# Infall Motions in the Hot Core Associated with Hyper-Compact HII Region
G345.0061+01.794 B

Toktarkhan Komesh Energetic Cosmos Laboratory, Nazarbayev University, Astana
010000, Kazakhstan  Institute of Experimental and Theoretical Physics, Al-
Farabi Kazakh National University, Almaty 050040, Kazakhstan  Guido Garay
Departamento de Astronomia, Universidad de Chile, Camino el Observatorio 1515,
Las Condes, Santiago, Chile  Christian Henkel  Max-Planck-Institut für
Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany  Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, Urumqi 830011, PR China  Astronomy
Department, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi
Arabia  Aruzhan Omar  Institute of Experimental and Theoretical Physics, Al-
Farabi Kazakh National University, Almaty 050040, Kazakhstan  Faculty of
Physics and Technology, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Robert Estalella  Departament de Física Quàntica i Astrofísica,
Institut de Ciències del Cosmos, Universitat de Barcelona,
IEEC-UB, Martí i Franquès 1, 08028 Barcelona, Spain  Zhandos Assembay
Institute of Experimental and Theoretical Physics, Al-Farabi Kazakh National
University, Almaty 050040, Kazakhstan  Faculty of Physics and Technology, Al-
Farabi Kazakh National University, Almaty 050040, Kazakhstan  Dalei Li
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011,
PR China  Andrés Guzmán  National Astronomical Observatory of Japan, National
Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Jarken Esimbek  Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, Urumqi 830011, PR China  Jiasheng Huang  Chinese Academy of Sciences
South America Center for Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100101, PR China  Yuxin He  Xinjiang
Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, PR China
Nazgul Alimgazinova  Energetic Cosmos Laboratory, Nazarbayev University,
Astana 010000, Kazakhstan  Institute of Experimental and Theoretical Physics,
Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan  Faculty of
Physics and Technology, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Meiramgul Kyzgarina  Energetic Cosmos Laboratory, Nazarbayev
University, Astana 010000, Kazakhstan  Institute of Experimental and
Theoretical Physics, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Faculty of Physics and Technology, Al-Farabi Kazakh National
University, Almaty 050040, Kazakhstan  Shukirgaliyev Bekdaulet  Energetic
Cosmos Laboratory, Nazarbayev University, Astana 010000, Kazakhstan  Heriot-
Watt International Faculty, Zhubanov University, 263 Zhubanov brothers str,
030000 Aktobe, Kazakhstan  Department of Computation and Data Science, Astana
IT University, 55/11 Mangilik El ave, 010000 Astana, Kazakhstan  Fesenkov
Astrophysical Institute, 23 Observatory str., 050020 Almaty, Kazakhstan
Nurman Zhumabay  Energetic Cosmos Laboratory, Nazarbayev University, Astana
010000, Kazakhstan  Institute of Experimental and Theoretical Physics, Al-
Farabi Kazakh National University, Almaty 050040, Kazakhstan  Institute of
Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical
University, Almaty 050010, Kazakhstan  Arailym Manapbayeva  Institute of
Experimental and Theoretical Physics, Al-Farabi Kazakh National University,
Almaty 050040, Kazakhstan  Faculty of Physics and Technology, Al-Farabi Kazakh
National University, Almaty 050040, Kazakhstan

###### Abstract

We report high angular resolution observations, made with the Atacama Large
Millimeter Array in band 6, of high excitation molecular lines of $\rm
CH_{3}CN$ and $\rm SO_{2}$ and of the H29$\alpha$ radio recombination line
towards the G345.0061+01.794 B HC H ii region, in order to investigate the
physical and kinematical characteristics of its surroundings. Emission was
detected in all observed components of the J=14$\rightarrow$13 rotational
ladder of $\rm CH_{3}CN$ and in the $30_{4,26}-30_{3,27}$ and
$32_{4,28}-32_{3,29}$ lines of $\rm SO_{2}$. The peak of the velocity
integrated molecular emission is located $\sim$0$\,.\\!\\!^{\prime\prime}$4
northwest of the peak of the continuum emission. The first-order moment images
and channel maps show a velocity gradient, of 1.1 km s-1 arcsec-1, across the
source, and a distinctive spot of blueshifted emission towards the peak of the
zero-order moment. The rotational temperature is found to decrease from
252$\pm 24$ Kelvin at the peak position to 166$\pm 16$ Kelvin at its edge,
indicating that our molecular observations are probing a hot molecular core
that is internally excited. The emission in the H29$\alpha$ line arises from a
region of 0$\,.\\!\\!^{\prime\prime}$65 in size, where its peak coincides with
that of the dust continuum. We model the kinematical characteristics of the
”central blue spot” feature as due to infalling motions, suggesting a central
mass of 172.8$\pm 8.8M_{\odot}$. Our observations indicate that this HC H ii
region is surrounded by a compact structure of hot molecular gas, which is
rotating and infalling toward a central mass, that is most likely confining
the ionized region. The observed scenario is reminiscent of a ”butterfly
pattern” with an approximately edge-on torus and ionized gas roughly parallel
to its rotation axis.

ISM: molecules —ISM: clouds —ISM: cores — stars: formation —stars: massive
—ISM: kinematics and dynamics

††facilities: ALMA††software: astropy (Astropy Collaboration et al., 2013,
2018), CASA (McMullin et al., 2007), MADCUBA (Martín et al., 2019)

##  1 Introduction

The formation of high-mass stars begins inside dense and massive molecular
cores where high-mass protostellar objects accrete at rates between 10-5 and
10-3 M⊙ yr-1 (Tan et al., 2014). These objects finish their Kelvin–Helmholtz
(K-H) contraction very rapidly and reach the main sequence (Norberg & Maeder,
2000; Keto & Wood, 2006). At this point, the star radiates extreme ultraviolet
(UV) photons that ionize its surroundings, producing very small regions of
ionized gas, observationally characterized by sizes $\leq 0.03$ pc, densities
n${}_{e}>$ 106 cm-3, and emission measures $>$ 108 pc cm-6 (Kurtz, 2000).
These hyper-compact (HC) regions are thought to signpost an early stage of the
evolutionary path of High Mass Young Stellar Objects (HMYSOs).

Theoretical calculations show that almost half of the mass of O-type stars is
accreted after the K-H contraction and the onset of ionizing radiation
(Hosokawa & Omukai, 2009; Zhang et al., 2014). How high-mass stars keep
accreting despite the onset of the ionizing radiation is not well established.
Theoretical works have shown that under steady spherical accretion, radiation
pressure inhibits the growth of the protostars. An effective way to circumvent
the radiation and ionized gas pressure is accretion from a disk, allowing the
accreting material to reach the young high-mass stars much more easily by
flowing inward, mainly through the plane perpendicular to the angular momentum
vector of the system (Nakano, 1989; Kuiper et al., 2011). Accretion through a
disk may only choke the ionized region near the disk plane, allowing for H ii
region development in the polar regions (Keto, 2007). In this scenario, an HC
H ii region should consist of an ionized biconical cavity confined by a
rotating and contracting predominantly neutral molecular core.

How does accretion proceed after the onset of the ionizing radiation? How does
the envelope material avoid being ionized and blown away by its own pressure?
To answer these questions we undertook ALMA Band 6 observations towards a set
of luminous embedded HMYSOs associated with HC Hii regions in order to
simultaneously observe molecular emission in highly excited transitions of
$\rm SO_{2}$ and $\rm CH_{3}CN$ and emission from the ionized gas in the
H29$\alpha$ hydrogen recombination line. The two molecules have been used to
trace velocity gradients, indicative of rotation, towards hot molecular cores
around luminous young high-mass stars in several cases (e.g., Beltrán et al.,
2014; Guzmán et al., 2014; Sánchez-Monge et al., 2013). Our molecular
observations are intended to assess whether or not HC H ii regions are
associated with rotating hot molecular cores on scales of 3000 AU, as well as
to detect inflow motions from the surrounding gas. Our goal is to find
evidence of disk accretion and to settle the question as to whether or not
accretion onto the HMYSO is maintained after stellar contraction and UV photon
injection.

In this work, we present observations towards the HC H ii region
G345.0061+01.794 B (hereafter G345.01 B, Guzmán et al., 2012) associated with
IRAS 16533-4009. The Spitzer-GLIMPSE survey shows that it is associated with a
bright compact mid-infrared source prominent in the 4.5 $\mu$m band. The
G345.01 B HC H ii region has the kinematic near and far distances of 1.7 and
14.7 kpc (Urquhart et al., 2007), respectively. The ambiguity can be resolved
through spectrophotometric estimation (e.g., Guzmán et al., 2020). In this
study, we adopt the spectrophotometric distance of 2.38 kpc (Moisés et al.,
2011). Choosing the far distance not only results in an unrealistically
massive cluster but also places it at an impractical distance above the
Galactic plane.

The paper is organized as follows: in §2 we describe the observations
performed with the Atacama Large Millimeter Array (ALMA); in §3 we present the
observational results; in §4 we discuss the analysis of the data, including
the physical relationship between the hot molecular core and the HC H ii
region; and in §5 we present a summary of the main points addressed in this
paper.

##  2 Observations

We observed, using ALMA in Band 6 between 256.3 and 259.6 GHz, the dust
continuum and molecular line emission towards the HC H ii region G345.01 B
(see Table 1). The observations were carried out, as part of ALMA Cycle 3,
during 21 May 2016, using the 12-m array. The ALMA field of view at this
wavelength is $\sim 22\arcsec$, defined as the FWHM of the primary beam. The
phase center of the array was (RA, Dec, J2000) = (16h56m47s,
–40∘14′25$\arcsec$). We observed four spectral windows in dual polarization
mode. The first window was centered at the frequency of 256.302035 GHz, has a
bandwidth of 1875.00 MHz and a spectral resolution of 1.129 MHz. This setup
was chosen to map the H29$\alpha$ radio recombination line (RRL) emission from
the HC II region. The second and third windows were centered, respectively, at
the frequencies of 258.388716 and 259.599448 GHz each with 234.38 MHz
bandwidths and 488.281 kHz ($\sim 0.564$ km s-1) channels. These two setups
were chosen to observe the emission from the purported hot core in two high
excitation temperature lines of $\rm SO_{2}$. The fourth window, centered at
the frequency of 257.325000 GHz, has a bandwidth of 468.75 MHz and a
resolution of 488.281 kHz. This setup was chosen to observe emission of CH3CN,
a good temperature probe of both the large scale diffuse and small scale dense
gas, in the J=14–13 ladder. Zapata et al. (2015) point out that, based on the
analysis of IRAS 16547-4247, to detect emission from the inner regions of the
rotating core it is important to use molecular transitions with high upper
energy levels ($>$ 300 Kelvin). The selected $\rm SO_{2}$ transitions,
$30_{4,26}-30_{3,27}$ and $32_{4,28}-32_{3,29}$, have upper level temperatures
of 471 and 531 Kelvin, respectively, and those of the lines in the $\rm
CH_{3}CN$ 14–13 ladder range between 92 and 670 Kelvin.

J1427-4206 was used as bandpass calibrator, J1717-3342 was observed as phase
calibrator, and J1617-5848 was measured as flux calibrators. Table 1 lists the
parameters of each spectral window, the synthesized beams and the rms noise
achieved. The integration time on source was 35 minutes. Calibration and
reduction of these data were done using the Common Astronomy and Software
Applications (CASA, McMullin et al., 2007).

The continuum was subtracted by selecting the line free channels from the
visibilities using the CASA task uvcontsub. These line-free channels in turn
were used to create the continuum images directly in tclean.

Figure 1: Spectra of the methyl cyanide emission, integrated over a region of 0$\,.\\!\\!^{\prime\prime}$5, centered on the G345.01 B HC H ii region. K components of the $\rm CH_{3}CN$ 14–13 transition are marked with red dashed lines ($V_{\mathrm{LSR}}$=–14 km s-1). In addition, some other lines detected in this spectral window are marked with blue dashed lines. Figure 2: Images of the velocity integrated intensity (moment 0; upper panels) and intensity weighted velocities (moment 1; lower panels) in the $\rm CH_{3}CN$ J=14–13 K=2, 3, 4, 6, 7 and 8 transitions toward the G345.01 B HC H ii region. Superimposed are contours of the continuum emission. Contour levels are 10$\sigma$, 20$\sigma$, 40$\sigma$, 100$\sigma$, 200$\sigma$, 400$\sigma$ and 800$\sigma$, where $\sigma$ is 0.3 mJy/beam. The black dashed line shown in the lower left panel indicates the position of the PV cut mentioned in section 3.1.1. The black ellipse shown at the bottom right corner of the lower left panel indicates the beam size. Figure 3: Channel maps for the K=3 component of the $\rm CH_{3}CN$ J=14–13 transition. The contours provide the continuum emission already presented in Fig. 2. The white ellipse shown in the lower right corner of the bottom left panel indicates the beam size. Table 1: Observational parameters.  Transition | Center Freq. | Bandwidth | Vel. res. | Synthesized beam | P.A. of the synthesized beams | rms noise
---|---|---|---|---|---|---
| GHz | GHz | km s-1 | ″ | ∘ | mJy/beam
H29$\alpha$ | 256.302035 | 1.875 | 1.320 | 0.52$\times$0.40 | –65.3 | 3.8
$\rm SO_{2}$ v=0 30(4,26)–30(3,27) | 259.599448 | 0.23438 | 0.564 | 0.51$\times$0.40 | –64.9 | 4.4
$\rm SO_{2}$ v=0 32(4,28)–32(3,29) | 258.388716 | 0.23438 | 0.566 | 0.51$\times$0.40 | –66.1 | 3.9
$\rm CH_{3}CN$ v=0 14–13 ladder | 257.325000 | 0.46875 | 0.569 | 0.52$\times$0.40 | –65.2 | 4.0
Table 2: O

bservational parameters of $\rm CH_{3}CN$ J = 14 $\rightarrow$ 13 rotational
lines and other lines detected in this spectral window.  Species Transition
Frequency $\rm E_{u}/k$ (GHz) (K) $\rm CH_{3}CN$ $14_{0}\rightarrow 13_{0}$
257.527 92.70 $14_{1}\rightarrow 13_{1}$ 257.522 99.85 $14_{2}\rightarrow
13_{2}$ 257.508 121.28 $14_{3}\rightarrow 13_{3}$ 257.483 156.99
$14_{4}\rightarrow 13_{4}$ 257.448 206.98 $14_{5}\rightarrow 13_{5}$ 257.404
271.23 $14_{6}\rightarrow 13_{6}$ 257.349 349.72 $14_{7}\rightarrow 13_{7}$
257.285 442.45 $14_{8}\rightarrow 13_{8}$ 257.210 549.38 $14_{9}\rightarrow
13_{9}$ 257.127 670.50 13CH3OH 15( 3,13)– 15( 2,14) +– 257.422 321.79 CH3OH
18(3,16)–18(2,17) +– 257.402 446.53 CH${}_{3}^{13}$CN 14(1)–13(1), F=14–13
257.393 99.80 14(2)–13(2), F=14–13 257.380 121.23 14(3)–13(3), F=14–13 257.355
156.94 14(4)–13(4), F=14–13 257.321 206.93 (CH3)2CO 24(3,22)–23(3,21) EE
257.373 165.68 24(3,22)–23(2,21) AE 257.338 165.75 CH3CH2CN 30(0.30)–29(0,29)
257.310 193.54 CH3OCHO 21(9,13)–20(9,12)A 257.298 377.19 23(2,22)–22(2,21)E
257.242 342.23 20(5,15)–19(5,14)E 257.227 142.79 CH3OCHO 20( 5,15)– 19( 5,14)A
257.253 142.79 29SiC2 11(4,8)–10(4,7) 257.276 105.23 SO2 7(3,5)– 7(2,6)
257.100 47.84

##  3 Results

###  3.1 Molecular emission

####  3.1.1 $\rm CH_{3}CN$

Figure 1 presents the spectrum of the emission in the J=14$\rightarrow$13
rotational transition of $\rm CH_{3}CN$ integrated over a region of
0$\,.\\!\\!^{\prime\prime}$5 in diameter, centered on the G345.01 B HC H ii
region. Other lines are also present. The stronger ones are indicated in Fig.
1 and briefly addressed in Sect. 4.3. Their transitions, line frequencies and
upper state energy levels are given in Table 2. The J=14$\rightarrow$13
rotational transition of $\rm CH_{3}CN$ consists of 14 $K$ components ($K$=0,
1, …13; $K$ being the projection of the total angular momentum of the molecule
onto its principal rotation axis) of which ten lie within the observed
spectral window (red dotted lines).

Figure 2 displays images of the zero-order (upper panels) and first-order
moments (lower panels) of the emission in the K=2, 3, 4, 6, 7 and 8 components
of the J=14–13 ladder of $\rm CH_{3}CN$. Moments of the K=0, 1, 5, 9
components are not shown since they are blended with each other or with other
molecular lines (see Fig. 1). Superimposed are contours of the continuum
emission. The peak of the velocity integrated intensity $\rm CH_{3}CN$
emission is located
$\sim$0$\,.\\!\\!^{\prime\prime}$4$\pm$0$\,.\\!\\!^{\prime\prime}$1 northwest
of the peak of the continuum emission. The first-order moment images show a
velocity gradient from roughly east to west with average velocities
preferentially blueshifted on the western side and redshifted on the eastern
side. A spot of blueshifted emission is seen towards the peak of the zero-
order moment. The ”blue spot” feature is present in all K components shown in
Figure 2, confirming that its detection is a robust result.

Fig. 3 presents channel maps of the emission in the K=3 component, which
clearly exhibits the shift in velocity, from blueshifted velocities in the
West to redshifted velocities in the East. Additionally, a ”butterfly pattern”
is also noteworthy, which may indicate the presence of accretion disks around
high-mass stars.

The ”blue spot” feature and a velocity change are illustrated in Fig. 4, which
presents position-velocity diagrams of the emission in the $K$=0, 1, 2, 3
components along a direction from the red part to the blue part, with a
position angle (P.A.) of 255° passing through the continuum peak. There is a
clear change in velocity across the source of 4.3 km s-1 over a region of
3$\,.\\!\\!^{\prime\prime}$8 (equivalent to 98 km s-1 pc-1 at a distance of
2.38 kpc). If this velocity gradient is due to gravitationally bound rotation,
it implies a dynamical mass within a 0.023 pc radius of 51 $M_{\odot}$, less
than half the mass of the central object as derived in Sec. 4.3. We conclude
that the hot molecular gas is likely bound and rotating around the central
object.

####  3.1.2 $\rm SO_{2}$

In addition to the high excitation lines of $\rm SO_{2}$ observed on purpose
in this work, the spectral window of the RRL encompasses four additional
transitions of $\rm SO_{2}$ all of which connect levels with less than 50
Kelvin above the ground state. The transitions and their parameters are listed
in Table 3; col.(2) gives the frequency, col.(3) the energy of the upper
state, col.(4) the Einstein A coefficient, and col.(5) the statistical weight
of the upper state.

Figure 5 shows images of the velocity integrated intensity (upper panels) and
intensity-weighted velocity (moment 1; lower panels) in all six observed $\rm
SO_{2}$ lines, in order of increasing excitation temperature. The peak
position of the integrated intensity in $\rm SO_{2}$ is similar
($<$0$\,.\\!\\!^{\prime\prime}$01) to that in the $\rm CH_{3}CN$ lines. The
”blue spot” signature is also present in the $\rm SO_{2}$ moment one maps.

Table 3: Parameters of the observed SO2 transitions.  Transitions | Frequency | $\rm E_{u}/k$ | $A_{\mathrm{ul}}$ | $g_{\mathrm{u}}$
---|---|---|---|---
| GHz | Kelvin | 10-4cm-1 |
_High excitation lines_ |  |  |
$30_{4,26}-30_{3,27}$ | 259.599 | 471.50 | 2.07 | 61
$32_{4,28}-32_{3,29}$ | 258.389 | 531.10 | 2.1 | 65
_Low excitation lines_ |  |  |
$3_{3,1}-3_{2,2}$ | 255.958 | 27.62 | 0.66 | 7
$4_{3,1}-4_{2,2}$ | 255.553 | 31.29 | 0.93 | 9
$5_{3,3}-5_{2,4}$ | 256.247 | 35.89 | 1.07 | 11
$7_{3,5}-7_{2,6}$ | 257.100 | 47.84 | 1.22 | 15
Figure 4: Position-velocity diagrams of K=0, 1, 2 and 3 components of the $\rm
CH_{3}CN$ J=14–13 transition along a direction from the red to the blue part,
with a position angle (P.A.) of 255° passing through the continuum peak (see
Fig.2). Contour levels are 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80% and 90% of
the peak value of 0.7 Jy/beam. Figure 5: Same as Fig. 2 for
$30_{4,26}-30_{3,27}$,
$32_{4,28}-32_{3,29},3_{3,1}-3_{2,2},4_{3,1}-4_{2,2},5_{3,3}-5_{2,4}$ and
$7_{3,5}-7_{2,6}$ $\rm SO_{2}$ emissions Figure 6: Left panel: an image of the
velocity integrated H29$\alpha$ RRL emission along with the dust continuum
contours (see Fig.2). Right panel: a spectrum of the H29$\alpha$ RRL emission
integrated over the source. A Gaussian fit to the line profile gives a
linewidth of 33.7$\pm$2.3 km s-1 and a line center velocity of
$V_{\mathrm{LSR}}$=–18.1$\pm$0.9 km s-1.  Figure 7:  Rotational diagrams of
$\rm CH_{3}CN$ derived from the peak position (”blue spot”, in lower panels of
Fig. 2), and in half-rings going from the peak position towards the southeast
(position angle: 135∘) with different radii (R1 to R5) from the peak position
using MADCUBA. The rings are centered at the peak position and have widths of
0$\,.\\!\\!^{\prime\prime}$15\. The inner ring (R1) has an inner radius of
0$\,.\\!\\!^{\prime\prime}$33\. The rotational temperature decreases from the
peak position to the edge of the source ($\sim$1″) from 252$\pm 24$ to 166$\pm
16$ Kelvin.

###  3.2 Ionized gas: H29$\alpha$ RRL emission

Since at Band 6 frequencies the continuum is likely to be dominated by dust
emission, hydrogen recombination lines (HRLs) become the most direct way to
trace the ionized gas. Figure 6 (left panel) shows an image of the velocity
integrated H29$\alpha$ emission along with the dust continuum contours.
Intensities were integrated from $V_{\mathrm{LSR}}$=–44 to 8 km s-1. The
position of the peak in the velocity integrated line emission, of 18.7 Jy/beam
km s-1, is coincident with that of the dust continuum (RA, Dec) (J2000) =
(16h56m47.6$\pm 0.1$s, –40∘14′26$\,.\\!\\!^{\prime\prime}$07$\pm{0.10}$).

A Gaussian fit to the observed H29$\alpha$ brightness distribution indicates
that the HC HII region has a deconvolved angular size (FWHM)
$\theta_{s}=\sqrt{0\hbox{$\,.\\!\\!^{\prime\prime}$}75\times
0\hbox{$\,.\\!\\!^{\prime\prime}$}56}\approx
0\hbox{$\,.\\!\\!^{\prime\prime}$}65$, corresponding to the geometrical mean
of the deconvolved major and minor axes. At the distance of 2.38 kpc this
implies a diameter of 0.0075 pc.

Figure 6, right panel, shows a spectrum of the H29$\alpha$ RRL emission
integrated over the source. A Gaussian fit to the line profile gives a
linewidth of 33.7$\pm$2.3 km s-1 and a line center velocity of
$V_{\mathrm{LSR}}$=–18.1$\pm$0.9 km s-1. This velocity differs from those of
the $\rm CH_{3}CN$ and $\rm SO_{2}$ molecular lines. It may be due to the fact
that ionized gas and molecular cloud are slightly displaced.

The electron temperature $T^{*}_{e}$ can be derived from the following
expression (Gordon & Sorochenko, 2002) assuming local thermodynamic
equilibrium (LTE) conditions:

$\begin{split}T^{*}_{e}=\Bigg{[}\bigg{(}\frac{6985}{\alpha(v,T_{e})}\bigg{)}\bigg{(}\frac{\Delta
V_{\mathrm{H29\alpha}}}{\mathrm{km\,s}^{-1}}\bigg{)}^{-1}\bigg{(}\frac{S_{\mathrm{ff}}}{S_{\mathrm{H29\alpha}}}\bigg{)}\bigg{(}\frac{\nu}{\mathrm{GHz}}\bigg{)}^{1.1}\\\
\times\bigg{(}1+\frac{N(\mathrm{He^{+}})}{N(\mathrm{H^{+}})}\bigg{)}^{-1}\Bigg{]}^{0.87},\end{split}$
(1)

where $S_{\mathrm{ff}}$ is the free-free continuum flux density,
${S_{\mathrm{H29\alpha}}}$ is the H29$\alpha$ peak flux density,
$\alpha(v,T_{e})$ $\sim$ 1 is a slowly varying function of frequency (Mezger &
Henderson, 1967), and $N(\mathrm{He^{+}})/N(\mathrm{H^{+}})$ is the $He^{+}$
to $H^{+}$ abundance ratio. The free-free flux density, $S_{\mathrm{ff}}$,
cannot be derived from the continuum emission at 256 GHz because of the
contribution of dust emission at this frequency. We estimate it using the
parameters of the HC HII region (Emission Measure (EM) and size) derived by
Guzmán et al. (2012), from a fit to the observed radio continuum spectra at
lower frequencies, obtaining a value of $S_{\mathrm{ff}}$=325$\pm$65 mJy.
Using the observed values of ${S_{\mathrm{H29\alpha}}}$=883$\pm$47 mJy and
$\Delta V_{\mathrm{H29\alpha}}$=33.7$\pm$2.3 km s-1 and adopting a value of
0.096 for the $He^{+}$ to $H^{+}$ abundance ratio (Mehringer, 1994), we get an
electron temperature $T^{*}_{e}$=8094$\pm$1534 Kelvin.

Further parameters of the region of ionized gas can be computed using the
equations presented in Mezger & Henderson (1967) and Mezger et al. (1974).
Assuming that the HII region is spherical and homogeneous, using the values of
the continuum flux density at 256 GHz (325$\pm$65 mJy), the angular size
(0$\,.\\!\\!^{\prime\prime}$65), electron temperature (8094$\pm$1534 Kelvin)
and distance (2.38 kpc), we determine an electron density of
(2.1$\pm$0.2$)\times$105 cm-3, an emission measure of (4.6$\pm$0.9)$\times$108
pc cm-6, a mass of ionized gas (3.6$\pm$0.3)$\times$10-3 $M_{\odot}$ and the
number of ionizing photons required to excite the HII region becomes
(2.6$\pm$0.6$)\times$1047s-1. The errors do neither include the distance
uncertainty nor potentially strong density gradients as suggested by Kurtz et
al. (2000). If we consider the near and far distances of 1.7 and 14.7 kpc, the
inferred Lyman continuum flux suggests the presence of a massive star
equivalent to a zero-age main-sequence star of type B0 and O5.5, respectively,
within the G345.01 B HC H II region (Panagia, 1973) which corresponds to a
stellar object with a mass of 10 $M_{\odot}$ to 30 $M_{\odot}$.

###  3.3 Dust continuum

Continuum contours are overlaid on all moment maps depicting $\rm CH_{3}CN$ ,
$\rm SO_{2}$, and H29$\alpha$ RRL in Figs. 2, 3, 5, and 6a. Continuum contour
levels are set at 10$\sigma$, 20$\sigma$, 40$\sigma$, 100$\sigma$,
200$\sigma$, 400$\sigma$ and 800$\sigma$, with $\sigma$ being 0.3 mJy/beam.
The peak position, measuring 0.234 Jy/beam, aligns
($<$0$\,.\\!\\!^{\prime\prime}$01) with that of the ionized gas (H29$\alpha$
RRL), while the peak of the velocity-integrated intensity of $\rm CH_{3}CN$
and $\rm SO_{2}$ emissions is situated (as mentioned before) approximately
0$\,.\\!\\!^{\prime\prime}$4 northwest of the continuum emission peak. We
estimate the dust continuum flux of 565$\pm$112 mJy by subtracting the
contribution of the free-free emission of 325$\pm$65 mJy from the total flux
of the continuum map of 890$\pm$47 mJy.

The mass of the core (central star) is obtained from the following equation
using the dust continuum flux:

$M_{\text{dust}}=\frac{S_{\mathrm{1.1mm}}\times
D^{2}}{k_{\mathrm{1.1mm}}\times B_{\mathrm{1.1mm}}(T_{\text{dust}})},$ (2)

where $S_{\text{1.1mm}}$ is the dust continuum flux, $D$ is the distance to
the object, and $T_{\text{1.1mm}}$ is the dust temperature.
$B_{\mathrm{1.1mm}}(T_{\text{dust}})$ represents the Planck function (black
body radiation) at the dust temperature $T_{\text{dust}}$, and we adopt a
grain emissivity, $k_{\mathrm{1.1mm}}$ = 0.0078 cm2 g-1, which is the value
calculated by Ossenkopf & Henning (1994) for bare grains and dense gas, where
a gas-to-dust ratio of 100 is assumed. Adopting $T_{\text{dust}}$ = 250 K (see
Section 4.1), we derive a dust mass of $\sim$4 M⊙ (for a distance of 2.38
kpc). This dust mass is significantly smaller than the mass of the central
object as derived in Sec. 4.3, while its gas mass is larger. We conclude that
the gravitational field is significantly influenced by the central star.

##  4 Discussion

###  4.1 Rotational temperature of $\rm CH_{3}CN$

Rotational temperature and total column density of $\rm CH_{3}CN$ were derived
from the MAdrid Data CUBe Analysis (MADCUBA111MADCUBA is a software developed
in the Madrid Center of Astrobiology (INTA-CSIC) which enables to visualise
and analyse single spectra and data cubes: https://cab.inta-csic.es/madcuba/.,
Martín et al., 2019) software assuming LTE.

Figure 7 displays rotational diagrams of the emission at the peak position
(”blue spot” in Fig. 2) and in half-rings going from the peak position towards
the South-East (diameters at position angles of 135°) with different radii (R1
to R5) from the peak position. The rings are centered at the peak position and
have widths of 0$\,.\\!\\!^{\prime\prime}$15\. The inner ring (R1) has an
inner radius of 0$\,.\\!\\!^{\prime\prime}$33\. The rotational temperature
decreases outwards with distance from the ”blue spot” (exciting star) being
252$\pm 24$ Kelvin at the peak position and 166$\pm 16$ Kelvin at the edge of
the molecular ($\rm CH_{3}CN$) structure ($\sim$0.01 pc). Figure 8 plots the
computed rotational temperature versus the projected distance from the ”blue
spot”. A power law fit to the observed dependence gives
$T_{\mathrm{rot}}=154\,r^{-0.35\pm 0.19}$. This result suggests that the
molecular gas is heated via collisional excitation with hot dust, which in
turn is heated by the absorption of radiation emitted by the central star
(Scoville & Kwan, 1976). Using expression (11) in Garay & Lizano (1999), we
infer that the power-law index of dust emissivity at far infrared wavelengths,
$\beta$, is 0.55 with an emmisivity, $f$, of 0.08 and that the luminosity of
the central object is 1.1$\times$104 L⊙. This luminosity is in good agreement
with the luminosity of the star ionizing the HC HII region as determined by
Guzmán et al. (2012) from radio continuum observations.

Figure 8: Rotational temperature of the $\rm CH_{3}CN$ versus projected
distance from the ”blue spot” (see the lower panels in Fig. 2). A power law
fit to the observed dependence gives $T_{\mathrm{rot}}=154\,r^{-0.35\pm
0.19}$.

###  4.2 Rotational temperature of $\rm SO_{2}$

From the emission of the four low excitation lines of $\rm SO_{2}$ shown in
Table 3 we obtain the rotational temperature of the envelope with MADCUBA.
Figure 9 plots a rotational diagram of emission from the whole region. A
linear fit to the observed trend implies a temperature of 40$\pm$6 Kelvin.
Since the data are well matched by a linear fit, all $\rm SO_{2}$ lines appear
to be optically thin.

Figure 9: Rotational diagram using the emission from the low excitation lines
of $\rm SO_{2}$ integrated over the whole source. A linear fit to the observed
trend implies an excitation temperature of 40$\pm$6 Kelvin. Figure 10:
Observed line profiles of several K components of the $\rm CH_{3}CN$ J=14–13
transition (marked with red dashed lines) toward the ”blue spot” position (the
angular size is $\sim 0$$\,.\\!\\!^{\prime\prime}$4, VLSR=–17 km s-1). Figure
11: G345.01 first-order moment for the $\rm CH_{3}CN$ transitions as a
function of angular distance, measured for full rings of width
0$\,.\\!\\!^{\prime\prime}$05 centered on the average position of the peak of
the ”blue spot”, (RA, Dec) (J2000) = (16h56m47.6s,
–40∘14′25$\,.\\!\\!^{\prime\prime}$98), and average radius $\theta$. The error
bars are the standard deviation of uncertainties of the velocity inside each
ring. The black solid line shows the best fit to the data. The best fit is
obtained for an infall radius much larger than the beam size, an ambient gas
velocity $V_{\mathrm{LSR}}$=–12.49$\pm$0.16 km s-1, and a central mass of
172.8$\pm 8.8M_{\odot}$ (for a distance of 2.38 kpc).

###  4.3 Infall motions

Within the spectral window covered by the ALMA data (see Fig. 1), additional
spectral features are also detected. Not only the line profiles of $\rm
CH_{3}CN$ and $\rm SO_{2}$ but also the lines of other species such as CH3OCHO
and 13CH3OH, integrated over a region of 0$\,.\\!\\!^{\prime\prime}$5 in size
and centered on the G345.01 B HC HII region, show a blue asymmetry in their
line profiles. Fig. 10 shows the observed line profiles of several K
components of the $\rm CH_{3}CN$ J=14–13 transition (marked with red dashed
lines) toward the ”blue spot” position. These spectra also show a blue
asymmetry in their line profiles. The appearance of the line profiiles is
consistent with the detection of the ”classic” asymmetric line profile infall
signature (Narayanan et al., 1998). Furthermore, Mayen-Gijon et al. (2014) and
Estalella et al. (2019) suggested that the ”blue spot” feature mentioned above
(see §3.1) is a clear signature of infall. The observation of a blue spot,
coupled with the absence of a corresponding red spot, aligns well with the
proposed infall scenario. This discrepancy can be attributed to the
obscuration of the putative red spot, which lies adjacent to and directly in
front of the stellar object, by more extended redshifted gas situated in the
foreground. The central region of the first-order map appears blueshifted
because the blueshifted emission, arising from gas close to and mainly behind
the central stellar object, is stronger than the redishifted emission from gas
farther away and in front of the stellar object. This asymmetry is produced
when the optical depth is high enough so that at a given line-of-sight
velocity the gas facing the observer hides the emission from the gas behind it
(Anglada et al., 1987, 1991).

At larger distances from the center, the integrated intensity decreases, the
blue and redshifted intensities become similar, and the intensity-weighted
mean velocity approaches the systemic velocity of the cloud. Therefore, the
first-order moment of an infalling envelope is characterized by a compact spot
of blueshifted emission toward the position of the zeroth-order moment peak.

In order to determine the infall velocity, central mass and infall radius we
use the hallmark model of Estalella et al. (2019). The value of the first-
order moment as a function of the angular distance was obtained for the
unblended K components of the J=14–13 transition of $\rm CH_{3}CN$ by
averaging the first-order moment in concentric rings of width
0$\,.\\!\\!^{\prime\prime}$05 centered on the average position of the peak of
the ”blue spot”, (RA, Dec) (J2000) = (16h56m47.6s,
–40∘14′25$\,.\\!\\!^{\prime\prime}$98). The first-order moment profiles of the
different components are presented in Figure 11. They seem to belong to two
separate groups, with the first-order moment values for the K=2, 3, 4
components being higher than those of the K=6, 7, 8 components, especially
near the peak position. However, the K=7 component follows the K=2, 3, 4
components beyond 0$\,.\\!\\!^{\prime\prime}$5 distance from the peak. The
difference is likely due to the higher K lines probing the hotter gas close to
the HC H ii region. The best fit is obtained for an infall radius much larger
than the beam size, an ambient gas velocity of
$V_{\mathrm{LSR}}$=–12.49$\pm$0.16 km s-1, and a central mass of 172.8$\pm
8.8M_{\odot}$ (for a distance of 2.38 kpc and the error does not include the
distance uncertainty). To ensure accuracy and assess sensitivity of the model
to distance variations, we also fit this model using the near distance of 1.7
kpc, and we get an ambient gas velocity of $V_{\mathrm{LSR}}$=–12.46$\pm$0.16
km s-1, and a central mass of 126.0$\pm 8.7M_{\odot}$. This central mass is
much larger than that derived from the number of Lyman photons presented in
Section 3.2. This difference could be attributed to a forming very massive
star, which is still undergoing infall, emitting fewer ionizing photons than a
main sequence star of the same mass. The clear detection of the ”central blue
spot” signature in the G345.0061+01.794B HC H ii region indicates that infall
motions play a fundamental role in the gas kinematics of this source.

###  4.4 Butterfly pattern

In the channel maps (in Figure 3) a ”butterfly pattern” like structure is
observed. The body of the butterfly, a potentially inflated rotating torus,
showing redshifted emission in the east and blueshifted one in the west,
extends across the center of the source. Roughly parallel to the rotation axis
of this torus we find ionized gas on both sides, i.e. in the north and south.
Extent and thickness of the torus are similar and the ionized gas in the north
is oriented slightly eastwards, while that one in the south is also oriented
somewhat eastwards. Hence the butterfly pattern is not perfectly fulfilled,
but the observational data come close to it. Apparently, we see the source
from a suitable viewing angle, i.e. the torus is seen approximately edge-on.

##  5 Conclusion

We carried out high angular resolution observations, using ALMA, of emission
in highly excited molecular lines of $\rm CH_{3}CN$ and $\rm SO_{2}$ and in
the H29$\alpha$ radio recombination line towards the G345.0061+01.794 B HC H
ii region. The main results and conclusions are summarized as follows:

  1. 1.

Emission was detected in all ten observed K components of the
J=14$\rightarrow$13 rotational ladder of $\rm CH_{3}CN$ and in the
$30_{4,26}-30_{3,27}$ and $32_{4,28}-32_{3,29}$ lines of $\rm SO_{2}$. The
peak of the velocity integrated molecular line intensity is located slightly
NW (about 0$\,.\\!\\!^{\prime\prime}$4$\pm$0$\,.\\!\\!^{\prime\prime}$1) of
the peak of the continuum emission.

  2. 2.

The first-order moment images of the molecular emission show a central spot of
blueshifted emission, with respect to the systemic velocity of the cloud,
located at the peak of the zero-order moment, seen in all K-components of $\rm
CH_{3}CN$ , in the $\rm SO_{2}$ lines and in transitions of other species
serendipitously detected in the measured frequency band.

  3. 3.

Rotational diagrams of the emission of the methyl cyanide lines show that the
rotational temperature has a peak value of 252$\pm 24$ Kelvin at the position
of the ”blue spot” and decreases outwards reaching a value of 166$\pm 16$
Kelvin at the edge ($\sim$1″, 0.01 pc for a distance of 2.38 kpc) of the
molecular structure, indicating that our observations are probing a hot
molecular core that is internally excited. In addition, from the emission of
four low excitation lines of $\rm SO_{2}$ we estimate a rotational temperature
of 40$\pm$6 Kelvin for the envelope integrated over the whole source.

  4. 4.

The first-order moment images and channel maps of the molecular emission show
a velocity gradient from roughly east to west with average velocities
preferentially blueshifted on the western and redshifted on the eastern side.
The change in velocity amounts to 4.3 km s-1 over a region of
3$\,.\\!\\!^{\prime\prime}$8 (equivalent to 98 km s-1 pc-1 at the distance of
2.38 kpc).

  5. 5.

Emission was detected in the H29$\alpha$ line, having a line center velocity
of $V_{\mathrm{LSR}}$=–18.1$\pm$0.9 km s-1 and a linewidth (FWHM) of
33.7$\pm$2.3 km s-1. The position of the peak in the velocity integrated
emission is coincident with that of the dust continuum. The radio
recombination line observations indicate that the ionized gas emission arises
from a region having a radius of 0.0037 pc, a mass of ionized gas of
(3.6$\pm$0.3)$\times$10-3 $M_{\odot}$, an electron temperature of
8094$\pm$1534 Kelvin, an emission measure of (4.6$\pm$0.9)$\times$108 pc cm-6,
and an electron density (2.1$\pm$0.2$)\times$105 cm-3 (for a distance of 2.38
kpc).

  6. 6.

We modeled the kinematical characteristics of a ”central blue spot” feature as
due to infalling motions, deriving a central mass of 172.8$\pm 8.8M_{\odot}$
(for a distance of 2.38 kpc). We conclude that the HC H ii region is
surrounded by a compact structure of hot molecular gas, which is rotating and
infalling toward a central mass of 172.8$\pm 8.8M_{\odot}$, that is most
likely confining the region of ionized gas.

  7. 7.

The properties for the source are reminiscent of the theoretically proposed
”butterfly pattern”, with the rotating torus seen almost edge-on and the
ionized gas extending roughly perpendicular to it.

The authors would like to express their sincere gratitude to Eric Keto for his
critical evaluation of the paper. This research was funded by the Science
Committee of the Ministry of Science and Higher Education of the Republic of
Kazakhstan (Grant Nos. AP13067768 and AP14870504) and sponsored (in part) by
the Chinese Academy of Sciences (CAS), through a grant to the CAS South
America Center for Astronomy (CASSACA) in Santiago, Chile. GG acknowledges
support from ANID BASAL project FB210003. RE acknowledges partial financial
support from the grants PID2020-117710GB-I00 and CEX2019-000918-M funded by
MCIN/ AEI /10.13039/501100011033. JE acknowledges support from the National
Key R&D Program of China under grant No.2022YFA1603103 and the Regional
Collaborative Innovation Project of Xinjiang Uyghur Autonomous Region grant
2022E01050. DL acknowledges support from National Natural Science Foundation
of China (NSFC) through grant No. 12173075 and support from Youth Innovation
Promotion Association CAS. YH acknowledges support from the CAS ”Light of West
China” Program under grant No. 2020-XBQNXZ-017 and the Xinjiang Key Laboratory
of Radio Astrophysics under grant No. 2023D04033. This paper makes use of the
following ALMA data: ADS/JAO.ALMA#2015.1.01371.S. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with
NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ.

## References

  * Anglada et al. (1991) Anglada, G., Estalella, R., Rodriguez, L. F., & Lopez, J. C. R. 1991, A&A, 252, 639
  * Anglada et al. (1987) Anglada, G., Rodriguez, L. F., Canto, J., Estalella, R., & Lopez, R. 1987, A&A, 186, 280
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Beltrán et al. (2014) Beltrán, M. T., Sánchez-Monge, Á., Cesaroni, R., et al. 2014, A&A, 571, A52, doi: 10.1051/0004-6361/201424031
  * Estalella et al. (2019) Estalella, R., Anglada, G., Díaz-Rodríguez, A. K., & Mayen-Gijon, J. M. 2019, A&A, 626, A84, doi: 10.1051/0004-6361/201834998
  * Garay & Lizano (1999) Garay, G., & Lizano, S. 1999, PASP, 111, 1049, doi: 10.1086/316416
  * Gordon & Sorochenko (2002) Gordon, M. A., & Sorochenko, R. L. 2002, Radio Recombination Lines. Their Physics and Astronomical Applications, doi: 10.1007/978-0-387-09604-9
  * Guzmán et al. (2012) Guzmán, A. E., Garay, G., Brooks, K. J., & Voronkov, M. A. 2012, ApJ, 753, 51, doi: 10.1088/0004-637X/753/1/51
  * Guzmán et al. (2020) Guzmán, A. E., Sanhueza, P., Zapata, L., Garay, G., & Rodríguez, L. F. 2020, ApJ, 904, 77, doi: 10.3847/1538-4357/abbe09
  * Guzmán et al. (2014) Guzmán, A. E., Garay, G., Rodríguez, L. F., et al. 2014, ApJ, 796, 117, doi: 10.1088/0004-637X/796/2/117
  * Hosokawa & Omukai (2009) Hosokawa, T., & Omukai, K. 2009, ApJ, 691, 823, doi: 10.1088/0004-637X/691/1/823
  * Keto (2007) Keto, E. 2007, ApJ, 666, 976, doi: 10.1086/520320
  * Keto & Wood (2006) Keto, E., & Wood, K. 2006, ApJ, 637, 850, doi: 10.1086/498611
  * Kuiper et al. (2011) Kuiper, R., Klahr, H., Beuther, H., & Henning, T. 2011, ApJ, 732, 20, doi: 10.1088/0004-637X/732/1/20
  * Kurtz et al. (2000) Kurtz, S., Cesaroni, R., Churchwell, E., Hofner, P., & Walmsley, C. M. 2000, in Protostars and Planets IV, ed. V. Mannings, A. P. Boss, & S. S. Russell, 299–326
  * Kurtz (2000) Kurtz, S. E. 2000, in Revista Mexicana de Astronomia y Astrofisica Conference Series, Vol. 9, Revista Mexicana de Astronomia y Astrofisica Conference Series, ed. S. J. Arthur, N. S. Brickhouse, & J. Franco, 169–176
  * Martín et al. (2019) Martín, S., Martín-Pintado, J., Blanco-Sánchez, C., et al. 2019, A&A, 631, A159, doi: 10.1051/0004-6361/201936144
  * Mayen-Gijon et al. (2014) Mayen-Gijon, J. M., Anglada, G., Osorio, M., et al. 2014, MNRAS, 437, 3766, doi: 10.1093/mnras/stt2172
  * McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & D. J. Bell, 127
  * Mehringer (1994) Mehringer, D. M. 1994, ApJS, 91, 713, doi: 10.1086/191953
  * Mezger & Henderson (1967) Mezger, P. G., & Henderson, A. P. 1967, ApJ, 147, 471, doi: 10.1086/149030
  * Mezger et al. (1974) Mezger, P. G., Smith, L. F., & Churchwell, E. 1974, A&A, 32, 269
  * Moisés et al. (2011) Moisés, A. P., Damineli, A., Figuerêdo, E., et al. 2011, MNRAS, 411, 705, doi: 10.1111/j.1365-2966.2010.17713.x
  * Nakano (1989) Nakano, T. 1989, ApJ, 345, 464, doi: 10.1086/167919
  * Narayanan et al. (1998) Narayanan, G., Walker, C. K., & Buckley, H. D. 1998, ApJ, 496, 292, doi: 10.1086/305363
  * Norberg & Maeder (2000) Norberg, P., & Maeder, A. 2000, A&A, 359, 1025.  https://arxiv.org/abs/astro-ph/0005535
  * Ossenkopf & Henning (1994) Ossenkopf, V., & Henning, T. 1994, A&A, 291, 943
  * Panagia (1973) Panagia, N. 1973, AJ, 78, 929, doi: 10.1086/111498
  * Sánchez-Monge et al. (2013) Sánchez-Monge, Á., Cesaroni, R., Beltrán, M. T., et al. 2013, A&A, 552, L10, doi: 10.1051/0004-6361/201321134
  * Scoville & Kwan (1976) Scoville, N. Z., & Kwan, J. 1976, ApJ, 206, 718, doi: 10.1086/154432
  * Tan et al. (2014) Tan, J. C., Beltrán, M. T., Caselli, P., et al. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 149, doi: 10.2458/azu_uapress_9780816531240-ch007
  * Urquhart et al. (2007) Urquhart, J. S., Busfield, A. L., Hoare, M. G., et al. 2007, A&A, 474, 891, doi: 10.1051/0004-6361:20078025
  * Zapata et al. (2015) Zapata, L. A., Palau, A., Galván-Madrid, R., et al. 2015, MNRAS, 447, 1826, doi: 10.1093/mnras/stu2527
  * Zhang et al. (2014) Zhang, Y., Tan, J. C., & Hosokawa, T. 2014, ApJ, 788, 166, doi: 10.1088/0004-637X/788/2/166


$\omega_{\Lambda}$ extends to a conformal mapping between the closed Riemann
surfaces $\mathbb{C}\diagup\Lambda$ and $Y(\Lambda)$.

We describe now the lifts of the mapping classes of $\mathbb{P}^{1}$ with
distinguished points $BL=\mathring{BL}(\Lambda)\cup\infty$ to mapping classes
of $Y(\Lambda)$.

Notice that for each lattice $\Lambda$ the mapping
${\mathbb{C}}\ni\zeta\to-\zeta$ maps $\Lambda$ onto itself. This mapping
descends to an involution of ${\mathbb{C}}\diagup\Lambda$, i.e. to a self-
homeomorphism $\iota_{\Lambda}$ of ${\mathbb{C}}\diagup\Lambda$ such that
$\iota_{\Lambda}^{2}={\rm id}$.

Formula (50) implies the following equality

$\left(\wp_{\Lambda}\left(-\zeta\right),\wp^{\prime}_{\Lambda}\left(-\zeta\right)\right)=\left(\wp_{\Lambda}\left(\zeta\right),-\wp^{\prime}_{\Lambda}\left(\zeta\right)\right)\,.$
(62)

Conjugate the restriction
$\iota_{\Lambda}\mid(\mathbb{C}\setminus\Lambda)\diagup\Lambda$ by the
conformal mapping $\omega_{\Lambda}^{-1}$. We obtain a self-homeomorphism of
$\mathring{Y}({\Lambda})$ which we denote by $\iota$. The mapping $\iota$
satisfies the equality

$\iota(z,w)=(z,-w)\,.$ (63)

The involution $\iota$ extends to an involution of ${Y}({\Lambda})$, denoted
also by $\iota$. By (63) the involution $\iota$ fixes the projection to
${\mathbb{P}}^{1}$ of each point of the double branched covering space
$Y(\Lambda)$ and interchanges the sheets over each point. Hence, it fixes each
of the three finite branch points, it also fixes $\infty$, and it does not fix
any other point.

Let $\widetilde{\varphi}$ be any real linear self-homeomorphism of
${\mathbb{C}}$ which maps $\Lambda$ onto itself, and let $\varphi$ be the
induced mapping on ${\mathbb{C}}\diagup\Lambda$. Then $\varphi$ commutes with
$\iota_{\Lambda}$. Indeed,
$\widetilde{\varphi}(-\zeta)=-\widetilde{\varphi}(\zeta)$,
$\zeta\in{\mathbb{C}}$.

Each mapping class ${\mathfrak{m}}$ in
$\mathfrak{M}({\mathbb{C}}\diagup\Lambda;\,0\diagup\Lambda,\emptyset)$ can be
represented by a self-homeomorphism of ${\mathbb{C}}\diagup\Lambda$ which
commutes with $\iota_{\Lambda}$. Indeed,
$\mathfrak{M}({\mathbb{C}}\diagup\Lambda;\;0\diagup\Lambda,\,\emptyset)$
contains a mapping $\varphi$ that lifts to a real linear self-map
$\widetilde{\varphi}$ of ${\mathbb{C}}$ which maps $\Lambda$ onto itself. This
mapping $\varphi$ commutes with $\iota_{\Lambda}$.

As a corollary, each mapping class ${\mathfrak{m}}$ in
$\mathfrak{M}(Y(\Lambda);\,s^{\infty},\,\emptyset)\cong\mathfrak{M}({\mathbb{C}}\diagup\Lambda;\,0\diagup\Lambda,\emptyset)$
is a lift of a mapping class
$\mathfrak{m}_{1}\in\mathfrak{M}(\mathbb{P}^{1};\\{\infty\\},\mathring{E})$
for the set $\mathring{E}\stackrel{{\scriptstyle
def}}{{=}}\mathring{BL}(\Lambda)\subset
C_{3}(\mathbb{C})\diagup\mathcal{S}_{3}$. Indeed, if
$\varphi\in\mathfrak{m}\in\mathfrak{M}({\mathbb{C}}\diagup\Lambda;\,0\diagup\Lambda,\,\emptyset)$
commutes with $\iota_{\Lambda}$, then
$\psi=\omega_{\Lambda}\circ\varphi\circ\omega_{\Lambda}^{-1}$ commutes with
$\iota$ and represents the mapping class in
$\mathfrak{M}(Y(\Lambda);\,s^{\infty},\,\emptyset)$ that corresponds to
$\mathfrak{m}$. Denote by $\mathring{\psi}$ the restriction of $\psi$ to
$\mathring{Y}$.

In coordinates $(z,w)$ on $\mathring{Y}(\Lambda)$ we write

$\mathring{\psi}(z,w)=(\mathring{\psi}_{1}(z,w),\mathring{\psi}_{2}(z,w))\,.$
(64)

Since $\mathring{\psi}$ commutes with $\iota$ we obtain by (63)

$(\mathring{\psi}_{1}(z,-w),\mathring{\psi}_{2}(z,-w))=\mathring{\psi}\circ\iota(z,w)=\iota\circ\mathring{\psi}(z,w)=(\mathring{\psi}_{1}(z,w),-\mathring{\psi}_{2}(z,w))\,.$
(65)

Hence $\mathring{\psi}_{1}(z,w)=\mathring{\psi}_{1}(z,-w)$. Since each point
in $(z,w)\in\mathring{Y}_{\Lambda}$ is determined by $z$ and the sign of $w$,
this means that $\mathring{\psi}_{1}(z,w)$ depends only on the coordinate
$z\in{\mathbb{C}}$, not on the sheet (determined by $w$). Further, if $\iota$
fixes $(z,w)$, then $w=0$, and by (65) for $w=0$,
$\mathring{\psi}_{2}(z,w)=\mathring{\psi}_{2}(z,-w)=-\mathring{\psi}_{2}(z,w)$.
In other words, if $\iota$ fixes $(z,w)$, then it also fixes
$\mathring{\psi}(z,w)$. Hence, $\mathring{\psi}$ maps the set of finite branch
points (the preimage of $\mathring{BL}({\Lambda})$ under the branched covering
map) onto itself, and its extension $\psi$ maps the preimage $s^{\infty}$ of
$\infty$ to itself. We saw, that ${\psi}$ induces a self-homeomorphism ${\sf
P}\psi$ of ${\mathbb{P}}^{1}$ in the class
${\mathfrak{M}}({\mathbb{P}}^{1};\\{\infty\\},\mathring{BL}({\Lambda}))$,
${\sf P}\psi|\mathbb{C}={\sf P}\mathring{\psi}$, ${\sf
P}\mathring{\psi}(z)=\mathring{\psi}_{1}(z,w),\,(z,w)\in\mathbb{C}^{2}$. We
call ${\sf{P}}(\psi)$ the projection of $\psi$. The mapping class
$\mathfrak{m}$ is a lift of the mapping class $\mathfrak{m}_{1}$ of
$\psi_{1}$. Notice that the provided arguments imply also the following fact.
For two mapping classes
$\mathfrak{m}^{1},\mathfrak{m}^{2}\in\mathfrak{M}(Y(\Lambda);s^{\infty},\emptyset)$
the equality ${\sf P}(\mathfrak{m}^{1}\mathfrak{m}^{2})={\sf
P}(\mathfrak{m}^{1}){\sf P}(\mathfrak{m}^{2})$ holds. Indeed, take
representing maps $\psi^{1}$ and $\psi^{2}$, that commute with the involution
$\iota$. Using the arguments given above, the equality ${\sf
P}(\psi^{1}\circ\psi^{2})={\sf P}(\psi^{1})\circ{\sf P}(\psi^{2})$ follows.

Let again $\mathring{E}$ be an element of
$C_{3}(\mathbb{C})\diagup\mathcal{S}_{3}$ and
$E=\mathring{E}\cup\\{\infty\\}$. Any mapping class
$\mathfrak{m}_{1}\in\mathfrak{M}(\mathbb{P}^{1};\\{\infty\\},\mathring{E})$
has exactly two lifts
$\mathfrak{m}_{\pm}\in\mathfrak{M}(Y_{E};s^{\infty},\emptyset)$. Here $Y_{E}$
is the double branched covering of $\mathbb{P}^{1}$ with finite branch locus
$\mathring{E}$ and $s^{\infty}$ is the branch point over $\infty$. Indeed, for
any representative $\psi_{1}$ of $\mathfrak{m}_{1}$ there are exactly two
self-homeomorphisms of the double branched covering that lift $\psi_{1}$. They
are obtained as follows. Cut $\mathbb{P}^{1}$ along two disjoint simple curves
$\gamma_{1}$ and $\gamma_{2}$, that join disjoint pairs of points of $E$.
$\psi_{1}$ maps the pair of curves $\gamma_{1}$ and $\gamma_{2}$ to another
pair of curves $\gamma^{\prime}_{1}$ and $\gamma^{\prime}_{2}$ joining (maybe
different) disjoint pairs of points of $E$. The double branched covering over
$\mathbb{P}^{1}$ is obtained in two different ways, either gluing two sheets
of $\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$ crosswise together, or
gluing two sheets of
$\mathbb{P}^{1}\setminus(\gamma^{\prime}_{1}\cup\gamma^{\prime}_{2})$
crosswise together. $\varphi_{1}$ maps
$\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$ homeomorphically onto
$\mathbb{P}^{1}\setminus(\gamma^{\prime}_{1}\cup\gamma^{\prime}_{2})$.
Consider the mapping that takes the first sheet of
$\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$ onto the first sheet of
$\mathbb{P}^{1}\setminus(\gamma^{\prime}_{1}\cup\gamma^{\prime}_{2})$ and the
second sheet of $\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$ onto the
second sheet of
$\mathbb{P}^{1}\setminus(\gamma^{\prime}_{1}\cup\gamma^{\prime}_{2})$ and
lifts $\varphi_{1}\mid\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$. This
mapping extends to a self-homeomorphism $\varphi$ of the double branched
covering that lifts $\psi_{1}$. There is exactly one more self-homeomorphism
of the double branched covering that lifts $\psi_{1}$. This self-homeomorphism
maps the first sheet of $\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$
onto the second sheet of
$\mathbb{P}^{1}\setminus(\gamma^{\prime}_{1}\cup\gamma^{\prime}_{2})$ and the
second sheet of $\mathbb{P}^{1}\setminus(\gamma_{1}\cup\gamma_{2})$ onto the
first sheet of
$\mathbb{P}^{1}\setminus(\gamma^{\prime}_{1}\cup\gamma^{\prime}_{2})$. In
other words, there are two lifts of $\psi_{1}$ to the double branched covering
of ${\mathbb{P}}^{1}$ with branch locus $E$, and they differ by involution.

The following proposition relates Theorem 10 to the respective theorem for
elliptic bundles that will be formulated below.

######  Proposition 2.

Let $X$ be a Riemann surface (smooth surface, respectively) of genus $g$ with
${m}\geq 1$ holes with base point $x_{0}$ and curves $\gamma_{j}$ representing
a set of generators $e_{j}\in\pi_{1}(X,x_{0})$.
$(1)$ Each complex analytic (differentiable, respectively) family of Riemann
surfaces of type $(1,1)$ over $X$ is holomorphically (smoothly, respectively)
isomorphic to the double branched covering $\mathfrak{F}$ of a special
holomorphic (smooth, respectively) $(0,4)$-bundle $\mathfrak{F}_{1}$ over $X$.
The monodromy of the bundle $\mathfrak{F}$ along each $\gamma_{j}$ is a lift
of the respective monodromy of the bundle $\mathfrak{F}_{1}$.
$(2)$ Vice versa, for each special holomorphic (smooth, respectively)
$(0,4)$-bundle over $X$ and each collection $\mathfrak{m}^{j}$ of lifts of the
$2{g}+{m}-1$ monodromy mapping classes $\mathfrak{m}_{1}^{j}$ of the bundle
along the $\gamma_{j}$ there exists a double branched covering by a complex
analytic (differentiable, respectively) family of Riemann surfaces of type
$(1,1)$ with collection of monodromy mapping classes equal to the
$\mathfrak{m}^{j}$. For each given holomorphic (smooth, respectively) special
$(0,4)$-bundle over $X$ there are up to holomorphic (smooth, respectively)
isomorphisms exactly $2^{2{g}+{m}-1}$ holomorphic (smooth, respectively)
families of Riemann surfaces of type $(1,1)$ that lift the $(0,4)$-bundle.
$(3)$ A lift of a special $(0,4)$-bundle is reducible if and only if the
special $(0,4)$-bundle is reducible.

Proof. We start with the proof of the first statement of the proposition.
Consider a complex analytic (smooth, respectively) family of Riemann surfaces
of type $(1,1)$ over $X$. By Lemma 16 and Proposition 1 we may assume that the
family has the form
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
for a holomorphic (smooth, respectively) family of lattices $\Lambda(x),\,x\in
X$.

We consider the complex (smooth, respectively) submanifold
$\mathring{\mathbold{BL}}(\Lambda)$ of $X\times\mathbb{C}$ that intersects
each fiber $\\{x\\}\times\mathbb{C}$ along the set
$\\{x\\}\times\mathring{BL}(\Lambda(x))$ (see equations (56) and (61)) and the
complex (smooth, respectively) submanifold $\mathbold{BL}(\Lambda)$ of
$X\times\mathbb{P}^{1}$ that intersects each fiber
$\,\\{x\\}\times\mathbb{P}^{1}\,$ along the set $\,\\{x\\}\times
BL(\Lambda(x))\,$ with
$BL(\Lambda(x))\,=\,\mathring{BL}(\Lambda(x))\cup\\{\infty\\}$. The
holomorphic (smooth, respectively) family
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
is holomorphically isomorphic (isomorphic as a smooth family of Riemann
surfaces, respectively) to the canonical double branched covering
$(\mathcal{Y}_{\mathbold{BL}(\Lambda)},\mathcal{P},\mathbold{s}^{\infty},X)$
(see Section 10) of the special $(0,4)$-bundle $(X\times\mathbb{P}^{1},{\rm
pr}_{1},\mathbold{BL}(\Lambda),X)$. To see this we recall that for each $x\in
X$ the mapping
$\omega_{\Lambda_{x}}:(\mathbb{C}\setminus\Lambda_{x})\diagup\Lambda_{x}\to\mathring{Y}_{{BL}(\Lambda_{x})}$
is a surjective conformal mapping that depends holomorphically (smoothly,
respectively) on $x$ (see equation (58)). Hence, the mapping defines a
holomorphic (smooth, respectively) homeomorphism from
$\mathcal{X}_{\Lambda}\setminus\mathbold{s}_{\Lambda}$ onto
$\mathring{\mathcal{Y}}_{\mathbold{BL}(\Lambda)}$, that maps the fiber over
$x$ of the first bundle conformally onto the fiber over $x$ of the second
bundle.

The set ${\mathcal{Y}}_{\mathbold{BL}(\Lambda)}$ is obtained as in Section 10
by ”adding a point to each fiber”. As in Section 10 the mapping extends to a
holomorphic (smooth, respectively) homeomorphism
$\mathcal{X}_{\Lambda}\to{\mathcal{Y}}_{\mathbold{BL}(\Lambda)}$, that maps
the fiber over $x$ conformally onto the fiber over $x$. The extension maps
$\mathbold{s}_{\Lambda}$ to $\mathbold{s}^{\infty}$. We proved that the
bundles
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
and
$(\mathcal{Y}_{\mathbold{BL}(\Lambda)},\mathcal{P},\mathbold{s}^{\infty},X)$
are holomorphically isomorphic (isomorphic as smooth families of Riemann
surfaces, respectively). Hence, each holomorphic (smooth, respectively)
$(1,1)$-family over a finite open Riemann surface is holomorphically
isomorphic (smoothly isomorphic), and in particular, isotopic to a double
branched covering of a holomorphic (smooth, respectively) $(0,4)$-bundle.

We identify the mapping class groups in the fiber over the base point $x_{0}$
of the bundles
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
and
$(\mathcal{Y}_{\mathbold{BL}(\Lambda)},\mathcal{P},\mathbold{s}^{\infty},X)$
using the described isomorphism. Having in mind this identification, we prove
now that the monodromy mapping class along $\gamma_{j}$ of the $(1,1)$-bundle
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
is a lift of the monodromy mapping class along $\gamma_{j}$ of the special
$(0,4)$-bundle $(X\times\mathbb{P}^{1},{\rm
pr}_{1},\mathbold{BL}(\Lambda),X)$. Parameterise $\gamma_{j}$ by the unit
interval $[0,1]$. Write $\Lambda_{t}\stackrel{{\scriptstyle
def}}{{=}}\Lambda(\gamma_{j}(t))=a(t)\mathbb{Z}+b(t)\mathbb{Z},\,t\in[0,1],$
for smooth functions $a$ and $b$ on $[0,1]$. For $t\in[0,1]$ we denote by
$\widetilde{\varphi}^{t}$ the real linear self-homeomorphism of $\mathbb{C}$
that maps $a(0)$ to $a(t)$ and $b(0)$ to $b(t)$. Let $\varphi^{t}$ be the
homeomorphism from the fiber over $\gamma_{j}(0)$ onto the fiber over
$\gamma_{j}(t)$ of the $(1,1)$-bundle that lifts to $\widetilde{\varphi}^{t}$.
Then ${\varphi}^{0}={\rm Id}$ and ${\varphi}^{1}$ represents the monodromy
mapping class of the $(1,1)$-bundle
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
along $\gamma_{j}$. Let
$\mathring{\psi}^{t}:\mathring{Y}(\Lambda_{0})\to\mathring{Y}(\Lambda_{t})$ be
obtained from the commutative diagram

$\textstyle{(\mathbb{C}\setminus\Lambda_{0})\diagup\Lambda_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathring{\varphi}^{t}}$$\scriptstyle{\omega_{\Lambda_{0}}}$$\textstyle{(\mathbb{C}\setminus\Lambda_{t})\diagup\Lambda_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{\Lambda_{t}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathring{Y}(\Lambda_{0})}}$$\scriptstyle{\mathring{\psi}^{t}}$$\textstyle{\mathring{Y}(\Lambda_{t})}$

where
$\mathring{\varphi}^{t}=\varphi^{t}|(\mathbb{C}\setminus\Lambda_{0})\diagup\Lambda_{0}$.

We use the restriction to $\mathring{Y}(\Lambda_{0})$ of the coordinates
$(z,w)$ on $\mathbb{C}^{2}$, and let $\iota_{0}$ be the involution on
$\mathring{Y}(\Lambda_{0})$ and $\iota_{t}$ the involution on
$\mathring{Y}(\Lambda_{t})$. Choose respective coordinates on
$\mathring{Y}(\Lambda_{t})$ and write
$\mathring{\psi}^{t}(z,w)=(\mathring{\psi}^{t}_{1}(z,w),\mathring{\psi}^{t}_{2}(z,w)),\;(z,w)\in\mathring{Y}(\Lambda_{0})$.
Since
$\widetilde{\varphi}^{t}(-\zeta)=-\widetilde{\varphi}^{t}(\zeta),\zeta\in\mathbb{C},$
we obtain

$\mathring{\psi}^{t}\,\iota_{0}=\iota_{t}\,\mathring{\psi}^{t}\,.$ (66)

In coordinates $(z,w)$ this means
$(\mathring{\psi}^{t}_{1}(z,-w),\mathring{\psi}^{t}_{2}(z,-w))=(\mathring{\psi}^{t}_{1}(z,w),-\mathring{\psi}^{t}_{2}(z,w))$.
Hence, $\mathring{\psi}^{t}_{1}(z,w)=\mathring{\psi}^{t}_{1}(z,-w)$, so that
$\mathring{\psi}^{t}_{1}$ depends only on the coordinate $z\in\mathbb{C}$.
Moreover, if $\iota_{0}$ fixes $(z,w)$ (equivalently, if
$z\in\mathring{BL}(\Lambda_{0})$ ), then $w=0$, hence
$\mathring{\psi}^{t}_{2}(z,w)=\mathring{\psi}^{t}_{2}(z,-w)=-\mathring{\psi}^{t}_{2}(z,w)$,
i.e. $\iota_{t}$ fixes $\mathring{\psi}^{t}(z,w)$ (equivalently,
$\mathring{\psi}^{t}_{1}(z,w)\in\mathring{BL}(\Lambda_{t})$). We saw that the
self-homeomorphism $\mathring{\psi}^{t}_{1}$ of $\mathbb{C}$ maps the set
$\mathring{BL}(\Lambda_{0})$ onto the set $\mathring{BL}(\Lambda_{t})$. Each
$\mathring{\psi}^{t}_{1}$ extends to a self-homeomorphism $\psi^{t}_{1}$ of
$\mathbb{P}^{1}$ that maps the distinguished points
$BL(\Lambda_{0})\stackrel{{\scriptstyle
def}}{{=}}\mathring{BL}(\Lambda_{0})\cup\\{\infty\\}$ onto the distinguished
points $BL(\Lambda_{t})\stackrel{{\scriptstyle
def}}{{=}}\mathring{BL}(\Lambda_{t})\cup\\{\infty\\}$. Hence, $\psi^{1}_{1}$
represents the monodromy mapping class along $\gamma_{j}$ of the special
$(0,4)$-bundle with set of finite distinguished points
$\mathring{BL}(\Lambda_{t})$ in the fiber over $\gamma_{j}(t)$. The mapping
$\mathring{\psi}^{1}=(\mathring{\psi}_{1}^{1},\mathring{\psi}_{2}^{1})$ is
equal to
$\omega_{\Lambda_{0}}\circ\mathring{\varphi}^{1}\circ\omega_{\Lambda_{0}}^{-1}$.
Identifying the mapping class groups of $\mathbb{C}\diagup\Lambda_{0}$ with
the mapping class group of $Y_{BL(\Lambda_{0})}$ by the isomorphism induced by
$\omega_{\Lambda_{0}}$, we may identify the monodromy of the bundle
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\,\mathbold{s}_{\Lambda}\,,X)\,$
along $\gamma_{j}$ with the mapping class of the extension $\psi^{1}$ of
$\mathring{\psi}^{1}=(\mathring{\psi}_{1}^{1},\mathring{\psi}_{2}^{1})$. The
mapping $\psi_{1}^{1}$ is the projection of $\psi^{1}$ under the double
branched covering from $\mathbb{C}\diagup\Lambda_{0}$ with distinguished point
$0\diagup\Lambda_{0}$ onto $\mathbb{P}^{1}$ with distinguished points
${BL}(\Lambda_{0})\cup\\{\infty\\}$. We proved that the monodromy mapping
class of the bundle
$(\mathcal{X}_{\Lambda},\mathcal{P}_{\Lambda},\mathbold{s}_{\Lambda},X)$ along
each $\gamma_{j}$ is a lift of the respective monodromy mapping class of the
bundle $(X\times\mathbb{P}^{1},{\rm pr}_{1},\mathbold{BL}(\Lambda),X)$.

We prove now the second statement. Consider the special holomorphic, or
smooth, respectively, $(0,4)$-bundle $\mathfrak{F}_{1}\stackrel{{\scriptstyle
def}}{{=}}\big{(}X\times\mathbb{P}^{1},{\rm pr}_{1},\mathbold{E},X\big{)}$.
The complex or smooth, respectively, submanifold $\mathbold{E}$ of
$X\times\mathbb{P}^{1}$ intersects each fiber $\\{x\\}\times\mathbb{P}^{1}$
along the set $\\{x\\}\times E_{x}$, with
$E_{x}=(\mathring{E}_{x}\cup\\{\infty\\})$, where $\mathring{E}_{x}\subset
C_{3}(\mathbb{C})\diagup\mathcal{S}_{3}$. In Section 10 we obtained the
canonical lift
$(\mathcal{\mathcal{Y}}_{\mathbold{E}},\mathcal{P},\mathbold{s}^{\infty},X)$.
Denote the monodromy mapping class along $\gamma_{j}$ of the canonical lift by
$\mathfrak{m}^{j}_{+}$. Recall that each mapping class $\mathfrak{m}_{1}^{j}$
has two lifts and they differ by involution. Let $\mathfrak{m}^{j}_{-}$ be the
mapping class whose representatives differ from those of
$\mathfrak{m}^{j}_{+}$ by composition with the involution $\iota$ of the fiber
over the base point.

Take any subset $J$ of $\\{1,2,\ldots,2{g}+{m}-1\\}$. The lift of the
$(0,4)$-bundle whose monodromy mapping class along $\gamma_{j}$ equals
$\mathfrak{m}^{j}_{+}$ if $j\notin J$, and $\mathfrak{m}^{j}_{-}$ if $j\in J$,
is obtained as follows. Let $\tilde{U}$ be the subset of the universal
covering $\tilde{X}\overset{{p}}{-\\!\\!\\!\longrightarrow}X$ that was used in
Section 5, and let $D$, and $\tilde{D}_{j},\,j=0,\ldots,$ be as in Section 5.
Consider the lift of the bundle
$(\mathcal{\mathcal{Y}}_{\mathbold{E}},\mathcal{P},\mathbold{s}^{\infty},X)$
to $\tilde{X}$ and restrict the lifted bundle to $\tilde{U}$. Denote the
obtained bundle on $\tilde{U}$ by $\tilde{\mathfrak{F}}$. For each point $x\in
D$ we let $\tilde{x}_{j}\in\tilde{D}_{j},\,j=0,\ldots,$ be the points with
$p(\tilde{x}_{j})=x$. We identify the fiber of $\tilde{\mathfrak{F}}$ over
$\tilde{x}_{j},\,j=0,\ldots,$ with the fiber ${Y}_{x}$, $x=p(\tilde{x}_{j})$,
of the canonical double branched covering of the $(0,4)$-bundle.
$\mathring{Y}_{x}$ is obtained from $Y_{x}$ by removing the point of
$\mathbold{s}^{\infty}$ from $Y_{x}$.

Glue for each $x\in D$ and each $j=1,\ldots,$ the fiber of
$\tilde{\mathfrak{F}}$ over $\tilde{x}_{j}$ to the fiber of
$\tilde{\mathfrak{F}}$ over $\tilde{x}_{0}$ using the identity if $j\notin J$
and the mapping $\iota$ if $j\in J$. More detailed, the gluing mapping of the
punctured fibers in the case $j\in J$ equals

$(\tilde{x}_{j},z,w)\to(\tilde{x}_{0},z,-w),\;\;x\in
D,\;(z,w)\in\mathring{Y}_{x}\,.$ (67)

Since the gluing mappings are holomorphic (smooth, respectively), we obtain a
complex analytic (smooth, respectively) family over $X$ of double branched
coverings of $\mathbb{C}$. Extend the family to a complex analytic (smooth,
respectively) family of double branched coverings of $\mathbb{P}^{1}$ over
$X$. This can be done as in Section 10 by considering the families
$\mathcal{Y}^{r_{n}}_{\mathbold{E}|{X}_{n}}$ for an exhausting sequence of
relatively compact open subsets $X_{n}$ of $X$ and a suitable sequence of real
numbers $r_{n}$. We obtained for each collection of lifts of the
$\mathfrak{m}^{j}_{1}$ a complex analytic (smooth, respectively) family of
Riemann surfaces of type $(1,1)$ with given fiber over the base point and with
monodromy mapping classes equal to this collection. It is easy to see that two
holomorphic (smooth, respectively) $(1,1)$-bundles that lift a given
$(0,4)$-bundle over a finite open Riemann surface (smooth surface,
respectively) are holomorphically isomorphic (smoothly isomorphic,
respectively), if and only if their monodromies coincide. Hence, for each
given holomorphic (smooth, respectively) special $(0,4)$-bundle over $X$ there
are up to holomorphic (smooth, respectively) isomorphism exactly
$2^{2{g}+{m}-1}$ holomorphic (smooth, respectively) families of Riemann
surfaces of type $(1,1)$ that lift the $(0,4)$-bundle.

It remains to prove the third statement. Let ${\sf
P}:(\mathcal{X},\mathcal{P},\mathbold{s},X)\to(X\times\mathbb{P}^{1},{\rm
pr}_{1},\mathbold{E},X)$ be a double branched covering of a $(0,4)$-bundle by
a $(1,1)$-family of Riemann surfaces. If the special $(0,4)$-bundle is
reducible, then there is a simple closed curve $\gamma$ that divides the fiber
$\mathbb{P}^{1}$ over the base point $x_{0}$ into two connected components,
each of which contains two distinguished points. The curve $\gamma$ is mapped
by each monodromy mapping class of the $(0,4)$-bundle to a class of isotopic
curves. The preimage of the closed curve $\gamma$ under the covering map
consists of two simple closed curves $\tilde{\gamma}_{1}$ and
$\tilde{\gamma}_{2}$, that are homotopic to each other in
$\mathcal{P}^{-1}(x)\setminus\mathbold{s}$, and each of the two curves cuts
the torus into an annulus. Each monodromy mapping class of the $(1,1)$-bundle,
being a lift of the respective monodromy mapping class of the $(0,4)$-bundle,
takes each $\tilde{\gamma}_{j}$ to a class of isotopic curves. Hence, the
$(1,1)$-bundle is also reducible.

Suppose now that a double branched covering
$\mathfrak{F}=(\mathcal{X},\mathcal{P},\mathbold{s},X)$ of a special
$(0,4)$-bundle ${\sf P}(\mathfrak{F})=(X\times\mathbb{P}^{1},{\rm
pr}_{1},\mathbold{E},X)$ is reducible and prove that the $(0,4)$-bundle is
reducible. Let $x_{0}\in X$ be the base point of $X$, and let the fiber of the
bundle $\mathfrak{F}$ over $x_{0}$ be $\mathbb{C}\diagup\Lambda$ with
distinguished point $0\diagup\Lambda$. Any admissible system of curves on the
torus $\mathbb{C}\diagup\Lambda$ with a distinguished point contains exactly
one curve.

Let $\gamma_{0}$ be a simple closed curve in
$(\mathbb{C}\setminus\Lambda)\diagup\Lambda$ that reduces all monodromy
mapping classes of the bundle $\mathfrak{F}$. The curve $\gamma_{0}$ is free
isotopic in $\mathbb{C}\diagup\Lambda$ to a curve $\gamma$ with base point
$0\diagup\Lambda$ that lifts under the covering map
$p:\mathbb{C}\to\mathbb{C}\diagup\Lambda$ to the straight line segment that
joins $0$ with another lattice point $\lambda$ in $\Lambda$. Since
$\gamma_{0}$ is free isotopic in $\mathbb{C}\diagup\Lambda$ to a simple closed
curve on the torus, $\lambda$ is a primitive element of the lattice. Indeed,
with $\Lambda=\\{na+mb:n,m\in\mathbb{Z}\\}$ for real linearly independent
complex numbers $a$ and $b$ we get $\lambda=n(\lambda)a+m(\lambda)b$, where
$n(\lambda)$ and $m(\lambda)$ are relatively prime integer numbers. Then there
is another element
$\lambda^{\prime}=n(\lambda^{\prime})a+m(\lambda^{\prime})b\in\Lambda$ such
that

$\begin{vmatrix}n(\lambda)&n(\lambda^{\prime})\\\
m(\lambda)&m(\lambda^{\prime})\\\ \end{vmatrix}=1\,.$

The two elements $\lambda$ and $\lambda^{\prime}$ generate $\Lambda$.
Multiplying $\Lambda$ by a non-zero complex number and changing perhaps
$\lambda^{\prime}$ to $-\lambda^{\prime}$, we may assume that $\lambda=1$ and
$\lambda^{\prime}=\tau$ with ${\rm Im}\tau>0$. In other words, after changing
$\mathbb{C}\diagup\Lambda$ to the conformally equivalent Riemann surface
$\mathbb{C}\diagup\Lambda_{\tau}$, we may assume that the fiber of
$\mathfrak{F}$ over $x_{0}$ equals $\mathbb{C}\diagup\Lambda_{\tau}$. After a
free isotopy in $(\mathbb{C}\setminus\Lambda_{\tau})\diagup\Lambda_{\tau}$ we
may assume that the reducing curve $\gamma_{0}$ lifts under the covering map
$p:\mathbb{C}\to\mathbb{C}\diagup\Lambda_{\tau}$ to the segment
$\frac{\tau}{2}+[0,1]\subset\mathbb{C}$. Denote by $\gamma$ the curve in
$\mathbb{C}\diagup\Lambda_{\tau}$ that lifts to $[0,1]$, and by
$\gamma^{\prime}$ the curve on the torus that lifts to $[0,\tau]$. $\gamma$
and $\gamma^{\prime}$ represent a pair of generators of the fundamental group
of the closed torus with base point $0\diagup\Lambda_{\tau}$.

The complement $(\mathbb{C}\diagup\Lambda_{\tau})\setminus\gamma_{0}$ of
$\gamma_{0}$ in the fiber over $x_{0}$ is a topological annulus with
distinguished point $0\diagup\Lambda_{\tau}$. Since $\gamma_{0}$ reduces all
monodromy mapping classes of the $(1,1)$-family $\mathfrak{F}$, each monodromy
mapping class has a representative that fixes $\gamma_{0}$ pointwise. These
representatives map the topological annulus
$(\mathbb{C}\diagup\Lambda_{\tau})\setminus\gamma_{0}$ homeomorphically onto
itself, fixing the distinguished point and fixing the boundary pointwise, or
fixing the boundary pointwise after an involution. Hence, each monodromy is a
power of a Dehn twist about $\gamma_{0}$, maybe composed with an involution.

The Dehn twist on ${\mathbb{C}}\diagup\Lambda_{\tau}$ about $\gamma_{0}$ can
be represented by a self-homeomorphism $\psi_{\tau}$ of
${\mathbb{C}}\diagup\Lambda_{\tau}$ which lifts under $p$ to the real linear
self-map $\widetilde{\psi}_{\tau}$ of ${\mathbb{C}}$ which maps $1$ to $1$ and
$\tau$ to $1+\tau$. This can be seen by looking at the action of
$\widetilde{\psi}_{\tau}$ on the lifts of the curves $\gamma$ and
$\gamma^{\prime}$. $\widetilde{\psi}_{\tau}$ takes $\widetilde{\gamma}=[0,1]$
to itself, and $\widetilde{\gamma}^{\prime}=[0,\tau]$ to $[0,\tau+1]$. Note
that the real $2\times 2$ matrix corresponding to $\widetilde{\psi}_{\tau}$ is
$\begin{pmatrix}1&({\rm Im}\,\tau)^{-1}\\\ 0&1\end{pmatrix}$.

We consider the double branched covering
${\sf{p}}_{\Lambda_{\tau}}:({\mathbb{C}}\setminus\Lambda_{\tau})\diagup\Lambda_{\tau}\to\mathbb{C}$
with branch locus
$\mathring{BL}(\Lambda_{\tau})=\\{e_{1}(\tau),e_{2}(\tau),e_{3}(\tau)\\}$
determined by the Weierstraß $\wp$-function,
${\sf{p}}_{\Lambda_{\tau}}=\wp_{\Lambda_{\tau}}\circ p$ (with $p$ being the
projection
$p:\mathbb{C}\setminus\Lambda_{\tau}\to(\mathbb{C}\setminus\Lambda_{\tau})\diagup\Lambda_{\tau}$)
and its extension
${\sf{p}}_{\Lambda_{\tau}}^{c}:{\mathbb{C}}\diagup\Lambda_{\tau}\to\mathbb{P}^{1}$.
The Weierstrass $\wp$-function extends to a mapping, also denoted by
$\wp_{\Lambda_{\tau}}$, $\wp_{\Lambda_{\tau}}:\mathbb{C}\to\mathbb{P}^{1}$
that takes $0$ to $\infty$, $\frac{1}{2}$ to $e_{1}$, $\frac{\tau}{2}$ to
$e_{2}$, and $\frac{1+\tau}{2}$ to $e_{3}$. The line segments
$[0,\frac{1}{2}]$, $[0,\frac{\tau}{2}]$, $[\frac{1}{2},\frac{1+\tau}{2}]$, and
$[\frac{\tau}{2},\frac{1+\tau}{2}]$ are mapped under ${\wp}_{\Lambda_{\tau}}$
to simple curves $\gamma_{\infty,e_{1}}$, $\gamma_{\infty,e_{2}}$,
$\gamma_{e_{1},e_{3}}$, and $\gamma_{e_{2},e_{3}}$ in $\mathbb{P}^{1}$, each
of which joins the first mentioned point with the last mentioned point. The
union of the four curves (each with suitable orientation) is a closed curve in
$\mathbb{P}^{1}$ that divides $\mathbb{P}^{1}$ into two connected components
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. The Weierstrass $\wp$-function
${\wp}_{\Lambda_{\tau}}$ maps the open parallelogram $R$ in the complex plane
with vertices $0$, $1$, $\frac{\tau}{2}$, $1+\frac{\tau}{2}$ conformally onto
its image. Indeed, the ”double” of this parallelogram, i.e. the parallelogram
with vertices $0$, $1$, $\tau$, $\tau+1$, is the interior of a fundamental
polygon for the covering $p:\mathbb{C}\to\mathbb{C}\diagup{\Lambda_{\tau}}$
and
${\wp}_{\Lambda_{\tau}}(\zeta)={\wp}_{\Lambda_{\tau}}(-\zeta+n+m\tau),\,n,m\in\mathbb{Z}$.
By the last equation we get in particular

${\wp}_{\Lambda_{\tau}}(\frac{\tau}{2}+t)={\wp}_{\Lambda_{\tau}}(\frac{\tau}{2}-t)\,,$
(68)

$t\in[0,\frac{1}{2}]$. After perhaps relabeling $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$, we may assume that ${\wp}_{\Lambda_{\tau}}$ maps the open
parallelogram $R_{-}$ with vertices
$0,\,\frac{1}{2},\frac{\tau}{2},\frac{1+\tau}{2},$ (the ”left half” of $R$)
onto $\mathcal{C}_{1}$, and then it maps the parallelogram $R_{+}$ with
vertices $\frac{1}{2},\,1,\,\frac{1+\tau}{2},1+\frac{\tau}{2},$ (the ”right
half” of $R$) onto $\mathcal{C}_{2}$.

The mapping $\tilde{\psi}_{\tau}$ fixes the segment $[0,\frac{1}{2}]$, and
translates the segment $[\frac{\tau}{2},\frac{1+\tau}{2}]$ by $\frac{1}{2}$.
It maps $R_{-}$ to the parallelogram $\tilde{\psi}_{\tau}(R_{-})\subset R$
whose horizontal sides are $[0,\frac{1}{2}]$ and
$[\frac{1+\tau}{2},\frac{2+\tau}{2}]$. ${\wp}_{\Lambda_{\tau}}$ maps
$[\frac{1}{2}+\frac{\tau}{2},1+\frac{\tau}{2}]$ to $\gamma_{e_{3},e_{2}}$
which equals $\gamma_{e_{2},e_{3}}$ with inverted orientation (see (68)). It
maps the segment $[0,\frac{1+\tau}{2}]$ to a curve $\gamma_{\infty,e_{3}}$
which is contained in $\mathcal{C}_{1}$ except of its endpoints and joins
$\infty$ and $e_{3}$, and the segment $[\frac{1}{2},1+\frac{\tau}{2}]$ is
mapped to a curve $\gamma^{\prime}_{e_{1},e_{2}}$ with initial point $e_{1}$
and terminating point $e_{2}$, which is contained in $\mathcal{C}_{2}$ except
its endpoints, so that the union of the curves $\gamma_{\infty,e_{1}}$,
$\gamma^{\prime}_{e_{1},e_{2}}$, $\gamma_{e_{2},e_{3}}$, and
$\gamma_{e_{3},\infty}$ is the oriented boundary of a domain
$\mathcal{C}^{\prime}_{1}$ (i.e. the domain $\mathcal{C}^{\prime}_{1}$ is ”on
the left” when walking along the oriented curve). Here $\gamma_{e_{3},\infty}$
equals $\gamma_{\infty,e_{3}}$ with inverted orientation. The mapping
${\wp}_{\Lambda_{\tau}}\circ\tilde{\psi}_{\tau}$ takes $R_{-}$ to the domain
$\mathcal{C}^{\prime}_{1}$. The mapping
$\varphi_{\tau,1}\stackrel{{\scriptstyle
def}}{{=}}{\wp}_{\Lambda_{\tau}}\circ\tilde{\psi}_{\tau}\circ({\wp}_{\Lambda_{\tau}}|R_{-})^{-1}:\mathcal{C}_{1}\to\mathcal{C}^{\prime}_{1}$
is homotopic on $\mathcal{C}_{1}$ to a mapping that fixes a (large)
neighbourhood of $\gamma_{\infty,e_{1}}$ in $\mathcal{C}_{1}$, maps a
neighbourhood of $\gamma_{e_{2},e_{3}}$ in $\mathcal{C}_{1}$ into a
neighbourhood of $\gamma_{e_{2},e_{3}}$ in $\mathbb{C}$, such that it rotates
(in suitable coordinates) a smaller neighbourhood of $\gamma_{e_{2},e_{3}}$ in
$\mathcal{C}_{1}$ by the angle $\pi$ around $e_{3}$, and rotates points in the
remaining set by an angle between $0$ and $\pi$.

A similar argument for the parallelogram with vertices $\pm\frac{1}{2}$,
$\frac{\tau}{2}\pm\frac{1}{2}$ and the domain $\mathcal{C}_{2}$ shows that
with $R_{+}-1\stackrel{{\scriptstyle def}}{{=}}\\{z-1:z\in R_{+}\\}$ the
mapping $\varphi_{\tau,2}\stackrel{{\scriptstyle
def}}{{=}}{\wp}_{\Lambda_{\tau}}\circ\tilde{\psi}_{\tau}\circ({\wp}_{\Lambda_{\tau}}|R_{+}-1)^{-1}$
is homotopic on $\mathcal{C}_{2}$ to a mapping with the same properties as
above for the role of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ interchanged.

The mapping $\varphi_{\tau}$ that is equal to $\varphi_{\tau,j}$ on
$\mathcal{C}_{j}$ extends to a self-homeomorphism of $\mathbb{P}^{1}$ whose
mapping class is conjugate to $\mathfrak{m}_{\sigma_{1}}$.

Since $\tilde{\psi}_{\tau}|R_{-}=(p|R)^{-1}\circ\psi_{\tau}\circ p|R_{-}$ and
${\wp}_{\Lambda_{\tau}}\circ\tilde{\psi}_{\tau}|R_{-}=\varphi_{\tau,1}\circ{\wp}_{\Lambda_{\tau}}|R_{-}$
we obtain
${\wp}_{\Lambda_{\tau}}\circ(p|R)^{-1}\circ\psi_{\tau}=\varphi_{\tau,1}\circ{\wp}_{\Lambda_{\tau}}\circ(p|R_{-})^{-1}$,
hence, since ${\wp}_{\Lambda_{\tau}}={\sf p}_{\Lambda_{\tau}}\circ p$, the
equality ${\sf p}_{\Lambda_{\tau}}\circ\psi_{\tau}=\varphi_{\tau,1}\circ{\sf
p}_{\Lambda_{\tau}}$ holds on $p(R_{-})$. By the same reason the equality
holds for $p(R_{+})$, and since $p(R_{-}\cup R_{+})$ is dense in
$\mathbb{P}^{1}$, it holds on $\mathbb{P}^{1}$. We proved that the projection
of $\psi_{\tau}$ is homotopic to a conjugate of $\mathfrak{m}_{\sigma_{1}}$,
which is a reducible mapping class. We proved that the $(0,4)$-bundle is
reducible. Proposition 2 is proved. $\Box$

Let ${\mathfrak{F}}$ be a $(\sf{g},\sf{m})$-fiber bundle over
$\partial{\mathbb{D}}$ and $\widehat{\mathfrak{F}}$ its isomorphism class. Let
$\mathfrak{m}_{\mathfrak{F}}$ be the monodromy mapping class of the bundle
$\mathfrak{F}$ and $\widehat{\mathfrak{m}_{\mathfrak{F}}}$ its conjugacy
class. By the bijective correspondence of Theorem 9 we will write also

${\mathcal{M}}(\widehat{{\mathfrak{m}}_{\mathfrak{F}}})={\mathcal{M}}(\widehat{\mathfrak{F}})\,$
(69)

for the conformal module ${\mathcal{M}}(\widehat{\mathfrak{F}})$ of the
bundle. Corollary 3 below shows that for $(1,1)$-bundles this invariant
descends to the conformal module of braids.

Let $\widehat{\mathfrak{F}}$ be a smooth isomorphism class of $(1,1)$-bundles
over the circle, and $\widehat{{\mathfrak{m}}_{\mathfrak{F}}}$ the associated
conjugacy class of mapping classes. For any representative
${\mathfrak{m}}_{\mathfrak{F}}\in\widehat{{\mathfrak{m}}_{\mathfrak{F}}}$ and
any mapping class $\mathfrak{m}$ the equality ${\sf
P}(\mathfrak{m}^{-1}{\mathfrak{m}}_{\mathfrak{F}}\mathfrak{m})={\sf
P}(\mathfrak{m})^{-1}{\sf P}({\mathfrak{m}}_{\mathfrak{F}}){\sf
P}({\mathfrak{m}})$ holds. Hence, ${\sf
P}(\widehat{{\mathfrak{m}}_{\mathfrak{F}}})$ is well-defined. Let ${\sf
P}(\widehat{\mathfrak{F}})$ be the smooth isomorphism class of special
$(0,4)$-bundles that corresponds to ${\sf
P}(\widehat{{\mathfrak{m}}_{\mathfrak{F}}})$. Recall that a representative of
the latter can be obtained as the projection of a smooth family of Riemann
surfaces of type $(1,1)$ that is in the class of $\widehat{\mathfrak{F}}$ and
is a double branched covering of a special $(0,4)$-bundle. The class ${\sf
P}(\widehat{\mathfrak{F}})$ defines a conjugacy class of $3$-braids $\hat{b}$
that is defined up to multiplication by $\Delta_{3}^{2k}$ for integers $k$.
For the conformal module $\mathcal{M}(\hat{b})$ of any such conjugacy class of
braids and the conformal module $\mathcal{M}(\widehat{\mathfrak{F}})$ of the
smooth isomorphism class $\widehat{\mathfrak{F}}$ the following corollary
holds.

######  Corollary 3.

$\displaystyle\mathcal{M}(\widehat{\mathfrak{F}})=\mathcal{M}(\hat{b})\,.$

Proof. $\mathcal{M}(\widehat{\mathfrak{F}})$ is the supremum of the conformal
modules of annuli on which there exists a holomorphic $(1,1)$-bundle that
represents $\mathfrak{F}$. $\mathcal{M}({{\sf P}(\widehat{\mathfrak{F}})})$ is
the supremum of the conformal modules of annuli on which there exists a
special holomorphic $(0,4)$-bundle that represents ${\sf
P}(\widehat{\mathfrak{F}})$. By [17] we have the equality
$\mathcal{M}(\hat{b})=\mathcal{M}(\widehat{b\Delta_{3}^{2k}})=\mathcal{M}({{\sf
P}(\widehat{\mathfrak{F}})})$ for any braid $b$ that is associated to a bundle
representing the class ${{\sf P}(\widehat{\mathfrak{F}})}$. Statement (1) of
Proposition 2 implies the inequality
$\mathcal{M}(\widehat{\mathfrak{F}})\leq\mathcal{M}({{\sf
P}(\widehat{\mathfrak{F}})})$, Statement (2) implies the opposite inequality.
The corollary is proved. $\Box$

In the following Theorem 11 we consider isotopies of smooth $(1,1)$-families
over a torus with a hole to complex analytic $(1,1)$-families. Theorem 11 is a
more comprehensive version of Theorem 4 from the introduction. (Theorem 4
states the first assertion of Theorem 11.)

######  Theorem 11.

$(1)$ Let $X$ be a smooth surface of genus one with a hole with base point
$x_{0}$, and with a chosen set $\mathcal{E}=\\{e_{1},e_{2}\\}$ of generators
of $\pi_{1}(X,x_{0})$. Define the set
$\mathcal{E}_{0}=\\{e_{1},e_{2},e_{1}e_{2}^{-1},e_{1}e_{2}^{-2},e_{1}e_{2}e_{1}^{-1}e_{2}^{-1}\\}$
as in Theorem 2. Consider a smooth $(1,1)$-bundle $\mathfrak{F}$ on $X$.
Suppose for each $e\in\mathcal{E}_{0}$ the restriction of the bundle
$\mathfrak{F}$ to an annulus representing $\hat{e}$ has the Gromov-Oka
property. Then the bundle $\mathfrak{F}$ has the Gromov-Oka property on $X$.
$(2)$ If a bundle $\mathfrak{F}$ as in $(1)$ is irreducible, then it is
isotopic to an isotrivial bundle, and hence, for each conformal structure
$\omega$ on $X$ the bundle $\mathfrak{F}_{\omega}$ is isotopic to a bundle
that extends to a holomorphic $(1,1)$-bundle on the closed torus
$\omega(X)^{c}$. In particular, $\mathfrak{F}$ is isotopic to a holomorphic
bundle for any conformal structure $\omega$ on $X$ (maybe, of first kind).
$(3)$ Any smooth reducible bundle $\mathfrak{F}$ on $X$ has a single
irreducible bundle component. This irreducible bundle component is isotopic to
an isotrivial bundle. There is a Dehn twist in the fiber over the base point
such that the $(1,1)$-bundle $\mathfrak{F}$ can be recovered from the
irreducible bundle component up to composing each monodromy by a power of the
Dehn twist. A smooth reducible bundle on $X$ is isotopic to a holomorphic
$(1,1)$-bundle for each conformal structure of second kind on $X$.
$(4)$ A reducible holomorphic $(1,1)$-bundle over a punctured Riemann surface
is locally holomorphically trivial.

Proof. By Lemma 16 and Proposition 2 we may assume that $\mathfrak{F}$ is a
double branched covering of a smooth special $(0,4)$-bundle, denoted by ${\sf
P}(\mathfrak{F})$. Suppose that for each $e\in\mathcal{E}_{0},$ the
restriction of the smooth $(1,1)$-bundle $\mathfrak{F}$ to an annulus
$A_{\hat{e}}$ representing $\hat{e}$ has the Gromov-Oka property. Then there
is a conformal structure $\omega_{e}:A_{\hat{e}}\to\omega_{e}(A_{\hat{e}})$
such that $\omega_{e}(A_{\hat{e}})$ has conformal module bigger than
$\frac{\pi}{2}\log(\frac{3+\sqrt{5}}{2})^{-1}$ and the pushed forward bundle
$(\mathfrak{F}|A_{\hat{e}})_{\omega_{e}}$ is isotopic to a holomorphic bundle
$\mathfrak{F}_{e}$ on $\omega_{e}(A_{\hat{e}})$. By Proposition 1 we may
assume that $\mathfrak{F}_{e}$ is of the form (37), and by Proposition 2 we
may assume that $\mathfrak{F}_{e}$ is a double branched covering of a
holomorphic special $(0,4)$-bundle on $\omega_{e}(A_{\hat{e}})$, denoted by
${\sf P}(\mathfrak{F}_{e})$. Since the monodromy homomorphisms of
$\mathfrak{F}|A_{\hat{e}}$ and $\mathfrak{F}_{e}$ are conjugate, the monodromy
homomorphisms of ${\sf P}(\mathfrak{F}|A_{\hat{e}})$ and ${\sf
P}(\mathfrak{F}_{e})$ are also conjugate. Hence, ${\sf
P}(\mathfrak{F}|A_{\hat{e}})$ and ${\sf P}(\mathfrak{F}_{e})$ are isotopic. We
saw that for each $e\in\mathcal{E}$ the restriction of the special
$(0,4)$-bundle ${\sf P}(\mathfrak{F})$ to an annulus representing $\hat{e}$ is
isotopic to a holomorphic bundle for a conformal structure on the annulus of
conformal module bigger than $\frac{\pi}{2}\log(\frac{3+\sqrt{5}}{2})^{-1}$.
By Theorem 10 the special $(0,4)$-bundle ${\sf P}(\mathfrak{F})$ is isotopic
to a holomorphic special $(0,4)$-bundle for any conformal structure of second
kind on $X$. Hence, the canonical double branched covering of ${\sf
P}(\mathfrak{F})$ is isotopic to a holomorphic $(1,1)$-bundle for each
conformal structure of second kind on $X$. Since the monodromy mapping class
of any double branched covering of the $(0,4)$-bundle along each loop $\gamma$
in $X$ with base point $x_{0}$ differs at most by involution from the
monodromy mapping class along $\gamma$ of the canonical double branched
covering, by Proposition 2 also the bundle $\mathfrak{F}$ is isotopic to a
holomorphic $(1,1)$-bundle for each conformal structure of second kind on $X$.
Statement (1) is proved.

Suppose a bundle $\mathfrak{F}$ as in (1) is irreducible. Then by Proposition
2 the special $(0,4)$-bundle ${\sf{P}}(\mathfrak{F})$ is also irreducible.
Moreover, the $(0,4)$-bundle ${\sf{P}}(\mathfrak{F})$ is isotopic to an
isotrivial bundle, i.e. its lift to a finite covering $\hat{X}$ is isotopic to
the trivial bundle, and therefore, for each conformal structure $\omega$ on
$X$ (maybe, of first kind) the $(0,4)$-bundle
${\sf{P}}(\mathfrak{F})_{\omega}$ is isotopic to a bundle that extends
holomorphically to the closed torus $\omega(X)^{c}$. Then also the canonical
double branched covering of the $(0,4)$-bundle is isotopic to an isotrivial
bundle. Hence, the bundle $\mathfrak{F}$ is isotopic to an isotrivial bundle.
In particular, for each conformal structure (maybe, of first kind) on $X$ the
bundle $\mathfrak{F}_{\omega}$ is isotopic to a holomorphic bundle on
$\omega(X)$. This gives statement (2).

Suppose now $\mathfrak{F}$ is any smooth reducible bundle on $X$. Then ${\sf
P}(\mathfrak{F})$ is reducible. By Theorem 10 the special $(0,4)$-bundle ${\sf
P}(\mathfrak{F})$ is isotopic to a holomorphic special $(0,4)$-bundle for any
conformal structure of second kind on $X$. Hence, for any conformal structure
of second kind on $X$ the bundle $\mathfrak{F}$ is isotopic to a holomorphic
$(1,1)$-bundle which is a double branched covering.

Each admissible set of curves on a punctured torus consists of exactly one
curve. Hence, there is an admissible closed curve $\gamma$ in the punctured
fiber $\mathcal{P}^{-1}(x_{0})\setminus\mathbold{s}^{\infty}$ over the base
point $x_{0}$, that is mapped by the monodromy mapping class along each curve
$\gamma_{j}$ representing an element $e_{j}\in\mathcal{E}$ to a curve that is
isotopic to $\gamma$ in the punctured fiber over $x_{0}$. Choose for each
monodromy mapping class a representative that maps the curve $\gamma$ onto
itself.

The complement of $\gamma$ on the punctured torus
$\mathcal{P}^{-1}(x_{0})\setminus\mathbold{s}^{\infty}$ is connected and
homeomorphic to the thrice punctured Riemann sphere. This implies first that
there is a single irreducible component of $\mathfrak{F}$. Further, the
restrictions of suitable representatives of the monodromy mapping classes to
the complement of $\gamma$ on the punctured torus are conjugate to self-
homeomorphisms of the thrice punctured Riemann sphere. The conjugated
homeomorphisms extend to self-homeomorphisms of the Riemann sphere with three
distinguished points. The self-homeomorphisms may interchange the two points
that come from different edges of $\gamma$, but each of them fixes the third
distinguished point. Except the identity there is only one isotopy class of
such mappings and its square is the identity. Hence, the monodromy mapping
classes of the irreducible bundle component are powers of a single periodic
mapping class. In other words, the irreducible bundle component is isotopic to
an isotrivial bundle.

The monodromy mapping classes of the original bundle $\mathfrak{F}$ are powers
of Dehn twists about $\gamma$, maybe, composed with the involution, and up to
powers of Dehn twists about $\gamma$ in the fiber $\mathcal{P}^{-1}(x_{0})$
the bundle can be recovered from the irreducible bundle component. Statement
(3) is proved.

We prove now statement (4). Consider a reducible holomorphic (1,1)-bundle
$\mathfrak{F}$ over a punctured Riemann surface of genus $1$. By Proposition 2
it is holomorphically isomorphic to the double branched covering of a
holomorphic special $(0,4)$-bundle ${\sf P}(\mathfrak{F})$ over a punctured
Riemann surface. By Proposition 2 the special holomorphic$(0,4)$-bundle ${\sf
P}(\mathfrak{F})$ is also reducible. By Theorem 10 the bundle ${\sf
P}(\mathfrak{F})$ is holomorphically trivial. Then the double branched
covering $\mathfrak{F}$ of ${\sf P}(\mathfrak{F})$ is locally holomorphically
trivial. Theorem 11 is proved. $\Box$

## References

  * [1] Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological Entropy. Trans. Amer Math. Soc. 114, 309–319 (1965).
  * [2] Ahlfors, L.: Lectures on Quasiconformal Mappings. Van Nostrand, Princeton (1966).
  * [3] Arnold, V.: On some Topological Invariants of Algebraic Functions. Trudy Moskov. Mat. Obsc. 21, 27–46 (1970). Engl. Transl.: Trans. Moscow Math. Soc. 21, 30–52 (1970).
  * [4] Bers, L. : An Extremal Problem for Quasiconformal Mappings and a Theorem by Thurston. Acta Math. 141, 73–98 (1978).
  * [5] Birman, J.: Braids, Links and Mapping Class Groups. Ann. Math. Studies 82, Princeton Univ. Press (1975).
  * [6] Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series,49. Princeton University Press, Princeton, NJ, (2012).
  * [7] Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces. Soc. Math. France, Paris, 1979, Astérisque 66–67 (1991).
  * [8] Forster, O.: Lectures on Riemann surfaces. Translated from the 1977 German original by Bruce Gilligan. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, (1991).
  * [9] Forstneric, F.: Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis. A Series of Modern Surveys in Mathematics 56. Springer, Heidelberg, (2011).
  * [10] Forstneric, F., Slapar, M.: Stein structures and holomorphic mappings. Math. Z. 256 , no. 3, 615–646 (2007).
  * [11] Forstneric, F.: Runge Approximation on Convex Sets Implies the Oka Property. Ann. of Math. (2) 163, no. 2, 689–707 (2006).
  * [12] Fischer, W., and Grauert, H., Lokal-triviale Familien Kompakter Komplexer Mannigfaltigkeiten (German) Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 89–94 (1965).
  * [13] Gromov, M.: Oka’s Principle for Holomorphic Sections of Elliptic Bundles. J. Amer. Math. Soc. 2, no. 4, 851–897 (1989).
  * [14] Gorin, E. and Lin, V., On Separable Polynomials over Commutative Banach Algebras. Dokl. Akad. Nauk SSSR 218, no 3, 505-508 (1974). Engl. Transl.: Soviet Math. Dokl. 15, no 5, 1357–1361 (1974).
  * [15] Hansen, V.L.: Polynomial Covering Spaces and Homomorphisms into the Braid Groups. Pacif. J. Math. 81\. 2, 399-410 (1979).
  * [16] Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Third edition. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, (1990).
  * [17] Jöricke, B.: Braids, Conformal Module and Entropy,. C.R. Acad. Sci. Paris, Ser. I, 351, 289-293 (2013).
  * [18] Jöricke, B.: Fundamental Groups, Slalom Curves and Extremal Length, in Operator Theory: Advances and Applications, Vol. 261, 307-315, Springer 2018.
  * [19] Jöricke, B.: Fundamental Groups, 3-Braids, and Effective Estimates of Invariants. Math. Z. 294, 1553–1609 (2020).
  * [20] Jöricke, B.: Riemann Surfaces of Second Kind and Finiteness Theorems. arXiv:2102.02139.
  * [21] Kassel, C. and Turaev, V.: Braid Groups, Graduate Texts in Mathematics 247, Spinger (2008).
  * [22] Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Grundlehren der Mathematischen Wissenschaften, 283, Springer-Verlag, New York, (1986).
  * [23] Penner, R.: Bounds on Least Dilatations. Proc. Amer. Math. Soc. 113 2, 443–450 (1991).
  * [24] Song, W., and Ko,K., and Los,J.: Entropy of Braids. J. Knot Theory Ramifications, 11 4, 647–666 (2002).
  * [25] Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Company, 1966.
  * [26] Steenrod, N.: The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951.
  * [27] Stout, E.L.: Bounded holomorphic funtions on finite Riemann surfaces. Trans. Amer. Math. Soc. bf 120, 2, 255-285 (1965).
  * [28] Wielandt, H.: Finite Permutation Groups. Translated from the German by R. Bercov, Academic Press, New York-London (1964).
  * [29] Zjuzin, Ju.: Irreducible Separable Polynomials with Holomorphic Coefficients on a Certain Class of Complex Spaces. (Russian), Mat. Sb. (N.S.) 102(144) 4, 569–591, 632 (1977).