††thanks: Juan Carlos Torres Fellow

,††thanks: ESA Research Fellow

# Testing Self-Organized Criticality Across the Main Sequence
using Stellar Flares from TESS

Adina D. Feinstein  NSF Graduate Research Fellow<EMAIL_ADDRESS>Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL
60637, USA  Darryl Z. Seligman  Department of the Geophysical Sciences,
University of Chicago, Chicago, IL 60637, USA  Maximilian N. Günther
Department of Physics, and Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139,
USA  European Space Agency (ESA), European Space Research and Technology
Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands  Fred C.
Adams  Physics Department, University of Michigan, Ann Arbor, MI 48109
Astronomy Department, University of Michigan, Ann Arbor, MI 48109

###### Abstract

Stars produce explosive flares, which are believed to be powered by the
release of energy stored in coronal magnetic field configurations. It has been
shown that solar flares exhibit energy distributions typical of self-organized
critical systems. This study applies a novel flare detection technique to data
obtained by NASA’s TESS mission and identifies $\sim 10^{6}$ flaring events on
$\sim 10^{5}$ stars across spectral types. Our results suggest that magnetic
reconnection events that maintain the topology of the magnetic field in a
self-organized critical state are ubiquitous among stellar coronae.

Introduction — The theory of self-organized criticality (Bak  _et al._ , 1988)
describes a class of dissipative dynamical systems which remain at a critical
point with no intrinsic length or time scale. The existence of the critical
state requires a local instability, which occurs when some parameter exceeds
its critical value and results in a dissipative transport process where this
same parameter increases at neighboring sites. A simple physical analogy is a
sand pile. As sand particles are added, the difference in height between
neighboring sites on the pile increases. When the additional sand particles
make the new height difference exceed a critical threshold, avalanche events
occur. This system maintains a critical slope, representing a dynamical
attractor that is insensitive to the initial conditions. This critical state
is maintained via nonlinear avalanche events spanning all length scales
triggered by perturbations.

Figure 1: A color-magnitude diagram (Gaia $B_{\textrm{p}}-R_{\textrm{p}}$ and
M${}_{\textrm{G}}$) for our sample colored by flare rate. The flare rate was
calculated by weighting each flare by its output “probability” from stella.
Here, we accounted for all identified flares, not just flares with
probabilities $\geq 0.9$. Stars towards the top of the main sequence tend to
have higher flare rates. This trend could be due to being young and metal-rich
or being binaries.

While the sand pile analogy is simplistic by construction, self-organized
criticality naturally manifests in a variety of physical systems. Applications
have been found in hydrodynamical turbulence, forest fires and other
percolation systems (Turcotte, 1999), landslides (Bak  _et al._ , 1990;
Turcotte  _et al._ , 2002), neuroscience (Ribeiro  _et al._ , 2010; Hesse and
Gross, 2014), climate fluctuations (Grieger, 1992), rainfall (Andrade  _et
al._ , 1998), accretion disks (Dendy  _et al._ , 1998), traffic jams (Nagel
and Herrmann, 1993), evolution (Bak and Sneppen, 1993), extinction (Newman,
1996), financial markets (Bak  _et al._ , 1997) and even Conway’s game of Life
(Bak  _et al._ , 1989), to name a non-comprehensive list. The theory also
explains the Gutenberg-Richter (Gutenberg and Richter, 1956) law for the
distribution of earthquake energies (Bak and Tang, 1989; Sornette and
Sornette, 1989; Olami  _et al._ , 1992),

$\frac{dN}{dE}\sim E^{-\alpha}\,,$ (1)

where $N$ is the number of earthquakes, $E$ is the energy released in the
earthquake (where the earthquake magnitude $m\propto\log E$), and the power-
law exponent $1.25<\alpha\sim 1.5$. A 3-dimensional slip-stick model of
tectonic events produces a critical exponent of $\alpha=1.35$ (Bak and Tang,
1989), in reasonable agreement with the observed law.

Even the solar coronal magnetic field may reside in a self-organized critical
state (Lu and Hamilton, 1991). This hypothesis naturally explains the power-
law dependence of the magnitude of solar flares, which is of the same form as
Equation (1), where $\alpha\approx 1.4$ (Lu and Hamilton, 1991). This
characteristic exponent and the subsequent temporal clustering are universal
between earthquakes and solar flares (de Arcangelis  _et al._ , 2006).

Explosive events on the Sun are believed to be driven by the energy stored in
twisted coronal magnetic field lines (Parker, 1989). Such field configurations
are generated through dynamo action in the fluid-dominated interior
(Charbonneau, 2010), and through convective and coriolis driven vortical
subsurface plasma flows (Parker, 1955; Moffatt, 1978; Longcope  _et al._ ,
1998; Seligman  _et al._ , 2014). The concentration of magnetic field lines
can lead to the release and subsequent dissipation of energy via the process
of magnetic reconnection (Sturrock  _et al._ , 1984). These reconnection
events could be triggered when the angle, $\theta$, between neighboring
magnetic field vectors is greater than a critical value, $\theta_{c}$. The
reconnection event changes the angles for neighboring field lines and can
trigger additional events (Sturrock  _et al._ , 1984; Parker, 1988; Sturrock
_et al._ , 1990). When $\theta<\theta_{c}$, magnetic reconnection proceeds
slowly, allowing for the buildup of magnetic energy in the form of twisted
field lines. Thus, this configuration satisfies all the requirements for a
self-organized critical system, if given a source for energy injection.

Recent studies have begun to explore if flaring events on stars across all
spectral types and evolutionary stages exhibit the same power-law distribution
as in the Sun. This finding would suggest that other stars also maintain self-
organized critical states in their coronal magnetic fields. However, this
hypothesis has previously been difficult to test due to (i) the lack of a long
observational baseline to capture a sufficient number of flaring events, (ii)
the lack of observations of a large number of stars across spectral types, and
(iii) the difficulty in observing, identifying and characterizing flares,
especially those with low amplitudes relative to instrumental noise. Extra-
solar surveys such as Kepler/K2 (Borucki and Others, 2010) and the Transiting
Exoplanet Survey Satellite (TESS; (Ricker and Others, 2015)) have provided
solutions to (i) and (ii). These missions have provided unprecedented, high
precision, long-baseline light curves for hundreds of thousands of stars.

Aschwanden and Güdel (2021) identified a power-law dependence with
$\alpha=2.09\pm 0.24$ for flaring events observed in a set of stars across
stellar types observed with Kepler. This strongly suggests that the self-
organized critical state observed in the Solar corona is ubiquitous. However,
the published Kepler catalogs tend to be biased towards high-amplitude flares
because of the inefficiency at identifying low-amplitude events of implemented
flare-detection algorithms. Moreover, these catalogs are contaminated with
other features easily mistaken for flaring events, such as rapidly rotating or
variable stars (Davenport, 2016). Further complicating matters, the flares
that are both observed and identified correctly in Kepler are not fully
temporally resolved due to the 30-minute cadence of the observations. The TESS
mission has recently provided 2-minute cadence light curves for $\sim$200,000
of the nearest and brightest stars. This data provides a full sample of
spectral types across the main sequence to search for temporally-resolved
flaring events (Günther  _et al._ , 2020).

Additionally, machine learning techniques curated to identify flares in TESS
short cadence data have been developed to rectify issue (iii). These novel
techniques are capable of identifying low-amplitude flares with high fidelity
(Feinstein  _et al._ , 2020a; Vida  _et al._ , 2021). This combination of
unprecedented data and efficient identification techniques provides a new
opportunity to expand the hypothesis of self-organized criticality in the
solar corona to a sample of stars representative of the galactic census of
spectral types and ages. In this letter, we provide a complementary search for
indications of self-organized criticality in flaring events observed by TESS,
using our newly created catalog of stellar flares from two years of data
(Günther et al, in prep).

Table 1:  Sample Summary Statistics

$B_{p}-R_{p}$ | Mass (M⊙) | $N_{\textrm{stars}}$ | $N_{\textrm{flares, prob}\geq 0.5}$
---|---|---|---
$[2.78,4.86]$ | $[0.05,0.3)$ | 9,241 | 59,150
$[2.13,2.78)$ | $[0.3,0.5)$ | 20,124 | 108,963
$[1.21,2.13)$ | $[0.5,0.8)$ | 17,914 | 139,445
$[0.327,1.21)$ | $[0.8,1.7)$ | 85,609 | 571,556
$[-0.12,0.327)$ | $[1.7,3.0]$ | 3,770 | 20,447
Red Giant Branch | [0.8, 1.8] | 5,157 | 10,965

11footnotetext: Gaia $B_{p}-R_{p}$ to stellar mass conversion for stars along
the main sequence were completed using “A Modern Mean Dwarf Stellar Color and
Effective Temperature Sequence” (version 2021.03.02).22footnotetext: Red giant
branch stellar masses are adopted from (Wu  _et al._ , 2019).

Observations with TESS — NASA’s five-year TESS mission is currently performing
time-series photometry of $\sim 90\%$ of the sky (Ricker and Others, 2015).
The survey observes in $24^{\circ}\times 96^{\circ}$ sectors for $\sim$ 27
days at a time. During its primary 2-year mission, TESS observed $\sim$200,000
pre-selected stars at 2-minute cadence across both ecliptic hemispheres. TESS
has been providing an unprecedented data set at high temporal resolution with
which to identify flaring events across stellar spectral types and ages. The
light curves used in this study were processed by the Science Processing
Operations Center (SPOC) pipeline operated at the NASA Ames Research Center,
which performs optimized aperture selection and systematics detrending
(Jenkins  _et al._ , 2016). We applied pipeline-assigned quality flags to mask
contaminated111Contamination can originate from cosmic rays, reaction wheel
desaturation events, and other spacecraft sources. For more information see
Table 28 in Tenenbaum and Jenkins (2018). regions of the light curves.

Flare Identification — As mentioned previously, flare identification in time
series photometric data has proven challenging. Previous methods of flare
identification have relied on detrended, or “cleaned”, light curves and have
heuristics for outlier detections. However, detrending often removes low-
amplitude flares entirely, and the outlier-heuristics (Chang  _et al._ , 2015)
are biased towards the identification of only the highest amplitude events.
Conversely, more lenient heuristics produce a significant number of false
positive events.

Neural networks are a class of supervised machine learning algorithm optimized
for visualization problems such as identifying features in 1-dimensional time
series (e.g. Ansdell  _et al._ (2018); Pearson  _et al._ (2018); Vida  _et
al._ (2021)) or images (e.g. Tanoglidis  _et al._ (2021)). Here, we used the
convolutional neural networks (CNN) developed in Feinstein  _et al._
(2020a)222The pre-trained CNNs are available online:
https://archive.stsci.edu/hlsp/stella., which are accompanied by the open-
source software package stella (Feinstein  _et al._ , 2020b), and are easily
scalable to large data sets. stella is unique in that it provides a
“probability” that any given cadence in a light curve is associated with a
flaring event. We ran all light curves through 10 pre-trained stella models
and averaged the probability outputs for our final catalog of flaring events.
We require a probability $\geq 0.9$ for an event, meaning the event has a 90%
probability of being a true flare. We provide upper and lower limits by only
including flaring events with a probability $\geq 0.50$ and $\geq 0.99$,
respectively.

While stella accurately recovers a high percentage of real flaring events, it
was trained on a relatively small data set with limited examples of
contaminating features such as variable and eclipsing binary stars, and noisy
light curves. In Günther et al. (in prep) and this study, we therefore apply
four quality control filters to the stella outputs, mitigating the risk of
false positives:

  1. 1.

Signal-to-Noise Ratio Filter: This filter removes any false positives
originating from the increased photon noise of faint targets. To do this, we
estimated the root-mean-square (RMS) noise of a given light curve. We first
remove any variability using a biweight filter with a window size of 20
minutes and then take the RMS of the flattened light curve. We require stellar
flare candidates to have an amplitude of $\geq 3\times$ RMS noise.

  2. 2.

Outliers: This filter removes flaring events that appear in limited data
points. We removed any flares with a fitted duration of $\leq 4$ minutes which
correspond to 2 TESS data points.

  3. 3.

Eclipsing Binaries (EBs): This filter removes false positives originating from
EBs, where the ingress and egress of the eclipse events could be mis-
classified as flare candidates. For this purpose, we use the entire TESS
threshold-crossing-event (TCE) catalog provided by the SPOC pipeline. We check
if the flare peak times are associated with known eclipses. If a flare
candidate falls within the eclipse window (approximated as 3 times the TCE
duration), we remove the flare from the catalog.

  4. 4.

Variability/Rotation: This filter removes false positives originating from the
peaks of fast-variable and fast-rotating targets. If the target has $\geq 10$
flares, we compute a Lomb-Scargle periodogram of the target light curve and of
the affiliated probability time series. If the variability detected from each
data sets are within 2 days and have a false-alarm probability $<0.05$, all
flares from this target are removed.

After these filters are applied, we are left with a catalog of $N_{\rm
flares}=958,659$ originating from $N_{\rm stars}=161,836$ for our analysis. A
summary of our catalog is presented in Table 1 and the full catalogs are made
available in Günther et al. (in prep.). We calculate the flare rate per star,
$\beta_{\textrm{star}}$, as

$\beta_{\textrm{star}}=\frac{1}{\tau_{\textrm{obs}}}\,\bigg{(}\,\sum_{i=0}^{N}p_{i}\bigg{)}\,,$
(2)

where $N$ is the number of flares for a given star, $p_{i}$ is the stella
probability for a given flare indexed by $i$, and $\tau_{\textrm{obs}}$ is the
total observed time. An overview of the flare rates for all stars in our
sample is presented in Figure 1. It is evident that stars with the highest
flare rates ($\geq 0.3$ day-1) fall along the upper edge of the main sequence
(meaning they are brighter, i.e. have a lower absolute magnitude, than other
stars of that given color). This trend could be indicative of binary star
systems (Hurley and Tout, 1998) or young, metal-rich stars (Kotoneva  _et al._
, 2002). For our analysis, we only account for stars along the main sequence
and the red giant branch (RGB; stars that fork to the right at 1 $\leq$ Gaia
Bp-Rp and M${}_{\textrm{G}}\leq 2.5$).

Figure 2: The cumulative flare frequency distributions (FFDs) in our sample of
stars binned by the flare amplitude and subdivided into different mass bins;
the slope, $\alpha$, and error is given in the upper-right corner of each
subpanel. The bins are the FFD for flares with a probability $\geq 0.9$. The
upper and lower errors on the FFD are defined as flares with probability $\geq
0.99$ and $\geq 0.5$. All bins exhibit clear power-laws, although some bins
are incomplete for low-amplitude flares (e.g., $0.05\leq M/M_{\odot}\leq 0.3)$
or high-amplitude flares (e.g., Red Giant Branch) .

Measured Flare Rates — We present our measured flare frequency distributions
(FFDs) as a function of stellar mass and flare amplitude (Figure 2). The mass
bins were selected based on spectral types/changes in interior stellar
structure: stars with masses $M/M_{\odot}\lesssim 0.3$ are fully convective
Chabrier and Baraffe (1997); Dorman  _et al._ (1989), stars with masses
$0.3<M/M_{\odot}\lesssim 1.7$ have convective exteriors and radiative
interiors, and stars with masses $M/M_{\odot}>1.7$ have radiative exteriors
and convective interiors Heger  _et al._ (2000). We present these results in
flare amplitude space, compared to energy, to remove any additional errors
resulting from estimating stellar luminosities. In theory, the flare energy is
more directly relevant to the predicted self-organized critical state.
However, in practice, the uncertainty in mass and luminosity of each star
makes the amplitude a more reliable quantity.

We measure the slope of each distribution, $\alpha$, and associated error
using a Markov Chain Monte Carlo (MCMC) approach with emcee (Goodman and
Weare, 2010; Foreman-Mackey  _et al._ , 2013). Our MCMC fit was initialized
with 300 walkers and ran for 5000 steps. After visual inspection, we removed
the first 800 burn-in steps and verified our chains converged following the
method of (Geweke, 1992). We present our FFDs and measured slope with error in
Figure 2.

Figure 3: Comparison of measured flare rate slopes, $\alpha$, as a function of
stellar mass. Our rates are plotted as stars. Literature values are plotted as
circles and colored by the number of flares in the given sample, which range
from single to $10^{4}$ stars Shibayama  _et al._ (2013); Günther  _et al._
(2020); Ilin  _et al._ (2019); Lin  _et al._ (2019); Howard  _et al._ (2019);
Yang and Liu (2019); Feinstein  _et al._ (2020a); Raetz  _et al._ (2020); Ilin
_et al._ (2021); Aschwanden and Güdel (2021). The highest mass stars have
higher flare rate indices (light green) than previously measured. Our RGB star
(yellow) slope is within $1\sigma$ to that of main sequence stars in the same
mass range. The remaining data points fall within the expected range of self-
organized critical systems. We estimate the masses of our RGB stars using
results from (Wu  _et al._ , 2019).

We find that the FFDs for stars with masses $0.3\leq M/M_{\odot}\lesssim 3.0$
appear as featureless power-laws. The measured slopes are consistent with
models of flaring activity as self-organized critical systems ($\alpha\approx
1.4$, (Lu and Hamilton, 1991)) and with that measured for the Sun
($\alpha=1.65\pm 0.1$; de Arcangelis  _et al._ , 2006).

Our lowest mass bin ($0.05\leq M/M_{\odot}\leq 0.3$) and our sample of RGB
stars do not follow the same trends. Stars with $0.05\leq M/M_{\odot}\leq 0.3$
show a featureless power-law for flare amplitudes $\geq 5\%$; while power-laws
are indicative of a self-organized critical state, the difference in interior
structure may result in a shallower flare rate (here, $\alpha\sim 1$). The
most energetic flares have historically been observed on low-mass stars in
this bin Feinstein  _et al._ (2020a). It is possible that because these stars
are fully convective and therefore have larger convective cells than other
main sequence stars, they produce more energetic flares. Similarly, red giants
have larger convective cells and display even shallower slopes in their FFD.
We also note that our bins of stars with $1.7<M/M_{\odot}\leq 3.0$ and RGB
stars have the fewest number of flares (Table 1) and while we are able to
measure the slopes of these FFDs, they are incomplete to the lowest- and
highest-amplitude flares.

Finally, we compare our measured FFD slopes to those presented by previous
authors in Figure 3. Previous studies used an order of magnitude fewer flaring
events than ours, and tend to report steeper slopes ($\alpha\sim 2$) than the
ones measured here. The variation in measured slopes arises based on sample
size/selection, and the cadence of the analyzed data (from 1- to 30-minute
cadence). This discrepancy may also be explained by fewer high-amplitude
events in the samples, given their rarity. In this study, we also extended our
measurement of flare rates to include a broader stellar mass and evolutionary
stage range.

Conclusions and Future Work — In this letter, we analyzed the light curves of
161,836 stars that were observed by TESS at a 2-minute cadence. These light
curves were processed and searched for flares using a novel machine learning
algorithm (Feinstein  _et al._ , 2020a, b), resulting in a catalog of $\sim
10^{6}$ flaring events (Günther et al., in prep.). In Figure 1, we show how
the flare rate changes as a function of location on a color-magnitude diagram.
In Figure 2, we show the flare frequency distribution of stars as a function
of stellar mass and compare it to that found in the literature (Figure 3).
Main sequence stars with $M/M_{\odot}\geq 0.3$ exhibit power-law distributions
of flare rates with slopes that are characteristic of self-organized critical
systems ($\alpha\approx 1.4$). The resulting indices are somewhat smaller for
stars with $M/M_{\odot}<0.3$ and for red giant stars (which have slopes
$\alpha\sim 1$). This discrepancy may be due to differences in the interior
structure of these stars, which have larger convective cells compared to other
main sequence stars.

Although the measured slopes of the flare rates show some scatter, the results
of this paper indicate a high degree of universality. In the working picture
that emerges, subsurface convective regions efficiently inject sufficient
energy in the form of twisted magnetic fields near the stellar surface.
Subsequent reconnection events then act to produce flares and maintain the
magnetic field topology in a self-organized critical state. For completeness,
we note that power-law distributions can arise through a variety of mechanisms
(Newman, 2005), not only via self-organized criticality, so that additional
theoretical modeling of these systems is indicated.

TESS will continue to observe $\sim 90\%$ of the sky for the next three years.
The full-frame images that observe $\sim 10^{6}$ stars per month will soon
increase to a higher cadence, making flare identification more feasible for an
order of magnitude more stars than presented herein. This increase should
further complete the distribution of the rarest, high-energy events. The
timing of sympathetic flares after a “main” flare event (defined by some
amplitude threshold), can be used to further investigate if these systems
maintain a self-organized critical state, as was observed for the Sun and for
earthquake aftershocks (de Arcangelis  _et al._ , 2006). Due to the short
baseline of TESS ($\sim 1$ month), this study may only be truly complete for
stars within TESS’s continuous viewing zone, those that get $\sim 1$ year of
continuous observations. Such a study could also be completed with the already
available Kepler light curves, which have a baseline of 4-years to search for
the timing of sympathetic flares.

###### Acknowledgements.

We thank Fausto Cattaneo, Adrian Price-Whelan, Jacob Bean, Leslie Rogers and
Konstantin Batygin for insightful conversations. ADF acknowledges support from
the National Science Foundation Graduate Research Fellowship Program under
Grant No. (DGE-1746045). MNG acknowledges support from MIT’s Kavli Institute
as a Juan Carlos Torres Fellow and from the European Space Agency (ESA) as an
ESA Research Fellow. This research has made use of NASA’s Astrophysics Data
System Bibliographic Services. This work made use of the following open-source
Python packages (Harris  _et al._ , 2020; Astropy Collaboration  _et al._ ,
2013, 2018; Virtanen  _et al._ , 2020; Hunter  _et al._ , 2007).

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# On the topology of the Milnor boundary for real analytic singularities

R. Araújo dos Santos  ,  A. Menegon  ,  M. Ribeiro  ,  J. Seade  and  I. D.
Santamaria Guarín

###### Abstract.

We study the topology of the boundaries of the Milnor fibers of real analytics
map-germs $f:(\mathbb{R}^{M},0)\to(\mathbb{R}^{K},0)$ and $f_{I}:=\Pi_{I}\circ
f:(\mathbb{R}^{M},0)\to(\mathbb{R}^{I},0)$ that admit Milnor’s tube
fibrations, where $\Pi_{I}:({\mathbb{R}}^{K},0)\to({\mathbb{R}}^{I},0)$ is the
canonical projection for $1\leq I<K.$ For each $I$ we prove that the Milnor
boundary $\partial F_{I}$ is given by the double of the Milnor tube fiber
$F_{I+1}.$ We prove that if $K-I\geq 2$, then the pair $(\partial
F_{I},\partial F_{f})$ is a generalized $(K-I-1)$-open-book decomposition with
binding $\partial F_{f}$ and page $F_{f}\setminus\partial F_{f}$ \- the
interior of the Milnor fibre $F_{f}$ (see the definition below). This allows
us to prove several new Euler characteristic formulae connecting the Milnor
boundaries $\partial F_{f},$ $\partial F_{I},$ with the respectives links
$\mathcal{L}_{f},\mathcal{L}_{I},$ for each $1\leq I<K,$ and a Lê-Greuel type
formula for the Milnor boundary.

##  1\. Introduction

One of the most active and challenging areas in singularity theory is the
study of non-isolated singularities of complex spaces. For instance, if
$f:(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ is a holomorphic germ of function with
non-isolated critical point, the degeneration process of the non-critical
levels to the non-isolated singularity hypersurface defined by $f$ is still
not well-understood, unlike the isolated singularity case.

One approach to this problem is to study such degeneration over a small sphere
around the origin. In other words, one tries to understand the topology of the
boundary of the Milnor fiber and how it degenerates to the link of $f$. This
problem has been attacked by several authors like Siersma [40, 41], Nemethi-
Szilard [33], Michel-Pichon [29, 30, 31], Bobadilla-Menegon [16], Menegon-
Seade [28] and Aguilar-Menegon-Seade [1].

The corresponding understanding for real analytic singularities is still very
poor. Although one can define a Milnor fibration for many classes of real
analytic germs of mapping $f:(\mathbb{R}^{M},0)\to(\mathbb{R}^{K},0)$, not
much is known about the topology of the corresponding Milnor fiber or the link
of $f$ (see [34, 20, 28, 25, 21] for some results), and even less about the
boundary of such objects.

The first part of this paper aims to introduce a new perspective to deal with
such problem, inspired mainly by [14, 3, 27]. The idea is to relate the
topology of the boundary of the Milnor fiber of $f$, denoted by $\partial
F_{f}$, with the boundary of the Milnor fiber of the composition $f_{I}$ of
$f$ with some projection $(\mathbb{R}^{K},0)\to(\mathbb{R}^{I},0)$, which we
denote by $\partial F_{I}$. As a result, in Section 3 we prove that for
$K-I\geq 2$ there is a generalized open-book decomposition

$\dfrac{f_{K-I}}{\|f_{K-I}\|}:\partial F_{I}\setminus\partial F_{f}\to
S^{K-I-1}\,,$

where $f_{K-I}$ is the composition of $f$ with the projection
$(\mathbb{R}^{K},0)\to(\mathbb{R}^{K-I},0)$. The particular case $0\leq
K-I\leq 1$ and the case of a complex ICIS
$({\mathbb{C}}^{M+K},0)\to({\mathbb{C}}^{K},0)$ are analyzed.

On the other hand, the understanding of the topology of the boundary of the
Milnor fiber of the function-germ $f_{I}$ also provides a tool to better
understanding the topology of the Milnor fiber of the map-germ $f$ itself. In
fact, in Section 4 we use the aforementioned open-book decomposition to obtain
some formulae relating the Euler characteristics of $\partial F_{I}$, $F_{f}$
and the link $\mathcal{L}_{I}$ of $f_{I}$, for $I=1,\dots,K$.

Finally, in the last section of the article we use those Euler characteristic
formulae to get a hint on the possible topological behaviour of real analytic
map-germs on an odd number of variables and how similar or different it can be
when compared with the complex setting.

##  2\. Notations and basics definitions

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),f=(f_{1},\ldots,f_{K})$ be
an analytic map germ and consider the following diagram

(1)
${({\mathbb{R}}^{M},0)}$${({\mathbb{R}}^{K},0)}$${({\mathbb{R}}^{I},0)}$${({\mathbb{R}}^{K-I},0)}$$\scriptstyle{f}$$\scriptstyle{f_{K-I}}$$\scriptstyle{f_{I}}$$\scriptstyle{\Pi_{K-I}}$$\scriptstyle{\Pi_{I}}$

where the projections $\Pi_{I}(y_{1},\ldots,y_{K})=(y_{1},\ldots,y_{I})$ and
$\Pi_{K-I}(y_{1},\ldots,y_{K}):=(y_{I+1},\ldots,y_{K}),$ $f_{I}=\Pi_{I}\circ
f$ and $f_{K-I}=\Pi_{K-I}\circ f.$

Basic notations and definitions:

The zero locus of $f$ is defined and denoted by $V(f):=\\{f=0\\},$
respectively, $V(f_{I})=\\{f_{I}=0\\}$ and $V(f_{K-I})=\\{f_{K-I}=0\\}.$
Hence,

$V(f_{I})\supseteq V(f)\subseteq V(f_{K-I}).$

The singular set of $f$, denoted by ${\rm{Sing\hskip 2.0pt}}f$, is defined to
be the set of points $x\in({\mathbb{R}}^{M},0)$ such that the rank of the
Jacobian matrix $df(x)$ is lower than $K$. Analogously, we define the singular
sets ${\rm{Sing\hskip 2.0pt}}f_{I}$ and ${\rm{Sing\hskip 2.0pt}}f_{K-I}$ of
$F_{I}$ and $F_{K-I}$, respectively. The discriminant set of $f$ is then
defined by

${\rm{Disc\hskip 2.0pt}}f:=f({\rm{Sing\hskip 2.0pt}}f)\,.$

The polar set of $f$ relative to $g(x):=\|x\|^{2}$ is defined and denoted by
${\rm{Sing\hskip 2.0pt}}(f,g)$. Analogously, we define ${\rm{Sing\hskip
2.0pt}}(f_{I},g)$ and ${\rm{Sing\hskip 2.0pt}}(f_{K-I},g)$.

The next diagram relates the singular and the polar sets:

(2) ${{{\rm{Sing\hskip 2.0pt}}(f_{I})}}$${{{\rm{Sing\hskip
2.0pt}}(f_{I},g)}}$${{{\rm{Sing\hskip
2.0pt}}\left(\dfrac{f_{I}}{\|f_{I}\|},g\right)}}$${{{\rm{Sing\hskip
2.0pt}}(f)}}$${{{\rm{Sing\hskip 2.0pt}}(f,g)}}$${{{\rm{Sing\hskip
2.0pt}}\left(\dfrac{f}{\|f\|},g\right)}}$${{{\rm{Sing\hskip
2.0pt}}(f_{k-I})}}$${{{\rm{Sing\hskip 2.0pt}}(f_{k-I},g)}}$${{{\rm{Sing\hskip
2.0pt}}\left(\dfrac{f_{k-I}}{\|f_{k-I}\|},g\right)}}$

######  Definition 2.1.

We say that a map germ
$f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),f=(f_{1},\ldots,f_{K})$ is
tame, or satisfies the transversality condition at the origin if

$\overline{{\rm{Sing\hskip 2.0pt}}(f,g)\setminus V(f)}\cap{\rm{Sing\hskip
2.0pt}}f\subseteq\\{0\\}$

as a germ of set at the origin.

######  Lemma 2.2.

Let $1\leq I\leq K-1.$ If $f$ is tame, then $f_{I}$ and $f_{K-I}$ are tame as
well.

It is well known that the tameness conditions for $f,$ $f_{I}$ and $f_{K-I}$
induce the following fibrations on the boundary of the closed ball
$S_{\epsilon}^{M-1}:=\partial B_{\epsilon}^{M}$:

(3) $f_{|}:S_{\epsilon}^{M-1}\cap f^{-1}(B_{\eta_{1}}^{K}\setminus\\{0\\})\to
B_{\eta_{1}}^{K}\setminus\\{0\\}$

(4) ${f_{I}}_{|}:S_{\epsilon}^{M-1}\cap
f^{-1}_{I}(B_{\eta_{2}}^{I}\setminus\\{0\\})\to
B_{\eta_{2}}^{I}\setminus\\{0\\}$

(5) ${f_{K-I}}_{|}:S_{\epsilon}^{M-1}\cap
f^{-1}_{K-I}(B_{\eta_{3}}^{K-I}\setminus\\{0\\})\to
B_{\eta_{3}}^{K-I}\setminus\\{0\\}$

Moreover, under the extra conditions ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}$ there
also exists the Milnor tube’s fibration is the following sense: there exists
$\epsilon_{0}>0$ small enough such that for all $0<\epsilon\leq\epsilon_{0}$
there exists $0<\eta_{1}\ll\epsilon$ such that the restriction map

(6) $f_{|}:B_{\epsilon}^{M}\cap f^{-1}(B_{\eta_{1}}^{K}\setminus\\{0\\})\to
B_{\eta_{1}}^{K}\setminus\\{0\\}$

is a locally trivial smooth fibration, where $B_{\epsilon}^{M},$ respectively
$B_{\epsilon}^{K},$ stand for the closed ball in ${\mathbb{R}}^{M}$ with
radius $\epsilon,$ centered at origin, respectively in ${\mathbb{R}}^{K}$ with
radius $\eta.$

Hence, for the same reason, we conclude the existence of the Milnor tube
fibrations for $f_{I}$ and $f_{K-I}:$

(7) ${f_{I}}_{|}:B_{\epsilon}^{M}\cap
f^{-1}_{I}(B_{\eta_{2}}^{I}\setminus\\{0\\})\to
B_{\eta_{2}}^{I}\setminus\\{0\\}$

(8) ${f_{K-I}}_{|}:B_{\epsilon}^{M}\cap
f^{-1}_{K-I}(B_{\eta_{3}}^{K-I}\setminus\\{0\\})\to
B_{\eta_{3}}^{K-I}\setminus\\{0\\}$

From now on denote by $F_{f}$, $F_{I}$ and $F_{K-I}$ the Milnor fibers of the
fibrations (6), (7) and (8), respectively, by $\partial F_{f}$, $\partial
F_{I}$ and $\partial F_{K-I}$ the fibers of (3), (4) and (5).

Consider the Milnor tube fibration ${f_{I}}_{|}:B_{\epsilon}^{M}\cap
f^{-1}_{I}(B_{\eta_{2}}^{I}\setminus\\{0\\})\to
B_{\eta_{2}}^{I}\setminus\\{0\\}$ and $z\in B_{\eta_{2}}^{I}\setminus\\{0\\}.$
Thus the fiber $F_{I}=f^{-1}(\Pi_{I}^{-1}(z)).$

Denote by $D^{K-I}=\Pi^{-1}_{I}(z)\cap(B_{\eta_{1}}^{K}\setminus\\{0\\})$ the
$(K-I)-$dimensional closed disc given by the intersection of the fiber of
projection $\Pi_{I}:{\mathbb{R}}^{K}\to{\mathbb{R}}^{I}$ with the closed ball
$B_{\eta_{1}}^{K}\setminus\\{0\\}$ on the target space of fibration (6).

Thus $F_{I}=f^{-1}(D^{K-I})$ and one may consider the restriction map
$f:F_{I}\to D^{K-I}$ which is a smooth surjective proper submersion, and hence
a smooth trivial fibration.

Therefore, the following homeomorphism follows $F_{I}\approx F_{f}\times
D^{K-I},$ which is a diffeomorphism after smoothing the corners. On the
boundary of the Milnor fiber the following diffeomorphism holds true:

(9) $\partial F_{I}\approx(\partial F_{f}\times D^{K-I})\cup(F_{f}\times
S^{K-I-1}).$

We remark that the next Proposition is in the same vein as [12, Corollary 4].

######  Proposition 2.3.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ be a tame
map germ with ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}$ and for $1\leq I<K$ consider
the composition map $f_{I}=\Pi_{I}\circ f$ where
$\Pi_{I}:{\mathbb{R}}^{K}\to{\mathbb{R}}^{I}$ is the projection map. Then the
boundary $\partial F_{I}$ of the Milnor fiber $F_{I}$ is obtained (up to
homeomorphism) by the gluing together two disjoint copies of the Milnor fiber
$F_{I+1}$ along the common boundary $\partial F_{I+1}.$

###### Proof.

The proof follows from the composition

${({\mathbb{R}}^{M},0)}$${({\mathbb{R}}^{I+1},0)}$${({\mathbb{R}}^{I},0)}$$\scriptstyle{f_{I+1}}$$\scriptstyle{f_{I}=\widehat{\Pi}_{I}\circ
f_{I+1}}$$\scriptstyle{\widehat{\Pi}_{I}}$

where $\widehat{\Pi}_{I}(y_{1},\ldots,y_{I+1})=(y_{1},\ldots,y_{I})$ and the
fact that $f$ being tame implies the same to $f_{I+1}$ and $f_{I}.$ In the
same manner ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}$ implies ${\rm{Disc\hskip
2.0pt}}f_{I+1}=\\{0\\}$, ${\rm{Disc\hskip 2.0pt}}f_{I}=\\{0\\}.$

Hence, $F_{I}\approx F_{I+1}\times[-1,1]$ and the boundary

(10) $\partial F_{I}\approx(\partial
F_{I+1}\times[-1,1])\cup(F_{I+1}\times\\{-1,1\\})$

Figure 1.

Now it is easy to see that the closed manifold $\partial F_{I}$ is obtained by
the gluing the two disjoint copies of $F_{I+1}$ given by
$F_{I+1}\times\\{-1\\}\cup F_{I+1}\times\\{1\\}$ along the boundaries of the
cylinder $\partial F_{I+1}\times[-1,1].$ See the Figure (1). Therefore the
result follows. ∎

##  3\. Fibration structure on the boundary of the Milnor fiber

From now on we will consider $f$ tame, ${\rm{Disc\hskip 2.0pt}}f=\\{0\\},$ and
$V(f)\neq\\{0\\}.$ One may adjust the radii $\eta_{1},$ $\eta_{2}$ and
$\eta_{3}$ in the fibrations (3), (4) and (5) such that $\partial F_{I}\subset
S_{\epsilon}^{M-1}\cap f^{-1}(B_{\eta_{1}}^{K}\setminus\\{0\\})$ and the
restriction map $f_{K-I}:\partial F_{I}\to{\mathbb{R}}^{K-I}$ is well defined,
for any $1\leq I<K.$

######  Lemma 3.1.

The restriction map $f_{K-I}:\partial F_{I}\to{\mathbb{R}}^{K-I}$ is a smooth
submersion.

###### Proof.

For $y\in B_{\eta_{2}}^{I}\setminus\\{0\\}$ consider $\partial
F_{I}=S_{\epsilon}^{M-1}\cap f_{I}^{-1}(y)$ and the matrix

$A(x):=\left[\begin{array}[]{c}{\rm{d}}f_{I}(x)\\\ {\rm{d}}f_{K-I}(x)\\\
{\rm{d}}g(x)\end{array}\right].$

By the tameness of $f$ we have that for all $x\in\partial F_{I}$ the rank of
$A(x)$ is maximal. Hence $f_{K-I}$ is a smooth submersion. ∎

Now since $0\in{\mathbb{R}}^{K-I}$ is a regular value of $f_{K-I}:\partial
F_{I}\to{\mathbb{R}}^{K-I}$ by the compactness of $\partial F_{I}$ one may
choose $\tau>0$ small enough and a closed disc
$D^{K-I}_{\tau}\subset{\mathbb{R}}^{K-I}$ centered at the origin
$0\in{\mathbb{R}}^{K-I},$ such that all $y\in D^{K-I}_{\tau}$ is a regular
value of the restriction map $f_{K-I}.$

Hence the restriction map $f_{K-I}:\partial F_{I}\cap
f^{-1}_{K-I}(D^{K-I}_{\tau})\to D^{K-I}_{\tau}$ is a smooth onto submersion,
then a trivial fibration with the fiber diffeomorphic to $\partial
F_{f}=\partial F_{I}\cap f_{K-I}^{-1}(0).$ Therefore,

(11) $\partial F_{I}\cap f^{-1}_{K-I}(D^{K-I}_{\tau})\approx\partial
F_{f}\times D_{\tau}^{K-I}.$

Denote by $T_{\tau}(\partial F_{f}):=\partial F_{f}\times D_{\tau}^{K-I}$ the
closed tubular neighbourhood of the embedded submanifold $\partial
F_{f}\longhookrightarrow\partial F_{I}.$ See the Figure (2).

Figure 2.

Then, by (9) it follows that the complement

(12) $\partial F_{I}\setminus int(T_{\tau}(\partial F_{f}))\approx F_{f}\times
S^{K-I-1}$

In this section we will prove that the embedded submanifold $\partial
F_{f}\longhookrightarrow\partial F_{I}$ yields on the boundary $\partial
F_{I}$ an interesting structure. For that, let
$f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),M>K\geq 2$ and take $1\leq
I\leq K-2$ as in the beginning of section 2.

We are now ready to introduce the main result of this section. Before that, we
introduce an appropriate definition which fits with the type of fibration
structure we are able to prove on the boundary of the Milnor fibration.

Following H. Winkelnkemper in [42], A. Ranicki [39]111including the appendix
”The history and applications of open books”, by H. E. Winkelnkemper, E.
Looijenga [18], see also [15] and [4, section 3], given $M$ a smooth manifold
$M$ and $N\subset M$ a submanifold of codimension $k\geq 2$ in $M,$ suppose
that for some trivialization $t:T(N)\to N\times B^{k}$ of a tubular
neighbourhood $T(N)$ of $N$ in $M,$ the fiber bundle defined by the
composition $\pi\circ t$ in the diagram below

${T(N)\setminus
N}$${N\times(B^{k}\setminus\\{0\\})}$${S^{k-1}}$$\scriptstyle{t}$$\scriptstyle{\pi\circ
t}$$\scriptstyle{\pi}$

where $\pi(x,y):=\dfrac{y}{\|y\|},$ extends to a smooth locally trivial fiber
bundle $p:M\setminus N\to S^{k-1};$ e.i., $p_{|_{T(N)\setminus N}}=\pi\circ
t.$

In such a case the pair $(M,N)$ above will be called a generalized
$(k-1)$-open-book decomposition on $M$ with binding $N$ and page the fiber
$p^{-1}(y),y\in S^{k-1}.$

The main result of this section is:

######  Theorem 3.2.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ be a tame
map germ with ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}.$ Then, for each $1\leq I<K$
such that $K-I\geq 2,$ the pair $(\partial F_{I},\partial F_{f})$ is a
generalized $(K-I-1)$-open-book decomposition, with binding $\partial F_{f}$
and page $F_{f}\setminus\partial F_{f}$\- the interior of the Milnor fiber
$F_{f}.$

###### Proof.

Consider the restriction map $f_{K-I}:\partial F_{I}\to{\mathbb{R}}^{K-I}.$ It
follows by Lemma 3.1 that we may adjust the radii of the fibrations (3), (4)
and (5) such that for a small enough radius $\tau$ in the diagram below

(13) ${\partial F_{I}\cap
f^{-1}_{K-I}(D^{K-I}_{\tau}-\\{0\\})}$${D^{K-I}_{\tau}-\\{0\\}}$${S^{K-I-1}}$$\scriptstyle{f_{K-I}}$$\scriptstyle{\dfrac{f_{K-I}}{\|f_{K-I}\|}}$$\scriptstyle{\Pi_{R}(z)=\frac{z}{\|z\|}}$

the projection $\dfrac{f_{K-I}}{\|f_{K-I}\|}$ is a (trivial) locally fiber
bundle, where $\Pi_{R}$ is the radial projection.

It induces the trivial fibration on the diagonal projection

(14) ${\partial F_{I}\cap
f^{-1}_{K-I}(S^{K-I-1}_{\tau})}$${S^{K-I-1}_{\tau}}$${S^{K-I-1}}$$\scriptstyle{f_{K-I}}$$\scriptstyle{\dfrac{f_{K-I}}{\|f_{K-I}\|}}$$\scriptstyle{\Pi_{R}(z)=\frac{z}{\|z\|}}$

Applying again the Lemma 3.1 in the diagram below we get that the horizontal
map is a locally trivial fibration over its image; and thus, the diagonal
projection is again a locally trivial smooth fibration.

(15) ${\partial F_{I}\setminus
f^{-1}_{K-I}(int(D^{K-I}_{\tau}))}$${{\mathbb{R}}^{K-I}\setminus\\{0\\}}$${S^{K-I-1}}$$\scriptstyle{f_{K-I}}$$\scriptstyle{\dfrac{f_{K-I}}{\|f_{K-I}\|}}$$\scriptstyle{\Pi_{R}(z)=\frac{z}{\|z\|}}$

Now we may glue the fibrations (13) and (15) along the fibration (14) to get a
smooth projection of a locally trivial fiber bundle

(16) $\dfrac{f_{K-I}}{\|f_{K-I}\|}:\partial F_{I}\setminus\partial F_{f}\to
S^{K-I-1}.$

Now see that the diffeomorphism of (11) says that the trivialization in the
horizontal map of the diagram (9) is given by

$\partial F_{I}\cap f^{-1}_{K-I}(D^{K-I}_{\tau}-\\{0\\})\approx\partial
F_{f}\times(S_{\tau}^{K-I-1}\times(0,\tau]).$

Hence the fiber of the diagonal fibration in the diagram (13) should be
diffeomorphiuc to $\partial F_{f}\times(0,\tau].$

Figure 3.

On the other hand, the diffeomorphism (12) assures that the fiber in the
diagonal projection of the diagram (15) should be diffeomorphic to $F_{f}.$
The fiber of the boundary trivial fibration (14) is clearly diffeomorphic to
$\partial F_{f}.$ Therefore, we conclude that the fiber of fibration (16) must
be diffeomorphic to the gluing (using the identity diffeomorphism on the
boundary) $F_{f}\cup_{\partial F_{f}}(\partial
F_{f}\times(0,\tau])=F_{f}\setminus\partial F_{f}.$ See Figure (3) and the
proof is finished for $K-I-1\geq 1,$ i.e., $K-I\geq 2.$ ∎

######  Remark 3.3.

Notice that for $K-I=2$, a generalized open-book decomposition is an open-book
in the usual sense. We also remark that in Theorem 3.2 we assumed $K-I\geq 2;$
If we consider the case $K=I$ then by convention $f_{K-I}:=f_{0}\equiv 0$ and
${\mathbb{R}}^{0}=\\{0\\},$ so there is nothing to be said. For $I=K-1,$ then
the study of the restriction function $f_{K-I}:\partial F_{I}\to\mathbb{R}$
reduces to that of Proposition 2.3 and the construction above leads to a
"fibration" over $S^{0}=\\{-1,1\\}.$

###  3.1. The case of ICIS holomorphic map germ

Let us consider now a holomorphic map germ
$f=(f_{1},\ldots,f_{K}):({\mathbb{C}}^{M+K},0)\to({\mathbb{C}}^{K},0),K\geq
2.$ For $1\leq I<K$ consider the complex projections
$\Pi_{I}:({\mathbb{C}}^{K},0)\to({\mathbb{C}}^{I},0),\Pi_{I}(z_{1},\ldots,z_{K})=(z_{1},\ldots,z_{I}),$
and
$\Pi_{K-I}:({\mathbb{C}}^{K},0)\to({\mathbb{C}}^{K-I},0),\Pi_{K-I}(z_{1},\ldots,z_{K})=(z_{I+1},\ldots,z_{K}).$
Thus, the compositions as in the diagram (1) becomes $f_{I}:=\Pi_{I}\circ
f=(f_{1},\ldots,f_{I})$ and $f_{K-I}:=\Pi_{K-I}\circ
f=(f_{I+1},\ldots,f_{K}).$

If we assume further that $f$ is ICIS, it is known that ${\rm{Disc\hskip
2.0pt}}f:=f({\rm{Sing\hskip 2.0pt}}f)$ is a complex hypersurface in
${\mathbb{C}}^{K},0$ and then for all $\delta>0$ small enough the space
$B_{\delta}^{2K}\setminus{\rm{Disc\hskip 2.0pt}}f,$ where $B_{\delta}^{2K}$
stand for the open ball in ${\mathbb{C}}^{K}\equiv{\mathbb{R}}^{2K}.$ Then,
the space $B_{\delta}^{2K}\setminus{\rm{Disc\hskip 2.0pt}}f$ is a connected
space.

In fact, it was proved by H. Hamm, Lê D. Tràng and by E. Looijenga in [17]
that there exists $\epsilon_{0}>0$ small enough such that for each
$0<\epsilon\leq\epsilon_{0}$ there exists $0<\delta\ll\epsilon$ such that the
projection map

$f:\overline{B}_{\epsilon}^{2M+2K}\setminus f^{-1}({\rm{Disc\hskip
2.0pt}}f)\to B_{\delta}^{2K}\setminus{\rm{Disc\hskip 2.0pt}}f$

is a smooth locally trivial fibration. Thus, by the connectedness property of
the base space the Milnor fiber $F_{f}$ is uniquely defined, up to
diffeomorphism.

In addition to that, the ICIS condition is equivalent to the condition
${\rm{Sing\hskip 2.0pt}}f\cap V_{f}=\\{0\\}.$ Hence, $f$ is tame according to
the Definition 2.1. We may also use the argument of Looijenga in [17] for a
"good representative" to guarantee that, up to a linear coordinate change in
${\mathbb{C}}^{K},$ it follows that either map germ $f_{I}$ and $f_{K-I}$ are
ICIS as well.

The following result is an interesting application of our Theorem 3.2 and also
it provides an extension of [7, Proposition 3.2, p. 481]. See also [33,
Chapter 3] and compare with [7, Proposition 3.2, p. 481].

######  Proposition 3.4.

Let $f:(\mathbb{C}^{M+K},0)\to({\mathbb{C}}^{K},0),K>I\geq 1,$ be a germ of
ICIS such that $f_{I}$ and $f_{K-I}$ are ICIS as well. Then, the Milnor
projection $\dfrac{f_{K-I}}{\|f_{K-I}\|}:\partial F_{I}\setminus\partial
F_{f}\to S^{2K-2I-1}$ induces a generalized $(2K-2I-1)$-open-book
decomposition on the boundary $\partial F_{I}$ with binding $\partial F_{f}.$

###### Proof.

Since $K>I$ then $K-I\geq 1$ and the dimension of the sphere on the target
space is $2(K-I)-1\geq 1.$ Hence, the same ideas in the proof of Theorem 3.2
work in this case. ∎

######  Remark 3.5.

  1. (1)

We recall that if one has an isolated complex hypersurface singularity germ,
then its link has a canonical contact structure which is Stein fillable (see
for instance [35]). These statements extend naturally to the setting we
envisage in Proposition 3.4.

  2. (2)

Still the complex ICIS above, the generalized open-book decomposition also
extends with the same proof for the pair links
$(\mathcal{L}_{I},\mathcal{L}_{f}),$ where $\mathcal{L}_{I}:=f_{I}^{-1}(0)\cap
S_{\epsilon}^{M-1}$ and $\mathcal{L}_{f}:=f^{-1}(0)\cap S_{\epsilon}^{M-1},$
for all $\epsilon>0$ smal enough.

##  4\. The Euler characteristic formulae

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ be an
analytic map-germ. We will assume along the section that $f$ is tame and
${\rm{Disc\hskip 2.0pt}}f=\\{0\\}.$ Thus, $\dim F_{f}=M-K$ and $\dim\partial
F_{f}=M-K-1.$

Denote by $\widehat{F_{f}}=F_{f}\cup_{\partial F_{f}}F_{f}$ the closed
manifold built by gluing two copies of $F_{f}$ along the boundary $\partial
F_{f}$ using the identity diffeomorphism on $\partial F_{f}.$ By the additive
property of the Euler characteristic we have that
$\chi(\widehat{F_{f}})=2\chi(F_{f})-\chi(\partial F_{f}).$ Hence

(17) $\chi(\partial F_{f})=\begin{cases}0,&\quad\text{if M-K is even}.\\\
2\chi(F_{f}),&\quad\text{if M-K is odd}.\end{cases}$

Applying again the additive Euler characteristic to the diffeomorphism (9) we
get

$\chi(\partial F_{I})=\chi(\partial F_{f}\times D^{K-I})+\chi(F_{f}\times
S^{K-I-1})-\chi(\partial F_{f}\times S^{K-I-1})=$

$=\chi(\partial F_{f})+\chi(F_{f}).\chi(S^{K-I-1})-\chi(\partial
F_{f}).\chi(S^{K-I-1}).$

Thus, together with (17) it reduces to

(18) $\chi(\partial F_{I})=\begin{cases}\chi(F_{f}).\chi(S^{K-I-1}),&\text{if
M-K is even}.\\\ \chi(F_{f}).\chi(S^{K-I}),&\text{if M-K is odd}.\end{cases}$

We may consider the convention $\chi(S^{-1})=\chi(\emptyset)=0$ and the fact
that for the $0-$dimensional sphere $\chi(S^{0})=\chi(\\{-1,1\\})=2.$ Then all
the above discussion, including the special cases of $I=1$ and $I=K$ may be
summarized as below. By convention, for $I=1$ we denote $F_{1}:=F_{f}.$

######  Theorem 4.1.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ be an
analytic map-germ, tame with ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}.$ Then:

  1. $1)$

$\chi(\partial F_{I})=\chi(F_{f}).\chi(S^{M-I-1}),$ for any $1\leq I\leq K.$

  2. $2)$

Lê-Greuel’s type formula: $\chi(\partial F_{I+1})-\chi(\partial
F_{I})=2(-1)^{M-I}\chi(F_{f}),$ for any $1\leq I<K.$

  3. $3)$

$\chi(\partial F_{I})=\chi(\partial F_{I+2}),$ for any $1\leq I<K-1.$

###### Proof.

The item $1)$ follows from the identity (18). To prove the item $2)$ just
exchange $I$ by $I+1$ in the item $1)$ and take the difference. The item $3)$
is immediate from item $1).$ ∎

###  4.1. Relating the Euler characteristic of the links

Consider again $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ a
tame polynomial map-germ with ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}.$ For each
$1<I\leq K$ the map $f_{I}:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{I},0)$ admits
the Milnor tube fibrations (4) and by restriction it induces the fibrations
$f_{I}:B^{M}_{\epsilon}\cap f_{I}^{-1}(S_{\eta}^{I-1})\to S_{\eta}^{I-1}$ with
Milnor fibers $F_{I}$ and the fibration $f_{I}:S^{M-1}_{\epsilon}\cap
f_{I}^{-1}(S_{\eta}^{I-1})\to S_{\eta}^{I-1}$ with fiber $\partial F_{I}.$

Denote by $T_{\eta}(F_{I}):=B^{M}_{\epsilon}\cap f_{I}^{-1}(S_{\eta}^{I})$ the
Milnor tube of $f_{I}$ and by $\mathcal{L}_{I}:=f_{I}^{-1}(0)\cap
S_{\epsilon}^{M-1}$ the respective link.

We may consider $\eta$ small enough such that the sphere $S_{\epsilon}^{M-1}$
is homeomorphic to the gluing $T_{\eta}(f_{I})\cup_{\partial
T_{\eta}(f_{I})}N_{\eta}(f_{I}),$ where $\mathcal{L}_{I}\subset
N_{\eta}(f_{I}):=f_{I}^{-1}(B^{I}_{\eta})$ is a semi-algebraic neighbourhood
that retract to the link $\mathcal{L}_{I},$ as proved by A. Durfee in [13].

Thus,

$\chi(S_{\epsilon}^{M-1})=\chi(T_{\eta}(f_{I}))+\chi(N_{\eta}(f_{I}))-\chi(\partial
T_{\eta}(f_{I}))=\chi(F_{I})\chi(S^{I-1})+\chi(\mathcal{L}_{I})-\chi(\partial
F_{I})\chi(S^{I-1}).$

Hence,

(19)
$\chi(\mathcal{L}_{I})=\chi(S^{M-1})-\chi(F_{f})\chi(S^{I-1})+\chi(\partial
F_{I})\chi(S^{I-1})$

######  Lemma 4.2.

The following holds true:

$\chi(\mathcal{L}_{I})=\chi(S^{M-1})+(-1)^{M-I-1}\chi(F_{f})\chi(S^{I-1}).$

###### Proof.

The proof follows from Proposition 18 and equation (19). ∎

The next result provides in particular a second proof of [14, Proposition 7.1,
p. 4861].

######  Proposition 4.3.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ be a tame
polynomial map-germ with ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}.$ Then:

  1. $1)$

$\chi(\mathcal{L}_{I+1})-\chi(\mathcal{L}_{I})=2(-1)^{M-I}\chi(F_{f}),$ for
each $1\leq I<K.$

  2. $2)$

$\chi(\mathcal{L}_{I+2})=\chi(\mathcal{L}_{I}),$ for each $1\leq I<K-1.$

###### Proof.

In the equation (19) just exchange $I$ by $I+1$ and take the difference. Then
we have

$\chi(\mathcal{L}_{I+1})-\chi(\mathcal{L}_{I})=\chi(F_{f})\chi(S^{I-1})-\chi(\partial
F_{I})\chi(S^{I-1})-\chi(F_{f})\chi(S^{I})+\chi(\partial F_{I+1})\chi(S^{I}).$

Now we may apply Proposition 4.1, item $1),$ to get

$\chi(\mathcal{L}_{I+1})-\chi(\mathcal{L}_{I})=(-1)^{M-I}\chi(F_{f})\chi(S^{I-1})+(-1)^{M-I}\chi(F_{f})\chi(S^{I})=2(-1)^{M-I}\chi(F_{f}).$

This ends the proof of item $1).$ Item $2)$ is trivial. ∎

######  Remark 4.4.

We point out that the Lê-Greuel type formula obtained in the Theorem 4.1, item
(2), is somehow similar to that obtained in [11, Theorem 1, p. 3], but with
the difference that in [11] the authors worked with the Euler number of the
Milnor fibers, instead of its boundary.

In view of the Proposition 4.1 and the Proposition 4.3, we can see that for
all $1\leq I<K$ we have $\chi(\partial F_{I+1})-\chi(\partial
F_{I})=\chi(\mathcal{L}_{I+1})-\chi(\mathcal{L}_{I}).$ Thus, $\chi(\partial
F_{I+1})-\chi(\mathcal{L}_{I+1})=\chi(\partial
F_{I})-\chi(\mathcal{L}_{I})=\cdots=\chi(\partial
F_{2})-\chi(\mathcal{L}_{2})=\chi(\partial F_{1})-\chi(\mathcal{L}_{1}).$
Hence, it suggests the following definition.

######  Definition 4.5.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ $M>K\geq 2,$ be a tame
polynomial map-germ with ${\rm{Disc\hskip 2.0pt}}f=\\{0\\}$. The degree of
degeneracy on the Milnor boundary of $f$ is defined as the number

$DB(f):=\chi(\partial F_{1})-\chi(\mathcal{L}_{1})\,.$

Clearly, if $f$ has an isolated singularity at the origin one has that
$DB(f)=0$.

##  5\. On the boundaries of the Milnor fibers and the links on each stage I.

In the real setting we do not expect to prove theorems regarding the degree of
connectivity of the Milnor fibers, its boundaries nor the respective links of
$f_{I},$ on each stage $I$. Notwithstanding, in the case where the dimension
$M$ of the source space is even, for all $I,$ $1\leq I\leq K,$ we may write
$\chi(\partial F_{I})=\chi(S^{I+1})\chi(F_{I})$ and as an application of Lemma
4.2 we conclude that $\chi(\partial F_{I})=\chi(\mathcal{L}_{I}),$ and hence
$DB(f)=0.$ However, if the source space $M$ is odd-dimensional some
interesting relations between the boundaries of the Milnor fiber, the links of
the singularities and the Milnor fibers on the Milnor tubes come up on each
stage $I$, and it provides a way to distinguish between the homotopy type of
the Milnor boundary and the link of the singularities $f_{I}$ for each $1\leq
I\leq K,$ as described below.

We first remind that for odd dimension $M\geq 2$ the equation (19) becomes

$(*):~{}\chi(\mathcal{L}_{I})=2-\chi(F_{f})\chi(S^{I-1})+\chi(\partial
F_{I})\chi(S^{I-1}).$

This allows us to prove the below result whose proof we left as an exercise.

######  Lemma 5.1.

Let $M$ odd and $I$ such that $M>K\geq I\geq 2.$ Then, for each $I$ the
following conditions hold for the links $\mathcal{L}_{I}$ and the boundaries
$\partial F_{I}$ of the Milnor fibers $F_{I}:$

  1. (1)

if $I$ is even then $\chi(\mathcal{L}_{I})=2,$ by equation $(*).$ Moreover,
since $\dim F_{I}=M-I$ is odd then $\chi(\partial
F_{I})=2\chi(F_{I})=2\chi(F_{f});$

  2. (2)

if $I$ is odd then $\chi(\mathcal{L}_{I})=2-2\chi(F_{f}),$ by equation $(*).$
Moreover, since $\dim\partial F_{I}=M-I-1$ is odd then $\chi(\partial
F_{I})=0.$

Now we are ready to state the main result of this section.

######  Theorem 5.2.

Consider $M$ odd, $M>K\geq I\geq 2.$ Let
$f:(\mathbb{R}^{M},0)\to(\mathbb{R}^{K},0)$ and
$f_{I}:(\mathbb{R}^{M},0)\to(\mathbb{R}^{I},0)$ be real analytic map germs as
in the diagram (1). Then, $\chi(F_{f})=1$ if and only if $\chi(\partial
F_{I})=\chi(\mathcal{L}_{I})$ in some stage $I.$ Moreover, if the last
equality holds true on any stage $I$ it also will holds true on all stages
$I,$ $2\leq I\leq K<M.$

###### Proof.

The proof follows from Lemma 5.1.

For the "if" case, we can see that $\chi(F_{I})=\chi(F_{f})=1$ implies that
the two quantities $\chi(\partial F_{I})=\chi(\mathcal{L}_{I})$ in the either
cases of Lemma 5.1. Moreover, the equality in some stage $I$ clearly implies
that on all $I,$ $2\leq I\leq K<M.$

For the "only if" case, if we suppose that in some stage (even or odd) $I$ the
equality $\chi(\partial F_{I})=\chi(\mathcal{L}_{I})$ holds true, then again
by Lemma 5.1 we conclude that $2\chi(F_{I})=2.$ Therefore, $\chi(F_{f})=1.$ ∎

The next result provides a natural class of map germs where one of two
conditions above holds true. Beside that, it also provides another proof of
[3, Proposition 3, item ii), p. 71].

######  Corollary 5.3.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0)$, with $M>K\geq 2$ and $M$
odd, be a real analytic map-germ with an isolated critical point at the
origin. Then, for each $I,$ $M>K\geq I\geq 1,$ we have that $\chi(F_{I})=1.$

###### Proof.

The proof might be left as exercise, but we will screatch it below for the
sake of convenience.

Since $f$ have an isolated singular point at the origin, one may apply the
diagram (2) to get that ${\rm{Sing\hskip 2.0pt}}(f_{I})\subseteq\\{0\\}$ for
each fixed $I.$ It is enough to consider the case ${\rm{Sing\hskip
2.0pt}}(f_{I})=\\{0\\},$ because the case ${\rm{Sing\hskip
2.0pt}}(f_{I})=\emptyset$ the result follows as an easy application of the
Inverse Function Theorem version for map germs.

Now, if we assume further that $M$ is odd then for each $I$ the link
$\mathcal{L}_{I}$ must be not empty, and it is in fact a smooth manifold
diffeomorphic to $\partial F_{I}$ and thus $\chi(\partial
F_{I})=\chi(\mathcal{L}_{I}).$ Therefore, one may apply the Theorem 5.2 and
conclude that $\chi(F_{I})=1,$ for each $1\leq I\leq K<M.$ ∎

######  Remark 5.4.

For the existence of map germ $({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0),$ M
odd, $M>K\geq 2,$ with isolated critical point at the origin, the reader may
consult [4, section 5.2, p. 101].

######  Corollary 5.5.

Let $M>K\geq I\geq 1$ and $f$ be as in Theorem 5.2. If $\chi(F_{f})\neq 1$
then at all stages $I,$ the Milnor boundary $\partial F_{I}$ and the
respective link $\mathcal{L}_{I}$ of $f_{I}$ cannot be homotopically
equivalent.

###### Proof.

It is now trivial because $\chi(F_{I})=\chi(F_{f})\neq 1,$ on each stage $I.$
Therefore, by Theorem 5.2 the respectives Milnor boundary $\partial F_{I}$ and
the link $\mathcal{L}_{I}$ can not be homotopically equivalent. ∎

The next example shows that for odd-dimension $M$ it is easy to construct a
family of map germ where the Euler characteristic of the Milnor fiber is not
equal to one.

######  Example 5.6.

Let $f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0)$ be an analytic map germ
$M>K\geq 2,$ with ${\rm{Sing\hskip 2.0pt}}f=\\{0\\},$ and
$g:({\mathbb{R}}^{K},0)\to({\mathbb{R}}^{K},0)$ be an analytic ramified
covering map branched along $\\{0\\}$ with t-sheets, $t\geq 2.$ Then, for all
fixed $z\in({\mathbb{R}}^{K},0),$ $0<\|z\|\ll 1,$ and all $x\in g^{-1}(z)$ the
map $g$ is a local diffeomorphism and the fiber $g^{-1}(z)$ consists of a
finite number of points, and we set $t:=\\#g^{-1}(z).$ Thus, ${\rm{Sing\hskip
2.0pt}}g=\\{0\\}$ and the composition map germ $h=g\circ
f:({\mathbb{R}}^{M},0)\to({\mathbb{R}}^{K},0)$ safisfies that ${\rm{Sing\hskip
2.0pt}}h=f^{-1}(0)\subseteq V_{h}.$ Since $f$ is tame, it is not hard to see
that $h$ is tame as well. Then the map $h$ admits a Milnor tube fibration with
Milnor fiber $F_{h}=\sqcup_{i=1}^{t}F_{f}$ (t-disjoint copies of $F_{f}$).
Therefore we have that $\chi(F_{h})=t.\chi(F_{f})=t\geq 2,$ where we use that
$\chi(F_{f})=1$ by Corollary 5.3.

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