# Cohomological connectivity of perturbations of map-germs

Yongqiang Liu, Guillermo Peñafort Sanchis, Matthias Zach  Yongqiang Liu: The
Institute of Geometry and Physics, University of Science and Technology of
China, 96 Jinzhai Road, Hefei 230026 P.R. China<EMAIL_ADDRESS>Guillermo
Peñafort Sanchis: Mathematics Department, Universitat de València, Dr.
Moliner, 50, 46100 Burjassot - Spain<EMAIL_ADDRESS>Matthias Zach:
Institut für Algebraische Geometrie, Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover - Germany<EMAIL_ADDRESS>
###### Abstract.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$ be a finite map-germ with
$n<p$ and $Y_{\delta}$ the image of a small perturbation $f_{\delta}$. We show
that the reduced cohomology of $Y_{\delta}$ is concentrated in a range of
degrees determined by the dimension of the instability locus of $f$. In the
case $n\geq p$ we obtain an analogous result, replacing finiteness by
$\mathcal{K}$-finiteness and $Y_{\delta}$ by the discriminant
$\Delta(f_{\delta})$. We also study the monodromy associated to the
perturbation $f_{\delta}$.

###### 2010 Mathematics Subject Classification:

58K15, 58K60, 32S30, 57R45

The first author is partially supported by the starting grant KY2340000123
from University of Science and Technology of China, National Natural Science
Funds of China (Grant No. 12001511) and the project “Analysis and Geometry on
Bundles" of Ministry of Science and Technology of the People‘s Republic of
China.
The second author is partially supported by the ERCEA 615655 NMST Consolidator
Grant and by the Basque Government through the BERC 2014-2017 program, the
Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa
excellence accreditation SEV-2013-0323 and by Programa de Becas Posdoctorales
en la UNAM, DGAPA, Instituto de Matemáticas, UNAM

##  1\. Introduction

In this paper we establish bounds for the vanishing cohomology for images and
discriminants of map-germs with non-isolated instabilities. As will become
clear, these results are parallel to the classical bounds of Kato and
Matsumoto on the cohomology of the Milnor fiber of non-isolated hypersurface
singularities [KM75].

It is also our intention to illustrate that perverse sheaves are a powerful
tool in the study of singular mappings. For the reader who is unfamiliar with
the machinery of perverse sheaves, [Dim04] provides a quick introduction. Here
we have included only the information that is necessary for our applications,
and have avoided technical definitions. For those who are already well-versed
in the topic, some non-trivially perverse sheaves related to the alternating
cohomology of multiple points are introduced in Section 5. The perversity of
these sheaves was discovered by Houston [Hou00] .

In this paper, all homology and cohomology groups are assumed to have complex
coefficients, unless otherwise specified.

###  1.1. The Milnor fibration

Consider a germ of a non-constant holomorphic function
$g\colon(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ and a representative of $g$
thereof, defined on some open neighborhood of a closed ball $B_{r}$ of
sufficiently small radius $r$ centered at the origin. Consider a punctured
open disc $D^{*}$, centered at the origin in $\mathbb{C}$ and of radius
$\delta>0$ with $\delta$ sufficiently small with respect to $r$. This gives
rise to a locally trivial fibration

$g\colon B_{r}\cap g^{-1}(D^{*})\to D^{*},$

called the _Milnor fibration_ , whose generic fibre

$M=M_{g}(0)=g^{-1}(\delta)\cap B_{r}.$

is known as the _Milnor fibre_ of $g$ at $0\in\mathbb{C}^{n+1}$. Milnor showed
[Mil68] that if $g^{-1}(0)$ has an isolated singularity at $0$, then $M$ is
homotopy equivalent to a bouquet of $n$-dimensional spheres. In particular,
the reduced cohomology of $M$—that is the vanishing cohomology of $g$ at
$0$—is concentrated in the middle degree. Later, Kato and Matsumoto [KM75]
established their results for functions with non-isolated singularities: If
the critical locus $\Sigma(g)$ has dimension $d$, then the Milnor fiber $M$ is
at least $(n-d-1)$-connected. Consequently, the reduced cohomology
$\tilde{H}^{p}(M)$ is concentrated in the range of degrees $n-d\leq p\leq n$.
This concentration of reduced cohomology is what we refer to as _cohomological
connectivity_.

Besides the cohomological version of Kato’s and Matsumoto’s connectivity
result, we wish to also study the monodromy transformations. Since our setting
will be slightly different from that of the Milnor fibration, we briefly
review the general notion of monodromy for a topologically locally trivial
fibration with fiber $F$

$E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}D^{*}$

over a punctured disk $D^{*}$. Let $\exp\colon S\to D^{*}$ be the universal
cover of $D^{*}$ by an infinite strip $S$. Since $S$ is simply connected, we
may choose a global trivialization of the the fiber product

$E^{\prime}:=E\times_{D^{*}}S\cong F\times S$

and deliberately identify the fiber $F_{\delta}=\pi^{-1}(\delta)$ over a point
$\delta\in D^{*}$ with any fiber of $E^{\prime}$ over a point
$\delta^{\prime}$ mapping to $\delta\in D$. The chosen global trivialization
of $E^{\prime}$ over $S$ furnishes a notion of parallel transport of the fiber
$F$ over $D^{*}$ which is unique up to homotopy and independent of the choices
made. The monodromy operator is defined to be the homeomorphism

$h\colon F\to F$

obtained from parallel transport along a closed loop in $D^{*}$ passing once
counterclockwise around the origin. It follows that the induced maps on
cohomology

$h^{i}\colon H^{i}(F)\to H^{i}(F)$

are well defined automorphisms.

Let us recall the classical monodromy theorem in the Milnor fibration setting,
see [Dim92, Theorem 3.1.20]:

######  Theorem 1.1.

Let $h^{i}\colon H^{i}(M)\to H^{i}(M)$ be the $i$-th monodromy operator on the
Milnor fibre associated to a germ
$g\colon(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ (not necessarily with isolated
singularities). Then

  1. (1)

the eigenvalues of $h^{i}$ are roots of unity.

  2. (2)

the size of the Jordan blocks of $h^{i}$ is at most $i+1$.

It is worth noting that the first item is valid in the more general setting of
the fibration induced by any analytic map-germ
$g\colon(X,0)\to(\mathbb{C},0)$, where $X$ is an arbitrary analytic space
[Lê78].

###  1.2. The fibration associated to an unfolding

Rather than looking at germs of hypersurfaces, we will be looking at
singularities that arise from multigerms of mappings

$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0).$

Here $S$ is a _finite_ subset of $\mathbb{C}^{n}$, mapped to $0$ by $f$.
Moreover, we always assume $f$ to be $\mathcal{K}$-finite (for its definition,
see Definition 2.1).

Whenever $p>n$, our attention will be directed towards the image of $f$,
denoted by

$(Y,0)=(\operatorname{Im}f,0)\subset(\mathbb{C}^{p},0).$

For obvious reasons, the analytic space $(Y,0)$ is sometimes called a
_parametrizable singularity_.

Whenever $n\geq p$, $\mathcal{K}$-finite mappings are surjective and there is
no interest in studying the image of $f$ in this case. The attention is
directed towards the discriminant instead: the _critical locus_ of $f$ is the
germ

$\Sigma(f)=\\{x\in\mathbb{C}^{n}\mid\mathrm{d}f_{x}\text{ is not
surjective}\\}$

and the discriminant of $f$ is defined to be the image

$\Delta(f)=f(\Sigma(f)).$

Observe that the study of discriminants comprises the case of parametrizable
singularities in the sense that, for $p>n$, the differential cannot be
surjective and hence $\Sigma(f)=\mathbb{C}^{n}$ and $Y=\Delta(f)$.

Let us summarize how the classical Milnor fibration is replaced by the
fibration determined by a one-parameter family in the context of map germs: A
_one-parameter unfolding_ $F=(f_{t},t)$ of $f$ is a germ

$F\colon(\mathbb{C}^{n}\times\mathbb{C},S\times\\{0\\})\to(\mathbb{C}^{p}\times\mathbb{C},0),$

of the form $F(x,t)=(f_{t}(x),t)$ and such that $f_{0}=f$. In the case $n<p$,
the projection from the image $Y=\operatorname{Im}F$ to the parameter space
gives a fibration, the generic fiber of which is $Y_{\delta}$, the image of a
perturbation $f_{\delta}$. In the case $n\geq p$ we consider the fibration
defined on the discriminant $\Delta(F)$ whose generic fibre is
$\Delta(f_{\delta})$. These fibrations depend not only on the map germ $f$,
but also on the chosen unfolding $F$. Special attention is paid to the case
where the perturbations $f_{\delta}$ are stable. In this case, the generic
fibre $Y_{\delta}$ (or $\Delta(f_{\delta})$) plays a role closer to that of
the classical Milnor fibre. In the case $n<p$, the image $Y_{\delta}$ of a
stable perturbation is called a _disentanglement_ of $f$. For the precise
definition of these terms see Section 3.1.

Unfoldings play the role of deformations and are still local objects. To
define perturbations—the nearby objects—we need a well chosen representative.
It is customary to absorb all associated technicalities in the definition of a
good representative.

######  Definition 1.2.

Let $V\subset\mathbb{C}^{p}$ and $T\subset\mathbb{C}$ be open neighborhoods of
the origin and

$F\colon W\to V\times T$

a representative of a one parameter unfolding defined on an open subset
$W\subset\mathbb{C}^{n}\times\mathbb{C}$. We call $F$ a _good representative_
if it is a $\mathcal{K}$-finite map and moreover satisfies the following
conditions:

  1. (1)

the family $\pi\colon\Delta(F)\to T$ is a locally trivial fibration over
$T\setminus\\{0\\}$,

  2. (2)

the central fiber $\pi^{-1}(0)$ is contractible,

  3. (3)

the space $\Delta(F)$ retracts onto the central fiber.

For any fixed nonzero value $\delta\in T\setminus\\{0\\}$ in the parameter
space of a good representative the map

$f_{\delta}\colon W_{\delta}\to V$

on $W_{\delta}:=W\cap(\mathbb{C}^{n}\times\\{\delta\\})$ is called a
_perturbation_ of $f$. With no risk of confusion, we also write $f$ for the
representative $f_{0}\colon W_{0}\to V$.

######  Remark 1.3.

A good representative of a $\mathcal{K}$-finite germ can be obtained from an
arbitrary representative $F\colon U\times T\to V\times T$ as follows: The
discriminant $\Delta(F)$ in $V\times T$ is a closed complex analytic set and
the projection $\pi\colon V\times T\to T$ a holomorphic function on it. The
Milnor-Lê fibration asserts that for sufficiently small ball $B_{r}\subset V$
around the origin and a subsequently chosen disc $D_{\delta}\subset T$ with
$r\gg\delta>0$ the restriction

$\pi\colon\Delta(F)\cap(B_{r}\times D_{\delta})\to D_{\delta},$

is a map satisfying the three properties mentioned above. Now the good
representative is furnished by choosing $B_{r}\times D_{\delta}$ small enough
such that the restriction

$F|_{F^{-1}(B_{r}\times D_{\delta})}\colon F^{-1}(B_{r}\times D_{\delta})\to
B_{r}\times D_{\delta}$

is a $\mathcal{K}$-finite map, replacing $V\times T$ by $B_{r}\times
D_{\delta}$, and finally setting $W:=F^{-1}(B_{r}\times D_{\delta})$.

Parametrizable singularities and even their disentanglements are usually
highly singular. This is due to the fact that the image of a map $f$ may be
singular even if $f$ is stable, that is when $f$ does not admit any non-
trivial unfoldings; see the examples below. Thus, the singular locus of
$(Y,0)$ is not well suited to be the analogue of the critical locus of $g$
from the classical Kato-Matsumoto result. Instead, we will be considering the
_instability locus_

$\mathrm{Inst}(f)\subseteq\Delta(f)$

which is an analytic subset of the discriminant of $f$ (see Section 3.1). The
bound on the vanishing cohomology of the disentanglement will be given in
terms of the dimension $d$ of $\mathrm{Inst}(f)$. We give some examples to
illustrate the situation.

######  Example 1.4.

Consider the function

$g\colon\mathbb{C}^{3}\to\mathbb{C},\quad(x,y,z)\mapsto xy$

and the associated hypersurface $X=g^{-1}(0)$. Since $z$ is a dummy
coordinate, the Milnor fiber $M_{g}(0)$ has the homotopy type of a circle.
This is consistent with Kato and Matsumoto’s theorem, because the critical
locus of $g$, and hence also the singular locus of $X$, has dimension one, see
Figure 1.

Figure 1. The singularity $X=\\{xy=0\\}$ and its Milnor fibre
$\\{xy=\delta\\}$. The dashed green line is the singular locus of $X$.

The situation changes when we think of $(X,0)$ as a parametrized singularity
given by a bi-germ

$f\colon(\mathbb{C}^{2},\\{p,q\\})\to(\mathbb{C}^{3},0)$

with the two obvious branches, see Figure 2.

Figure 2. The same singularity as in Figure 1, now regarded as the image of a
transverse double point bi-germ.

This map germ is known as a “transverse double point”. It is stable, meaning
that it cannot be perturbed by any unfolding, up to analytic isomorphism.
Therefore, a sufficiently small representative of the image of $f$ coincides
with its disentanglement. We see that, unlike the Milnor fiber $M_{g}(0)$ of
$g$, the disentanglement is a singular space with two smooth branches crossing
transversally and it has trivial reduced cohomology.

######  Example 1.5.

We will now consider map germs $f$ with _isolated instability_ , i.e. those
for which $d=\operatorname{dim}\mathrm{Inst}(f)=0$. This property turns out to
be equivalent to $f$ being _finitely determined_ [Mat68]. The families
$S_{k},B_{k}$ and $H_{k}$ of Mond [Mon85] are examples of such finitely
determined germs $(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$. Here we consider
the germ

$(x,y)\mapsto(x^{2},y^{2},x^{3}+y^{3}+xy),$

depicted in Figure 3.

Figure 3. A germ $f$ with isolated instability and its perturbation
$f_{\delta}$. The red thick dot at the vertex of the cone represents the
instability locus $\mathrm{Inst}(f)$.

Mond showed that the disentanglement of a finitely determined germ
$(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$ has the homotopy type of a bouquet
of $n$-dimensional spheres [Mon91]. Moreover, he showed that the number of
spheres is independent of the chosen stabilization. Moreover, all finitely
determined map-germs $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$ with
$n\leq 14$ admit perturbations to stable mappings111 The pairs $(n,p)$ of
dimensions where every map-germ $(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$
admits a stable perturbation are called Mather’s nice dimensions. These
comprise all pairs $(n,n+1)$ with $n\leq 14$, cf. [MN20, Section 5.2.2]).
$f_{\delta}$.

A similar connectivity result holds for discriminants of perturbations of
finitely determined map-germs
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$, when $n\geq p$ [DM91]. The
discriminant $\Sigma(f_{\delta})$ has the homotopy type of a bouquet of
spheres of dimension $p-1$.

######  Example 1.6.

The cuspidal edge $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$, given by

$(x,y)\mapsto(x,y^{2},y^{3}),$

parametrizes the cartesian product of a usual cusp and the complex line, see
Figure 4.

It is not finitely determined, because it has a line of instabilities. Maps
with non-isolated instability locus may admit more than one disentanglement:
The cuspidal edge can, for example, be perturbed into a cuspidal node

$(x,y)\mapsto(x,y^{2},y^{3}-\delta y)$

which is stable and has the homotopy type of a circle. Another perturbation of
$f$ is

$(x,y)\mapsto(x,y^{2},y^{3}+\delta y(x^{2}-\delta)).$

The image of this last one has the homotopy type of a bouquet of two-
dimensional spheres.

Figure 4. The cuspidal edge and two different perturbations. The red dashed
line represents the instability locus $\mathrm{Inst}(f)$.

The main point of this article is to show that despite the fact that different
unfoldings may lead to different disentanglements with possibly distinct
homotopy types, there is _always_ a bound on the degrees of the nontrivial
reduced cohomology groups of _any_ given disentanglement. As the three
examples above suggest, this bound is not related to the dimension of the
singular locus of the image (equal to one for all the germs
$(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ in our examples), but it is
controlled by the dimension of the instability locus (empty, zero-dimensional
and one-dimensional, respectively in the examples).

Before stating our results, we shall introduce a construction that connects
the Milnor fibration to disentanglement fibration. Indeed, we will show that
the study of the Milnor fibration of germs of hypersurfaces in
$\mathbb{C}^{n}$ is equivalent to that of disentanglements of bi-germs of
immersions $(\mathbb{C}^{n},\\{p,q\\})\to(\mathbb{C}^{n+1},0)$.

######  Example 1.7.

To any hypersurface $X=V(g)\subseteq(\mathbb{C}^{n},0)$, not necessarily with
isolated singularity, we are going to associate a bi-germ of immersion
$(\mathbb{C}^{n},0)\sqcup(\mathbb{C}^{n},0)\to\mathbb{C}^{n+1}$. For
convenience, label the two copies of $(\mathbb{C}^{n},0)$ as $U_{1}$ and
$U_{2}$, then let $f\colon U_{1}\sqcup U_{2}\to(\mathbb{C}^{n+1},0)$ be the
map of the form

$x\mapsto\begin{cases}(x,g(x))&\textnormal{ if }x\in U_{1}\\\
(x,0)&\textnormal{ if }x\in U_{2}\end{cases}$

A different choice $g^{\prime}$ of a generator of the ideal $\langle g\rangle$
gives rise to a different map $f^{\prime}$, but there is a change of
coordinates in $(\mathbb{C}^{n+1},0)$ turning $f$ into $f^{\prime}$. In other
words, $f$ and $f^{\prime}$ are $\mathcal{A}$-equivalent and, consequently,
the study of their disentanglements is equivalent.

Conversely, every bi-germ of immersion
$(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ arises—up to
$\mathcal{A}$-equivalence—by this construction: Given such a bi-germ, we can
take a change of coordinates turning the second branch into $x\mapsto(x,0)$,
and so that the normal vector to the first branch at the origin has a nonzero
last coordinate in $\mathbb{C}^{n+1}$. This makes the first branch locally
into a graph, as desired. A more direct way to invert the process is as
follows: The two branches of a bi-germ $f$ of immersion are two germs
$f^{(1)},f^{(2)}\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$. Take a
reduced equation $L=0$ for the image of the first branch. Applying the above
contruction to the function germ $g=L\circ
f^{(2)}\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ gives rise to a bi-germ
which is $\mathcal{A}$-equivalent to the original immersion $f$.

Having explained the construction and its inverse, let us describe the
relation between the hypersurface $X$ and the immersion $f$: First and most
obvious, the intersection of the two branches is

$\operatorname{Im}(f|_{U_{1}})\cap\operatorname{Im}(f|_{U_{2}})=X\times\\{0\\}.$

Moreover, the two branches cross transversally, except on the singularities of
$X$. It is well known that the instabilities of an immersion are located
precisely at points where the branches do not intersect in general position
(this is a particular case of Theorem 3.3 in [MN20], taking into account that
monogerms of immersions are stable). In particular, it follows that

$\mathrm{Inst}(f)=\operatorname{Sing}X\times\\{0\\}.$

The Milnor fibre of $X$ has the form $M=g^{-1}(\\{\delta\\})\cap B_{r}$, for
suitable choices of $r\gg\delta>0$. The same discussion about singularities
and stabilities shows that a stabilization of $f$ is given by the bi-germ
$F=(f_{t},t)$, with

$f_{t}|_{U_{1}}=f|_{U_{1}}\quad\text{and}\quad f_{t}|_{U_{2}}=(x,t).$

A stable perturbation of $f$ is given by $f_{\delta}\colon B_{r}\sqcup
B_{r}\to\mathbb{C}^{n+1}$ and the Milnor fibre $M$ of $g$ is recovered as the
intersection of the two branches of $f_{\delta}$.

Figure 5. The singularity $\\{xy=0\\}\subseteq\mathbb{C}^{2}$ and its Milnor
fibre can be realized as the intersection of the branches of a bi-germ and its
stable perturbation. The singular locus of $\\{xy=0\\}$ corresponds to the
instability locus of the bi-germ.

Our claim that the study of the Milnor fibration of hypersurfaces is the same
as the study of disentanglements of bi-germs of immersions also carries over
to the vanishing topology in the following sense:

######  Remark 1.8.

Let $M$ be the Milnor fibre of a hypersurface $X=V(g)$ and
$Y_{\delta}=\operatorname{Im}f_{\delta}$ be the disentanglement of the bi-germ
associated to $g$, as in Example 1.7. Then, there is an isomorphism

$\tilde{H}^{i}(M)\cong\tilde{H}^{i+1}(Y_{\delta}),$

compatible with the monodromy actions on $M$ and $Y_{\delta}$. In fact,
$Y_{\delta}$ is homotopy equivalent to the suspension of $M$ and we will come
back to this isomorphism on Example 5.13 and Remark 5.14 to illustrate the
more general approach to the computation of the cohomology of mappings.

##  2\. Main Results

In this paper, we always assume that
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$ is $\mathcal{K}$-finite:

######  Definition 2.1.

A map $f$ is _$\mathcal{K}$ -finite_ if the restriction $f|_{\Sigma(f)}$ is a
finite map.

This condition arises naturally in the study of contact equivalence of
singular map-germs [MN20, Section 4.4] and it is generally considered mild.
What appears here as its definition is usually regarded as a geometric
characterization [MN20, Proposition 4.3]. Observe that, for $p>n$, the
condition for a germ $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$ to be
$\mathcal{K}$-finite is just that $f$ is finite.

The $\mathcal{K}$-finiteness condition is helpful in at least three ways:
First of all, it entails that for dimensions $p\leq n+1$ the discriminant
$\Delta(f)$ is a hypersurface in $\mathbb{C}^{p}$ (Proposition 3.3). Second,
the instability locus of $\mathcal{K}$-finite mappings is analytic
(Proposition 3.2). Lastly, $\mathcal{K}$-finiteness ensures the existence of
stable unfoldings which are used in the definition of multiple point spaces in
Section 5.

######  Theorem 2.2.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$ be a $\mathcal{K}$-finite
map-germ, with $p\leq n$, and let $d=\operatorname{dim}(\mathrm{Inst}(f))$.
The reduced cohomology of the discriminant of any perturbation $f_{\delta}$
satisfies

$\tilde{H}^{q}(\Delta(f_{\delta}))=0,\text{ for any }q\notin[p-1-d,p-1].$

The previous result also holds for disentanglements, as they are particular
cases of perturbations.

######  Theorem 2.3.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a finite map-germ
and let $d=\operatorname{dim}(\mathrm{Inst}(f))$. The reduced cohomology of
the image of any perturbation $f_{\delta}$ satisfies

$\tilde{H}^{q}(Y_{\delta})=0\text{, for any }q\notin[n-d,n].$

In the case of germs $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$, apart
from the image $Y$, it is common to study the _source double point space_
$D(f)\subseteq\mathbb{C}^{n}$. For a map $f\colon X\to Z$ between complex
manifolds, it consists of the set of points $x\in X$ such that the germ
$f\colon(X,f^{-1}(f(x)))\to(\operatorname{Im}f,f(x))$ is not an isomorphism,
that is,

$D(f)=\\{x\in X\mid f^{-1}(f(x))\neq\\{x\\}\\}\cup\\{x\in X\mid f\text{ is not
immersive at $x$}\\}.$

A point $x\in D(f)$ is called a _double point_ of $f$, even in the case that
$f^{-1}(\\{f(x)\\})=\\{x\\}$. Details on the analytic structure of $D(f)$ are
found in Section 4. For now, it is enough to know the following: Whenever
$f\colon X\to Z$ is a finite mapping between complex manifolds with
$\operatorname{dim}Z=\operatorname{dim}X+1$ and
$\operatorname{dim}(\mathrm{Inst}(f))<\operatorname{dim}X$, then $D(f)$ is a
hypersurface in $X$.

######  Theorem 2.4.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a finite map-germ
and let $d=\operatorname{dim}(\mathrm{Inst}(f))$. The reduced cohomology of
the source double point space of any perturbation $f_{\delta}$ satisfies

$\tilde{H}^{q}(D(f_{\delta}))=0\text{, for any }q\notin[n-1-d,n-1].$

######  Remark 2.5.

Just as the disentanglements and discriminants of Example 1.5, the source
double points space $D(f_{\delta})$ of a finitely determined map germ
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ has the homotopy type of a
bouquet of spheres of dimension $n-1$. This observation is due to R. Giménez
Conejero and J.J. Nuño-Ballesteros [GN20].

######  Example 2.6.

Let us come back to the bi-germ of immersion

$f\colon U_{1}\sqcup U_{2}\to\mathbb{C}^{n+1}$

associated to a hypersurface $X\subseteq\mathbb{C}^{n}$, described in Example
1.7. Since each of the branches $f|_{U_{i}}$ is injective and immersive, the
space $D(f)$ is the preimage by $f$ of the intersection of the the two
branches $Y_{i}=\operatorname{Im}f|_{U_{i}}$. Therefore, the source double
point space consist of two disjoint copies

$D(f)=X\sqcup X.$

The same argument shows that the double point space of the stable perturbation
of $f$ is

$D(f_{\delta})=M\sqcup M$

where $M$ is the Milnor fibre of $X$. From this point of view the
cohomological version of Kato’s and Matsumoto’s connectivity result is a
particular case of Theorem 2.4. The classical connectivity result due to
Milnor appears as a particular case of Remark 2.5.

######  Corollary 2.7.

Let $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$ be a finite map-germ
and assume that $\operatorname{dim}(\mathrm{Inst}(f))<n-1$. Then, the singular
locus $\operatorname{Sing}Y_{\delta}$ of the image of any perturbation of $f$
is connected.

###### Proof.

Theorem 2.4 implies immediately that $D(f_{\delta})$ is connected. Moreover,
from the set-theoretical description of the source double point space given
above, it follows that the singular locus of the image $Y$ of a finite and
generically one-to-one map $f\colon X\to Z$ between complex manifolds is, as a
set, the image of the double point space. In other words, finite and
generically one-to-one maps satisfy $\operatorname{Sing}Y=f(D(f))$, which
clearly implies our claim. Now observe that, for dimensions
$\operatorname{dim}Y>\operatorname{dim}X$, stable mappings are generically
one-to-one. Therefore, mappings satisfying
$\operatorname{dim}(\mathrm{Inst}(f))<n-1$ (hence generically stable) are
necessarily generically one-to-one as well. ∎

In order to obtain cohomological connectivity results for germs
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with arbitrary $p>n+1$,
something stronger than $\mathcal{K}$-finiteness is required: A map $f\colon
U\to V$ between complex manifolds of dimensions $n=\operatorname{dim}U$ and
$p=\operatorname{dim}V$ is _dimensionally correct_ if its multiple point
spaces $D^{k}(f)$ are empty or have the minimal dimension $kn-(k-1)p$ (see
Section 5 for the definition of multiple points).

######  Theorem 2.8.

Let $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ be a dimensionally
correct map-germ and $p>n+1$. Then, for any perturbation $f_{\delta}$, all
possibly non-trivial reduced cohomologies $\tilde{H}^{q}(Y_{\delta})$ are
concentrated in the degrees $q$ satisfying

$q=kn-(k-1)(p-1)-s,$

for some $1<k\leq\left\lfloor\frac{p}{p-n}\right\rfloor$ and $0\leq s\leq
d=\operatorname{dim}\mathrm{Inst}(f)$.

As the following example shows, Theorem 2.8 does not hold if the dimensionally
correct hypothesis is omitted.

######  Example 2.9.

By adding a zero coordinate function to the cuspidal edge of Example 1.6, we
obtain the germ $f^{\prime}\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{4},0)$
given by

$(x,y)\mapsto(x,y^{2},y^{3},0).$

Similarly, adding a zero coordinate to the perturbation of Example 1.6 gives a
perturbation $f^{\prime}_{\delta}$, given by

$(x,y)\mapsto(x,y^{2},y^{3}+\delta y(x^{2}-\delta),0),$

whose image is, obviously, isomorphic to that of the mentioned example. This
image has non-trivial cohomology in degree two. If $f^{\prime}$ was
dimensionally correct, then Theorem 2.8 would allow for nontrivial vanishing
cohomology only in degrees $0$ and $1$. Indeed, this condition on $f^{\prime}$
is violated since the source double point space $D(f^{\prime})$ contains the
whole line $\\{y=0\\}$ ($f^{\prime}$ is not immersive there) whereas, for
germs $(\mathbb{C}^{2},0)\to(\mathbb{C}^{4},0)$, the minimal dimension for
double points is $2\cdot 2-1\cdot 4=0$.

######  Remark 2.10.

A first version of Theorems 2.3 and 2.8 was proven in [SZ18], under the extra
assumptions that $f_{\delta}$ is stable, that $f$ has corank one and—in the
case of $p=n+1$—that $f$ is a dimensionally correct monogerm. Two examples of
map-germs that only the new version can handle are
$(x,y)\mapsto(x^{2},y^{2},x^{3}+y^{3}+xy)$ (which has corank two) and
$(x,y)\mapsto(x,y^{3},y^{4})$ (which is not dimensionally correct, since it
has a line of triple points, but satisfies the hypotheses of Theorem 2.3).

Finally, let us return to the study of monodromy. Let $F$ be a one-parameter
unfolding of a finite map-germ
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$, with $p>n$. The projection
from the image $\mathcal{Y}$ of a good representative of $F$ to the
deformation parameter restricts to a locally trivial fibration

$\mathcal{Y}^{*}\to D^{*}$

over the punctured disk. Since the disentanglement $Y_{\delta}$ is the fibre
of the previous fibration, for each $i$ there is corresponding monodromy
automorphism

$h^{i}\colon H^{i}(Y_{\delta},\mathbb{C})\to H^{i}(Y_{\delta},\mathbb{C}).$

On one hand, the fact that $f$ is finite implies that $\mathcal{Y}$ is
analytic, and thus it is clear that the eigenvalues of $h^{i}$ are roots of
the unity, cf. Remark 7.3. On the other hand $\mathcal{Y}$ is not smooth, (in
fact $\mathcal{Y}$ need not be a complete intersection when $p>n+1$) and thus
the classical result about the size of Jordan blocks does not apply in our
setting. As we will see, under the right hypothesis the classical bound for
the Jordan blocks and a spectral sequence argument can be combined to bound
the size of the Jordan blocks of the monodromy of perturbations.

######  Theorem 2.11.

Let $F$ be a stable one-parameter unfolding of a corank one map germ
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p>n$. Then the size of
the Jordan blocks of $h^{i}$ are at most $i(i+1)/2$. Moreover, if $f$ has only
isolated instability locus and $p>n+1$, the possibly non-trivial reduced
cohomologies are $\tilde{H}^{kn-(k-1)(p-1)}(Y_{\delta})$, for
$1<k\leq\left\lfloor\frac{p}{p-n}\right\rfloor$, and the size of the
corresponding Jordan block is at most $kn-(k-1)p+1$.

###  2.1. Some remarks on generality

In writing this paper, we decided to sacrifice some generality for the sake of
clarity. The following remarks list some improvements that ended up being left
behind:

######  Remark 2.12.

Throughout the text, the instability locus can be replaced by the _topological
instability locus_. Topological stability is defined by using homeomorphisms
instead of diffeomorphisms in the definition of trivial unfolding. In general,
the topological instability locus is smaller, giving rise to sharper
cohomological connectivity bounds.

######  Remark 2.13.

Theorem 2.8 is stated for monogerms, but it applies to multigerms as well.
However, the involved representations of the symmetric group made the
exposition more complicated and seemed off-topic.

######  Remark 2.14.

The “dimensionally correct” condition of Theorem 2.8 may be relaxed by
introducing _strict_ dimensional correctness and adding some extra hypotheses
(counterintuitively, strict dimensional correctness is less demanding than
dimensional correctness, see Definition 5.2). The cohomological connectivity
predicted by Theorem 2.8 also holds in the following situations:

  1. (1)

The unfolding $F$ underlying $f_{\delta}$ is strictly dimensionally correct.

  2. (2)

The germ $f$ is strictly dimensionally correct, and $Y_{\delta}$ is a
disentanglement (i.e. $f_{\delta}$ is a stable perturbation).

The second situation is, in fact, a particular case of the first, because
stable perturbations of strictly dimensionally correct germs arise only from
strictly dimensionally correct unfoldings. The reason not to include this
improvement is that it depends on the unfolding $F$, not only on $f$. Example
2.9 shows that perturbations $f_{\delta}$ of a strictly dimensionally correct
map-germ $f$ need not satisfy the cohomological connectivity of Theorem 2.8.

We thank David Massey, who inspired us to use perverse sheaves as a tool for
understanding perturbations of map-germs. We also thank Juan José Nuño
Ballesteros for useful discussions about $\mathcal{K}$-finiteness. The third
author would like to thank the Singularity group at BCAM in Bilbao for their
kind hospitality.

##  3\. Preliminaries

Here we include definitions related to stability, properties involving
$\mathcal{K}$-finiteness and the basic notions of the theory of perverse
sheaves which are used to show our results.

###  3.1. Stability and $\mathcal{K}$-finiteness

An unfolding
$F\colon(\mathbb{C}^{n}\times\mathbb{C}^{r},S\times\\{0\\})\to(\mathbb{C}^{p},0)$
of a germ $f$ is _trivial_ if there exist an unfolding $\Phi$ of
$(\operatorname{id}_{\mathbb{C}^{n}},S)$ and an unfolding $\Psi$ of
$(\operatorname{id}_{\mathbb{C}^{p}},0)$ such that the following diagram
commutes:

${(\mathbb{C}^{n}\times\mathbb{C}^{r},S\times\\{0\\})}$${(\mathbb{C}^{p}\times\mathbb{C}^{r},(0,0))}$${(\mathbb{C}^{n}\times\mathbb{C}^{r},S\times\\{0\\})}$${(\mathbb{C}^{p}\times\mathbb{C}^{r},(0,0)).}$$\scriptstyle{F}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$$\scriptstyle{f\times\operatorname{id}_{\mathbb{C}^{r}}}$

Note that these conditions make $\Phi$ and $\Psi$ into germs of
biholomorphism.

A germ $f$ is _stable_ if every unfolding of $f$ is trivial. A
$\mathcal{K}$-finite map $f\colon M\to N$ is stable at $q\in N$ if the germ of
$f$ at $S=\Sigma(f)\cap f^{-1}(q)$ is stable. We say that the map $f$ is
stable if it is $\mathcal{K}$-finite and it is stable at every $q\in N$.

A one-parameter unfolding $F$ is called a _stabilisation_ of $f$ if it admits
a good representative such that for every non-zero $\delta\in T$ the
perturbations $f_{\delta}$ are stable. In this case, $f_{\delta}$ is called a
_stable perturbation_ of $f$, and $Y_{\delta}=\operatorname{Im}f_{\delta}$ is
called a _disentanglement_ of $Y$.

The _instability locus_ of $f\colon M\to N$ is the support

$\mathrm{Inst}(f)=\operatorname{Supp}\frac{\theta(f)}{T\mathcal{A}_{e}(f)},$

where $\theta(f)$ is the sheaf of vector fields along $f$, and
$T\mathcal{A}(f)$ is the extended $\mathcal{A}$-tangent space of $f$, see
[Wal81]. What gives its name to the instability locus is the following result
from [MN20, Theorem 3.2]:

######  Proposition 3.1.

Let $f\colon M\to N$ be a $\mathcal{K}$-finite map, let $q\in N$ and
$S=f^{-1}(q)\cap\Sigma(f)$. If $q\notin\mathrm{Inst}(f)$, then the germ of $f$
at $S$ is stable.

Introducing the instability locus as the support of a sheaf allows us to
justify that it is an analytic space. The proof of the following proposition
can be extracted from that of [MN20, Proposition 4.2]:

######  Proposition 3.2.

If $f$ is $\mathcal{K}$-finite, then $\theta(f)/(T_{e}\mathcal{A}(f))$ is a
coherent $\mathcal{O}_{Y}$-module. In particular, $\mathrm{Inst}(f)$ is
analytic.

We finish this subsection with a result which is well known, but whose proof
we include for lack of a reference.

######  Proposition 3.3.

Let $f\colon X\to Z$ be a $\mathcal{K}$-finite map between complex manifolds,
let $n=\operatorname{dim}X$ and $p=\operatorname{dim}Z$. If $p\leq n+1$, then
the dimension of $\Delta(f)$ is $p-1$.

###### Proof.

The case $p=n+1$ is clear, because then $\Sigma(f)=X$, the map is finite and
$\Delta(f)=\operatorname{Im}f$. We may thus assume $q\leq n$. Since $f$ is
$\mathcal{K}$-finite, $\Delta(f)$ is an analytic space of the same dimension
as $\Sigma(f)$. On the one hand, Sard’s Theorem implies that $\Delta(f)$ is
not all of $Z$, hence its dimension is at most $p-1$. On the other hand,
$\Sigma(f)$ is defined as the vanishing locus of $p$-minors of an $n\times
p$-matrix with $p\leq n$. The results in [EN62] imply that the dimension of
any component of $\Delta(f)$ is greater or equal to $n-(n-p+1)=p-1$. ∎

###  3.2. Perverse sheaves

In this subsection, we will summarize those parts of the machinery of perverse
sheaves on complex analytic spaces which we shall need for the proof of our
theorems. We follow the standard reference [Dim04] and all the details can be
found there.

Throughout this section, $X$ stands for a complex analytic variety. For a
sheaf $\mathscr{F}$ on $X$ its sheaf cohomology groups will be denoted by

$H^{i}(X;\mathscr{F}).$

We write $\mathbb{C}_{X}$ for the constant sheaf on $X$ associated to the
field $\mathbb{C}$. Recall that the cohomology of $\mathbb{C}_{X}$ is
isomorphic to the singular cohomology with complex coefficients:

$H^{\bullet}(X;\mathbb{C}_{X})\cong H_{\mathrm{sing}}^{\bullet}(X).$

By $C(X)$ we denote the category of complexes of sheaves of
$\mathbb{C}_{X}$-modules on $X$:

$\mathscr{F}^{\bullet}:\qquad\cdots\to\mathscr{F}^{i-1}\to\mathscr{F}^{i}\to\mathscr{F}^{i+1}\to\cdots$

From any such complex of sheaves we obtain the collection of cohomology
sheaves, denoted by

$\mathcal{H}^{i}(\mathscr{F}^{\bullet}),\quad i\in\mathbb{Z}.$

A morphism of complexes of sheaves
$\varphi^{\bullet}\colon\mathscr{F}^{\bullet}\to\mathscr{G}^{\bullet}$ is
called a quasi-isomorphism, if the induced maps on the cohomology sheaves

$\varphi\colon\mathcal{H}^{i}(\mathscr{F}^{\bullet})\to\mathcal{H}^{i}(\mathscr{G}^{\bullet})$

is an isomorphism for all $i\in\mathbb{Z}$. By $\mathcal{F}^{\bullet}[d]$ we
will denote the shift by $d$ of the complex $\mathcal{F}$ which is given by
the terms

$\left(\mathcal{F}^{\bullet}[d]\right)^{k}=\mathcal{F}^{k-d}$

for every $k$ together with the appropriately shifted differentials from
$\mathcal{F}$.

By $D(X)$ we denote the derived category of sheaves of
$\mathbb{C}_{X}$-modules on $X$ which is obtained from $C(X)$ by localizing at
the set of quasi-isomorphisms. In particular this construction entails that
two complexes of sheaves $\mathscr{F}^{\bullet}$ and $\mathscr{G}^{\bullet}$
are isomorphic in $D(X)$ if and only if they are quasi-isomorphic in $C(X)$.

A complex of sheaves $\mathscr{F}^{\bullet}$ is in the bounded derived
category $D^{b}(X)$ if its nontrivial cohomology sheaves
$\mathcal{H}^{i}(\mathscr{F}^{\bullet})$ are confined to a bounded range of
indices $i$. For any such complex of sheaves $\mathscr{F}^{\bullet}\in
D^{b}(X)$ we can define the _support_ as

$\operatorname{Supp}\mathscr{F}^{\bullet}=\overline{\\{x\in
X\mid\mathcal{H}^{i}(\mathscr{F}^{\bullet})_{x}\neq 0\text{ for some
}i\in\mathbb{Z}\\}}$

where we denote by $\mathcal{H}^{i}(\mathscr{F}^{\bullet})_{x}$ the stalk of
the cohomology sheaf $\mathcal{H}^{i}(\mathscr{F}^{\bullet})$ at the point
$x\in X$.

A sheaf $\mathscr{F}$ on $X$ can be regarded as a complex of sheaves
concentrated in degree 0. This is a fully faithful embedding of the category
of sheaves on $X$ into the bounded derived category, cf. [Dim04, Proposition
1.3.3 iii]. This allows us to simplify notation and write $\mathscr{F}$
instead of $\mathscr{F}^{\bullet}$ for the complex of sheaves associated to a
sheaf $\mathscr{F}$.

Under this embedding into the bounded derived category, classical sheaf
cohomology reappears as hypercohomology. More generally, for a continuous map
$f\colon X\to Y$ of topological spaces and a sheaf $\mathscr{F}$ on $X$ one
has

$R^{i}f_{*}\mathscr{F}\cong\mathcal{H}^{i}(Rf_{*}\mathscr{F})$

where the left hand side denotes the $i$-th derived pushforward of a single
sheaf and

$Rf_{*}\colon D^{b}(X)\to D^{b}(Y)$

denotes the derived pushforward in the derived categories, cf. [Dim04, Section
2.3]. For the special case of a projection $p\colon X\to\\{pt\\}$ to a point,
one obtains

$H^{i}(X,\mathscr{F})=R^{i}p_{*}\mathscr{F}=\mathcal{H}^{i}(Rp_{*}\mathscr{F})=\mathbb{H}^{i}(X,\mathscr{F})$

where

$\mathbb{H}^{i}(X,-)\colon D^{b}(X)\to\mathbb{C}\text{-Vect}$

is the $i$-th hypercohomology functor.

Again for a map $f\colon X\to Y$ and a complex of sheaves $\mathscr{F}$ on $X$
the hypercohomology functors satisfy the following fundamental property:

$\mathbb{H}^{i}(Y,Rf_{*}\mathscr{F})=\mathbb{H}^{i}(X,\mathscr{F}),$

the Leray spectral sequence, see [Dim04, Proposition 2.3.4].

Similar translations of the above identifications between the category of
$\mathbb{C}_{X}$-modules and the derived category exist for relative
cohomology for pairs of spaces; see [Dim04] for details.

We shall say that a complex of sheaves $\mathscr{F}\in D^{b}(X)$ on a complex
analytic space $X$ is _constructible_ if there exists a complex analytic
stratification of $X$ with locally closed _complex analytic_ strata
$S_{\alpha}\subset X$ such that

  1. (1)

The restriction of $\mathscr{F}$ to each one of the $S_{\alpha}$ has locally
constant cohomology sheaves,

  2. (2)

All stalks are finite dimensional $\mathbb{C}$-vector spaces.

The category of bounded constructible complexes of sheaves on $X$ will be
denoted by $D_{c}^{b}(X)$.

From now on, we will mostly be interested in a very special type of bounded
complexes of constructible sheaves: the perverse sheaves for the middle
perversity. Note that a perverse sheaf need not be a sheaf, but a complex of
sheaves. It is our intention for the reader to accept “being perverse” as a
good property which can be used without knowing its details. We proceed by
stating the necessary properties and results for the intended usage.

######  Proposition 3.4.

[Dim04, Remark 5.1.19] Let $\mathscr{F}$ be a perverse sheaf on a complex
analytic variety $X$, and let $d$ be the dimension of the support of
$\mathscr{F}$. For any point $x\in X$, the stalk cohomology groups satisfy

$\mathcal{H}^{i}(\mathscr{F})_{x}=0\textnormal{, for all }i\notin[-d,0].$

Note that since by definition any perverse sheaf is constructible, its support
is always a closed _complex analytic_ subset and it makes sense to speak of
its (complex) dimension.

######  Proposition 3.5.

[Dim04, Theorem 5.1.20] If $X$ is a locally complete intersection of complex
dimension $d$, then the shifted constant sheaf $\mathbb{C}_{X}[d]$ is
perverse.

Let $g:\mathcal{X}\to\mathbb{C}$ be a holomorphic function defined on a
complex analytic variety and set $X=g^{-1}(0)$. Associated to $g$, there is
the _vanishing cycle functor_

$\phi_{g}:D^{b}_{c}(\mathcal{X})\to D^{b}_{c}(X).$

For any point $x\in X$ and any $\mathscr{F}\in D^{b}_{c}(\mathcal{X})$, the
stalk cohomology of $\phi_{g}\mathscr{F}$ can be computed as follows (see e.g.
[Dim04, page 106 (4.1)]):

(1) $\mathcal{H}^{i}(\phi_{g}\mathscr{F})_{x}=\mathbb{H}^{i+1}(B_{r},B_{r}\cap
g^{-1}(\\{\delta\\});\mathscr{F}),$

where $B_{r}$ is a sufficiently small open ball in $\mathcal{X}$ centered at
$x$ and $\delta\in\mathbb{C}\setminus\\{0\\}$ is small enough with respect to
$r$. The last result we need is essential, as it allows to produce new
perverse sheaves from old:

######  Theorem 3.6.

[Dim04, Theorem 5.2.21] The vanishing cycle functor shifted by one
$\phi_{g}[-1]\colon D^{b}_{c}(\mathcal{X})\to D^{b}_{c}(X)$ takes perverse
sheaves on $\mathcal{X}$ to perverse sheaves on $X$.

The ingredients we have introduced can be put together to give a simple proof
of the cohomological connectivity of Milnor fibers:

Let $g$ be a suitable representative of a holomorphic map germ
$(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ defined on some open subset
$\mathcal{X}\subset\mathbb{C}^{n+1}$. Take the perverse sheaf
$\mathbb{C}_{\mathcal{X}}[n+1]$ and apply the shifted vanishing cycle functor
associated to $g$ to obtain a perverse sheaf on $X=g^{-1}(0)$. The stalk
cohomology at a point $x\in X$ recovers the reduced cohomology of the Milnor
fibre of the germ of $X$ at $x$ as follows:

$\displaystyle\mathcal{H}^{i}(\phi_{g}\mathbb{C}_{\mathcal{X}}[n])_{x}$
$\displaystyle=$ $\displaystyle\mathbb{H}^{i+1}(B_{r},B_{r}\cap
g^{-1}(\delta);\mathbb{C}_{\mathcal{X}}[n])$ $\displaystyle=$ $\displaystyle
H^{i+n+1}(B_{r},B_{r}\cap g^{-1}(\delta);\mathbb{C}_{\mathcal{X}})$
$\displaystyle=$ $\displaystyle H_{\mathrm{sing}}^{i+n+1}(B_{r},B_{r}\cap
g^{-1}(\delta))$ $\displaystyle=$
$\displaystyle\tilde{H}_{\mathrm{sing}}^{i+n}(B_{r}\cap g^{-1}(\delta)).$

Since the Milnor fibre of $g$ at a regular point of the function has trivial
reduced cohomology, we deduce that the support of
$\phi_{g}\mathbb{C}_{\mathcal{X}}[n]$ is contained in $\operatorname{Sing}X$.
Letting $d=\operatorname{dim}(\operatorname{Sing}X)$, Proposition 3.4 gives a
bound on the stalk cohomologies at the origin

$\mathcal{H}^{i}\left(\phi_{g}\mathbb{C}_{\mathcal{X}}[n]\right)_{0}\cong\tilde{H}^{i+n}_{\text{sing}}(B_{r}\cap
g^{-1}(\delta))=0\quad\text{ for all }i\notin[-d,0]$

which turns out to be the desired cohomological connectivity result for the
Milnor fiber of $g$.

As the reader will see, the proofs of our results follow the same pattern:
Find an appropriate perverse sheaf and apply the vanishing cycle functor
associated to the projection to the parameter space. By virtue of Proposition
3.3, an estimate on the dimension of the support of the sheaf will directly
translate to a bound on the non-zero degrees of the reduced cohomology of a
nearby object.

##  4\. Proof of Theorems 2.2, 2.3 and 2.4

Recall that, for $p=n+1$, finiteness and $\mathcal{K}$-finiteness are the same
and $Y=\Delta(f)$. Consequently, Theorem 2.2 and Theorem 2.3 can be considered
two instances of the same result for $\mathcal{K}$-finite germs
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p\leq n+1$. For a
$\mathcal{K}$-finite germ $f$, any unfolding $F$ is $\mathcal{K}$-finite and,
since $p\leq n+1$, the discriminant $\Delta(F)$ is a hypersurface in
$\mathbb{C}^{p}\times\mathbb{C}$, by Proposition 3.3. In particular, the
shifted constant sheaf $\mathbb{C}_{\Delta(F)}[p]$ is perverse. Projecting on
the unfolding parameter gives a family

$\Delta(F)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}T,$

and one checks the fiber over $\delta\in T$ to be $\Delta(f_{\delta})$.
Applying the associated vanishing cycle functor gives a perverse sheaf
$\phi_{\pi}\mathbb{C}_{\Delta(F)}[p-1]$ on $\Delta(f)$.

######  Lemma 4.1.

The support of $\phi_{\pi}\mathbb{C}_{\Delta(F)}[p-1]$ is contained in
$\mathrm{Inst}(f)$.

###### Proof.

Given a point away from the instability locus
$y\in\Delta(f)\setminus\mathrm{Inst}(f)$, $S:=f^{-1}(y)\cap\Sigma(f)$ is a
finite set because $f$ is $\mathcal{K}$-finite. By Proposition 3.1, the germ
$f\colon(U,S)\to(V,y)$ of $f$ at $S$ is stable and thus the unfolding

$F\colon(U\times T,S\times\\{0\\})\to(V\times T,(y,0))$

is a trivial unfolding. Consequently, there exist unfoldings $\Phi$ and $\Psi$
of the identity mappings $\operatorname{id}_{U}$ and $\operatorname{id}_{V}$,
respectively, such that the following diagram commutes:

${(U\times T,S\times\\{0\\})}$${(V\times
T,(y,0))}$${(U,S)\times(T,0)}$${(V,y)\times(T,0).}$$\scriptstyle{F}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$$\scriptstyle{f\times\operatorname{id}_{T}}$

The stalk at $y$ of the sheaf of vanishing cycles of $\pi$ on $\Delta(F)$ is

$\displaystyle\mathcal{H}^{i}(\phi_{\pi}\mathbb{C}_{\Delta(F)}[p-1])_{y}$
$\displaystyle=$ $\displaystyle\mathbb{H}^{i+1}(\Delta(F)\cap
B_{r},\pi^{-1}(\delta)\cap B_{r};\mathbb{C}_{\Delta(F)}[p-1])$
$\displaystyle\cong$
$\displaystyle\mathbb{H}^{i+1}((\Delta(f)\times\mathbb{C})\cap
B_{r},\Delta(f)\cap B_{r};\mathbb{C}_{\Delta(f)\times\mathbb{C}}[p-1])$
$\displaystyle=$ $\displaystyle H^{i+p}((\Delta(f)\times\mathbb{C})\cap
B_{r},\Delta(f)\cap B_{r};\mathbb{C})$ $\displaystyle=$ $\displaystyle 0,$

which finishes the proof. ∎

###### Proof of Theorems 2.2 and 2.3.

From the inclusion
$\operatorname{Supp}(\phi_{\pi}\mathbb{C}_{\Delta(F)}[p-1])\subseteq\mathrm{Inst}(f)$
and the hypothesis that $\operatorname{dim}\mathrm{Inst}(f)\leq d$, applying
Proposition 3.4 we obtain that

$\mathcal{H}^{i}(\phi_{\pi}\mathbb{C}_{\Delta(F)}[p-1])_{0}=0\textnormal{ for
}i\notin[-d,0].$

Theorems 2.2 and 2.3 follow, because for a good representative the stalk at
the origin is precisely

$\displaystyle\mathcal{H}^{i}(\phi_{\pi}\mathbb{C}_{\Delta(F)}[p-1])_{0}$
$\displaystyle=$
$\displaystyle\mathbb{H}^{i+1}(\Delta(F),\Delta(f_{\delta});\mathbb{C}_{\Delta(F)}[p-1])$
$\displaystyle=$ $\displaystyle\tilde{H}^{i+p-1}(\Delta(f_{\delta})),$

where the last equality is due to the fact that $\Delta(F)$ is contractible. ∎

The proof of Theorem 2.4 is very similar and thus will only be sketched.
Before that, we discuss some subtleties of the analytic structure of $D(f)$.
In order to avoid pathologies related to unfoldings, the source double point
space $D(f)$ is given an analytic structure which need not be reduced. For our
purposes, we do not need to know the details of this construction [MNP12,
Definition 2.2], but only the following two properties:

  1. (1)

For any unfolding $F=(f_{t},t)\colon U\times T\to V\times T$, the fiber over
$\delta\in T$ of the family

$D(F)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}T$

is the complex space $D(f_{\delta})$.

  2. (2)

If $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ is finite and
generically one-to-one, then the space $D(f)$ is a hypersurface.

The first statement follows from [MNP12, Lemma 4.2]. For the second statement,
the proof of [MNP12, Lemma 2.3] shows that $D(f)$ is a Cohen Macaulay subspace
of $\mathbb{C}^{n}$ of codimension one, hence a hypersurface.

###### Proof of Theorem 2.4.

First of all, observe that the statement of Theorem 2.4 is trivial unless the
dimension $d$ of the instability locus is smaller than $n-1$. Consequently, we
may assume $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ to be a finite
map-germ with $\operatorname{dim}(\mathrm{Inst}(f))<n-1$.

The map $f$ is generically one-to-one, because stable mappings are generically
one-to-one, and the conditions of finiteness and
$\operatorname{dim}(\mathrm{Inst}(f))<n-1$ imply that the preimage of the
instability locus is nowhere dense.

Any unfolding $F$ of $f$ is also finite and generically one-to-one, hence
$D(F)$ is a hypersurface in $\mathbb{C}^{n+1}$ and the sheaf
$\mathbb{C}_{D(F)}[n]$ is perverse. The shifted vanishing cycle functor
associated to the projection $\pi\colon D(F)\to T$ gives a perverse sheaf
$\phi_{\pi}\mathbb{C}_{D(F)}[n-1]$ on $D(f)$. The same argument used in Lemma
4.1 shows the inclusion

$\operatorname{Supp}\phi_{\pi}\mathbb{C}_{D(F)}\subseteq
f^{-1}(\mathrm{Inst}(f)),$

where the dimension of $f^{-1}(\mathrm{Inst}(f))$ is equal to $d$, because $f$
is finite. Then the result follows from the computation of the stalk

$\mathcal{H}^{i}(\phi_{\pi}\mathbb{C}_{D(F)}[n-1])_{0}\cong\tilde{H}^{n-1+i}(\Delta(f_{\delta})).\qed$

##  5\. Multiple point spaces

For finite germs $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with
$p>n+1$, the image $Y$ is no longer a hypersurface, and may even fail to be a
complete intersection, as shown by the well known twisted cubic:

$(\mathbb{C}^{2},0)\to(\mathbb{C}^{4},0),\quad(s,t)\mapsto(s^{3},s^{2}t,st^{2},t^{3}).$

Since the constant sheaf on non complete intersections is not necessarily
perverse, we cannot follow the same reasoning as in the proof of Theorems 2.3
and 2.2. Instead, we study the cohomology of the disentanglement via the image
computing spectral sequence due to Goryunov and Mond [GM93, Proposition 2.3]:

$E^{p,q}_{1}=H^{q}_{\operatorname{Alt}}(\mathscr{D}^{p+1}(f))\Rightarrow
H^{p+q}(Y).$

and the perversity of the sheaves

$\operatorname{Alt}R\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}[kn-(k-1)p]$

on the image $Y$, discovered by Houston [Hou00, Theorem 2.9]. This involves
the strict multiple point spaces $\mathscr{D}^{k}(f)$ and their alternating
cohomology which we will now discuss.

Unfortunately, there is no common agreement on the definition of multiple
point spaces and different notions are in circulation. What we will refer to
as the strict multiple points $\mathscr{D}^{k}(f)$ is the definition used by
Goryunov, Mond [GM93], and Houston [Hou97a], [Hou97b], [Hou00]. As Example 5.4
shows, the strict multiple point spaces are badly behaved in deformations:
They do not specialize to fibers. To remedy this fact, there is another, more
subtle definition of multiple point spaces $D^{k}(f)$ due to Gaffney [Ga83]
which we will describe in Section 5.2. These are what we will refer to as
simply the multiple point spaces of $f$. Fortunately, the results about
$\mathscr{D}^{k}(f)$ we want to use can be adapted to the spaces $D^{k}(f)$
without difficulties.

###  5.1. The strict multiple point spaces $\mathscr{D}^{k}(f)$

Let $f\colon X\to Z$ be a complex analytic map between holomorphic manifolds.
For $k\geq 1$, the $k$-th strict multiple point space of $f$ is defined to be

$\mathscr{D}^{k}(f)=\textnormal{closure}\\{(x_{1},\dots,x_{k})\in X^{k}\mid
f(x_{i})=f(x_{j}),x_{i}\neq x_{j}\text{ for all }i\neq j\\}.$

Note that $\mathscr{D}^{1}(f)=X$.

We recall the construction of the alternating complex due to Goryunov and Mond
[GM93]. $\mathscr{D}^{k}(f)$ are symmetric in the sense that group $S_{k}$
acts on them by permutating the points $x_{1},\dots,x_{k}$. This action
preserves the fibers of the maps

$\varepsilon^{k}\colon\mathscr{D}^{k}(f)\to Z,\quad(x_{1},\dots,x_{k})\mapsto
f(x_{1})=\dots=f(x_{k}).$

As a consequence, there is an action of $S_{k}$ on the pushforward sheaf
$\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$ on $Z$. For each
$\sigma\in S_{k}$, we write the associated automorphism as

$\sigma^{*}\colon\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}\to\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$

and define the _alternating operator_
$\operatorname{Alt}\colon\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}\to\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$
by the formula

$\operatorname{Alt}=\frac{1}{k!}\sum_{\sigma\in
S_{k}}\operatorname{sign}(\sigma)\sigma^{*}.$

The image of $\operatorname{Alt}$ defines a subsheaf, which we denote by
$\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$.

It is clear that the map $X^{k}\to X^{k-1}$ which forgets the $j$-th
coordinate takes a $k$-multiple point to a $(k-1)$-multiple point. Hence, we
have maps

$\varepsilon^{k,j}\colon\mathscr{D}^{k}(f)\to\mathscr{D}^{k-1}(f).$

These maps induce morphisms
$\varepsilon^{k,j}_{*}\colon\varepsilon^{k-1}_{*}\mathbb{C}_{\mathscr{D}^{k-1}(f)}\to\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$,
and one can see that there is a well defined differential

$\partial^{k}\colon\operatorname{Alt}\varepsilon^{k-1}_{*}\mathbb{C}_{\mathscr{D}^{k-1}(f)}\to\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)},$

of the form

$\partial^{k}=\sum_{i=1}^{k}(-1)^{i}\varepsilon^{k,i}_{*}.$

One can also check the equality $\partial^{k+1}\circ\partial^{k}=0$, so that
we obtain a complex of sheaves
$\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{\mathscr{D}^{\bullet}(f)},\partial^{\bullet}\right)$,
the alternating complex.

######  Proposition 5.1.

([GM93, Proposition 2.1]) The augmented complex

$0\to\mathbb{C}_{Y}\to\operatorname{Alt}\varepsilon^{1}_{*}\mathbb{C}_{\mathscr{D}^{1}(f)}\to\operatorname{Alt}\varepsilon^{2}_{*}\mathbb{C}_{\mathscr{D}^{2}(f)}\to\cdots$

is exact.

Goryunov and Mond have already argued that one has an isomorphism

(2)
$H^{i}(Y,\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)})\cong
H^{i}_{\operatorname{Alt}}(\mathscr{D}^{k}(f))$

where the term on the right hand side is the alternating cohomology

$\left\\{[c]\in
H^{i}_{\mathrm{sing}}(\mathscr{D}^{k}(f)):\sigma^{*}[c]=\operatorname{sign}(\sigma)\cdot[c]\text{
for all }\sigma\in S_{k}\right\\},$

a subspace of the singular cohomology of $\mathscr{D}^{k}(f)$. These
considerations were taken to the derived category in [Hou00]. Note that, since
$\varepsilon^{k}$ is finite, the associated pushforward of sheaves is an exact
functor and thus in particular

$\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}=R\varepsilon_{*}^{k}\mathbb{C}_{\mathscr{D}^{k}(f)}$

as complexes of sheaves on $Y$ in the derived category. The same holds for
their respective alternating subsheaves.

######  Definition 5.2.

Let $f\colon X\to Z$ be a complex analytic mapping between two complex
analytic manifolds with $n=\operatorname{dim}X$ and $p=\operatorname{dim}Z$.
Then $f$ is called strictly dimensionally correct if for all $k\geq 2$ the
strict multiple point space $\mathscr{D}^{k}(f)$ is either empty, or has
dimension $kn-(k-1)p$.

With these notations gathered we may cite the key result due to Houston, cf.
[Hou00, Theorem 2.9], slightly adapted to our setup:

######  Theorem 5.3.

Suppose $f\colon X\to Z$ is a strictly dimensionally correct complex analytic
map between complex manifolds of dimensions
$n=\operatorname{dim}X<\operatorname{dim}Z=p$. Then
$\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}[kn-(k-1)p]$
is a perverse sheaf.

###  5.2. The multiple point spaces $D^{k}(f)$

As we already mentioned before, the strict multiple point spaces discussed in
the previous section are not well behaved in families. This is illustrated by
the following example.

######  Example 5.4.

The cuspidal edge $f\colon(x,y)\mapsto(x,y^{2},y^{3})$ of Example 1.6 and the
similar germ $f^{\prime}\colon(x,y)\mapsto(x,y^{2},y^{3},0)$ of Example 2.9
are both strictly dimensionally correct because the maps are injective and
therefore the multiple point spaces are empty.

However, the map
$F^{\prime}\colon(x,y,t)\mapsto(x,y^{2},y^{3}+ty(x^{2}-1),0,t)$ from the
unfolding in Example 2.9 is _not_ strictly dimensionally correct: Its multiple
point spaces $\mathscr{D}^{1}(F)$ have dimension two while the expected
dimension is one.

Without the dummy variable, i.e. for the map $f$ as in Example 1.6, not only
$f$, but also the unfolding

$F\colon(x,y,t)\mapsto(x,y^{2},y^{3}+t\cdot y(x^{2}-t))$

is strictly dimensionally correct with expected dimension two for the double
point spaces. In this example we encounter another problem: Observe that the
strict double points of $f$ are empty while those of $F$ are not. This shows
the failure of specialization

$\mathscr{D}^{2}(F)\cap\\{t=0\\}\neq\mathscr{D}^{2}(f)$

and thereby illustrates the necessity to use Gaffney’s multiple point spaces.

The pathological behaviour exhibited in this example can be avoided by taking
a different definition of multiple points due to Gaffney [Ga83]. Our
exposition follows [NP17], where the reader can find proofs and details
omitted here.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$ be a finite multi-germ.
Since $f$ is finite, hence $\mathcal{K}$-finite, it admits a stable unfolding
[MN20, Theorem 7.2]

$F=(f_{t},t)\colon(\mathbb{C}^{n}\times\mathbb{C}^{r},S\times\\{0\\})\to(\mathbb{C}^{p}\times\mathbb{C}^{r},(0,0))$

where $t$ is the coordinate of the parameter space $(\mathbb{C}^{r},0)$. It is
clear that a point in $\mathscr{D}^{k}(F)$ has $t_{i}=t_{j}$, for all $1\leq
i,j\leq k$. Therefore $\mathscr{D}^{k}(F)$ can be embedded in
$(\mathbb{C}^{n})^{k}\times\mathbb{C}^{r}$ rather than in
$(\mathbb{C}^{n}\times\mathbb{C}^{r})^{k}$. The multiple point spaces of $f$
are defined as

$D^{k}(f):=\mathscr{D}^{k}(F)\cap\\{t=0\\}.$

The space $D^{k}(f)$ does not depend on the chosen stable unfolding $F$.
Multiple point spaces of finite mappings between complex manifolds are defined
by patching the multiple point spaces of their corresponding multi-germs.

To compare $\mathscr{D}^{k}(f)$ to $D^{k}(f)$, we write the _the $k$-th fat
diagonal of $X$_ as

$\Delta^{k}=\\{(x_{1},\dots,x_{k})\in X^{k}\mid x_{i}=x_{j}\text{ for some
}i\neq j\\}.$

######  Proposition 5.5.

(Properties of the multiple point spaces) Let $f\colon X\to Z$ be a finite
mapping between complex manifolds. Then

  1. (1)

the spaces $D^{k}(f)$ and $\mathscr{D}^{k}(f)$ satisfy the relation

$\mathscr{D}^{k}(f)=\overline{D^{k}(f)\setminus\Delta^{k}};$

  2. (2)

if $f$ is stable, then $D^{k}(f)=\mathscr{D}^{k}(f)$;

  3. (3)

unlike $\mathscr{D}^{k}(f)$, the spaces $D^{k}(f)$ behave well under
deformations in the sense that, for any unfolding $F\colon X\times T\to
Z\times T$, the family $\pi\colon D^{k}(F)\to T$ satisfies

$D^{k}(f_{\delta})=\pi^{-1}(\delta),\text{ for any }\delta\in T.$

Just as in the case of $\mathscr{D}^{k}(f)$, the spaces $D^{k}(f)$ are
$S_{k}$-invariant spaces endowed with finite maps $\varepsilon^{k,j}\colon
D^{k}(f)\to D^{k-1}(f)$, giving rise to a complex

$\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{D^{\bullet}(f)},\partial^{\bullet}\right).$

We wish to replace $\mathscr{D}^{k}(f)$ by our favourite $D^{k}(f)$ in the
study of $\mathbb{C}_{Y}$. While the spaces $D^{k}(f)$ and
$\mathscr{D}^{k}(f)$ and hence also the sheaves
$\varepsilon_{*}^{k}\mathbb{C}_{D^{k}(f)}$ and
$\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$ may differ, the
corresponding alternating sheaves do not:

######  Lemma 5.6.

For any finite map $f$ between complex manifolds, the complexes
$\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{D^{\bullet}(f)},\partial^{\bullet}\right)$
and
$\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{\mathscr{D}^{\bullet}(f)},\partial^{\bullet}\right)$
are equal.

Note that with this lemma we may replace
$\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{\mathscr{D}^{k}(f)}$ by
$\operatorname{Alt}\varepsilon_{*}^{k}\mathbb{C}_{D^{k}(f)}$ in Proposition
5.1. The idea behind the proof of Lemma 5.6 is not new and appears already in
[GM93], as well as [Hou00] and [Hou97b]. Due to the lack of a citeable
reference, we give a self contained argument.

###### Proof.

We verify that the stalks of both sheaves coincide at an arbitrary point $y\in
Y$. Fix one such point and let

$P:=\left(\varepsilon^{k}\right)^{-1}(y)\subset D^{k}(f)$

be its preimage under the finite map $\varepsilon^{k}\colon D^{k}(f)\to Y$.
Set $P^{\prime}:=\mathscr{D}^{k}(f)\cap P$ to be the preimage in the subspace
$\mathscr{D}^{k}(f)$. According to Proposition 5.5, every point $x\in
P\setminus P^{\prime}$ has to be contained in the fat diagonal $\Delta^{k}$.
Now the pushforward of $\mathbb{C}_{D^{k}(f)}$ along $\varepsilon^{k}$
decomposes as

$\displaystyle\varepsilon_{*}^{k}\mathbb{C}_{D^{k}(f)}$ $\displaystyle=$
$\displaystyle\left(\bigoplus_{x\in
P^{\prime}}\mathbb{C}_{D^{k}(f),x}\right)\oplus\left(\bigoplus_{x\notin
P^{\prime}}\mathbb{C}_{D^{k}(f),x}\right).$

The alternating operator clearly respects this decomposition since
$P^{\prime}$ is itself an $S_{k}$-invariant subspace. Note that the first
summand is nothing but the stalk of
$\varepsilon_{*}^{k}\mathbb{C}_{\mathscr{D}^{k}(f)}$ by definition of
$P^{\prime}$. Thus, we may conclude the proof by showing that the second
summand has no alternating part.

To see the latter, let $x=(x_{1},\dots,x_{k})\in P\setminus P^{\prime}$ be an
arbitrary point and $c\in\mathbb{C}_{D^{k}(f),x}$ an element of its associated
stalk. Since $x$ has to be a diagonal point, there exists one pair of indices
$0<i<j\leq k$ such that $x_{i}=x_{j}$. Let $\tau\in S_{k}$ be the
corresponding transposition. Then we have

$-\operatorname{Alt}c=\tau*\operatorname{Alt}c=\operatorname{Alt}(\tau*c)=\operatorname{Alt}c,$

because $\operatorname{Alt}c$ is alternating and $\tau$ takes the point $x$ to
itself. Thus, $\operatorname{Alt}c=0$ for every element $c$ of any stalk of
$\mathbb{C}_{D^{k}(f)}$ at points outside $\mathscr{D}^{k}(f)$. ∎

######  Remark 5.7.

Parallel to (2) we have the equality

$H^{i}(Y,\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(f)})=H^{i}_{\operatorname{Alt}}(D^{k}(f),\mathbb{C}).$

This shows indirectly that $\mathscr{D}^{k}(f)$ and $D^{k}(f)$ have the same
alternating cohomology, thanks to the equality
$\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{D^{\bullet}(f)},\partial^{\bullet}\right)=\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{\mathscr{D}^{\bullet}(f)},\partial^{\bullet}\right)$
proved in Lemma 5.6.

As in the case of $\mathscr{D}^{k}(f)$, if $f\colon X\to Z$ is a mapping
between two complex analytic manifolds with $n=\operatorname{dim}X$ and
$p=\operatorname{dim}Z$, then every irreducible component of $D^{k}(f)$ has
dimension at least $kn-(k-1)p$. The use of $D^{k}(f)$ rather than
$\mathscr{D}^{k}(f)$ calls for the following adaptation:

######  Definition 5.8.

A finite map-germ $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{p},0)$ with $p>n$
is _dimensionally correct_ if for all $k\geq 2$ the multiple point space
$D^{k}(f)$ is either empty or has dimension $kn-(k-1)p$.

It is clear that every unfolding $F$ of a finite map-germ $f$ must be finite
as well. Moreover, any stable unfolding of $F$ is a stable unfolding of $f$ as
well. From this observation, one concludes easily the following result:

######  Proposition 5.9.

If a map-germ $f$ is dimensionally correct, then every unfolding $F$ of $f$ is
dimensionally correct.

######  Remark 5.10.

The previous assertion does not hold if one replaces the property of being
dimensionally correct by that of being strictly dimensionally correct (see
Example 5.4). Being dimensionally correct implies being strictly dimensionally
correct, but the converse is not true, as shown again by the germ
$(x,y)\mapsto(x,y^{2},y^{3},0)$.

###  5.3. The image computing spectral sequence

Let $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ be a finite map-germ with
$p>n$ and let $F$ be a one-parameter unfolding of $f$. We wish to compute the
reduced cohomology of the disentanglement $Y_{\delta}$ of a good
representative of $F$. Goryunov and Mond have shown [GM93] how to do this in
terms of the alternating complex. We are basically going to rephrase their
argument in the derived category so that we can bring the perversity of the
alternating sheaves, Theorem 5.3, into the picture.

In the derived category, Proposition 5.1 and Lemma 5.6 assert that the
constant sheaf $\mathbb{C}_{\mathcal{Y}}$ on the image $\mathcal{Y}$ of the
unfolding $F$ is quasi-isomorphic to the alternating complex
$\left(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{D^{k}(F)},\partial^{\bullet}\right)$
and we may thus replace $\mathbb{C}_{\mathcal{Y}}$ by this complex.

If we let $\pi\colon\mathcal{Y}\to T$ be the projection onto the unfolding
parameter, then the reduced cohomology of the disentanglement $Y_{\delta}$ are
the vanishing cycles of $\pi$ at the origin:

$\displaystyle\tilde{H}^{i}(Y_{\delta})$ $\displaystyle\cong$
$\displaystyle\mathcal{H}^{i}(\phi_{\pi}\left(\mathbb{C}_{\mathcal{Y}}\right))_{0}$
$\displaystyle\cong$
$\displaystyle\mathcal{H}^{i}(\left(\phi_{\pi}(\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{D^{\bullet}(F)})\right)_{0}.$

The nearby and the vanishing cycles of $\pi\colon\mathcal{Y}\to T$ are
computed using the well known diagram

(3)
$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathcal{Y}\setminus
Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{Y_{\delta}\times
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\\{0\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$$\textstyle{T^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$

for a good representative $F$ of a $1$-parameter family and its image
$\mathcal{Y}$. Here, $\exp\colon S\to T^{*}$ is the universal cover of the
punctured unit disc and $Y_{\delta}\times S$ the trivialized fiber product
over the infinite strip $S$, cf. [Dim04, Chapter 4.2].

######  Lemma 5.11.

For a good representative $F$ of a $1$-parameter unfolding of a
$\mathcal{K}$-finite map $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with
$n<p$ there is a spectral sequence with first page

$\displaystyle E^{i,j}_{1}$ $\displaystyle=$
$\displaystyle\mathcal{H}^{i}\left(\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{j+1}_{*}\mathbb{C}_{D^{j+1}(F)}\right)\right)_{0}$

converging to the reduced cohomology of the disentanglement $Y_{\delta}$ of
$f$

$\tilde{H}^{i+j}(Y_{\delta})=\mathcal{H}^{i+j}(\phi_{\pi}\mathbb{C}_{\mathcal{Y}})_{0}.$

Here $\pi\colon\mathcal{Y}\to T$ is the projection of the image $\mathcal{Y}$
of $F$ to the deformation parameter.

###### Proof.

The standard procedure to obtain a quasi-isomorphism of the complex
$\operatorname{Alt}\varepsilon^{\bullet}_{*}\mathbb{C}_{D^{\bullet}(F)}$ with
a complex of injectives is to construct a double complex of injectives
$I^{\bullet,\bullet}$ where each column $I^{\bullet,j}$ resolves the sheaf
$\operatorname{Alt}\varepsilon_{*}^{j+1}\mathbb{C}_{D^{j+1}(F)}$, cf. [GM93,
Section 2]. Then the original complex is quasi-isomorphic to the total complex
$\mathrm{Tot}(I^{\bullet,\bullet})$ of this double complex
$I^{\bullet,\bullet}$.

The nearby cycles of $\pi$ are defined as

$\psi_{\pi}\left(\operatorname{Alt}\varepsilon_{*}^{\bullet}\mathbb{C}_{D^{\bullet}(F)}\right):=i^{-1}R(\rho\circ
j)_{*}(\rho\circ
j)^{-1}\left(\operatorname{Alt}\varepsilon_{*}^{\bullet}\mathbb{C}_{D^{\bullet}(F)}\right).$

This is a composition of three functors. The first one, $(\rho\circ j)^{-1}$,
takes injective sheaves on $\mathcal{Y}$ to injective sheaves on the open
subset $\mathcal{Y}\setminus Y_{0}$, see [KS94, Proposition 2.4.1] and the
third one, $i^{-1}$, is an exact functor. To compute the nearby cycles of
$\operatorname{Alt}\varepsilon_{*}^{\bullet}\mathbb{C}_{D^{\bullet}(F)}$—which
involves the derived pushforward in the middle—we may therefore apply the
functor $i^{-1}(\rho\circ j)_{*}(\rho\circ j)^{-1}$ to
$\mathrm{Tot}(I^{\bullet,\bullet})$. Note that applying this functor to a
seperate column $I^{\bullet,j}$ yields the nearby cycles of the single sheaf
$\operatorname{Alt}\varepsilon_{*}^{j+1}\mathbb{C}_{D^{j+1}(F)}$ considered as
a complex of sheaves concentrated in a single degree.

The vanishing cycles $\phi_{\pi}\mathbb{C}_{\mathcal{Y}}$ of $\pi$ are defined
as the mapping cone of the comparison morphism $c$

$\textstyle{i^{-1}\mathbb{C}_{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\psi_{\pi}\mathbb{C}_{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi_{\pi}\mathbb{C}_{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[+1]}$.

Following Definition 1.2), the image $\mathcal{Y}$ of a good representative is
connected, hence the stalk of $i^{-1}\mathbb{C}_{\mathcal{Y}}$ at the origin
is simply the vector space $\mathbb{C}$, concentrated in degree zero. It is
easy to check that on this single nontrivial term the comparison morphism $c$
is always an inclusion which makes the vanishing cycles coincide with the
reduced cohomology of the disentanglement, cf. (1).

We may again replace $\mathbb{C}_{\mathcal{Y}}$ by the alternating complex
$\operatorname{Alt}\varepsilon_{*}^{\bullet}\mathbb{C}_{D^{\bullet}(F)}$ and
subsequently by $\mathrm{Tot}(I^{\bullet,\bullet})$ in the distinguished
triangle of the comparison morphism:

$\textstyle{i^{-1}\mathrm{Tot}(I^{\bullet,\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{i^{-1}(\rho\circ
j)_{*}(\rho\circ
j)^{-1}\mathrm{Tot}(I^{\bullet,\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi_{\pi}\mathrm{Tot}(I^{\bullet,\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[+1]}$.

Now note that the functor $i^{-1}(\rho\circ j)_{*}(\rho\circ j)^{-1}$ commutes
with taking the total complex in a natural way. Thus we may identify the
distinguished triangle of the comparison morphism $c$ with the total complex
of the following “comparison morphism of double complexes”

$\textstyle{i^{-1}I^{\bullet,\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{c}}$$\textstyle{i^{-1}(\rho\circ
j)_{*}(\rho\circ
j)^{-1}I^{\bullet,\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Cone}(\tilde{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[+1]}$

which turns the vanishing cycles $\phi_{\pi}\mathrm{Tot}(I^{\bullet,\bullet})$
into the total complex of the double complex $\text{Cone}(\tilde{c})$.

The spectral sequence in question is now the spectral sequence of this double
complex $\text{Cone}(\tilde{c})$. A column-wise inspection of this double
complex reveals that we indeed find the vanishing cycles of the alternating
sheaves on the first page. ∎

######  Remark 5.12.

From the fact that the spaces $D^{k}(F)$ are empty for $k$ big enough, it
follows that the spectral sequence collapses at a certain page. Since we chose
a field for the coefficients in cohomology, the infinity page determines the
cohomology of the total complex. To be precise, there is an isomorphism of
vector spaces

$\mathcal{H}^{\ell}(\phi_{\pi}\mathbb{C}_{\mathcal{Y}})_{0}\cong\bigoplus_{i}E^{i,\ell-i}_{\infty}.$

######  Example 5.13.

Here we come back to the immersion bi-germ
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ associated to a
hypersurface $X=V(g)$, introduced in Example 2.6, where we find a particularly
simple spectral sequence. First of all, there are no triple or higher
multiplicity points. This is because the stable unfolding $F$ of $f$ (or any
unfolding of $f$ for that matter) is also a bi-germ of an immersion, hence it
cannot map more than two points to one. This gives
$\mathscr{D}^{k}(F)=\emptyset$, and thus $D^{k}(f)=\emptyset$, for all $k\geq
3$ . Consequently, the nonzero terms of the first page
$E_{1}^{\bullet,\bullet}$ are concentrated in the second column, corresponding
to the alternating cohomologies
$H^{i}_{\operatorname{Alt}}(D^{2}(f_{\delta}))$.

The fact that there are no triple points implies also that the maps
$D^{2}(F)\to D(F)$ and $D^{2}(f_{\delta})\to D(f_{\delta})$ are
homeomorphisms. Recall also from Example 2.6 that $D(f_{\delta})=M_{1}\sqcup
M_{2}$, where $M_{1}$ and $M_{2}$ are copies of the Milnor fibre of $X=V(g)$.
The homeomorphism $D^{2}(f_{\delta})\to D(f_{\delta})$ transforms the action
of the generator $\sigma\in S_{2}$ into the map taking a point $x\in M_{1}$ to
the same point in $M_{2}$, and vice-versa. In particular, after identifying
$D^{2}(f_{\delta})$ with $M_{1}\sqcup M_{2}$, we can choose a system of
generators of the cohomology $H^{i}(D^{2}(f_{\delta}))$ consisting of some
cocycles $c_{i}$ generating the cohomology of $M_{1}$ and their corresponding
cocycles $\sigma\cdot c_{i}$, which generate the cohomology of $M_{2}$. It now
follows immediately that the alternating part is generated by the cocycles
$c_{i}-\sigma\cdot c_{i}\in H^{i}_{\operatorname{Alt}}(D^{2}(f_{\delta}))$.
The linear map extending $c_{i}\mapsto c_{i}-\sigma\cdot c_{i}$ gives an
isomorphism

$H^{i}(M)\cong H^{i}_{\operatorname{Alt}}(D^{2}(f)).$

Finally, since $E_{1}^{\bullet,\bullet}$ contains a single nonzero column, the
spectral sequence collapses at page one, that is,
$E_{\infty}^{\bullet,\bullet}=E_{1}^{\bullet,\bullet}$. Now, the isomorphism
$\mathcal{H}^{\ell}(\phi_{\pi}\mathbb{C}_{\mathcal{Y}})_{0}\cong\bigoplus_{i}E^{i,\ell-i}_{\infty}$
takes the form

$\tilde{H}^{i}(M)\cong\tilde{H}^{i+1}(Y_{\delta}),$

as Remark 1.8 claimed.

######  Remark 5.14.

The previous example has a more visual version in homology (Figure 6 depicts
the case of $g=x^{2}+y^{2}$ which, after a change of coordinates, is the same
as in Figure 5, but with a real picture better suited for the present
discussion). We know that $D(f_{\delta})$ consists of two copies $M_{1}$ and
$M_{2}$ of $M$, contained in two copies $U_{1}$ and $U_{2}$ of an open ball
$U\in\mathbb{C}^{n}$. Now let $c$ be a cycle in $M$ and let $c_{i}$ be the
corresponding copies in $U_{i}$ (the two circles one the left side in Figure
5). Since $U$ is contractible, there are chains $a_{i}$ in $U_{i}$ with
boundary $\delta a_{i}=c_{i}$(the green and blue disks). Observe that
$\partial a_{i}$ is supported on $M_{i}$, which is the double point space of
$f_{\delta}$, and that $f_{\delta}$ glues $a_{1}$ and $a_{2}$ along the
boundaries $\partial a_{i}$. After changing the sign of $a_{2}$ if necessary,
we observe that $f_{*}(a_{1}+a_{2})$ is an $(i+1)$-dimensional cycle on
$Y_{\delta}$ (the blue and green cycle on the right side). The desired
isomorphism is determined by $c\mapsto f_{*}(a_{1}+a_{2})$.

Figure 6. The disentanglement of a bi-germ of immersion has the homology of a
two-point suspension.

##  6\. Proof of Theorem 2.8

In this section we will use the notation of Theorem 2.8 and impose the
hypothesis found there, that is,
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ is a dimensionally correct
map-germ with $p>n$ and instability locus $\mathrm{Inst}(f)$ of dimension $d$,
$F\colon W\to V\times T$ is a good representative of a one-parameter unfolding
of $f$ and $\mathcal{Y}=\operatorname{Im}F$ is the image of $F$.

We intend to control the stalk cohomology of
$\phi_{\pi}\mathbb{C}_{\mathcal{Y}}$, i.e. the reduced cohomology of the
disentanglement, by means of the first page of the spectral sequence in Lemma
5.11. Thus, we have to bound the vanishing cohomology of the alternating
sheaves $\operatorname{Alt}\varepsilon_{*}^{j+1}\mathbb{C}_{D^{j+1}(F)}$ on
the image and we will do so by exploiting the perversity of their respective
shifts and estimating the dimension of the support of their vanishing cycles.

######  Lemma 6.1.

The support of
$\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}\right)$
is contained in $\mathrm{Inst}(f)$.

###### Proof.

Take a point away from the instability locus $y\in Y\setminus\mathrm{Inst}(f)$
and let $S=f^{-1}(y)$. Since $f$ is stable at $y$, the unfolding
$(F,S\times\\{0\\})$ of the germ $(f,S)$ is trivial. Therefore, there exist an
unfolding $\Phi$ of $(\operatorname{id}_{\mathbb{C}^{n}},S)$ and an unfolding
$\Psi$ of $(\operatorname{id}_{\mathbb{C}^{p}},0)$, such that

$(F,S\times\\{0\\})=\Psi^{-1}\circ((f,S)\times\operatorname{id}_{(T,0)})\circ\Phi.$

We choose representatives $F,f,\Phi$ and $\Psi$ satisfying the equality

$F=\Psi^{-1}\circ(f\times\operatorname{id}_{T})\circ\Phi.$

As one may easily check, this makes the multiple points $D^{k}(F)$ and
$D^{k}(f\times\operatorname{id}_{T})$ isomorphic via
$\Phi^{-1}\times\stackrel{{\scriptstyle k}}{{\dots}}\times\Phi^{-1}$.
Moreover, the multiple points in $D^{k}(f\times\operatorname{id}_{T})$ are of
the form $((x_{1},t),\dots,(x_{k},t))$. Forgetting all but one of the copies
of $t$ gives an isomorphism [NP17, Proposition 3.8]

$\Omega\colon D^{k}(f\times\operatorname{id}_{T})\to D^{k}(f)\times T.$

In turn, the spaces $\mathcal{Y}$ and $Y\times T$ are isomorphic via $\Psi$.

There are two geometric considerations where the $k$-fold product structure of
$\Phi^{-1}\times\stackrel{{\scriptstyle k}}{{\dots}}\times\Phi^{-1}$ plays a
role: First of all, the isomorphisms and the $\varepsilon^{k}$ mappings are
compatible in the sense that there is a commutative diagram

${\phantom{Aaa}D^{k}(F)\phantom{Aaa}}$${\phantom{Aaa}D^{k}(f)\times
T\phantom{Aaa}}$${\phantom{Aaa}\mathcal{Y}\phantom{Aaa}}$${\phantom{Aaa}Y\times
T.\phantom{Aaa}}$$\scriptstyle{\Omega\circ(\Phi^{-1}\times\stackrel{{\scriptstyle
k}}{{\dots}}\times\Phi^{-1})}$$\scriptstyle{\varepsilon^{k}}$$\scriptstyle{\varepsilon^{k}\times\operatorname{id}_{T}}$$\scriptstyle{\Psi}$

Second, the map $\Omega\circ(\Phi^{-1}\times\stackrel{{\scriptstyle
k}}{{\dots}}\times\Phi^{-1})$ is compatible with the action of the symmetric
group $S_{k}$ on both $D^{k}(F)$ and on $D^{k}(f)\times T$ (the latter induced
by the action of $S_{k}$ on $D^{k}(f)$). This makes the construction of the
$\operatorname{Alt}$ operators on both sides equivalent and the pushforward
$\Psi_{*}$ takes the sheaf
$\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}$ on $\mathcal{Y}$
to the sheaf
$\operatorname{Alt}(\varepsilon^{k}\times\operatorname{id}_{T})_{*}\mathbb{C}_{D^{k}(f)\times
T}$ on $Y\times T$.

We can now show that $y$ is not in the support of
$\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}\right)$
by checking the vanishing of the stalk cohomology: Take a suitable ball
$B_{r}$ in $V$ around $y$. Then, for $\delta$ sufficiently small we find

$\displaystyle\mathcal{H}^{i}\left(\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}\right)\right)_{y}$
$\displaystyle=$ $\displaystyle\mathbb{H}^{i+1}\left(\mathcal{Y}\cap
B_{r},\mathcal{Y}\cap\pi^{-1}(\delta)\cap
B_{r};\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}\right)$
$\displaystyle\cong$ $\displaystyle\mathbb{H}^{i+1}\left((Y\times T)\cap
B_{r},(Y\times\\{\delta\\})\cap
B_{r};\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(f)\times
T}\right)$ $\displaystyle=$ $\displaystyle 0.$

The vanishing in the last line follows from the fact that due to the product
structure clearly $(Y\times T)\cap B_{r}$ retracts onto the fiber
$(Y\times\\{\delta\\})\cap B_{r}$ for $r\gg\delta>0$. ∎

Now we can determine where the non-trivial entries of the first page of the
spectral sequence in Lemma 5.11 are concentrated.

######  Proposition 6.2.

Let $F$ be a dimensionally correct one-parameter unfolding of a germ
$(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p>n$. For every integer $k\geq
2$, the nonzero cohomologies

$\mathcal{H}^{i}\left(\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}\right)\right)_{0}$

are concentrated in degrees $i\geq 0$ with

$kn-(k-1)p-d\leq i\leq kn-(k-1)p.$

In the case $k=1$ the above vanishing cycles are all zero.

###### Proof.

Since $F$ is dimensionally correct, the sheaves

$\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}[kn-(k-1)p+1]$

are perverse, and, by virtue of Theorem 3.6, so are the sheaves

$\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{k}_{*}\mathbb{C}_{D^{k}(F)}\right)[kn-(k-1)p]$

on $Y_{0}$. Since these sheaves are supported on a space of dimension at most
$d$, it follows from Proposition 3.4 that the non trivial cohomologies of
their stalks at the origin are concentrated in degrees $i$ with $-d\leq i\leq
0$.

Shifting everything back by $kn-(k-1)p$, we obtain the desired bounds for
$k\geq 2$. Note that, since the alternating cohomology

$\displaystyle\mathcal{H}^{i}\left(\phi_{\pi}\left(\operatorname{Alt}\varepsilon_{*}^{k}\mathbb{C}_{D^{k}(F)}\right)\right)_{0}$
$\displaystyle=$
$\displaystyle\tilde{H}^{i}_{\operatorname{Alt}}(D^{k}(f_{\delta}))\subset\tilde{H}^{i}(D^{k}(f_{\delta}))$

is a subspace of the singular reduced cohomology of the disentanglement, there
can be no contributions in negative degrees.

For $k=1$ the disentanglement always provides a product structure

$(D^{1}(F),(0,0))\cong(X\times T,(0,0))$

and therefore

$\mathcal{H}^{i}\left(\phi_{\pi}\left(\operatorname{Alt}\varepsilon^{1}_{*}\mathbb{C}_{D^{1}(F)}\right)\right)_{0}=H^{i}(X\times
T,X\times\delta)=0.$

This finishes the proof. ∎

###### Proof of Theorem 2.8.

The proof follows by the same spectral sequence argument as in [SZ18, Theorem
1.1]: The concentration of cohomology follows immediately from the isomorphism

$\mathcal{H}^{\ell}(\phi_{\pi}\mathbb{C}_{\mathcal{Y}})_{0}\cong\bigoplus_{i}E^{i,\ell-i}_{\infty}$

from Remark 5.12 and the vanishing of the entries of the first page coming
from Proposition 6.2. ∎

Let us give an example which illustrates the general situation:

######  Example 6.3.

Let $f\colon(\mathbb{C}^{16},0)\to(\mathbb{C}^{21},0)$ be a dimensionally
correct map whose instability locus has dimension $d=2$. According to
Proposition 6.2, the possibly nonzero entries in the first page of the
spectral sequence $E_{1}^{i,j}$ from Lemma 5.11 are the following:

$\begin{array}[]{ccccccccccccccccccccccccccccc}\lx@intercol\hfil
i\uparrow\hfil\lx@intercol\vrule\lx@intercol&&&&\\\
\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&&&&\\\ \lx@intercol\hfil
11\hfil\lx@intercol\vrule\lx@intercol&&H^{11}_{\operatorname{Alt}}(D^{2}(f_{\delta}))&&&\\\
\lx@intercol\hfil
10\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&H^{10}_{\operatorname{Alt}}(D^{2}(f_{\delta}))&&\\\
\lx@intercol\hfil
9\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&H^{9}_{\operatorname{Alt}}(D^{2}(f_{\delta}))&&\\\
\lx@intercol\hfil 8\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&&\\\
\lx@intercol\hfil 7\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&&\\\
\lx@intercol\hfil
6\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&H^{6}_{\operatorname{Alt}}(D^{3}(f_{\delta}))&\\\
\lx@intercol\hfil
5\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&H^{5}_{\operatorname{Alt}}(D^{3}(f_{\delta}))&\\\
\lx@intercol\hfil
4\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&H^{4}_{\operatorname{Alt}}(D^{3}(f_{\delta}))&\\\
\lx@intercol\hfil 3\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&&\\\
\lx@intercol\hfil 2\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&&\\\
\lx@intercol\hfil
1\hfil\lx@intercol\vrule\lx@intercol&\phantom{\vdots}&&&H^{1}_{\operatorname{Alt}}(D^{4}(f_{\delta}))\\\
\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&\phantom{H^{10}_{\operatorname{Alt}}(D^{2}(f_{\delta}))}&\phantom{\vdots}&&H^{0}_{\operatorname{Alt}}(D^{4}(f_{\delta}))\\\
\hline\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&0&1&2&3&\underset{j}{\longrightarrow}\end{array}$

In this case, the sequence necessarily collapses on this first page. In
general, the differentials between the nonzero entries on the first page can
lead to cancellations for the following ones and on the limit page. In either
case, the positions of the nonzero entries on the first page give a bound on
the nonzero entries of
$\bigoplus_{i}E_{\infty}^{i,k-i}\cong\mathcal{H}^{k}(\phi_{\pi}\mathbb{C}_{\mathcal{Y}})_{0}$.
For this particular example we see that the non-trivial cohomologies of a
disentanglement $Y_{\delta}$ are concentrated on the degrees
$\ell\in\\{3,4\\}\cup\\{6,7,8\\}\cup\\{10,11,12\\}.$

##  7\. Monodromy for disentanglements

Our study of the monodromy for disentanglements in Theorem 2.11 will be based
on the monodromy on the multiple point spaces and their alternating
cohomology. This requires an adaptation of the definitions to fibrations with
group actions.

Note that any stable one parameter unfolding

$F\colon(\mathbb{C}^{n},0)\times(\mathbb{C},0)\to(\mathbb{C}^{p},0)\times(\mathbb{C},0)$

of a finite map germ $f$ with $p>n$ as in Theorem 2.11 gives rise to analytic
fibrations

(4) $\pi\circ\varepsilon^{k}\colon D^{k}(F)\cap(\varepsilon^{k})^{-1}(B\times
D\setminus\\{0\\})\to D\setminus\\{0\\}$

for suitable choices of a representative $F$, a ball $B\subset\mathbb{C}^{p}$
in the target of $f$, and a disc $D\subset\mathbb{C}$ in the parameter space.
By construction, the symmetric group $\mathfrak{S}_{k}$ acts fiberwise on the
multiple point space $D^{k}(F)\cap(\varepsilon^{k})^{-1}(B\times D)$, for each
$k>0$.

As already noted by K. Houston in [Hou97a, Lemma 2.14], there exist
equivariant local trivializations. We may therefore suppose that the
trivialization of the pullback of the fibration (4) to the universal cover
$\exp\colon S\to D\setminus\\{0\\}$ is $\mathfrak{S}_{k}$-equivariant and that
the parallel transport of the fiber

$h\colon D^{k}(f_{\delta})\to D^{k}(f_{\delta})$

along a closed, counterclockwise loop in $D^{*}$ around the origin commutes
with the $\mathfrak{S}_{k}$-action on $D^{k}(f_{\delta})$. It is easy to see
that this implies that also the maps induced by $h$ on cohomology

$h^{i}\colon H^{i}(D^{k}(f_{\delta}))\to H^{i}(D^{k}(f_{\delta}))$

restrict to operators on the alternating cohomology
$H^{i}_{\operatorname{Alt}}(D^{k}(f_{\delta}))$. We therefore have a well
defined monodromy action

$h\colon\phi_{\pi}\left(\operatorname{Alt}\varepsilon_{*}^{k}\mathbb{C}_{D^{k}(F)}\right)\to\phi_{\pi}\left(\operatorname{Alt}\varepsilon_{*}^{k}\mathbb{C}_{D^{k}(F)}\right)$

on the vanishing cycles of the alternating sheaves. As we shall see now, this
gives the vanishing cycles of the alternating sheaves the structure of a
$\mathbb{C}[s,s^{-1}]$-module where $s$ is a formal variable standing for the
monodromy automorphism $h$.

To relate monodromy to our usual spectral sequence, it is useful to introduce
a point of view about eigenvalues and Jordan blocks slightly different from
that of the introduction: Consider the monodromy of a fibration
$E\xrightarrow{\pi}D^{*}$ over the punctured disk with fibre $F$, and assume
that the fibration was obtained as a small representative of an analytic germ
$(E,0)\to(D,0)$, as in [Lê78]. Consider the one-variable Laurent polynomial
ring $R:=\mathbb{C}[s,s^{-1}].$ The monodromy action gives each cohomology
$H^{i}(F)$ an $R$-module structure, by declaring that multiplication by $s$
acts on $H^{i}(F)$ by the monodromy automorphism $h^{i}\colon H^{i}(F)\to
H^{i}(F)$. It is well known that $F$ is a homotopy equivalent to a finite CW-
complex, and that each monodromy automorphism $h^{i}$ is idempotent. As a
consequence, each $H^{i}(F)$ is a finitely generated torsion $R$-module. Note
that $R$ is a principal ideal domain so that according to the structure
theorem, every finitely generated torsion $R$-module $A$ can be uniquely
decomposed as

$A\cong\oplus_{j=1}^{m}R/(s-\lambda_{j})^{a_{j}},$

where $\lambda_{j}$ are non-zero complex numbers and $a_{j}$ are positive
integers. It turns out that the set of eigenvalues of $h^{i}$ can be recovered
as

$\operatorname{Supp}(A):=\\{\lambda_{j}\\}_{1\leq j\leq m},$

and the maximal size of the Jordan blocks is

$J(A):=\max_{1\leq j\leq m}\\{a_{j}\\}.$

Now we list two simple facts related to finite generated torsion $R$-modules,
where the proofs are left to the reader.

######  Lemma 7.1.

Let $A\overset{u}{\to}B\overset{v}{\to}C$ be a complex of finitely generated
torsion $R$-modules. Then $\operatorname{Ker}v/\operatorname{Im}u$ is also a
finitely generated torsion $R$-module and we have that

  * (a)

$\operatorname{Supp}(\operatorname{Ker}v/\operatorname{Im}u)\subset\operatorname{Supp}(B)$,

  * (b)

$J(\operatorname{Ker}v/\operatorname{Im}u)\leq J(B)$.

######  Lemma 7.2.

Let $0\to A\to B\to C\to 0$ be a short exact sequence of finitely generated
torsion $R$-modules. Then we have that

  * (a)

$\operatorname{Supp}(B)=\operatorname{Supp}(A)\cup\operatorname{Supp}(C)$,

  * (b)

$\max\\{J(A),J(C)\\}\leq J(B)\leq J(A)+J(C)$.

###### Proof of Theorem 2.11.

The explanation in the beginning of this section shows that the spectral
sequence

$E_{1}^{i,j}=\mathcal{H}^{i}\left(\phi_{\pi}\left(\operatorname{Alt}\varepsilon_{*}\mathbb{C}_{D^{j+1}(F)}\right)\right)_{0}\Rightarrow\tilde{H}^{i+j}(Y_{\delta}).$

can be studied in the category of $R$-modules, where every entry in it is a
torsion $R$-module.

The assumption that $f$ has corank one implies that the one-parameter
unfolding $F$ of $f$ also has corank one. Since $F$ is stable by assumption,
the spaces $D^{k}(F)$ are smooth and have the right dimension. The generic
fibre $D^{k}(f_{\delta})$ coincides with the Milnor fiber of the germ of maps:

$D^{k+1}(F)\to T.$

Then the classical monodromy theorem about eigenvalues and size of Jordan
blocks apply to the monodromy automorphisms $H^{i}(D^{k}(f_{\delta}))\to
H^{i}(D^{k}(f_{\delta}))$. Hence, by Lemma 7.2, the size of the Jordan blocks
of the monodromy automorphism
$H^{i}_{\operatorname{Alt}}(D^{k}(f_{\delta}))\to
H^{i}_{\operatorname{Alt}}(D^{k}(f_{\delta}))$ is at most $i+1$, i.e.,

$J(E^{i,k-1}_{1})\leq i+1$

Since the page $E^{\bullet,\bullet}_{r+1}$ is computed from
$E^{\bullet,\bullet}_{r}$ by taking cohomologies, we conclude from Lemma 7.1
that the infinity page satisfies the above property as well. Note that
$E_{1}^{i,k-1}=0$ when $k\leq 1$. By the isomorphism
$H^{\ell}(Y_{\delta})\cong\bigoplus_{i}E^{i,\ell-i}_{\infty}$ and Lemma 7.2,
we obtain that

$J(H^{\ell}(Y_{\delta}))\leq\sum_{0\leq
i<\ell}J(E^{i,\ell-i}_{\infty})\leq\sum_{0\leq i<\ell}(i+1)=\ell(\ell+1)/2.$

When $f$ has only isolated instability locus and $p>n+1$, then Proposition 6.2
gives us that $E_{1}^{i,k-1}$ are concentrated in degrees $i=kn-(k-1)p$. In
particular, $p>n+1$ implies that the spectral sequence degenerates at
$E_{1}$-page. Hence the possibly non-trivial reduced cohomologies are

$\tilde{H}^{kn-(k-1)(p-1)}(Y_{\delta})\cong\tilde{H}^{kn-(k-1)p}_{\operatorname{Alt}}(D^{k}(f_{\delta}))$
for $1<k\leq\left\lfloor\frac{p}{p-n}\right\rfloor$

with the corresponding size of Jordan block at most $kn-(k-1)p+1$. ∎

######  Remark 7.3.

Observe that the proof of Theorem 2.11 can be used to show that the
eigenvalues of $h^{i}\colon H^{i}(Y_{\delta})\to H^{i}(Y_{\delta})$ are roots
of unity, directly from the classical result about eigenvalues of the
monodromy of hypersurfaces (that is, with no resort to the more general result
of Lê for fibrations on analytic spaces). Moreover, it shows that the
eigenvalues of the monodromy on $H^{i}(Y_{\delta})$ are a subset of the union
of eigenvalues of the monodromies on
$H^{i}_{\operatorname{Alt}}(D^{k}(f_{\delta})),k\geq 0$.

######  Remark 7.4.

We saw earlier in Remark 5.14 how the study of Milnor fibers of hypersurfaces
$g\colon(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ was equivalent to the study of
disentanglements of bi-germs
$f\colon(\mathbb{C}^{n},\\{p,q\\})\to(\mathbb{C}^{n+1},0)$. The above
considerations show that this also comprises the monodromy: Since all the
ingredients involved in the spectral sequence admit an $R$-module structure,
the isomorphism $\tilde{H}^{i}(M)\cong\tilde{H}^{i+1}(Y_{\delta})$ is an
isomorphism of $R$-modules and therefore the monodromies on both sides are
compatible by the considerations in Remark 1.8.

## References

  * [DM91] J. Damon and D. Mond, _A-codimension and the vanishing topology of discriminants._ , Invent. Math. 106 (1991), no. 2, 217–242.
  * [Dim92] A. Dimca, Singularities and Topology of Hypersurfaces. Universitext. Springer-Verlag, New York 1992.
  * [Dim04] A. Dimca, Sheaves in topology. Universitext. Springer-Verlag, Berlin, 2004. xvi+236 pp.
  * [EN62] J. A. Eagon, D. G. Northcott, _Ideals defined by matrices and a certain complex associated with them_ , Proc. Royal Soc. 269 (1962), 188–204.
  * [Ga83] T. Gaffney, _Multiple points and associated ramification loci_ , Proc. Symp. Pure Math. (American Mathematical Society, Providence) 40 (1983), pp. 429–437.
  * [GN20] R. Giménez Conejero and J.J. Nuño-Ballesteros, _On Whitney equisingular unfoldings of corank 1 germs_. Preprint.
  * [GM93] V. V. Goryunov, D. Mond, _Vanishing cohomology of singularities of mappings_ , Compositio Math. 89 (1993), pp. 45–80.
  * [Hou97a] K. Houston, _Local topology of images of finite complex analytic maps_ , Topology 36, 5, (1997), pp. 1007–1121.
  * [Hou97b] K. Houston, _Global topology of images of finite complex analytic maps_ , Math. Proc. Camb, Phil. Soc. (1997) 122, pp. 491–502.
  * [Hou00] K. Houston, _Perverse Sheaves on Image Multiple Point Spaces_ , Compositio Mathematica 123, (2000), pp. 117–130.
  * [dJP00] T. de Jong and G. Pfister, _Local analytic geometry, basic theory and applications_ , Springer, 2000.
  * [KS94] M. Kashiwara, P. Shapira, _Sheaves on manifolds_ , Grundlehren der Mathematischen Wissenschaften 292, Springer Verlag (1994).
  * [KM75] M. Kato, and Y. Matsumoto, _On the connectivity of the Milnor fiber of a holomorphic function at a critical point_ , Proc. Internat. Conf., Tokyo, Univ. Tokyo Press, (1975).
  * [Lê78] T. D. Lê, _The geometry of the monodromy theorem. C. P. Ramanujan–a tribute._ Tata Inst. Fund. Res. Studies in Math. Springer, Berlin-New York (1978), no. 8, pp. 157–173.
  * [MM89] W. L. Marar and D. Mond, _Multiple point schemes for corank $1$ maps_, J. London Math. Soc. 2 (1989), no. 3, 553–567.
  * [MNP12] W. L. Marar, J.J. Nuño-Ballesteros and G. Peñafort Sanchis, _Double point curves for corank 2 map germs from $\mathbb{C}^{2}$ to $\mathbb{C}^{3}$_. Topology Appl. 2 (2012), 526–536.
  * [Mat68] J.N. Mather, _Stability of $C^{\infty}$ mappings IV: Finitely determined map germs_, Publications Mathématiques de l’I.H.É.S. (1968), no. 35, 279–308.
  * [Mat69] J.N. Mather, _Stability of $C^{\infty}$ mappings IV: Classification of stable germs by $\mathbb{R}$-algebras_, Publications Mathématiques de l’I.H.É.S. (1969), no. 37, 223–248.
  * [Mat71] by same author, _Stability of $C^{\infty}$ mappings VI: The nice dimensions._, Lecture Notes in Math. Springer, Berlin 192 (1971), 207–253.
  * [Mil68] J. Milnor. _Singular points of complex hypersurfaces_ (AM-61). Princeton University Press, 1968.
  * [Mon85] D. Mond, _On the classification of germs of maps from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$_, Proc. London Math. Soc. 50 (1985), no. 2, 333–369.
  * [Mon91] by same author, _Vanishing cycles for analytic maps, singularity theory and applications (Warwick 1989)_ , vol. 1462, Springer, New York, 1991.
  * [MN20] D. Mond and J.J. Nuño-Ballesteros, _Singularities of mappings_. Grundlehren der mathematischen Wissenschaften Vol 357. Springer, Cham. 2020.
  * [NORW16] T. Nishimura, R. Oset Sinha, M. A. S. Ruas, and R. Wik Atique, _Liftable vector fields over corank one multigerms_ , Mathematische Annalen 366 (2016), no. 1, 573–611.
  * [NP17] J.J. Nuño-Ballesteros and G. Peñafort Sanchis, _Multiple point spaces of finite holomorphic maps_ , The Quarterly Journal of Mathematics 68 (2017), no. 2, 369–390.
  * [SZ18] G. Peñafort Sanchis, M. Zach, _Kato-Matsumoto-type results for disentanglements_ , Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 151, (2021), no. 1, 1–27.
  * [Wal81] C. T. C. Wall, _Finite determinacy of smooth map germs_ , Bull. London Math. Soc. 13 (1981), 481–539.
  * [Wei94] C. Weibel, _An introduction to homological algebra_ (Cambridge Studies in Advanced Mathematics). Cambridge University Press. 1994.


of our proposed method, has been implemented, and the results are presented
below. While AutoAttack suggests using different attack strategies, in our
case, we combine different attack strategies matched with different losses to
make the pool of attacks more strong and more diversified.

Table 23: The proposed method on AutoAttack (Mead setting). The attacks are
APGD-CE, APGD-DLR, FAB, SA.

| CIFAR10
---|---
| Ours
| AUROC$\uparrow$% |  FPR$\downarrow_{95\%}$%
Norm L1 |  |
|  |
$\varepsilon=5$ | 57.1 | 88.4
$\varepsilon=10$ | 67.1 | 75.7
$\varepsilon=15$ | 72.2 | 66.7
$\varepsilon=20$ | 72.7 | 65.2
$\varepsilon=25$ | 72.8 | 65.6
$\varepsilon=30$ | 73.4 | 64.0
$\varepsilon=40$ | 73.6 | 64.0
Norm L2 |  |
|  |
$\varepsilon=0.125$ | 67.4 | 81.0
$\varepsilon=0.25$ | 58.0 | 89.0
$\varepsilon=0.3125$ | 58.1 | 88.8
$\varepsilon=0.5$ | 69.4 | 74.7
$\varepsilon=1$ | 75.1 | 61.6
$\varepsilon=1.5$ | 76.1 | 60.7
$\varepsilon=2$ | 76.1 | 60.5
Norm L∞ |  |
|  |
$\varepsilon=0.03125$ | 75.7 | 61.0
$\varepsilon=0.0625$ | 76.0 | 60.7
$\varepsilon=0.125$ | 76.8 | 60.3
$\varepsilon=0.25$ | 76.8 | 60.0
$\varepsilon=0.3125$ | 78.6 | 57.6
$\varepsilon=0.5$ | 76.1 | 60.3