# Extreme Audio Time Stretching using Neural Synthesis

###### Abstract

A deep neural network solution for time-scale modification (TSM) focused on
large stretching factors is proposed, targeting environmental sounds.
Traditional TSM artifacts such as transient smearing, loss of presence, and
phasiness are heavily accentuated and cause poor audio quality when the TSM
factor is four or larger. The weakness of established TSM methods, often based
on a phase vocoder structure, lies in the poor description and scaling of the
transient and noise components, or nuances, of a sound. Our novel solution
combines a sines-transients-noise decomposition with an independent WaveNet
synthesizer to provide a better description of the noise component and an
improve sound quality for large stretching factors. Results of a subjective
listening test against four other TSM algorithms are reported, showing the
proposed method to be often superior. The proposed method is stereo compatible
and has a wide range of applications related to the slow motion of media
content.

##  1 Introduction

Time-scale modification (TSM) refers to a change in the duration or the
playback speed of a sound that does not affect its spectral characteristics,
such as pitch, timbre, and brightness [1, 2, 3]. If an audio signal is simply
played at a different sample rate, the frequency content is deemed to be
changed as the formants of the sound are moved. TSM methods are applied to
avoid this phenomenon, aiming to preserve or retrieve the original spectral
characteristics of the sound.

TSM has been long used in speech, e.g. in audio books and language–learning
services [4, 5, 6], music and remixing [7], broadcasting services [2], and
streaming platforms [8]. The ratio between the modified and the original time
support is controlled by the TSM factor $\alpha$, defining time stretching for
$\alpha>1$ and time compression for $\alpha<1$. All of the speech TSM
applications typically involve small TSM factors (0.25 $\leq\alpha\leq$ 4),
and do not allow for more extreme time-scaling operations, as state-of-the-art
TSM algorithms present poor audio quality at large stretch factors [9, 10]. In
such cases, the sounds typically present strong phasiness and heavily smeared
transients, both known artifacts in phase-vocoder-based TSM implementations
[11, 9]. There is however an interest for extreme audio time-stretching in
applications such as slow motion [12] and ambient sound generation [13, 14].

Recent work by Fierro and Välimäki [15] showed that the quality of a TSM
algorithm can be improved by providing a better description of its sines,
transient, and noise (STN) components, which can be separated and individually
processed according to their classification. It was also hinted that the
potential weakness of the fuzzy phase vocoder [9], which received the highest
overall score both on average and for the largest tested $\alpha$ in a recent
comparison of audio TSM methods [10], lies in the poor description and scaling
of the “noisy” component of percussive events, whose smearing is not countered
by the phase randomization and that would not benefit from a full preservation
as the time support of the time-stretched event would be incorrect.

The audio synthesis landscape changed since the introduction of WaveNet, a
deep generative model capable of synthesising raw audio waveforms [16]. The
WaveNet synthesizer can be used for TSM by adjusting the number of audio
samples generated per frame of the local conditioning signal, similarly to how
the hop size between consecutive frames is adjusted from analysis to synthesis
in traditional DSP methods. This idea was first proposed by Huang et al. [17],
although the TSM performance of such a model was not evaluated as time-
stretching was out of its scope.

In this work, we propose a deep learning based method for TSM capable of
producing state-of-the-art results for real-world environmental sounds when
large TSM factors are used. The proposed solution combines established DSP
algorithms and deep learning to produce a hybrid TSM method, whose novelty
lies in the neural resynthesis of the noise component, which is separated from
sines and transient using the STN decomposition.

The rest of this paper is structured as follows. Sec. 2 summarizes the STN
decomposition technique. Sec. 3 describes the proposed WaveNet architecture,
used to synthesize the stretched noise component. Sec. 4 details the TSM
pipeline. Sec. 5 evaluates the proposed method against four previous
techniques, and Sec. 6 concludes.

##  2 STN Decomposition

The STN separation method proposed by Fierro and Välimäki [15] decomposes a
sound into three abstract classes: sines (tonal content), transients
(impulsive events), and noise (sound nuances). The decomposition is achieved
through soft spectral masks that are derived from the signal spectrogram and
allow for perfect reconstruction.

To derive the class masks for an audio signal $x(n)$, median filtering is
first applied to its magnitude spectrum $X(m,k)$ to highlight vertical
(frequency) and horizontal (time) structures [18]:

$X_{\textrm{v}}(m,k)\\\
=\textrm{med}\Big{[}|X(m,k-\frac{L_{\textrm{v}}}{2}+1)|,...,|X(m,k+\frac{L_{\textrm{v}}}{2})|\Big{]}$
(1)

and

$X_{\textrm{h}}(m,k)\\\
=\textrm{med}\Big{[}|X(m-\frac{L_{\textrm{h}}}{2}+1,k)|,...,|X(m+\frac{L_{\textrm{h}}}{2},k)|\Big{]},$
(2)

where $\textrm{med}[\cdot]$ is the median function, and $X_{\textrm{v}}$ and
$X_{\textrm{h}}$ are the resulting horizontally- and vertically-enhanced
magnitude spectrograms, respectively. Parameters $L_{\textrm{h}}$ and
$L_{\textrm{v}}$ are the median filter lengths (in samples) in the time and
frequency directions, respectively.

Matrices $X_{\textrm{h}}$ and $X_{\textrm{v}}$ are then used to extract the
tonalness $R_{\textrm{s}}$ and transientness $R_{\textrm{t}}$ matrices with
the following elements [18]:

$R_{\textrm{s}}(m,k)=\frac{X_{\textrm{h}}(m,k)}{X_{\textrm{h}}(m,k)+X_{\textrm{v}}(m,k)}$
(3)

and

$R_{\textrm{t}}(m,k)=1-R_{\textrm{s}}(m,k)=\frac{X_{\textrm{v}}(m,k)}{X_{\textrm{h}}(m,k)+X_{\textrm{v}}(m,k)},$
(4)

respectively. Finally, the soft masks are obtained as follows:

$\displaystyle S(m,k)=f$ $\displaystyle\left(R_{\textrm{s}}(m,k)\right),$ (5)
$\displaystyle T(m,k)=f$ $\displaystyle\left(R_{\textrm{t}}(m,k)\right),$ (6)
$\displaystyle N(m,k)=1$ $\displaystyle-S(m,k)-T(m,k),$ (7)

where

$\displaystyle f($ $\displaystyle a)=\begin{cases}1,&\mbox{if
}a\geq\beta_{\textrm{U}}\\\
\sin^{2}{\Big{(}\dfrac{\pi}{2}\dfrac{a-\beta_{\textrm{L}}}{\beta_{\textrm{U}}-\beta_{\textrm{L}}}}\Big{)},&\mbox{if
}\beta_{\textrm{L}}\leq a<\beta_{\textrm{U}}\\\
0,&\mbox{otherwise},\end{cases}$ (8)

which are consequently imposed onto $X(m,k)$ via element-wise multiplication
to obtain the separated components. A group of functions to determine soft STN
masks for $\beta_{\textrm{U}}$ = 0.8 and $\beta_{\textrm{L}}$ = 0.7 is
visualized in Fig. 1.

This decomposition process is repeated for two consecutive stages [15]. The
first implements a large analysis window for better frequency resolution,
separating the sines from the transient and noise residual mixture; the second
uses a short analysis window for better temporal resolution, extracting the
transients from the residual. An example of two-stage STN decomposition for a
violin and castanet sound mixture is shown in Fig. 2.

Fig. 1: Functions for determining soft spectral masks for STN separation, as
described in [15].

(a) Original

(b) Sines

(c) Transients

(d) Noise

Fig. 2:  STN decomposition of a castanet and violin mixture.

The STN decomposition enables the application of different TSM algorithms for
each component. While traditional signal processing techniques have been
optimized to deal with either the sines or the transients, the noise component
remains relatively unexplored and hence is suitable for a deep learning
approach.

##  3 Noise Time Stretching via Wavenet

This section introduces a WaveNet architecture to synthesize the time-
stretched noise component.

###  3.1 WaveNet synthesizer

WaveNet is an autoregressive generative model capable of synthesizing raw
audio waveforms [16]. It was initially proposed for speech synthesis, and the
model and variants thereof are still commonly used as part of audio synthesis
pipelines. It consists of a stack of dilated 1D-convolutional layers, with the
dilation factor increasing exponentially at each layer. The input to the
network is also a raw audio waveform, with the model being trained to predict
the subsequent sample in the sequence, given the prior audio samples.

Additionally, it is possible for the WaveNet model to incorporate a
conditioning signal, allowing control over the audio generated by the model.
There are broadly two types of conditioning information: global and local. The
first describes features that influence the entirety of the generated audio,
e.g. the speaker identity in a speech synthesizer. The latter is used to
represent time-variant features of the desired audio waveform, such as
spectrograms and other time-frequency representations. In this work, we only
consider local conditioning signals.

###  3.2 WaveNet-based TSM

Time stretching via WaveNet is achieved by adjusting the number of audio
samples generated per frame of the local conditioning signal, according to
$\alpha$. For example, if the conditioning signal is the spectrogram extracted
from a target audio signal with a hop size of 256 samples, then the re-
synthesis for $\alpha=2$ will generate 512 audio samples for each frame of
conditioning signal.

In the proposed approach, the WaveNet synthesizer is used to re-synthesize the
noise component of the target signal, extracted using the two-stage STN
decomposition, according to the desired TSM factor $\alpha$. Three different
time-frequency representations were tested as local conditioning signals for
the network: spectrogram, mel-spectrogram, and Constant-Q Transform (CQT)
spectrogram. Different models were trained using each of these
representations, and it was found that using the CQT-spectrogram produced the
highest quality results. Note that the choice of time-frequency representation
is fixed and is part of the model architecture, so it cannot be changed during
or after training.

The use of the CQT spectrogram as a conditioning signal for a WaveNet
synthesizer was first proposed in [17]. The CQT transform [19] is a time-
frequency analysis method in which the frequency bins are logarithmically
spaced. Previous work has shown that the CQT is a good choice for synthesis of
musical sounds [17] and also environmental sound classification [20]. In this
work, to extract the CQT we use a 5.8 ms hop size (256 samples at the 44.1-kHz
sample rate), a minimum frequency of 32.7 Hz, a maximum frequency of Nyquist
(22.05 kHz), and 48 CQT bins per octave. This results in a total of 451
frequency bins.

The network was trained at a sample rate of 44.1 kHz, and a 10-component
Mixture-of-Logistic distributions (MoL) sampling method was used to generate
raw 16-bit audio [21]. Training was run for a total of 1,900,000 iterations,
and took approximately 200 hours on a GPU. At inference time, a beam-search
algorithm [17] was used to remove spurious impulse events generated by the
probabilistic sampling method.

###  3.3 Dataset

To include a diverse range of sounds, a dataset was constructed from three
source datasets. The datasets used were the ESC-50 dataset [22], a labeled
collection of environmental audio recordings, the Freesound Loops dataset
[23], a collection of short musical clips including electronic and acoustics
instrument sounds, as well as speech sounds from the LJ-speech dataset [24].
Samples were randomly removed from the LJ-speech and Freesound loops datasets,
so the different classes of sounds were evenly represented in the combined
dataset. In total, the dataset used for training contained approximately 4
hours of audio.

##  4 Hybrid TSM Method

The proposed model combines the use of traditional digital signal processing
techniques and the Wavenet synthesizer to improve the time-scale modification
performance for extreme time-stretching factor, and it is designed and trained
to provide high quality stretching for environmental sounds. The TSM pipeline
of the proposed method described in this section is visualized in Fig. 3.

Fig. 3: Block diagram of the TSM pipeline, for a single-channel signal. A
thicker line indicates a time-stretched signal.

###  4.1 Processing the individual STN components

The input audio signal is decomposed as described in Sec. 2. Transition
regions for the STN masks are defined by $\beta_{\textrm{U}}$ = [0.8 0.85] and
$\beta_{\textrm{L}}$ = [0.7 0.75], where the two elements refer to the
different transition region in each stage of the decomposition [15].

The sines are processed via phase vocoder with identity phase locking (IPL)
[25], as used in [9]. Transients are preserved and relocated onto the new time
axis [26] according to detected peaks, which are isolated due to the nature of
the STN decomposition. The noise component is neurally resynthesized at the
desired TSM factor, as described in Sec. 3.

###  4.2 Post-processing

Before mixing the three components, the time-stretched sines and noise are
processed through an envelope (see Fig. 3), which reshapes them according to
the envelope of the original signal to compensate for the pre-echo effect that
is typical of spectrogram-based TSM methods [3]. Finally, the three components
are added and the time-stretched output is obtained, as also illustrated in
Fig. 3.

A time-stretching output example is shown in Fig. 4 for the Soda sound
detailed in Table 1 and a TSM factor $\alpha$ = 4. The “hiss” of the can
opening is correctly resynthesized by the network, together with the noisy
part of the later clicks which are imposed over the preserved transients. The
STN decomposition is particularly helpful in this situation, as the merging of
unaltered transients and the neurally-synthesized noise helps generate
realistic sounds that retain the punch of percussive events while avoiding
common artifacts, such as loss of presence, metallic tones, and phasiness.

###  4.3 Stereo compatibility

The process described so far can be individually applied to the two channels
of a stereo sound, as STN decomposition is transparent with respect to the
stereo field. When a sound is decomposed into its individual components and
then recomposed, the stereo balance and phase coherence between the left and
the right channel, computed according to [27] are preserved. This allows for
each stereo channel to be processed independently.

Fig. 4: (Left) Soda sound and (right) its four-times stretched version processed via the proposed model. Table 1: Audio samples used in the listening test. Name | Description
---|---
Fireworks | Two fireworks exploding in the air
Soda | Hiss and click sound from a can opening
PingPong | A clip from an amateur ping pong game
Saw | Handsaw sawing through wood
Sneeze | A person sneezing two times
Rooster | A rooster crowing in the morning

(a)

(b)

Fig. 5: Barplots relative to the results of the subjective listening test,
showing the average mean opinion scores and 95% confidence intervals of the
data for TSM factors (a) $\alpha$ = 4 and (b) $\alpha$ = 8. The proposed
method outperforms the opposing algorithms in both cases for most of the
samples under test.

##  5 Evaluation

The proposed method has been validated and compared with previous TSM methods
in a listening test, which is reported in this section.

###  5.1 Listening test design

A formal blind listening test was conducted on a selection of 14 experienced
listeners, all of which reported previous experience in test design and no
hearing impairment or other relevant medical condition. The test software, a
customized version of WebMushra [28], was run on a desktop machine running
MacOS 10.14.6, using a single pair of Sennheiser HD 650 headphones, inside a
sound–proof listening booth at the Aalto Acoustics Lab, Espoo, Finland. A set
of six audio samples, listed in Table 1, were selected. As the test involves
extreme stretching factors, of up to 8 times, a very short duration for each
sample (approximately 2 s) was necessary to ensure that even the longest time-
stretched sounds remained below 18 s long.

The proposed method (PROP) was evaluated against the fuzzy phase vocoder (FPV)
[9], harmonic-percussive TSM (HP) [29], the two-step phase vocoder with
identity phase locking [25] (IPL), and WSOLA [4], which was provided as the
low-quality anchor (ANC). The last three algorithms are included in the TSM
Toolbox [30]. All the processed sounds were loudness normalized according to
ITU-BS.1770 recommendation [31] to prevent loudness differences from affecting
the grades. In each trial of the test, subjects were presented with one of the
audio excerpts, referred to as reference, and were asked to blindly rate the
quality of the time-scaling operation. The original reference was not included
among the samples to be rated. Subjects were instructed to rate each sample on
a scale from 0 to 100, with no obligation of using the full scale, as the
concept of perfect TSM is undefined and it was anticipated that none of the
samples would be perceived to be an ideal processing.

The test was divided in two parts of twelve trials (six trials repeated twice
for consistency) each, the first presenting a TSM factor of 4 and the second a
TSM factor of 8, for a total of 24 trials and 72 audio samples under
investigation. Listeners were allowed a short training session before starting
the actual test to get acquainted with the interface, the keyboard shortcuts,
and the task itself. The results of the training session were not included in
the statistical analysis. Prior to the test, familiarity of the subjects with
the concepts of time-scale modification and transient smearing was assessed.

###  5.2 Results

Mean Opinion Scores (MOS) were computed from the ratings given by the subjects
to estimate the quality of the time-stretching for the methods under test. Bar
plots displaying the mean and 95% confidence intervals of the data for are
shown, in Fig. 5. The proposed method clearly outperforms the other algorithms
under test under both time-stretching conditions for all samples but Rooster.
The majority of the test participants commented on this audio excerpt that the
noisy nature and the lack of sharp transients of the reference made it hard to
generate an expectation for a time-stretched version.

##  6 Conclusion

In this paper, we present a novel algorithm for extreme TSM, which takes
advantage of the decomposition of sound into sines, transients, and noise to
individually resynthesize the noise component using a deep neural network.
Sines and transients are time-stretched using established signal processing
techniques which also benefit from the decomposition. The results of a
subjective listening test suggest that the proposed algorithm performs
significantly better than previous TSM methods for real-world environmental
recordings, when a large time-stretching factor is used. Future work involves
redesigning the neural synthesis for standard time-stretching factors,
involving a larger dataset of more relevant sounds.

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# Fundamental groupoids and homotopy types of non-compact surfaces

Sergiy Maksymenko  Institute of Mathematics of NAS of Ukraine
<EMAIL_ADDRESS>and  Oleksii Nikitchenko  National Technical University of
Ukraine ‘‘Igor Sikorsky Kyiv Polytechnic Institute’’<EMAIL_ADDRESS>
###### Abstract.

The paper contains an application of van Kampen theorem for groupoids to
computation of homotopy types of certain class of non-compact foliated
surfaces obtained by at most countably many strips $\mathbb{R}\times(0,1)$
with boundary intervals in $\mathbb{R}\times\\{\pm 1\\}$ along some of those
intervals.

##  1\. Introduction

The present paper is devoted to applications of van Kampen theorem for
groupoids to computation of homotopy types of a certain class of non-compact
foliated surfaces called _striped surfaces_.

It was mentioned by M. Morse that a smooth function $f$ with non-degenerate
critical points (a Morse function) on a compact manifold $Z$ ‘‘contains’’ a
lot of homological information about the manifold itself (the famous Morse
inequalities). In particular, if $\dim M=2$, so $Z$ is a ‘‘surface’’, one can
even determine the topological type of $Z$ by the numbers of critical points
of distinct indices via any Morse function $f:M\to\mathbb{R}$. Motivated by
study functions of complex variable, in particular, harmonic functions being
real and imaginary parts of holomorphic functions, Morse extended his
observations in the book [17] to pseudoharmonic functions $f$ defined on
compact domains $Z$ in the complex plane $\mathbb{C}$.

By definition, a pseudoharmonic function $f:\mathbb{C}\supset Z\to\mathbb{R}$
is locally a composition $g\circ h$, where $g$ is a harmonic function, and $h$
is a homeomorphism of $\mathbb{C}$. Such a function is continuous, all its
critical (in a proper sense) points belonging to the interior of $Z$ are
isolated and are not local extremes (due to maximum principle for holomorphic
functions). Moreover, it is assumed that the restriction of $f$ to $\partial
M$ has only finitely many local minimums and maximums. Absence of local
extremes in the interior of $Z$ implies (by Jordan curve theorem) that the
foliation of $Z$ into connected components of level sets of $f$ has no closed
curves.

W. Kaplan [9, 10] characterized such foliation for pseudoharmonic function on
$\mathbb{R}^{2}$. Namely, he shown that every foliation $\mathcal{F}$ on the
plane $\mathbb{R}^{2}$ has the following properties:

  1. (1)

every leaf $\omega$ of $\mathcal{F}$ is an image of a proper embedding
$\omega:\mathbb{R}\to\mathbb{R}^{2}$, so
$\lim\limits_{t\to\pm\infty}\omega(t)=\infty$;

  2. (2)

there exists at most countably many leaves $\\{\omega_{i}\\}_{i\in\Lambda}$ of
$\mathcal{F}$, such that
$\mathbb{R}^{2}\setminus\mathop{\cup}\limits_{i\in\Lambda}\omega_{i}$ is a
disjoint union of ‘‘open strips’’ $\mathbb{R}\times(0,1)$ foliated into lines
$\mathbb{R}\times t$, $t\in(0,1)$;

  3. (3)

there exists a pseudoharmonic function $f:\mathbb{R}^{2}\to\mathbb{R}$ without
critical points such that $\mathcal{F}$ is a partition of $\mathbb{R}^{2}$
into connected components of level-sets of $f$.

Property (3) is in a spirit of de Rham theory: if $\mathcal{F}$ is smooth,
then it is defined by some closed differential $1$-form, and since
$\mathbb{R}^{2}$ is contractible, $\omega=df$ for some function $f$ satisfying
(3).

It also gives a certain connection between foliations on surfaces and
pseudoharmonic functions on the plane. Let $\mathcal{F}^{\prime}$ be a
foliation on a connected surface $Z^{\prime}$ without boundary distinct from
$2$-sphere and projective plane, and $p:Z\to Z^{\prime}$ be the universal
covering map. Then we get a well-defined foliation $\mathcal{F}$ on $Z$ whose
leaves are connected components of the inverses under $p$ of leaves of
$\mathcal{F}^{\prime}$, and the group $\pi_{1}Z^{\prime}$ of covering
transformations interchanges the leaves of $\mathcal{F}$. By [6, Corollary
1.8] $Z$ is homeomorphic to $\mathbb{R}^{2}$, whence $\mathcal{F}$ satisfies
(1)-(3), while $\mathcal{F}^{\prime}$ may loose all of those properties. One
may say that $\mathcal{F}^{\prime}$ is obtained from a foliation on
$\mathbb{R}^{2}$ into connected components of level sets of some
pseudoharminic function by some free and properly discontinuous action of
$\pi_{1}Z$. In general, such a function is not invariant with respect to the
action of $\pi_{1}Z$.

Notice also that property (2) proposes a certain classification of foliations
on $\mathbb{R}^{2}$ by studying the way in which such a foliation is glued
from open strips. This ‘‘combinatorics of gluing’’ would give certain
topological invariants of (pseudo)harmonic functions and potentially
information about symmetries of differential equations whose coefficients are
harmonic functions. Moreover, one could try to extend Kaplan’s technique to
arbitrary pseudoharmonic functions $f:\mathbb{R}^{2}\to\mathbb{R}$ just by
removing the set of singular points $\Sigma_{f}$ of $f$ and consider the
remained foliation on $\mathbb{R}^{2}\setminus\Sigma_{f}$. Such an approach
was used in the papers by W. Boothby [2, 3], M. Morse and J. Jenkins [7], M.
Morse [18].

Kaplan tried to minimize the total number of leaves, and for that reason his
construction was not ‘‘canonical’’. Namely, the choice of leaves
$\\{\omega_{i}\\}_{i\in\Lambda}$ (and thus a ‘‘cutting of the foliation
$\mathcal{F}$’’ into open strips) was not unique. Moreover, if $S$ is a
connected component of
$\mathbb{R}^{2}\setminus\mathop{\cup}\limits_{i\in\Lambda}\omega_{i}$, then
the homeomorphism $S\to\mathbb{R}\times(0,1)$ does not necessarily extend to
an embedding $\overline{S}\to\mathbb{R}\times[0,1]$.

In a series of papers by S. Maksymenko, Ye. Polulyakh [13, 12, 14, 15] and Yu.
Soroka [21, 22] it was studied a class of foliations on non-compact surfaces
$Z$ (called striped) glued from open strips in a certain ‘‘canonical’’ way. In
a joint paper [16] of the above three authors it was also described an
analogue of mapping class group for foliated homeomorphisms (sending leaves
into leaves) of such foliations $\mathcal{F}$ and proved that it is isomorphic
with an automorphism group of a certain graph (one-dimensional CW-complex) $G$
encoding an information about gluing a surface from strips. This graph is in a
certain sense _dual_ to the space of leaves of $\mathcal{F}$ .

The aim of the present paper is to prove that the connection between a striped
surface $Z$ and its graph $G$ is more deep: namely they are homotopy
equivalent, see Theorems 2.8 and 2.9. One of the difficulties of proving such
a result is that there is no canonical map $Z\to G$. We construct a continuous
injection $\varphi:G\to Z$, and then prove (using van Kampen theorem for
groupoids established by R. Brown and A. Salleh [5]) that $\varphi$ induces an
isomorphism of fundamental groupoids of $G$ and $Z$. This will imply that
$\varphi$ is a homotopy equivalence, since $G$ and $Z$ are aspherical. For
instance the ranks of their homology groups $H_{1}(Z,\mathbb{Z})$ and
$H_{1}(G,\mathbb{Z})$ are the same.

In fact, the result is rather simple when $Z$ is glued of finitely many
strips: in this case the above map $\varphi:G\to Z$ is an embedding and its
image $\varphi(G)$ is a strong deformation retract of $Z$. On the other hand,
if the number of strips is infinite, $G$ can be not a locally finite CW-
complex, having thus no countable local bases at some vertices. Therefore
there will be no embeddings of $G$ into a manifold $Z$, thus the image
$\varphi(G)$ will not be a strong deformation retract of $Z$. Nevertheless,
van Kampen theorem allows to accomplish the result.

Actually, the obtained result has no deal with a foliation itself but only
with a way in which a surface is glued from strips. Nevertheless, suppose we
are given a foliation $\mathcal{F}$ on a non-compact surface $Z$ whose leaves
are non-compact closed subsets of $Z$. Now, if $\mathcal{F}$ has ‘‘not so much
singular leaves’’, see Theorem 2.5 below and Figure 2.3, then it is a striped
surface. Therefore we get a partition into strips and our result shows that
the foliation ‘‘contains’’ an information about the homotopy type of the
underlying surface. Thus our statement could be viewed in the frame of Morse
theory in which the gradient lines connecting critical points of a Morse
function $f:Z\to\mathbb{R}$ (or equivalently decomposition of $Z$ into handles
in the sense of S. Smale corresponding to those critical points) determine a
CW-partition of $Z$ and this relates $f$ with homological and even topological
structure of $Z$.

The exposition of the paper is intended to be elementary in order to make it
accessible to a large audience of readers, and thus to propagate and
popularize usage of homotopy methods (like van Kampen theorem for groupoids)
to more applied problems.

##  2\. Striped surface and its graph

######  Definition 2.1 ([13]).

A subset $S\subset\mathbb{R}\times[-1,1]$ will be called a model strip if:

  1. 1)

$\mathbb{R}\times(-1,1)\subset S$;

  2. 2)

the intersection $S\cap(\mathbb{R}\times\\{\pm 1\\})$ is a union of at most
countably many mutually disjoint open intervals.

For instance, $\mathbb{R}\times(-1,1)$, $\mathbb{R}\times[-1,1]$,
$\bigl{(}\mathbb{R}\times(-1,1)\bigr{)}\cup\bigl{(}(-2,3)\times\\{1\\}\bigr{)}$
are model strips. Of course one can replace $[-1,1]$ with any other closed
segment $[a,b]\subset\mathbb{R}$.

Notice that condition 2) is equivalent to the assumption that $S$ is open as a
subset of $\mathbb{R}\times[-1,1]$. Define the following subsets of $S$:

$\displaystyle\ \partial_{-}S$
$\displaystyle:=S\cap(\mathbb{R}\times\\{-1\\}),$ $\displaystyle\
\partial_{+}S$ $\displaystyle:=S\cap(\mathbb{R}\times\\{1\\}),$
$\displaystyle\ \partial S$ $\displaystyle:=\partial_{-}S\cup\partial_{+}S.$

######  Definition 2.2 ([13]).

Let $Z$ be a two-dimensional manifold. A striped atlas on $Z$ is a map
$q:Z_{0}\to Z$ having the following properties:

  1. (1)

$Z_{0}=\mathop{\sqcup}\limits_{\alpha\in A}S_{\alpha}$ is at most countable
disjoint union of model strips;

  2. (2)

$q$ is a quotient map, so a subset $U\subset Z$ is open iff $q^{-1}(U)$ is
open in $Z_{0}$;

  3. (3)

there are two disjoint families $X=\\{X_{\beta}\\}_{\beta\in B}$,
$Y=\\{Y_{\beta}\\}_{\beta\in B}\subset\bigcup\limits_{\alpha\in A}\partial
S_{\alpha}$ of boundary intervals such that:

    1. a)

$q$ is an injective on $Z_{0}\setminus(X\cup Y)$;

    2. b)

$q(X_{\beta})=q(Y_{\beta})$ for each $\beta\in B$ and the restrictions
$q|_{X_{\beta}}:X_{\beta}\to q(X_{\beta})$ and $q|_{Y_{\beta}}:Y_{\beta}\to
q(Y_{\beta})$ are embeddings;

    3. c)

$q(X_{\beta})\cap q(X_{\beta^{\prime}})=\varnothing$ for each
$\beta\neq\beta^{\prime}\in B$;

    4. d)

$q(X_{\beta})\cap q(Z_{0}\setminus(X\cup Y))=\varnothing$ for each $\beta\in
B$.

Figures 2.1, 2.3, 2.4 contain examples of striped atlases.

The pair $(Z,q)$ will also be called a striped structure on $Z$. A striped
surface is a surface admitting a striped atlas. When talking about a striped
surface $Z$ we will also assume that some striped atlas $(Z,q)$ is fixed.
Notice that a striped surface is a non-compact two-dimensional manifold which
can be non-orientable and disconnected. Moreover, each connected component of
its boundary is an open interval.

###  2.3. Seams

Let $\beta\in B$. Then we have a homeomorphism $\gamma_{\beta}:Y_{\beta}\to
X_{\beta}$ given by $\gamma_{\beta}=(q|_{X_{\beta}})^{-1}\circ
q|_{Y_{\beta}}$. Therefore, a striped surface $Z$ can also be regarded as a
quotient space obtained by gluing some pairs of boundary intervals of model
strips via homeomorphisms $\\{\gamma_{\beta}\\}_{\beta\in B}$. The image

$\omega_{\beta}:=q(X_{\beta})=q(Y_{\beta})$

will be called a seam of $Z$ (as well as of $q$). In Figure 2.1 seams are
colored in red color.

Since $q^{-1}(\omega_{\beta})=X_{\beta}\cup Y_{\beta}$ is closed in $Z_{0}$,
and $q$ is a quotient map, it follows that each seam is a closed subset of
$Z$.

(a)

(b)

Figure 2.1. Striped atlases

###  2.4. Foliated characterization of striped surfaces

Though the notion of a striped surface looks rather restrictive, it
nevertheless covers a large class of surfaces equipped with foliations which
agree in a certain sense with a decomposition into strips. We recall here the
principal statement from [15]. It will not be used for the proof of our main
result, however we present it to describe the general picture.

By a foliated surface $(Z,\mathcal{F})$ we will mean a two-dimensional
manifold $Z$ equipped with a foliation $\mathcal{F}$. A saturation
$\mathrm{Sat}(U)$ of a subset $U\subset Z$ is the union of all leaves of
$\mathcal{F}$ intersecting $U$, that is

$\mathrm{Sat}(U):=\mathop{\cup}_{\gamma\cap U\not=\varnothing}\gamma$

A subset $U\subset Z$ is saturated whenever $U=\mathrm{Sat}(U)$ i.e. $U$ is a
union of leaves of $\mathcal{F}$.

Given an open subset $U\subset Z$, denote by $\mathcal{F}|_{U}$ the foliation
on $U$ consisting of connected components of the intersections of leaves of
$\mathcal{F}$ with $U$. We will call $\mathcal{F}|_{U}$ the restriction of
$\mathcal{F}$ to $U$.

A homeomorphism $h:Z\to Z^{\prime}$ between foliated surfaces
$(Z,\mathcal{F})$ and $(Z\,\mathcal{F}^{\prime})$ is foliated whenever for
each leaf $\gamma$ of $\mathcal{F}$ its image $h(\gamma)$ is a leaf of
$\mathcal{F}^{\prime}$.

Notice that each model strip $S$ admits a natural foliation into boundary
intervals and lines $\mathbb{R}\times\\{t\\}$, $t\in(-1,1)$. We will call this
foliation canonical. Now if $(Z,q)$ is a striped surface, then canonical
foliations on the corresponding model strips induce a foliation on all of $Z$
which we will also call the canonical foliation (of the striped atlas $q$).

Let $(Z,\mathcal{F})$ be a foliated surface. Say that a leaf
$\gamma\subset\mathrm{Int}(Z)$, resp. $\gamma\subset\partial Z$, of
$\mathcal{F}$ is regular if there exists an open saturated neighborhood $U$ of
$\gamma$ such that the pair $(\overline{U},U)$ is foliated homeomorphic to the
model strip $\bigl{(}\mathbb{R}\times[-1,1],\mathbb{R}\times(-1,1)\bigr{)}$,
resp. $\bigl{(}\mathbb{R}\times[0,1],\mathbb{R}\times[0,1)\bigr{)}$, via a
homeomorphism sending $\gamma$ to $\mathbb{R}\times 0$.

A leaf which is not regular will be called singular. For example, in the above
figures, the seams (red leaves) are precisely singular leaves.

For the case of striped surfaces and its canonical foliation, it is easy to
see that a leaf $\gamma$ is singular if and only if it satisfies one of the
following conditions:

  1. i)

$\gamma=q(\delta)$, for some boundary interval $\delta=(a,b)\times\varepsilon
1\subset\partial_{\varepsilon}S_{\alpha}$, where $\varepsilon\in\\{\pm\\}$ and
$\alpha\in A$;

  2. ii)

$\partial_{\varepsilon}S_{\alpha}$ contains some other boundary intervals
distinct from $\delta$.

######  Theorem 2.5 ([15, Theorem 4.4]).

Let $(Z,\mathcal{F})$ be a foliated surface such that every leaf $\gamma$ of
$\mathcal{F}$ is a non-compact closed subset of $Z$. Then the following
conditions are equivalent:

  1. 1)

$Z$ admits a striped atlas $q$ such that $\mathcal{F}$ coincides with the
canonical foliation of $q$;

  2. 2)

the collection of all singular leaves of $\mathcal{F}$ is locally finite.

In fact, not every foliation on the plane admits a striped atlas, see [16,
Examples 7.6, 7.7]. One of the reasons is that seams can converge to other
seams. We will briefly recall [16, Example 7.6]. Let $\mathcal{F}$ be the
foliation on $\mathbb{R}^{2}$ into parallel lines $\mathbb{R}\times t$,
$t\in\mathbb{R}$. Let also $\\{a_{n}\\}_{n\in\mathbb{N}}\mathbb{R}$ be a
strictly decreasing sequence of reals converging to some $a\in\mathbb{R}$, and
$K=\\{a\\}\cup\\{a_{n}\\}_{n\in\mathbb{N}}$. Consider the open subset
$Z=\mathbb{R}^{2}\setminus(0\times K)$ and let $\mathcal{F}|_{Z}$ be the
restriction of $\mathcal{F}$ to $Z$. Notice that every point $b\in K$ splits
$\mathbb{R}\times b$ into two arcs $\gamma^{-}_{b}=(-\infty,0)\times b$ and
$\gamma^{+}_{b}=(0,+\infty)\times b$ being the leaves of $\mathcal{F}|_{Z}$.
By property ii) the latter leaves are singular for $\mathcal{F}|_{Z}$ and they
converse to the leaves $\gamma^{-}_{a}$ and $\gamma^{+}_{a}$. Hence the family
of all singular leaves of $\mathcal{F}|_{Z}$ is not locally finite, and by
Theorem 2.5 $(Z,\mathcal{F}|_{Z})$ does not admit a striped atlas.

###  2.6. Graph of a striped surface

The above figures propose to consider a graph which encodes the gluing of
strips. Such a graph was introduced in [16, §6] and also takes to account
boundary components of striped surface which are not seams. We will consider
here a more simplified version of it.

Let $q:\mathop{\sqcup}\limits_{\alpha\in A}S_{\alpha}\to Z$ be a striped atlas
on a surface $Z$. Then one can associate to $q$ a one-dimensional CW-complex
(‘‘topological graph’’) $G$ whose vertices are strips of the atlas and the
edges correspond to gluing strips along boundary intervals. More precisely,

  1. 0)

$0$-skeleton of $G$ is $G^{(0)}=A$;

  2. 1)

Let $\beta\in B$, so we have a pair of boundary intervals
$\\{X_{\beta},Y_{\beta}\\}$ such that $X_{\beta}\subset\partial S_{\alpha}$
and $Y_{\beta}\subset\partial S_{\alpha^{\prime}}$ for some vertices
$\alpha,\alpha^{\prime}\in A$. Let also $I_{\beta}=[-1,1]$. Glue $I_{\beta}$
to $A$ via the following map:

$\chi_{\beta}:\partial I_{\beta}=\\{\pm 1\\}\to
A,\qquad\chi_{\beta}(-1)=\alpha,\qquad\chi_{\beta}(1)=\alpha^{\prime}.$

Then the resulting CW-complex:

$G=\Bigl{(}\,\mathop{\sqcup}\limits_{\beta\in
B}I_{\beta}\Bigr{)}\bigcup_{\chi_{\beta},\ \beta\in B}A.$

will be called the graph of the striped atlas $q$. We will also denote by the
same letter $\chi_{\beta}:I_{\beta}\to G$ the characteristic map of the
$1$-cell $I_{\beta}$. Thus

  * •

$\chi_{\beta}|_{(-1,1)}:(-1,1)\to G$ is an embedding and we will denote by
$e_{\beta}=\chi_{\beta}\bigl{(}(-1,1)\bigr{)}$ its image being an open
$1$-cells;

  * •

$\chi_{\beta}|_{\partial I_{\beta}}:\partial I_{\beta}\to A$ is the
corresponding gluing map.

###  2.7. Canonical injection $\varphi:G\to Z$

We will construct here a continuous injective map $\varphi:G\to Z$.

For each $\alpha\in A$ let $s_{\alpha}:=(0,0)\in\mathbb{R}\times(-1,1)\subset
S_{\alpha}$ be the origin. Define the map $\varphi:G^{(0)}\to Z$ by
$h(\alpha)=s_{\alpha}$.

Furthermore, for each $\beta\in B$ fix a point
$z_{\beta}\in\omega_{\beta}=q(X_{\beta})=q(Y_{\beta})$, and let

$\displaystyle(x_{\beta},\varepsilon)$ $\displaystyle=q^{-1}(z_{\beta})\cap
X_{\beta},$ $\displaystyle(y_{\beta},\varepsilon^{\prime})$
$\displaystyle=q^{-1}(z_{\beta})\cap Y_{\beta},$ (2.1)

where $\varepsilon,\varepsilon^{\prime}\in\\{\pm 1\\}$. Assume that
$X_{\beta}\subset S_{\alpha}$ and $Y_{\beta}\subset S_{\alpha^{\prime}}$ for
some $\alpha,\alpha^{\prime}\in A$.

Now define the following path $\varphi_{\beta}:I_{\beta}\equiv[-1,1]\to Z$ by
the following formula. see Figure 2.2:

$\varphi_{\beta}(t)=\begin{cases}q\bigl{(}2(1+t)x_{\beta},(1+t)\varepsilon\bigr{)},&t\in[-1,-\tfrac{1}{2}],\\\
q\bigl{(}x_{\beta},(1+t)\varepsilon\bigr{)},&t\in[-\tfrac{1}{2},0],\\\
q\bigl{(}y_{\beta},(1-t)\varepsilon^{\prime}\bigr{)},&t\in[0,\tfrac{1}{2}],\\\
q\bigl{(}2(1-t)y_{\beta},(1-t)\varepsilon^{\prime}\bigr{)},&t\in[\tfrac{1}{2},1].\end{cases}$

It easily follows that

$\displaystyle\varphi_{\beta}(-1)$ $\displaystyle=h(\alpha)=s_{\alpha},$
$\displaystyle\varphi_{\beta}(0)$ $\displaystyle=z_{\beta},$
$\displaystyle\varphi_{\beta}(1)$
$\displaystyle=h(\alpha^{\prime})=s_{\alpha^{\prime}}.$

Moreover, $\varphi_{\beta}$ is a simple path if $\alpha\not=\alpha^{\prime}$,
and a simple closed path if $\alpha=\alpha^{\prime}$.

Since the paths $\varphi_{\beta}$ agree with $\varphi$ at end-points, they
induce the required map $\varphi:G\to Z$.

Figure 2.2. The map $\varphi_{\beta}$ Figure 2.3. Striped structure and the
map $\varphi$ on $\mathbb{R}^{2}\setminus(0,0)$ for canonical foliation by
level sets of the function $f(x,y)=xy$ Figure 2.4.

The aim of the present paper is to prove the following

######  Theorem 2.8.

Let $(Z,q)$ be a striped surface and $G$ be its graph. Then for each $x\in G$
the above map $\varphi:G\hookrightarrow Z$ induces an isomorphism of the
fundamental groups

$\varphi_{*}:\pi_{1}(G,x)\to\pi_{1}\bigl{(}Z,\varphi(x)\bigr{)}.$

In particular, $\varphi$ is a homotopy equivalence.

In fact we will establish a more precise statement

######  Theorem 2.9.

Let $(Z,q)$ be a striped surface and $G$ be its graph. Then there exists a
subset $Z^{\prime}\subset Z$ such that the map $\varphi:G\hookrightarrow Z$
induces an isomorphism of the corresponding fundamental groupoids

$\varphi^{*}:\Pi_{1}(G,\varphi^{-1}(Z^{\prime}))\to\Pi_{1}(Z,Z^{\prime}).$

The proof of these theorems will be given in §4. They are based on application
of van Kampen theorem for groupoids, see Lemma 3.12.

Notice that when $Z_{0}$ has only finitely many seams, the result is rather
simple and in this case $\varphi(G)$ is a strong deformation retract of $Z$.

On the other hand, if the number of seams is infinite, the graph $G$ might be
not a locally finite CW-complex and $Z$ can not be deformed onto the image
$\varphi(G)$.

######  Example 2.10.

Let $Z=\mathbb{R}^{2}\setminus(\mathbb{Z}\times 0)$. Then $Z$ has an atlas
consisting of two model strips:

$\displaystyle S_{0}$ $\displaystyle=\mathbb{R}\times(-1,1)\ \bigcup\
\mathop{\cup}\limits_{n\in\mathbb{Z}}(n,n+1)\times\\{1\\},$ $\displaystyle
S_{1}$ $\displaystyle=\mathbb{R}\times(-1,1)\ \bigcup\
\mathop{\cup}\limits_{n\in\mathbb{Z}}(n,n+1)\times\\{-1\\}.$

and its graph has two vertices connected with countably many edges, see Figure
2.5. In this case $G$ is not locally finite at its vertices and therefore it
has no countable local base, whence $\varphi:G\to Z$ is not an embedding.

Figure 2.5.

Let us also mention the following statement describing certain topological
properties of striped surfaces.

######  Lemma 2.11.

Let $Z$ be a connected non-compact surface. Then $Z$ has the homotopy type of
an aspherical CW-complex, and $\pi_{1}Z$ is free.

###### Proof.

For the fact that every separable manifold has homotopy type of a CW-comples
see [11, Corollary 5.7.].

Further, let $p:\tilde{Z}\to Z$ be the universal covering of $Z$. Then by [6,
Corollary 1.8], the interior $\mathrm{Int}(\tilde{Z})$ is homeomorphic to
$\mathbb{R}^{2}$. Moreover, the inclusion
$\mathrm{Int}(\tilde{Z})\subset\tilde{Z}$ is a homotopy equivalence due to
existence of collars of the boundary of metrizable manifolds, [4, Theorem 2].
Hence $p$ induces isomorphisms
$0=\pi_{k}\mathrm{Int}(\tilde{Z})\cong\pi_{k}\tilde{Z}\cong\pi_{k}Z$, $k\geq
2$. This, by definition, means aspherity of $Z$.

Finally, the statement that $\pi_{1}Z$ is free, is proved in distinct sources,
e.g. [8] or [1, Theorem 44A]. Another proof is that since $Z$ is non-compact,
it’s cohomological dimension $cdZ\leq 1$, whence by [23, Corollary to Theorem
A], $\pi_{1}Z$ is free. See also [20] for discussions and [19]. ∎

##  3\. Fundamental groupoids

In this section we will briefly recall the notion of fundamental groupoid,
list some of its properties and formulate van Kampen theorem for groupoids.

###  3.1. Small categories

A small category $\mathfrak{C}$ is given by the following data:

  1. $\bullet$

a set $\mathrm{Ob}(\mathfrak{C})$, called set of objects;

  2. $\bullet$

a set $\mathrm{Hom}_{\mathfrak{C}}(X,Y)$ for each pair of objects
$X,Y\in\mathrm{Ob}(\mathfrak{C})$, called set of morphisms between $X$ and
$Y$;

  3. $\bullet$

for each pair of morphisms $f\in\mathrm{Hom}_{\mathfrak{C}}(X,Y)$,
$g\in\mathrm{Hom}_{\mathfrak{C}}(Y,Z)$ there defined a unique morphism $g\circ
f\in\mathrm{Hom}_{\mathfrak{C}}(X,Z)$, called composition of $f$ and $g$

such that the following axioms are satisfied:

  1. -

associativity: for any three morphisms $f\in\mathrm{Hom}_{\mathfrak{C}}(X,Y)$,
$g\in\mathrm{Hom}_{\mathfrak{C}}(Y,Z)$ and
$h\in\mathrm{Hom}_{\mathfrak{C}}(Z,W)$ we have

$h\circ(g\circ f)=(h\circ g)\circ f$

  2. -

identity: for each object $X\in\mathrm{Ob}(\mathfrak{C})$ there exists a
morphism $\mathrm{id}_{A}\in\mathrm{Hom}_{\mathfrak{C}}(X,X)$, called the
identity, such that for each $f\in\mathrm{Hom}_{\mathfrak{C}}(X,Y)$ we have

$f\circ\mathrm{id}_{X}=\mathrm{id}_{Y}\circ f=f.$

###  3.2. Functors

A functor $F\colon\mathfrak{C}\to\mathfrak{D}$ from category $\mathfrak{C}$ to
category $\mathfrak{D}$ is the following collection of maps (usually denoted
by the same letter $F$):

  1. 1)

a map $F:\mathrm{Ob}(\mathfrak{C})\to\mathrm{Ob}(\mathfrak{D})$, associating
to each object $X\in\mathfrak{C}$ an object $F(X)\in\mathfrak{D}$,

  2. 2)

a map
$F:\mathrm{Hom}_{\mathfrak{C}}(X,Y)\to\mathrm{Hom}_{\mathfrak{D}}(F(X),F(Y))$
for every pair of objects $X,Y\in\mathfrak{C}$,

such that the following conditions are satisfied:

  1. a)

$F(\mathrm{id}_{X})=\mathrm{id}_{F(X)}$ for each
$X\in\mathrm{Ob}(\mathfrak{C})$;

  2. b)

$F(g\circ f)=F(g)\circ F(f)$ for any morphisms $f,g$ in category
$\mathfrak{C}$.

###  3.3. Coequalizers

Let $f,g:X\to Y$ be two morphisms. Then a coequalizer of $f,g$ is a morphism
$h:Y\to Z$ such that

  1. 1)

$h\circ f=h\circ g$;

  2. 2)

for any other morphism $h^{\prime}:Y\to Z^{\prime}$ with $h^{\prime}\circ
f=h^{\prime}\circ g$ there exists a unique morphism $q:Z\to Z^{\prime}$ such
that $h^{\prime}=q\circ h$.

In other words, we have the following diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{h^{\prime}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{Z^{\prime}}$

which is commutative except for paths $f,g$ connecting $X$ and $Y$.

###  3.4. Groupoids

A morphism $f\in\mathrm{Hom}_{\mathfrak{C}}(X,Y)$ is called an isomorphism
whenever there exists $g\in\mathrm{Hom}_{\mathfrak{C}}(Y,X)$ such that $g\circ
f=\mathrm{id}_{X}$ and $f\circ g=\mathrm{id}_{Y}$.

A groupoid is a category in which all morphisms are isomorphisms. Moreover, a
morphism of groupoids is just a functor between the corresponding categories.

###  3.5. Example

For any set $X$ the cartesian product $X\times X$ has a natural groupoid
structure.

Evidently, if $f:X\to Y$ is a bijection of sets, then the induced mapping

$f\times f:X\times X\to Y\times Y,\qquad(f\times
f)(a,b)=\bigl{(}f(a),f(b)\bigr{)}$

is an isomorphism of groupoids.

###  3.6. Fundamental groupoid

Let $X$ be a topological space and $X^{\prime}\subset X$ be a subset. Let also
$I=[0,1]$, and $C\bigl{(}(I,\partial I),(X,X^{\prime})\bigr{)}$ be the set of
continuous paths with ends at $X^{\prime}$. Say that two paths
$\alpha,\beta\in C\bigl{(}(I,\partial I),(X,X^{\prime})\bigr{)}$ are
equivalent ($\alpha\sim\beta$) if

  1. (1)

$\alpha(0)=\beta(0)$, $\alpha(1)=\beta(1)$;

  2. (2)

$\alpha$ is homotopic to $\beta$ relatively $\\{0,1\\}$.

Then the set $\varPi_{1}(X,X^{\prime})=C\bigl{(}(I,\partial
I),(X,X^{\prime})\bigr{)}/\sim$ of the corresponding equivalence classes is
called the fundamental groupoid of pair $(X,X^{\prime})$ and the set
$X^{\prime}$ is the set of its base points.

The equivalence class of a path $\alpha\in C\bigl{(}(I,\partial
I),(X,X^{\prime})\bigr{)}$ will be denoted by $[\alpha]$. For a class
$q\in\varPi_{1}(X,X^{\prime})$ we will denote by $q(0)$ and $q(1)$ the common
start and end-points of all representatives of $q$.

Then $\varPi_{1}(X,X^{\prime})$ admits a natural ‘‘partial’’ operation of
composition of paths turning it into groupoid. Evidently, if $X^{\prime}$
consists of a unique point, then the multiplication is defined for all
elements of $\varPi_{1}(X,X^{\prime})$, and in this case
$\varPi_{1}(X,X^{\prime})$ is the same as the fundamental group of $X$ at the
point $X^{\prime}$.

Notice also that we have a natural map

$r_{X}:\varPi_{1}(X,X^{\prime})\to X^{\prime}\times X^{\prime},\qquad
r_{X}(q)=\bigl{(}q(0),q(1)\bigr{)},$ (3.1)

associating to each homotopy class of paths its start and end-points.

Moreover, every continuous map of pairs
$f\colon(X,X^{\prime})\to(Y,Y^{\prime})$ induces the following commutative
diagram:

$\textstyle{\varPi_{1}(X,X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\scriptstyle{r_{X}}$$\textstyle{\varPi_{1}(Y,Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{Y}}$$\textstyle{X^{\prime}\times
X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\times
f}$$\textstyle{Y^{\prime}\times Y^{\prime}.}$ (3.2)

The following simple lemma is left for the reader:

######  Lemma 3.7.

  1. (a)

The map (3.1) is a morphism of groupoids.

  2. (b)

If $X$ is path connected and simply connected, then $r_{X}$ is an isomorphism.

  3. (c)

Suppose $f:(X,X^{\prime})\to(Y,Y^{\prime})$ is a continuous map, $X$ and $Y$
are path connected and simply connected, and the restriction map
$f|_{X^{\prime}}\colon X^{\prime}\to Y^{\prime}$ is a bijection. Then all
morphisms in (3.2) are isomorphisms.

###  3.8. Coproducts

Let $\\{X_{\lambda}\\}_{\lambda\in\Lambda}$ be a collection of objects of a
category $\mathfrak{C}$. Then their coproduct is an object
$Y=\mathop{\sqcup}\limits_{\lambda\in\Lambda}X_{\lambda}$ together with a
collection of morphisms $i_{\lambda}:X_{\lambda}\to Y$ such that for any other
collection of morphisms $\\{f_{\lambda}:X_{\lambda}\to
Z\\}_{\lambda\in\Lambda}$, to the same object $Z$ there exists a unique
morphism $f\colon Y\to Z$ such that $f_{\lambda}=f\circ i_{\lambda}$. In other
words, for each $\lambda\in\Lambda$ the following diagram is commutative:

$\textstyle{X_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\lambda}}$$\scriptstyle{f_{\lambda}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}X_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f=\mathop{\sqcup}\limits_{\lambda\in\Lambda}f_{\lambda}}$$\textstyle{Z}$

Consider examples.

  1. a)

In the category of sets a coproduct is just a disjoint union of sets.

  2. b)

Let $\\{G_{\lambda}\\}_{\lambda\in\Lambda}$ be a collection of groupoids with
multiplications

$\mu_{\lambda}:G_{\lambda}\times G_{\lambda}\supset Q_{\lambda}\to
G_{\lambda},$

and sets of objects $E_{\lambda}$. Then their disjoint union

$G=\mathop{\sqcup}\limits_{\lambda\in\Lambda}G_{\lambda}$

has a natural groupoid structure in which the partial multiplication

$\mu:G\times G\supset\mathop{\sqcup}\limits_{\lambda\in\Lambda}Q_{\lambda}\to
G,$

is defined by $\mu(a,b)=\mu_{\lambda}(a,b)$, when $(a,b)\in Q_{\lambda}$. This
groupoid is the coproduct in the category of groupoids.

Notice also that if $\\{f_{\lambda}:G_{\lambda}\to H\\}_{\lambda\in\Lambda}$
is a collection of morphisms of groupoids, then the induced morphism
$f=\mathop{\sqcup}\limits_{\lambda\in\Lambda}f_{\lambda}:G\to H$ is given by
$f(a)=f_{\lambda}(a)$ if $a\in G_{\lambda}$.

  3. c)

Let $\\{(X_{\lambda},X^{\prime}_{\lambda})\\}_{\lambda\in\Lambda}$ be a
collection of pairs of topological spaces,
$X=\mathop{\sqcup}\limits_{\lambda\in\Lambda}X_{\lambda}$ and
$X^{\prime}=\mathop{\sqcup}\limits_{\lambda\in\Lambda}X^{\prime}_{\lambda}$ be
the corresponding coproducts of sets. For each $\lambda\in\Lambda$ the natural
inclusion of pairs
$j_{\lambda}:(X_{\lambda},X^{\prime}_{\lambda})\subset(X,X^{\prime})$ induces
the corresponding morphism of groupoids

$\Pi_{1}(j_{\lambda}):\varPi_{1}(X_{\lambda},X^{\prime}_{\lambda})\to\varPi_{1}(X,X^{\prime}).$

Hence by coproduct property, we have a unique morphism $\zeta$ making the
following diagram commutative:

$\textstyle{\varPi_{1}(X_{\lambda},X^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\lambda}}$$\scriptstyle{\Pi_{1}(j_{\lambda})}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(X_{\lambda},X^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta=\mathop{\sqcup}\limits_{\lambda\in\Lambda}\Pi_{1}(j_{\lambda})}$$\textstyle{\varPi_{1}(X,X^{\prime})}$
(3.3)

where $i_{\lambda}$ is a natural morphism into coproduct. We claim that
$\zeta$ is an isomorphism, i.e. it is bijective.

Let us show that $\zeta$ is surjective. Indeed, let $\alpha:(I,\partial
I)\to(X,X^{\prime})$ be a continuous map. We should show that
$[\alpha]=\zeta(q)$ for some
$q\in\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(X_{\lambda},X^{\prime}_{\lambda})$.
Since $I$ is path-connected and $X_{\lambda}\cap X_{\nu}=\varnothing$ for
$\lambda\not=\nu$ it follows that $\alpha(I)$ is contained in some
$X_{\lambda}$. Hence $\alpha=j_{\lambda}\circ\beta$ for some path
$\beta:(I,\partial I)\to(X_{\lambda},X^{\prime}_{\lambda})$. Therefore,
$[\alpha]=\Pi_{1}(j_{\lambda})\bigl{(}[\beta]\bigr{)}=\zeta\circ
i_{\lambda}\bigl{(}[\beta]\bigr{)}=\zeta\bigl{(}[i_{\lambda}\circ\beta]\bigr{)}$.

Let us prove that $\zeta$ is injective. Let
$p,q\in\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(X_{\lambda},X^{\prime}_{\lambda})$
be two classes such that $\zeta(p)=\zeta(q)$. By definition,
$p=i_{\lambda}([\alpha])$ and $q=i_{\nu}([\beta])$ for some paths

$\alpha:(I,\partial
I)\to(X_{\lambda},X^{\prime}_{\lambda}),\qquad\beta:(I,\partial
I)\to(X_{\nu},X^{\prime}_{\nu}).$

Then

$\Pi_{1}(j_{\lambda})\bigl{(}[\alpha]\bigr{)}=\zeta\bigl{(}i_{\lambda}([\alpha])\bigr{)}=\zeta(p)=\zeta(q)=\zeta\bigl{(}i_{\nu}([\beta])\bigr{)}=\Pi_{1}(j_{\nu})\bigl{(}[\beta]\bigr{)}$

which means that the paths
$j_{\lambda}\circ\alpha,j_{\lambda}\circ\beta:(I,\partial I)\to(X,X^{\prime})$
are homotopic relatively their ends. Hence $\lambda=\nu$. Moreover, let
$H:[0,1]\times[0,1]\to X$ be the corresponding homotopy between $H_{0}=\alpha$
and $H_{1}=\beta$. Since $[0,1]\times[0,1]$ is path-connected, it follows that
$H([0,1]\times[0,1])\subset X_{\lambda}$, and thus $H$ is a homotopy between
$\alpha$ and $\beta$ in $X_{\lambda}$ relatively their ends. This means that
$\lambda=\nu$,
$[\alpha]=[\beta]\in\varPi_{1}(X_{\lambda},X^{\prime}_{\lambda})$, whence
$p=i_{\lambda}([\alpha])=i_{\nu}([\beta])=q$. Thus $\zeta$ is injective.

###  3.9. van Kampen theorem for groupoids

Let $\Lambda$ be a set. It will be convenient to consider a category
$\mathfrak{C}$ whose objects are triples
$\bigl{(}X,X^{\prime},\mathcal{U}\bigr{)}$ where

  1. (1)

$X$ is a topological space;

  2. (2)

$X^{\prime}\subset X$ a subset;

  3. (3)

$\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ is an open cover of $X$
enumerated by the same set of indices $\Lambda$;

and morphisms between triples
$\bigl{(}X,X^{\prime},\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}\bigr{)}$
and
$\bigl{(}Y,Y^{\prime},\mathcal{V}=\\{V_{\lambda}\\}_{\lambda\in\Lambda}\bigr{)}\in\mathfrak{C}$
are continuous maps of pairs $f\colon(X,X^{\prime})\to(Y,Y^{\prime})$ such
that $f\bigl{(}V_{\lambda}\bigr{)}\subset U_{\lambda}$ for all
$\lambda\in\Lambda$.

Given a triple $\bigl{(}X,X^{\prime},\mathcal{U}\bigr{)}\in\mathfrak{C}$, for
each $n$-tuple $\nu=(\nu_{1},\ldots,\nu_{n})\in\Lambda^{n}$ put

$\displaystyle U_{\nu}$ $\displaystyle:=U_{\nu_{1}}\cap\cdots\cap
U^{\nu_{n}},$ $\displaystyle U^{\prime}_{\nu}$ $\displaystyle:=U_{\nu}\cap
X^{\prime}.$

In particular, for $n=2$ and $\nu=(j,k)\in A^{2}$ we have the following two
morphisms of fundamental groupoids:

$\displaystyle a_{jk}:\varPi_{1}(U_{j}\cap
U_{k},U^{\prime}_{(j,k)})\to\varPi_{1}(U_{j},U^{\prime}_{j}),$ $\displaystyle
b_{jk}:\varPi_{1}(U_{j}\cap
U_{k},U^{\prime}_{(j,k)})\to\varPi_{1}(U_{k},U^{\prime}_{k})$

induced by natural inclusions of $U_{j}\cap U^{k}$ into $U^{j}$ and $U^{k}$
respectively.

These morphisms yield morphisms of the corresponding coproducts:

$a,b:\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}\varPi_{1}(U_{(j,k)},U^{\prime}_{(j,k)})\to\bigsqcup\limits_{\lambda\in\Lambda}\varPi_{1}(U_{\lambda},U^{\prime}_{\lambda})$

Similarly, the inclusion $(U_{j},U^{\prime}_{j})\subset(X,X^{\prime})$ induces
a morphism

$c_{j}:\varPi_{1}(U_{j},U^{\prime}_{j})\to\varPi_{1}(X,X^{\prime}).$

Then the $\varPi_{1}$-diagram of the cover $\mathcal{U}$ is the following
diagram:

$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}\varPi_{1}(U_{(j,k)},U^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{\bigsqcup\limits_{\lambda\in\Lambda}\varPi_{1}(U_{\lambda},U^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\varPi_{1}(X,X^{\prime}).}$
(3.4)

######  Theorem 3.10 (van Kampen theorem for fundamental groupoids, [5]).

Suppose that an open cover $\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$
of $X$ has the following property:

  1. $(enumi)$

a subset $X^{\prime}\subset X$ meets each path-component of each non-empty
two-fold and three-fold intersection of distinct sets of $\mathcal{U}$.

Then in the $\varPi_{1}$-diagram (3.4) of the cover $\mathcal{U}$ the morphism
$c$ is a coequalizer for morphisms $a,b$ in the category of fundamental
groupoids.

###  3.11. $\varPi_{1}$-diagram for covers by simply connected sets

Let

$f:\bigl{(}X,X^{\prime},\\{U_{\lambda}\\}_{\lambda\in\Lambda}\bigr{)}\to\bigl{(}Y,Y^{\prime},\\{V_{\lambda}\\}_{\lambda\in\Lambda}\bigr{)}$

be a morphism in the category $\mathfrak{C}$. Then it is evident that $f$
induces the following diagram:

$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}(U_{(j,k)},U^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{f_{2}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}(U_{\lambda},U^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{f_{1}}$$\textstyle{(X,X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}(V_{(j,k)},V^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\scriptstyle{b^{\prime}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}(V_{\lambda},V^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{(Y,Y^{\prime})}$
(3.5)

where $f_{1}=\mathop{\sqcup}\limits_{\lambda\in\Lambda}f|_{V_{\lambda}}$ and
$f_{2}=\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}f|_{V_{(j,k)}}$. This
diagram is commutative except for the left square in which we have only the
following identities:

$\displaystyle f_{1}\circ a$ $\displaystyle=a^{\prime}\circ f_{2},$
$\displaystyle f_{1}\circ b$ $\displaystyle=b^{\prime}\circ f_{2}.$ (3.6)

Since $\varPi_{1}$ is a functor from the category of sets to the category of
groupoids, (3.5) implies the following diagram combined from two
$\varPi_{1}$-diagrams:

$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}\varPi_{1}(U_{(j,k)},U^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{f_{2}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(U_{\lambda},U^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{f_{1}}$$\textstyle{\varPi_{1}(X,X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}\varPi_{1}(V_{(j,k)},V^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\scriptstyle{b^{\prime}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(V_{\lambda},V^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{\varPi_{1}(Y,Y^{\prime})}$
(3.7)

Here we used the isomorphism of groupoids (3.3). Notice that diagram (3.7) is
also commutative except for left square in which only the identities (3.6)
hold.

The following lemma plays a key role in proving the basic theorem.

######  Lemma 3.12.

Let
$f:\bigl{(}X,X^{\prime},\\{U_{\lambda}\\}_{\lambda\in\Lambda}\bigr{)}\to\bigl{(}Y,Y^{\prime},\\{V_{\lambda}\\}_{\lambda\in\Lambda}\bigr{)}$
be a morphism in the category $\mathfrak{C}$. Suppose that

  1. (1)

for each $\lambda\in\Lambda$ spaces $U_{\lambda}$ and $V_{\lambda}$ are simply
connected;

  2. (2)

for each $\lambda\in\Lambda$ the restriction $f|_{U^{\prime}_{\lambda}}\colon
U^{\prime}_{\lambda}\equiv U_{\lambda}\cap
X^{\prime}\,\to\,V^{\prime}_{\lambda}\equiv V_{\lambda}\cap Y^{\prime}$ is a
bijection;

  3. (3)

the open covers $\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ and
$\\{V_{\lambda}\\}_{\lambda\in\Lambda}$ satisfy condition $(enumi)$ of van
Kampen Theorem 3.10, that is $X^{\prime}$ (resp. $Y^{\prime}$) meets each path
component of each non-empty two-fold and three-fold intersections of elements
of $\mathcal{U}$ (resp. $\mathcal{V}$).

Then the induced morphism
$f^{*}:\varPi_{1}(X,X^{\prime})\to\varPi_{1}(Y,Y^{\prime})$ of fundamental
groupoids from diagram (3.7) is an isomorphism.

###### Proof.

Conditions (1) and (2) mean that for each $\lambda\in\Lambda$ the restriction
map

$f|_{U_{\lambda}}:(U_{\lambda},U^{\prime}_{\lambda})\to(V_{\lambda},V^{\prime}_{\lambda})$

satisfies condition (c) of Lemma 3.7. Therefore the induced morphism of
groupoids
$f_{\lambda}:\varPi_{1}(U_{\lambda},U^{\prime}_{\lambda})\to\varPi_{1}(V_{\lambda},V^{\prime}_{\lambda})$
is an isomorphism. Hence so is the middle vertical morphism $f_{1}$ of diagram
(3.7).

Let
$g_{1}:\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(V_{\lambda},V^{\prime}_{\lambda})\to\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(U_{\lambda},U^{\prime}_{\lambda})$
be the isomorphism inverse to $f_{1}$. Denote $k=c\circ g_{1}$, see (3.8):

$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}\varPi_{1}(U_{(j,k)},U^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{f_{2}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(U_{\lambda},U^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{f_{1}}$$\textstyle{\varPi_{1}(X,X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathop{\sqcup}\limits_{(j,k)\in\Lambda^{2}}\varPi_{1}(V_{(j,k)},V^{\prime}_{(j,k)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\scriptstyle{b^{\prime}}$$\textstyle{\mathop{\sqcup}\limits_{\lambda\in\Lambda}\varPi_{1}(V_{\lambda},V^{\prime}_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\scriptstyle{g_{1}}$$\scriptstyle{k}$$\textstyle{\varPi_{1}(Y,Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$
(3.8)

Then $k\circ a^{\prime}=k\circ b^{\prime}$, whence by van Kampen Theorem 3.10
there exists a unique morphism
$g:\varPi_{1}(Y,Y^{\prime})\to\varPi_{1}(X,X^{\prime})$ such that $k=g\circ
c^{\prime}$. Since $f_{1}$ and $g_{1}$ are inverse each to other, it follows
that $f$ and $g$ must also be inverse each to other and therefore $f$ is an
isomorphism. ∎

##  4\. Proof of Theorem 2.9

Let $(Z,q)$ be a striped surface, $G$ its graph, and $\varphi:G\to Z$ the
continuous injective map defined in §2.7. We should construct a subset
$Z^{\prime}\subset Z$ such that the map $\varphi:G\hookrightarrow Z$ induces
an isomorphism of the corresponding fundamental groupoids
$\varphi_{*}:\Pi_{1}(G,\varphi^{-1}(Z^{\prime}))\to\Pi_{1}(Z,Z^{\prime})$.

In fact we will also define a special open cover $\mathcal{V}$ of $Z$, and
then consider a cover $\mathcal{U}=\varphi^{-1}(\mathcal{V})$ of $G$
consisting of inverse of elements of $\mathcal{V}$. Then $\varphi$ will
evidently induce a morphism of triples
$\varphi:(G,G^{\prime},\mathcal{U})\to(Z,Z^{\prime},\mathcal{V})$ in the
category $\mathfrak{C}$, and we will show that conditions (1)-(3) of Lemma
3.12 are satisfied. This will imply that $\varphi_{*}$ is an isomorphism.

### An open cover $\mathcal{V}$ of $Z$

Let $S$ be a model strip and $X=(a,b)\times\varepsilon\subset\partial S$,
$\varepsilon\in\\{\pm 1\\}$, be a boundary interval. Then the following open
neighborhood of $X$ in $S$:

$N_{X}=\begin{cases}(a,b)\times(0.8,1],&\varepsilon=1,\\\
(a,b)\times[-1,0.8),&\varepsilon=-1.\end{cases}$

will be called the standard neighborhood of $X$.

It is evident, that standard neighborhoods of boundary intervals are mutually
disjoint. Hence the family of boundary intervals of $S$ is discrete 111Recall
that a collections of subsets $\\{Q_{i}\\}_{i\in\Lambda}$ of a topological
space $X$ is called discrete, if for each $i\in\Lambda$ there exists an open
neighborhood $U_{i}$ of $Q_{i}$ such that $U_{i}\cap U_{j}=\varnothing$ for
$i\not=j\in\Lambda$. and in particular locally finite. Therefore, the union of
any number of boundary intervals is a closed set.

By a standard neighborhood $N_{\omega_{\beta}}$ of a seam $\omega_{\beta}$ we
will mean the union of images of standard neighborhoods of $X_{\beta}$ and
$Y_{\beta}$ (see Figure 2.2):

$N_{\omega_{\beta}}=q(N_{X_{\beta}})\cup q(N_{Y_{\beta}}).$

Then $N_{\omega_{\beta}}$ is open in $Z$. Moreover, $N_{\omega_{\beta}}\cap
N_{\omega_{\beta^{\prime}}}=\varnothing$ for $\beta\not=\beta^{\prime}$,
whence the family of seams is discrete and therefore locally finite. In
particular, the union of any collections of seams is also closed.

Furthermore, for $\alpha\in A$ put

$N_{S_{\alpha}}:=q\bigl{(}S_{\alpha}\setminus(X\cup Y)\bigr{)}.$

Thus $N_{S_{\alpha}}$ is image of a model strip $S_{\alpha}$ without boundary
intervals corresponding to seams. It follows that $N_{S_{\alpha}}$ is open in
$Z$. Hence we get the following open cover of $Z$.

$\mathcal{V}:=\\{N_{S_{\alpha}}\\}_{\alpha\in A}\ \cup\
\\{N_{\omega_{\beta}}\\}_{\beta\in B}.$

### The set $Z^{\prime}$

Notice that $N_{\omega_{\beta}}\setminus\omega_{\beta}$ consists of two
connected components each homeomorphic to an open rectangle and the image
$\varphi(G)$ intersects each of those components. Choose any two points

$\displaystyle
d_{\beta}\in\varphi(G)\cap\bigl{(}q(N_{X_{\beta}})\setminus\omega_{\beta}\bigr{)},$
$\displaystyle
d^{\prime}_{\beta}\in\varphi(G)\cap\bigl{(}q(N_{Y_{\beta}})\setminus\omega_{\beta}\bigr{)}.$

belonging to distinct components of
$N_{\omega_{\beta}}\setminus\omega_{\beta}$. In Figure 2.2 such points are
denoted by circles.

Let $J^{\prime}$ be the set of isolated vertices of $G$. Then every strip
$S_{\alpha}$ with $\alpha\in J$ is not glued to any other strips, and no
boundary intervals of $S$ are glued together. Let also
$K^{\prime}=\\{s_{\alpha}=(0,0)=\varphi(\alpha)\in S_{\alpha}\mid\alpha\in
J^{\prime}\\}\subset Z$ be the set of origins of such strips in $Z$. Put

$Z^{\prime}=K^{\prime}\ \cup\ \\{d_{\beta},d^{\prime}_{\beta}\mid\beta\in
B\\},$

and

$\displaystyle G^{\prime}$ $\displaystyle=\varphi^{-1}(Z^{\prime}),$
$\displaystyle L_{s_{\alpha}}$ $\displaystyle:=\varphi^{-1}(N_{S_{\alpha}}),$
$\displaystyle L_{\beta}$ $\displaystyle:=\varphi^{-1}(N_{\omega_{\beta}}).$

Thus $L_{\beta}$ is an open arc in some $1$-cell of $G$ containing two points
$\varphi^{-1}(d_{\beta})$ and $\varphi^{-1}(d^{\prime}_{\beta})$, while
$L_{s_{\alpha}}$ is a ‘‘star’’-neighborhood of the vertex $\alpha\in G^{0}=A$
such that each edge of $L_{s_{\alpha}}$ contains a unique point
$\varphi^{-1}(d_{\beta})$ or $\varphi^{-1}(d^{\prime}_{\beta})$ for some
$\beta\in B$.

It follows that

$\mathcal{U}:=\\{L_{s_{\alpha}}\\}_{\alpha\in A}\ \cup\
\\{L_{\beta}\\}_{\beta\in B}$

is an open cover of $G$, and
$\varphi:(G,G^{\prime},\mathcal{U})\to(Z,Z^{\prime},\mathcal{V})$ induces a
morphism in the category $\mathfrak{C}$.

### Verification of conditions of Lemma 3.12

(1) Evidently, the elements of $\mathcal{U}$ and $\mathcal{V}$ are even
contractible and therefore they are simply connected.

(2) Since $\varphi$ is injective, it follows that for any subset
$Q\subset\varphi(G)\subset Z$, we have that
$\varphi|_{\varphi^{-1}(Q)}:\varphi^{-1}(Q)\to Q$ is a bijection. In
particular, so are the restrictions

$\displaystyle\varphi:G^{\prime}\cap L_{s_{\alpha}}\to Z^{\prime}\cap
N_{S_{\alpha}},$ $\displaystyle\varphi:G^{\prime}\cap L_{\beta}\to
Z^{\prime}\cap N_{\omega_{\beta}}$

for each $\alpha\in A$ and $\beta\in B$.

(3) First notice that $Z^{\prime}$ (resp. $G^{\prime}$) meets every path
component of $Z$ (resp. $G$). Furthermore, $N_{S_{\alpha}}\cap
N_{S_{\alpha^{\prime}}}=N_{\omega_{\beta}}\cap
N_{\omega_{\beta^{\prime}}}=L_{s_{\alpha}}\cap
L_{s_{\alpha^{\prime}}}=L_{\beta}\cap L_{\beta^{\prime}}=\varnothing$ for
$\alpha\not=\alpha^{\prime}\in A$ and $\beta\not=\beta^{\prime}\in B$, which
implies that all three-fold intersections of elements of $\mathcal{V}$ and
$\mathcal{U}$ are empty.

Also, $N_{S_{\alpha}}\cap N_{\omega_{\beta}}\not=\varnothing$ iff either
$X_{\beta}$ or $Y_{\beta}$ or both of them are contained in $\partial
S_{\alpha}$. In this case each connected component of $N_{S_{\alpha}}\cap
N_{\omega_{\beta}}$ contains either $d_{\beta}$ or $d^{\prime}_{\beta}$. It
follows that each connected component of $L_{s_{\alpha}}\cap L_{\beta}$
contains either $\varphi^{-1}(d_{\beta})$ or
$\varphi^{-1}(d^{\prime}_{\beta})$. Thus $Z^{\prime}$ (resp. $G^{\prime}$)
meets all path components of all two-fold intersections of elements of
$\mathcal{V}$ (resp. $\mathcal{U}$). Hence all conditions of Lemma 3.12 holds,
whence $\varphi_{*}$ is an isomorphism of groupoids. ∎

##  5\. Proof of Theorem 2.8

By Theorem 2.9, $\varphi$ yields an isomorphism of fundamental groupoids. In
particular, if $x\in G^{\prime}$, then $\varphi$ also induces an isomorphism
of the fundamental groups $\pi_{1}(G,x)\to\pi_{1}(Z,\varphi(x))$. Since
$Z^{\prime}$ (resp. $G^{\prime}$) meets every path component of $Z$ (resp.
$G$), it follows that $\varphi$ induces isomorphism of fundamental groups are
each point $x\in G$.

Notice that every connected component of $G$ is covered by at most countable
tree, and therefore $G$ is aspherical. Moreover, $Z$ is also aspherical by
Lemma 2.11. Hence $\varphi$ induces isomorphisms between all the corresponding
homotopy groups of $G$ and $Z$. Moreover, since $Z$ has the homotopy type of
an infinite CW-complex, see Lemma 2.11, we have from the Whitehead theorem
that $\varphi$ is a homotopy equivalence between all corresponding connected
components of $G$ and $Z$. ∎

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