W_{j}\mid j\in J\}$ of $\mathbb{H}^{+}_{n}$ and a cover $\{V_{j}\mid j\in J\}$ of $\mathbb{H}^{+}_{n}$ by open subsets of $\mathbb{P}^{+}_{n}$ such that $\{W_{j}\mid j\in J\}$ refines $\{(r_{n,k})^{-1}(B)\mid B\in{\mathcal{B}}\}$ and such that each $V_{j}$ deformation retracts onto $W_{j}$. (See proof of Theorem 4.5.) Choose a partition $0=t_{0}<t_{1}<\cdots<t_{s}=1$ and indices $j_{1},j_{2},\dots,j_{s}\in J$ such that $\beta([t_{i-1},t_{i}])\subseteq V_{j_{i}}$ for $1\leqslant i\leqslant s$. Put $U_{i}=(d_{n-1}\circ d_{n-2}\circ\cdots\circ d_{1})^{-1}(V_{j_{i}})$ for $1\leqslant i\leqslant s$.

Now, let $\gamma:[0,1]\rightarrow\mathbb{P}$ be a loop with $\gamma(t_{i})=\alpha(t_{i})$ for all $0\leqslant i\leqslant s$ and $\gamma([t_{i-1},t_{i}])\subseteq U_{i}$ for all $1\leqslant i\leqslant s$. Then $\gamma:[0,1]\rightarrow\mathbb{P}$ is homotopic (relative to its endpoints) to a loop $\gamma^{\prime}:[0,1]\rightarrow\mathbb{H}^{+}_{n}\subseteq\mathbb{P}$ such that $r_{n,k}\circ\gamma^{\prime}:[0,1]\rightarrow\mathbb{H}^{+}_{n,k}$ and $r_{n,k}\circ\beta:[0,1]\rightarrow\mathbb{H}^{+}_{n,k}$ are $\mathcal{B}$-close. Hence, $[r_{n,k}\circ\gamma^{\prime}]=[r_{n,k}\circ\beta]\in\pi_{1}(\mathbb{H}^{+}_{n,k},b_{0})$, so that $g_{n,k}([\gamma^{\prime}])=h_{n,k}([r_{n,k}\circ\gamma^{\prime}])=h_{n,k}([r_{n,k}\circ\beta])=g_{n,k}([\beta])$. Choose $m\geqslant k$ sufficiently large, so that

[TABLE]

which, as a word in ${\mathcal{W}}^{+}_{n,n-1}$, has at least as many letters as

[TABLE]

Hence, $\chi_{n-1}([\gamma^{\prime}])=\omega_{n,n-1}([\gamma^{\prime}])$ is not the empty word. We conclude that $\chi([\gamma])=\chi([\gamma^{\prime}])$ is not trivial so that $[\gamma]\not=1\in\pi_{1}(\mathbb{P},b_{0})$. ∎

7. The 1-UV0 property

Definition 7.1** (1-UV0 [9]).**

We say that $X$ is 1-UV0 at $x\in X$ if for every neighborhood $U$ of $x$ in $X$, there is an open subset $V$ in $X$ with $x\in V\subseteq U$ such that for every map $f:D^{2}\rightarrow X$ from the unit disk with $f(S^{1})\subseteq V$ there is a map $g:D^{2}\rightarrow U$ with $f|_{S^{1}}=g|_{S^{1}}$.

We say that $X$ is 1-UV0 if $X$ is 1-UV0 at every point $x\in X$.

Proposition 7.2**.**

All one-dimensional spaces and all planar spaces are 1-UV0.

Proof.

In a one-dimensional space, every null-homotopic loop contracts within its own image [10, Lemma 2.2]. In a planar space, every null-homotopic loop has a contraction whose diameter equals that of the image of the loop [21, Lemma 13]. ∎

Theorem 7.3**.**

$\mathbb{P}$* is 1-UV0.*

Proof.

It suffices to show that $\mathbb{P}$ is 1-UV0 at $b_{0}$. Let $U$ be an open neighborhood of $b_{0}$ in $\mathbb{P}$.

Recall $U_{n}$ and $f^{-1}_{k}(U_{n})=L^{0}_{k,n}\cup L^{1}_{k,n}\cup L^{2}_{k,n}\subseteq\partial P_{k}$ from the proof of Theorem 4.5. Fix $m\in\mathbb{N}$ such that $b_{0}\in U_{m}\subseteq U\cap\mathbb{H}$. For each $k\in\mathbb{N}$, choose three pairwise disjoint open neighborhoods $M^{0}_{k},M^{1}_{k},M^{2}_{k}$ of $L^{0}_{k,m},L^{1}_{k,m},L^{2}_{k,m}$ in $P_{k}$, respectively, such that, for $i\in\{0,1,2\}$, $M^{i}_{k}\cap\partial P_{k}=L^{i}_{k,m}$, $M^{i}_{k}\cap P^{\circ}_{k}\subseteq U\cap P^{\circ}_{k}$, and $M^{i}_{k}$ deformation retracts onto $L^{i}_{k,m}$. Define $V^{\prime}_{k,m}=M^{0}_{k}\cup M^{1}_{k}\cup M^{2}_{k}$ and $V=\bigcup_{k\in\mathbb{N}}f_{k}(V^{\prime}_{k,m})$. Then $V$ is an open subset of $\mathbb{P}$ with $b_{0}\in V\subseteq U$.

Let $f:D^{2}\rightarrow\mathbb{P}$ be a map with $f(S^{1})\subseteq V$. We will show that $\alpha=f|_{S^{1}}$ contracts within $V$. Since $V$ is path connected, we may assume that $\alpha$ is a loop based at $b_{0}$ and show that $[\alpha]=1\in\pi_{1}(V,b_{0})$. Since $V$ deformation retracts onto $U_{m}\subseteq\mathbb{H}$, we may assume that $\alpha$ lies in $\mathbb{H}$. Since $[\alpha]=1\in\pi_{1}(\mathbb{P},b_{0})$, we have $[\alpha]=1\in\pi_{1}(\mathbb{H},b_{0})$ by Lemma 2.5. As $\mathbb{H}$ is one-dimensional, this implies that $\alpha$ contracts within its own image. ∎

8. Generalized covering projections and the homotopically Hausdorff property

In this section, we briefly review generalized covering projections. Let $X$ be a path-connected topological space and $H\leqslant\pi_{1}(X,x_{0})$. Even if there is no classical covering projection $p:(\widetilde{X},\widetilde{x})\rightarrow(X,x_{0})$ with $p_{\#}\pi_{1}(\widetilde{X},\widetilde{x})=H$ (see Remark 3.3), there might be a generalized one:

Definition 8.1** (Generalized covering projection [1, 22]).**

Let $X$ be a path-connected topological space. We call a map $q:\widehat{X}\rightarrow X$ a generalized covering projection if $\widehat{X}$ is nonempty, connected and locally path connected and if for every $\widehat{x}\in\widehat{X}$, for every connected and locally path-connected space $Y$, and for every map $f:(Y,y)\rightarrow(X,q(\widehat{x}))$ with $f_{\#}\pi_{1}(Y,y)\leqslant q_{\#}\pi_{1}(\widehat{X},\widehat{x})$, there is a unique map $g:(Y,y)\rightarrow(\widehat{X},\widehat{x})$ such that $q\circ g=f$.

Remark 8.2**.**

Suppose $q:(\widehat{X},\widehat{x})\rightarrow(X,x_{0})$ is a generalized covering projection. Then $q_{\#}:\pi_{1}(\widehat{X},\widehat{x})\rightarrow\pi_{1}(X,x_{0})$ is injective. If we put $K=q_{\#}\pi_{1}(\widehat{X},\widehat{x})$, then $q:\widehat{X}\rightarrow X$ is characterized as usual, up to equivalence, by the conjugacy class of $K$ in $G=\pi_{1}(X,x_{0})$. Moreover, $Aut(\widehat{X}\stackrel{{\scriptstyle q}}{{\rightarrow}}X)\cong N_{G}(K)/K$, where $N_{G}(K)$ denotes the normalizer of $K$ in $G$. (The standard arguments apply [29, 2.3.5 & 2.6.2].)

If it exists, a generalized covering projection can be obtained in the standard way: On the set of all paths $\alpha:([0,1],0)\rightarrow(X,x_{0})$ consider the equivalence relation $\alpha\sim\beta$ if and only if $\alpha(1)=\beta(1)$ and $[\alpha\cdot\beta^{-}]\in H$. Denote the equivalence class of $\alpha$ by $\left<\alpha\right>$ and denote the set of all equivalence classes by $\widetilde{X}_{H}$. Let $\widetilde{x}_{0}$ denote the class containing the constant path at $x_{0}$. We give $\widetilde{X}_{H}$ the topology generated by basis elements of the form $\left<\alpha,U\right>=\{\left<\alpha\cdot\gamma\right>\mid\gamma:([0,1],0)\rightarrow(U,\alpha(1))\}$, where $U$ is an open subset of $X$ and $\left<\alpha\right>\in\widetilde{X}_{H}$ with $\alpha(1)\in U$. Then $\widetilde{X}_{H}$ is connected and locally path connected and the endpoint projection $p_{H}:\widetilde{X}_{H}\rightarrow X$, defined by $p_{H}(\left<\alpha\right>)=\alpha(1)$, is a continuous surjection. Moreover, the map $p_{H}:\widetilde{X}_{H}\rightarrow X$ is open if and only if $X$ is locally path connected.

If $p_{H}:\widetilde{X}_{H}\rightarrow X$ has unique path lifting, then it is a generalized covering projection and, for every $\left<\alpha\right>\in\widetilde{X}_{H}$, $(p_{H})_{\#}:\pi_{1}(\widetilde{X}_{H},\left<\alpha\right>)\rightarrow\pi_{1}(X,\alpha(1))$ is a monomorphism onto $[\alpha^{-}]H[\alpha]$; in particular $(p_{H})_{\#}\pi_{1}(\widetilde{X}_{H},\widetilde{x}_{0})=H$ [22].

If $X$ admits a generalized covering projection $q:(\widehat{X},\widehat{x})\rightarrow(X,x_{0})$ such that $q_{\#}\pi_{1}(\widehat{X},\widehat{x})=H$, then there is a homeomorphism $h:(\widehat{X},\widehat{x})\rightarrow(\widetilde{X}_{H},\widetilde{x}_{0})$ with $p_{H}\circ h=q$ [1].

Remark 8.3**.**

For $p_{H}:\widetilde{X}_{H}\rightarrow X$ to have unique path lifting, every fiber $p_{H}^{-1}(x)$ with $x\in X$ must be $T_{1}$, but not necessarily discrete [22]. Moreover, T1 fibers are not sufficient [4, 30]. Note that these fibers are T1 if and only if for every $x\in X$,

[TABLE]

Definition 8.4** (Homotopically Hausdorff rel. $H$ [22]).**

We call $X$ homotopically Hausdorff relative to $H\leqslant\pi_{1}(X,x_{0})$, if every fiber of $p_{H}:\widetilde{X}_{H}\rightarrow X$ is $T_{1}$. We call $X$ homotopically Hausdorff if it is homotopically Hausdorff relative to $H=\{1\}$.

Remark 8.5**.**

If $X$ is strongly homotopically Hausdorff, then $X$ is homotopically Hausdorff. (Compare the formula of Remark 8.3 with that in Definition 4.1.)

Remark 8.6**.**

There are normal subgroups $H\trianglelefteqslant\pi_{1}(\mathbb{H},b_{0})$, such that $\mathbb{H}$ is not homotopically Hausdorff relative to $H$. For example, $\mathbb{H}$ is not homotopically Hausdorff relative to the commutator subgroup of $\pi_{1}(\mathbb{H},b_{0})$ [4, Example 3.10].

We abbreviate $\widetilde{X}_{\{1\}}$ by $\widetilde{X}$ and $p_{\{1\}}:\widetilde{X}_{\{1\}}\rightarrow X$ by $p:\widetilde{X}\rightarrow X$. Moreover, note that if $H=\{1\}$, then $\left<\alpha\right>=[\alpha]$.

Remark 8.7**.**

If $X$ is homotopically path Hausdorff, then $p:\widetilde{X}\rightarrow X$ has unique path lifting [20].

Remark 8.8**.**

If $X$ is path connected, 1-UV0 and metrizable, then $p:\widetilde{X}\rightarrow X$ has unique path lifting [4].

Theorem 8.9**.**

There exists a generalized covering projection $p:\widetilde{\mathbb{P}}\rightarrow\mathbb{P}$ with $\pi_{1}(\widetilde{\mathbb{P}},\widetilde{b}_{0})=\{1\}$.

Proof.

By Theorem 6.4 and Remark 8.7, $p:\widetilde{\mathbb{P}}\rightarrow\mathbb{P}$ has unique path lifting. Hence, $p:\widetilde{\mathbb{P}}\rightarrow\mathbb{P}$ is a generalized covering projection, $p_{\#}\pi_{1}(\widetilde{\mathbb{P}},\widetilde{b}_{0})=\{1\}$, and $p_{\#}:\pi_{1}(\widetilde{\mathbb{P}},\widetilde{b}_{0})\rightarrow\pi_{1}(\mathbb{P},b_{0})$ is injective. ∎

9. The discrete monodromy property

Let $X$ be a path-connected topological space and $H\leqslant\pi_{1}(X,x_{0})$. Even if the map $p_{H}:\widetilde{X}_{H}\rightarrow X$ does not have unique path lifting, for every path $\beta:[0,1]\rightarrow X$ and every $\left<\alpha\right>\in p_{H}^{-1}(\beta(0))$ there is a continuous standard path lift $\widetilde{\beta}:([0,1],0)\rightarrow(\widetilde{X}_{H},\left<\alpha\right>)$ with $p_{H}\circ\widetilde{\beta}=\beta$, defined by $\widetilde{\beta}(t)=\left<\alpha\cdot\beta_{t}\right>$, where $\beta_{t}(s)=\beta(ts)$.

Based on the standard path lift, we may define the standard monodromy for $p_{H}:\widetilde{X}_{H}\rightarrow X$, as follows. For a path $\beta:[0,1]\rightarrow X$ from $\beta(0)=x$ to $\beta(1)=y$, we define $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(y)$ by $\Phi_{\beta}(\left<\alpha\right>)=\left<\alpha\cdot\beta\right>$.

Clearly, $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(y)$ is a bijective function with inverse $\Phi_{\beta}^{-1}=\Phi_{\beta^{-}}$. However, $\Phi_{\beta}$ need not be continuous. (Such is the case for $(X,x_{0})=(\mathbb{H},b_{0})$ and $H=\{1\}$, although $p:\widetilde{X}\rightarrow X$ has unique path lifting. See [18] for a discussion.)

Remark 9.1**.**

Note that $\Phi_{\beta}$ depends only on the homotopy class $[\beta]$. Moreover, $\left<\alpha\right>\ast\left<\beta\right>:=\Phi_{\beta}(\left<\alpha\right>)$ is a well-defined group operation on $p_{H}^{-1}(x_{0})$ if and only if $H$ is a normal subgroup of $\pi_{1}(X,x_{0})$.

Definition 9.2** (Discrete monodromy).**

We say that $X$ has the discrete monodromy property relative to $H\leqslant\pi_{1}(X,x_{0})$ if for every $x,y\in X$ and for every path$\beta:[0,1]\rightarrow X$ from $\beta(0)=x$ to $\beta(1)=y$, the monodromy $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(y)$ is either the identity function or its graph is a discrete subset of $p_{H}^{-1}(x)\times p_{H}^{-1}(y)\subseteq\widetilde{X}_{H}\times\widetilde{X}_{H}$. We say that $X$ has the discrete monodromy property if it has the discrete monodromy property relative to $H=\{1\}$.

Remark 9.3**.**

Clearly, if every fiber of $p_{H}:\widetilde{X}_{H}\rightarrow X$ is discrete, then $X$ has the discrete monodromy property relative to $H$. However, the converse does not hold in general: $p:\widetilde{\mathbb{H}}\rightarrow\mathbb{H}$ has the discrete monodromy property (see Proposition 9.13), but $p^{-1}(b_{0})$ is not discrete.

Remark 9.4**.**

• (a)

$\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(y)$ is the identity function if and only if $x=y$ and $[\beta]\in[\alpha^{-}]H[\alpha]$ for all paths $\alpha$ in $X$ from $x_{0}$ to $x$.

• (b)

The graph of the identity function $id_{p_{H}^{-1}(x)}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(x)$ is discrete if and only if $p_{H}^{-1}(x)$ is discrete.

• (c)

$p_{H}^{-1}(x)$ is discrete if and only if for every path $\alpha$ in $X$ from $x_{0}$ to $x$, there is a $U\in{\mathcal{T}}_{x}$ such that $\pi(\alpha,U)\subseteq H$, i.e., $H\pi(\alpha,U)=H$.

Lemma 9.5**.**

Let $H\leqslant\pi_{1}(X,x_{0})$. The graph of $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(y)$ is discrete if and only if for every path $\alpha$ in $X$ from $x_{0}$ to $x$, there are $U\in{\mathcal{T}}_{x}$ and $V\in{\mathcal{T}}_{y}$ such that $H\pi(\alpha,U)\cap H\pi(\alpha\cdot\beta,V)=H$.

Proof.

First, observe that if $f:A\rightarrow B$ is an injective function between topological spaces, then its graph $\Gamma=\{(a,b)\in A\times B\mid f(a)=b\}$ is a discrete subset of $A\times B$ if and only if for every $a\in A$ there are $U\in{\mathcal{T}}_{a}$ and $V\in{\mathcal{T}}_{f(a)}$ such that $f(U)\cap V=\{f(a)\}$.

Now, suppose that the graph of $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(y)$ is discrete and let $\left<\alpha\right>\in p_{H}^{-1}(x)$. Choose $U\in{\mathcal{T}}_{x}$ and $V\in{\mathcal{T}}_{y}$ with

[TABLE]

Let $g\in H\pi(\alpha,U)\cap H\pi(\alpha\cdot\beta,V)$. Then $g=h_{1}[\alpha\cdot\gamma\cdot\alpha^{-}]$ for some $h_{1}\in H$ and some loop $\gamma$ in $U$, and $g=h_{2}[\alpha\cdot\beta\cdot\delta\cdot\beta^{-}\cdot\alpha^{-}]$ for some $h_{2}\in H$ and some loop $\delta$ in $V$. Hence, $\Phi_{\beta}(\left<\alpha\cdot\gamma\right>)=\left<\alpha\cdot\gamma\cdot\beta\right>=\left<\alpha\cdot\beta\cdot\delta\right>\in\left<\alpha\cdot\beta,V\right>\cap p_{H}^{-1}(y)$. Also, $\Phi_{\beta}(\left<\alpha\cdot\gamma\right>)\in\Phi_{\beta}(\left<\alpha,U\right>\cap p_{H}^{-1}(x))$. Therefore, $\left<\alpha\cdot\gamma\cdot\beta\right>=\Phi_{\beta}(\left<\alpha\cdot\gamma\right>)=\left<\alpha\cdot\beta\right>$, so that $g=h_{1}[\alpha\cdot\gamma\cdot\alpha^{-}]\in H$.

Conversely, let $\left<\alpha\right>\in p_{H}^{-1}(x)$ and suppose $U\in{\mathcal{T}}_{x}$ and $V\in{\mathcal{T}}_{y}$ are such that $H\pi(\alpha,U)\cap H\pi(\alpha\cdot\beta,V)=H$. Let

[TABLE]

Then $\widetilde{x}=\left<\alpha\cdot\gamma\cdot\beta\right>$ for some loop $\gamma$ in $U$, and $\widetilde{x}=\left<\alpha\cdot\beta\cdot\delta\right>$ for some loop $\delta$ in $V$. Then $[\alpha\cdot\gamma\cdot\alpha^{-}][\alpha\cdot\beta\cdot\delta^{-}\cdot\beta^{-}\cdot\alpha^{-}]\in H$, so that $[\alpha\cdot\gamma\cdot\alpha^{-}]\in H\pi(\alpha,U)\cap H\pi(\alpha\cdot\beta,V)=H$. Hence, $\widetilde{x}=\left<\alpha\cdot\gamma\cdot\beta\right>=\left<\alpha\cdot\beta\right>$. ∎

Remark 9.6**.**

In order to apply Lemma 9.5, there is no need to check every path $\alpha$: if $\alpha$ and $\alpha^{\prime}$ are two paths from $x_{0}$ to $x$ such that $[\alpha^{\prime}\cdot\alpha^{-}]H=H[\alpha^{\prime}\cdot\alpha^{-}]$, then we have $H\pi(\alpha,U)\cap H\pi(\alpha\cdot\beta,V)=H$ if and only if $H\pi(\alpha^{\prime},U)\cap H\pi(\alpha^{\prime}\cdot\beta,V)=H$.

Remark 9.7**.**

The space $(X,x_{0})=(\mathbb{H}\times[0,1],(b_{0},0))$ is the prototypical example of a space that does not have the discrete monodromy property, although $p:\widetilde{X}\rightarrow X$ has unique path lifting. Observe that for the path $\beta(t)=(b_{0},t)$ from $x=(b_{0},0)$ to $y=(b_{0},1)$, the graph of $\Phi_{\beta}:p^{-1}(x)\rightarrow p^{-1}(y)$ is not discrete.

Definition 9.8** (Locally quasinormal [18]).**

A subgroup $H\leqslant\pi_{1}(X,x_{0})$ is called locally quasinormal if for every $x\in X$, for every path $\alpha$ in $X$ from $\alpha(0)=x_{0}$ to $\alpha(1)=x$, and for every $U\in{\mathcal{T}}_{x}$, there is a $V\in{\mathcal{T}}_{x}$ such that $x\in V\subseteq U$ and $H\pi(\alpha,V)=\pi(\alpha,V)H$.

Remark 9.9**.**

Clearly, every normal subgroup of $\pi_{1}(X,x_{0})$ is locally quasinormal. Combining [23, Lemma 5.2] with Remark 9.4(c), we see that if $X$ is locally path connected, then every open subgroup of $\pi_{1}(X,x_{0})$ (in the topology of Remark 6.2) is locally quasinormal. For example, the nontrivial subgroup $K\leqslant\pi_{1}(\mathbb{H},b_{0})$ from [23] is open, while it does not contain any nontrivial normal subgroup of $\pi_{1}(\mathbb{H},b_{0})$.

The following is a straightforward variation on [18, Lemma 3.2]:

Lemma 9.10**.**

Let $H\leqslant\pi_{1}(X,x_{0})$, $x\in X$, $\alpha$ be a path in $X$ from $x_{0}$ to $x$, and $U\in{\mathcal{T}}_{x}$. Then the following are equivalent:

• (a)

$H\pi(\alpha,U)=\pi(\alpha,U)H$.

• (b)

For every $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(x)$ with $\Phi_{\beta}(\left<\alpha\right>)\in\left<\alpha,U\right>\cap p_{H}^{-1}(x)$, we have $\Phi_{\beta}(\left<\alpha,U\right>\cap p_{H}^{-1}(x))\subseteq\left<\alpha,U\right>\cap p_{H}^{-1}(x)$.

We include the proof for completeness.

Proof.

(i) First, assume that $H\pi(\alpha,U)=\pi(\alpha,U)H$ and $\Phi_{\beta}(\left<\alpha\right>)\in\left<\alpha,U\right>\cap p_{H}^{-1}(x)$. Then $\left<\alpha\cdot\beta\right>=\Phi_{\beta}(\left<\alpha\right>)=\left<\alpha\cdot\delta\right>$ for some loop $\delta$ in $U$. So, $[\beta]=[\alpha^{-}]h[\alpha\cdot\delta]$ for some $h\in H$. Let $\left<\gamma\right>\in\left<\alpha,U\right>\cap p_{H}^{-1}(x)$. Then $[\gamma]=h^{\prime}[\alpha\cdot\delta^{\prime}]$ for some $h^{\prime}\in H$ and some loop $\delta^{\prime}$ in $U$. Since $[\alpha\cdot\delta^{\prime}\cdot\alpha^{-}]h\in\pi(\alpha,U)H=H\pi(\alpha,U)$, we have $[\alpha\cdot\delta^{\prime}\cdot\alpha^{-}]h=h^{\prime\prime}[\alpha\cdot\delta^{\prime\prime}\cdot\alpha^{-}]$ for some $h^{\prime\prime}\in H$ and some loop $\delta^{\prime\prime}$ in $U$. Therefore, we have $[\gamma\cdot\beta]=h^{\prime}[\alpha\cdot\delta^{\prime}\cdot\alpha^{-}]h[\alpha\cdot\delta]=h^{\prime}h^{\prime\prime}[\alpha\cdot\delta^{\prime\prime}\cdot\alpha^{-}][\alpha\cdot\delta]=h^{\prime}h^{\prime\prime}[\alpha\cdot\delta^{\prime\prime}\cdot\delta]$. Hence, $\Phi_{\beta}(\left<\gamma\right>)=\left<\gamma\cdot\beta\right>=\left<\alpha\cdot\delta^{\prime\prime}\cdot\delta\right>\in\left<\alpha,U\right>\cap p_{H}^{-1}(x)$.

(ii) Now, assume that $\Phi_{\beta}(\left<\alpha,U\right>\cap p_{H}^{-1}(x))\subseteq\left<\alpha,U\right>\cap p_{H}^{-1}(x)$ whenever $\Phi_{\beta}(\left<\alpha\right>)\in\left<\alpha,U\right>\cap p_{H}^{-1}(x)$. It suffices to show that $\pi(\alpha,U)H\subseteq H\pi(\alpha,U)$. Let $[\tau]\in\pi(\alpha,U)H$. Then $[\tau]=[\alpha\cdot\delta\cdot\alpha^{-}][\gamma]$ for some loop $\delta$ in $U$ and some $[\gamma]\in H$. Put $\beta=\alpha^{-}\cdot\gamma\cdot\alpha$. Then $\Phi_{\beta}(\left<\alpha\right>)=\left<\alpha\right>$. Hence, $\left<\alpha\cdot\delta\cdot\beta\right>=\Phi_{\beta}(\left<\alpha\cdot\delta\right>)=\left<\alpha\cdot\delta^{\prime}\right>$ for some loop $\delta^{\prime}$ in $U$. Therefore, $[\tau]=[\alpha\cdot\delta\cdot\beta\cdot(\delta^{\prime})^{-}\cdot\alpha^{-}][\alpha\cdot\delta^{\prime}\cdot\alpha^{-}]\in H\pi(\alpha,U).$ ∎

Proposition 9.11**.**

Let $H\leqslant\pi_{1}(X,x_{0})$ be locally quasinormal. If $X$ has the discrete monodromy property relative to $H$, then every fiber of $p_{H}:\widetilde{X}_{H}\rightarrow X$ is T1.

Proof.

Suppose there is an $x\in X$ such that $p_{H}^{-1}(x)$ is not T1. Then there are $\left<\alpha\right>,\left<\gamma\right>\in p_{H}^{-1}(x)$ with $\left<\alpha\right>\not=\left<\gamma\right>$ such that for every $W\in{\mathcal{T}}_{x}$ we have $\left<\gamma\right>\in\left<\alpha,W\right>\cap p_{H}^{-1}(x)$. Let any $U\in{\mathcal{T}}_{x}$ be given. Choose $V\in{\mathcal{T}}_{x}$ with $x\in V\subseteq U$ and $H\pi(\alpha,V)=\pi(\alpha,V)H$. Put $\widetilde{V}=\left<\alpha,V\right>\cap p_{H}^{-1}(x)$ and $\beta=\alpha^{-}\cdot\gamma$. Then $\Phi_{\beta}(\left<\alpha\right>)=\left<\gamma\right>\in\widetilde{V}$. By Lemma 9.10, $\Phi_{\beta}(\left<\gamma\right>)\in\widetilde{V}$. Hence, $(\left<\alpha\right>,\Phi_{\beta}(\left<\alpha\right>))\not=(\left<\gamma\right>,\Phi_{\beta}(\left<\gamma\right>))$ are both elements of $\widetilde{V}\times\widetilde{V}$. We conclude that $\Phi_{\beta}:p_{H}^{-1}(x)\rightarrow p_{H}^{-1}(x)$ is not the identity function and that its graph is not discrete. ∎

Corollary 9.12**.**

If $X$ has the discrete monodromy property, then $X$ is homotopically Hausdorff.

Proof.

Since $H=\{1\}\leqslant\pi_{1}(X,x_{0})$ is locally quasinormal, this follows from Proposition 9.11. ∎

The proof of the following proposition is modelled on [14] and [8].

Proposition 9.13**.**

All one-dimensional metric spaces and all planar spaces have the discrete monodromy property.

Proof.

Let $x,y\in X$ and let $\beta:[0,1]\rightarrow X$ be a path from $\beta(0)=x$ to $\beta(1)=y$. If $\beta$ is a loop, we assume that it is essential. Let $\alpha$ be any path from $x_{0}$ to $x$. We wish to find open neighborhoods $U$ and $V$ of $x$ and $y$, respectively, such that $\pi(\alpha,U)\cap\pi(\alpha\cdot\beta,V)=\{1\}\leqslant\pi_{1}(X,x_{0})$.

(a) Suppose $X\subseteq\mathbb{R}^{2}$. If $x=y$, choose $\epsilon>0$ such that the loop $\beta$ cannot be homotoped within $X$ into $N_{\epsilon}(x)=\{z\in\mathbb{R}^{2}\mid\|x-z\|<\epsilon\}$, relative to its endpoints. (Here we use the fact that $X$ is homotopically Hausdorff; see Remark 8.3.) If $x\not=y$, choose any $\epsilon$ with $0<\epsilon<\|x-y\|/2$.

Suppose that $\partial N_{r}(x)\subseteq X$ for all $0<r<\epsilon$. Then $N_{\epsilon}(x)\subseteq X$. In this case, taking $U=N_{\epsilon}(x)$ gives $\pi(\alpha,U)=\{1\}$. So, by making $\epsilon$ smaller, if necessary, we may assume that $X\cap\partial N_{\epsilon}(x)\not=\partial N_{\epsilon}(x)$.

Put $U=X\cap N_{\epsilon}(x)$ and $V=X\cap N_{\epsilon}(y)$. Suppose, to the contrary, that there are essential loops $\delta$ and $\tau$ in $U$ and $V$, respectively, with $[\alpha\cdot\delta\cdot\alpha^{-}]=[\alpha\cdot\beta\cdot\tau\cdot\beta^{-}\cdot\alpha^{-}]$, i.e., $[\delta]=[\beta\cdot\tau\cdot\beta^{-}]$. Then there is a map $h:A\rightarrow X$ from an annulus $A$ whose boundary components $J_{1}$ and $J_{2}$ map to $\delta$ and $\tau$, respectively, along with a diametrical arc $a\subseteq A$ connecting $J_{1}$ to $J_{2}$ that maps to $\beta$.

If $x\not=y$, then $h^{-1}(X\cap\partial N_{\epsilon}(x))$ clearly separates $J_{1}$ from $J_{2}$ in $A$. However, this is also true if $x=y$; for otherwise there is an arc $a^{\prime}\subseteq A$ connecting $J_{1}$ to $J_{2}$ which $h$ maps to a path $\beta^{\prime}$ in $U$, so that $\beta$ is homotopic within $X$, relative to its endpoints, to the concatenation of an initial subpath $\delta^{\prime}$ of $\delta$, the path $\beta^{\prime}$, a terminal subpath $\tau^{\prime}$ of $\tau$, and a path of the form $\tau\cdot\tau\cdots\tau$ or $\tau^{-}\cdot\tau^{-}\cdots\tau^{-}$, all of which lie in $U$, violating the choice of $\epsilon$. Therefore, as in the proof of [8, Lemma 5.5], the loop $\delta$ contracts within $X$; a contradiction.

(b) Suppose $X$ is a one-dimensional metric space. We may assume that $\beta$ is a reduced non-degenerate path (possibly a loop). Choose open neighborhoods $U$ and $V$ of $x$ and $y$ in $X$, respectively, such that $\beta$ is not contained in $U\cup V$. Suppose, to the contrary, that there are essential loops $\delta$ and $\tau$ in $U$ and $V$, respectively, with $[\delta]=[\beta\cdot\tau\cdot\beta^{-}]$. We may assume that both $\delta$ and $\tau$ are reduced. Then $\beta([0,1])\subseteq\delta([0,1])\cup\tau([0,1])\subseteq U\cup V$ (see Lemma 4.4); a contradiction. ∎

Theorem 9.14**.**

$\mathbb{P}$* has the discrete monodromy property.*

Proof.

Let $1\not=[\beta]\in\pi_{1}(\mathbb{P},b_{0})$. In view of Remark 9.4(a), Lemma 9.5 and Remark 9.6, and since $\mathbb{P}$ locally contractible at every point other than $b_{0}$, it suffices to find an open neighborhood $V$ of $b_{0}$ in $\mathbb{P}$ such that $\pi(c_{b_{0}},V)\cap\pi(\beta,V)=\{1\}$, where $c_{b_{0}}$ denotes the constant path at $b_{0}$.

Choose $n\in\mathbb{N}$ such that $\beta([0,1])\cap P^{\circ}_{i}=\emptyset$ for all $i>n$. Put $\beta^{\prime}=d_{n}\circ d_{n-1}\circ\cdots\circ d_{1}\circ\beta$. Then $\beta^{\prime}([0,1])\subseteq\mathbb{H}^{+}_{n+1}$ and $[\beta]=[\beta^{\prime}]\in\pi_{1}(\mathbb{P},b_{0})$. So, we may assume that $\beta$ is a reduced loop in $\mathbb{H}^{+}_{n+1}$. Increasing $n$ if necessary, we may assume that $\beta$ traverses one of the circles $B_{i,j}$ with $i\in\{1,2,\dots,n\}$ and $j\in\{1,2\}$. As in the proof of Theorem 7.3, we may construct an open neighborhood $V$ of $b_{0}$ in $\mathbb{P}$ that does not fully contain $B_{i,j}$, such that $V$ deformation retracts onto $\mathbb{H}_{n+1}\subseteq\mathbb{H}$.

Suppose, to the contrary, that there are essential loops $\delta$ and $\tau$ in $V$ such that $[\delta]=[\beta\cdot\tau\cdot\beta^{-}]\in\pi_{1}(\mathbb{P},b_{0})$. We may assume that both $\delta$ and $\tau$ are reduced loops in $\mathbb{H}_{n+1}$. Let $F$ be a homotopy from $\delta$ to $\beta\cdot\tau\cdot\beta^{-}$ (relative to endpoints) within $\mathbb{P}$. Choose $k\geqslant n$ such that the image of $d_{k}\circ d_{k-1}\circ\cdots\circ d_{1}\circ F$ is contained in $\mathbb{H}^{+}_{k+1}$. Let $\beta^{\prime}$, $\delta^{\prime}$ and $\tau^{\prime}$ be the composition of $d_{k}\circ d_{k-1}\circ\cdots\circ d_{1}$ with $\beta$, $\delta$ and $\tau$, respectively. Then $\beta^{\prime}$, $\delta^{\prime}$ and $\tau^{\prime}$ are reduced loops in $\mathbb{H}^{+}_{k+1}$ such that $[\delta^{\prime}]=[\beta^{\prime}\cdot\tau^{\prime}\cdot\beta^{\prime-}]\in\pi_{1}(\mathbb{H}^{+}_{k+1},b_{0})$. However, neither $\delta^{\prime}$ nor $\tau^{\prime}$ traverses $B_{i,j}$, while $\beta^{\prime}$ does; a contradiction. (See Lemma 4.4.) ∎

Consider the subspace $w(X)$ of “wild” points of $X$, defined by

[TABLE]

The following is the main utility for spaces satisfying the discrete monodromy property, as implicitly used in [14] and [8]. The proof is given after Corollary 9.18 below.

Theorem 9.15**.**

Suppose both $X$ and $Y$ have the discrete monodromy property. If $f:X\rightarrow Y$ is a homotopy equivalence with homotopy inverse $g:Y\rightarrow X$, then $f$ maps $w(X)$ homeomorphically onto $w(Y)$, with inverse $g|_{w(Y)}$.

Remark 9.16**.**

In order to see the necessity of the assumptions in Theorem 9.15, consider $X=\mathbb{H}$ and $Y=\mathbb{H}\times[0,1]$. Then $X$ has the discrete monodromy property, $X$ and $Y$ are homotopy equivalent, but $w(X)=\{b_{0}\}$ and $w(Y)=\{b_{0}\}\times[0,1]$ are not homeomorphic.

For a path $\beta:[0,1]\rightarrow X$, we let $\varphi_{\beta}:\pi_{1}(X,\beta(0))\rightarrow\pi_{1}(X,\beta(1))$ be the base point changing isomorphism defined by $\varphi_{\beta}([\delta])=[\beta^{-}\cdot\delta\cdot\beta]$.

Lemma 9.17**.**

Suppose $Y$ has the discrete monodromy property. Let $f,g:X\rightarrow Y$ be maps such that $\varphi_{\beta}\circ f_{\#}=g_{\#}:\pi_{1}(X,x)\rightarrow\pi_{1}(Y,g(x))$ for some $x\in X$ and some path $\beta$ in $Y$ from $f(x)$ to $g(x)$. If $f(x)\not=g(x)$, then there is a $W\in{\mathcal{T}}_{x}$ such that $f_{\#}:\pi_{1}(W,x)\rightarrow\pi_{1}(Y,f(x))$ is trivial.

Proof.

Suppose $f(x)\not=g(x)$. Let $c_{f(x)}$ be the constant path at $f(x)$. By Lemma 9.5, there are $U\in{\mathcal{T}}_{f(x)}$ and $V\in{\mathcal{T}}_{g(x)}$ such that $\pi(c_{f(x)},U)\cap\pi(\beta,V)=\{1\}\leqslant\pi_{1}(Y,f(x))$. Choose $W\in{\mathcal{T}}_{x}$ with $f(W)\subseteq U$ and $g(W)\subseteq V$. Let $\ell$ be a loop in $W$, based at $x$. Then $f_{\#}([\ell])=[f\circ\ell]=[\beta\cdot(g\circ\ell)\cdot\beta^{-}]\in\pi(c_{f(x)},U)\cap\pi(\beta,V)=\{1\}$. ∎

Corollary 9.18**.**

Suppose $X$ is path connected and $Y$ has the discrete monodromy property. If $f,g:X\rightarrow Y$ are homotopic maps and $f_{\#}:\pi_{1}(X,x_{0})\rightarrow\pi_{1}(Y,f(x_{0}))$ is injective, then $f|_{w(X)}=g|_{w{(X)}}$.

Proof.

Let $F:X\times[0,1]\rightarrow Y$ be a map with $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x\in X$. Fix $x\in w(X)$ and let $\beta:[0,1]\rightarrow Y$ be given by $\beta(t)=F(x,t)$. Then $\varphi_{\beta}\circ f_{\#}=g_{\#}:\pi_{1}(X,x)\rightarrow\pi_{1}(Y,g(x))$. Since $x\in w(X)$ and since $f_{\#}:\pi_{1}(X,x)\rightarrow\pi_{1}(Y,f(x))$ is injective, its follows from Lemma 9.17 that $f(x)=g(x)$. ∎

Proof of Theorem 9.15.

Let $f:X\rightarrow Y$ and $g:Y\rightarrow X$ be a pair of homotopy inverses. Then, for every $x\in X$, $f_{\#}:\pi_{1}(X,x)\rightarrow\pi_{1}(Y,f(x))$ is an isomorphism; in particular, it is injective. Therefore, $f(w(X))\subseteq w(Y)$. Similarly, $g(w(Y))\subseteq w(X)$. Since $g\circ f$ is homotopic to the identity it follows from Corollary 9.18 that $g(f(x))=x$ for all $x\in w(X)$. Similarly, $f(g(y))=y$ for all $y\in w(Y)$. ∎

When working with spaces for which homomorphisms between fundamental groups are induced by continuous maps up to base point change, as is the case among all one-dimensional and planar Peano continua [8, 15, 25], the following provides additional utility:

Corollary 9.19**.**

Suppose both $X$ and $Y$ have the discrete monodromy property.Let $\phi:\pi_{1}(X,x_{0})\rightarrow\pi_{1}(Y,y_{0})$ be an isomorphism with $\phi=\varphi_{\alpha}\circ f_{\#}$ and $\phi^{-1}=\varphi_{\beta}\circ g_{\#}$ for some maps $f:X\rightarrow Y$ and $g:Y\rightarrow X$ and some paths $\alpha$ and $\beta$. Then $f$ maps $w(X)$ homeomorphically onto $w(Y)$, with inverse $g|_{w(Y)}$.

Proof.

For every $x\in X$, $f_{\#}:\pi_{1}(X,x)\rightarrow\pi_{1}(Y,f(x))$ is injective and for every $y\in Y$, $g_{\#}:\pi_{1}(Y,y)\rightarrow\pi_{1}(X,g(y))$ is injective. Therefore, $f(w(X))\subseteq w(Y)$ and $g(w(Y))\subseteq w(X)$. Let $x\in w(X)$. Choose a path $\gamma$ in $X$ from $x_{0}$ to $x$. Put $\delta=(g\circ f\circ\gamma)^{-}\cdot(g\circ\alpha)\cdot\beta\cdot\gamma$. Since $\varphi_{(g\circ\alpha)\cdot\beta}\circ(g\circ f)_{\#}=\varphi_{\beta}\circ g_{\#}\circ\varphi_{\alpha}\circ f_{\#}=id:\pi_{1}(X,x_{0})\rightarrow\pi_{1}(X,x_{0})$, we have $\varphi_{\delta}\circ(g\circ f)_{\#}=id:\pi_{1}(X,x)\rightarrow\pi_{1}(X,x)$. By Lemma 9.17, $g(f(x))=x$. Similarly, $f(g(y))=y$ for all $y\in w(Y)$. ∎