# On $k$-folding map-germs and hidden symmetries of surfaces in the Euclidean
3-space

Guillermo Peñafort Sanchis and Farid Tari

###### Abstract

Let $M$ be a smooth surface in $\mathbb{R}^{3}$ (or a complex surface in
$\mathbb{C}^{3}$) and $k\geq 2$ be an integer. At any point on $M$ and for any
plane in $\mathbb{R}^{3}$, we construct a holomorphic map-germ
$(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ of the form
$F_{k}(x,y)=(x,y^{k},f(x,y))$, called a $k$-folding map-germ. We study in this
paper the local singularities of $k$-folding map-germs and relate them to the
extrinsic differential geometry of $M$. More precisely, we

  * •

stratify the jet space of $k$-folding map-germs so that the strata of
codimension $\leq 4$ correspond to topologically equivalent
$\mathcal{A}$-finitely determined germs;

  * •

obtain the topological classification of $k$-folding map-germs on generic
surfaces in $\mathbb{R}^{3}$ (or $\mathbb{C}^{3}$);

  * •

generalise the work of Bruce-Wilkinson on folding maps ($k=2$);

  * •

recover, in a unified way, results obtained by considering the contact of
surfaces with lines, planes and spheres;

  * •

discover new robust features on smooth surfaces in $\mathbb{R}^{3}$.

0002010 Mathematics Subject classification: 58K05, 58K65, 53A05. 000Key Words
and Phrases. Singularities of mappings, surfaces, invariants, folding-maps,
stratification, robust features.

##  1 Introduction

The aim of this work is to study $k$-folding map-germs on complex surfaces in
$\mathbb{C}^{3}$ and relate them to the extrinsic differential geometry of
surfaces in $\mathbb{R}^{3}$.

The standard Whitney fold of order $k$ with respect to the plane $\pi_{0}:y=0$
in $\mathbb{C}^{3}$ is the map
$\omega_{k}\colon\mathbb{C}^{3}\to\mathbb{C}^{3},$ given by

$\omega_{k}(x,y,z)=(x,y^{k},z).$

The map $\omega_{k}$ ‘folds’ the space $\mathbb{C}^{3}$ along the plane
$\pi_{0}$, gluing the points $(x,y,z),$ $(x,\xi y,z),\dots,(x,\xi^{k-1}y,z),$
where $\xi=e^{2\pi i/k}$ is a primitive $k^{th}$-root of unity. The Whitney
fold of order $k$ with respect to any plane $\pi$, denoted by
$\omega_{k}^{\pi}$, is defined similarly in §2.

Let $M$ be a complex surface in $\mathbb{C}^{3}$. We call the restriction of
$\omega_{k}^{\pi}$ to $M$ the $k$-folding map on $M$ with respect to $\pi$. As
our study is local, given point $p$ on $M$ and a plane $\pi$ in
$\mathbb{C}^{3}$ through $p$, we choose a coordinate system so that $M$ is
locally the graph of a function $z=f(x,y)$ and $\pi=\pi_{0}:y=0$ (see Remarks
2.3(4)). Then the germ at $p_{0}$ of the $k$-folding map is represented in
standard form by the map-germ $F_{k}:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$,
given by

$F_{k}(x,y)=(x,y^{k},f(x,y)).$ (1)

For an analytic (resp. smooth) surface $M\subset\mathbb{R}^{3}$, the
$k$-folding map at a point $p$ on $M$ is constructed by complexifying $M$
(resp. a certain jet of a parametrisation of $M$) at $p$. The singularities of
a $k$-folding map-germ encode the local symmetries of $M$ with respect to the
(complex) reflection group of order $k$ whose hyperplane arrangement consists
of the single plane $\pi$.

The study of $2$-folding map-germs on surfaces in $\mathbb{R}^{3}$ was carried
out by Bruce and Wilkinson [6, 10, 32] (see [8, 13, 14, 15, 16, 32] for more
work on the subject), without resorting to complexification. The real map-
germs in [6, 10, 32] are called folding map-germs and our $2$-folding map-
germs are their complexifications. Complexifying does not give extra
information when $k=2$. For $k\geq 3$, per contra, $k$-folding maps reveal a
great deal of new geometric information. The local symmetries captured by
these map-germs cannot be seen in the real case, which is why we call them
_hidden symmetries_ of $M\subset\mathbb{R}^{3}$. The loci of their
singularities are visible on $M$ and capture extrinsic geometric information
of the surface.

Bruce and Wilkinson showed that folding maps capture the sub-parabolic and
ridge curves, as well as umbilic points and other special points on these
curves: these are robust features of the surface (i.e., they are special
geometric features that can be traced on an evolving surface; see §6.2 for
details). Passing to the complex setting, we show that the singularities of
$k$-folding maps, $k\geq 2$, capture in a unified way, known robust features
obtained by considering the contact of the surface with lines, planes and
spheres (parabolic, sub-parabolic, ridge and flecnodal curves, umbilic points,
$B_{3}$, $C_{3}$ and $S_{3}$-points, $A^{*}_{2}$-points, cusps of Gauss
(gulls-points) and butterfly-points). Our approach also reveals a new robust
feature on surfaces: when $k$ is divisible by $3$, we obtain a new curve,
called the $H_{3}$-curve. We also obtain new special points on previously
known curves as well as on the $H_{3}$-curve. This motivates the following
question: can the $H_{3}$-curve be obtained via the contact of the surface
with some special geometric object? Further work is also required for
understanding the link between local (hidden) symmetries of a surface and its
contact with lines and planes.

The paper is organised as follows. In §2, we set notation and give some
preliminaries. In §3, we obtain formulae for the invariants $C,$ $T,$
$\mu({\mathcal{D}})$ and $r({\mathcal{D}})$ of $k$-folding map-germs. These
are respectively, the number of cross-caps, the number of triple points, the
Milnor number and the number of branches of the double point curve. These
invariants determine the finite $\mathcal{A}$-determinacy and the topological
class of a $k$-folding map-germ. In §4, we produce a stratification of the
$l$-jet space of $k$-folding map-germs in standard form which is identified
with the $l$-jet space of germs of functions $J^{l}(2,1)$. The stratification
results are summarised as follows.

###### Theorem 1.1

For any integer $k\geq 2$, there is a stratification $\mathcal{S}_{k}$ of
$J^{11}(2,1)$ such that, for any stratum $S$ in $\mathcal{S}_{k}$ of
codimension $\leq 4$, all $k$-folding map-germs in standard form with
$11$-jets in $S$ are finitely $\mathcal{A}$-determined and are pairwise
topologically equivalent.

We relate in §6 the stratification of the jet space to the extrinsic
differential geometry of surfaces in $\mathbb{R}^{3}$. After clarifying what
it means for a surface to be generic, we deduce the following result about the
topological classes of $k$-folding map-germs.

###### Theorem 1.2

Let $k\geq 3$ be an integer and let $M$ be a generic smooth surface in
$\mathbb{R}^{3}$ (or a complex surface in $\mathbb{C}^{3}$). Then, at any
point $p$ on $M$ and for any plane $\pi$ through $p$, the $k$-folding map-germ
at $p$ with respect to $\pi$ is finitely $\mathcal{A}$-determined and is
topologically equivalent to one of the following map-germs:

$\begin{array}[]{cl}{\bf M}^{k}_{0}&(x,y)\mapsto(x,y^{k},y),\\\ {\bf
M}^{k}_{1}&(x,y)\mapsto(x,y^{k},xy+y^{2}),\\\ {\bf
M}^{k}_{l}&(x,y)\mapsto(x,y^{k},y^{2}+y^{3}+x^{l}y),\ l=2,3,4,\\\ {\bf
N}^{k}_{l}&(x,y)\mapsto(x,y^{k},y^{2}+x^{2}y+y^{2l-1}),\ l=3,4,\\\ {\bf
O}^{k}_{4}&(x,y)\mapsto(x,y^{k},y^{2}+x^{3}y+xy^{3}),\\\ {\bf
P}^{k}_{l}&(x,y)\mapsto(x,y^{k},xy+y^{3}+y^{3l-1}),\ l=2,3,4,\\\ {\bf
Q}^{k}_{3}&(x,y)\mapsto(x,y^{k},xy+y^{4}+y^{5}+y^{6}),\\\ {\bf
Q}^{k}_{4}&(x,y)\mapsto(x,y^{k},xy+y^{4}+y^{6}+y^{8}),\\\ \widetilde{{\bf
Q}}^{k}_{4}&(x,y)\mapsto(x,y^{k},xy+y^{4}+y^{5}+y^{7}),\\\ {\bf
R}^{k}_{4}&(x,y)\mapsto(x,y^{k},xy+y^{5}+y^{6}+y^{7}),\\\ {\bf
U}^{k}_{3}&(x,y)\mapsto(x,y^{k},x^{2}y+2xy^{2}+y^{3}+y^{4}),\\\ {\bf
U}^{k}_{4}&(x,y)\mapsto(x,y^{k},x^{2}y+2xy^{2}+y^{3}+y^{8}),\\\ {\bf
V}^{k,j,j^{\prime}}_{4}&(x,y)\mapsto(x,y^{k},x^{2}y+xy^{2}+{a_{j,j^{\prime}}}y^{3}+{b_{j,j^{\prime}}}x^{3}y+y^{4}),\\\
{\bf
W}^{k,j}_{4}&(x,y)\mapsto(x,y^{k},x^{2}y+xy^{2}+{c_{j}}y^{3}+4x^{2}y^{2}+y^{4}),\\\
{\bf W}^{3p,p}_{4}&(x,y)\mapsto(x,y^{k},x^{2}y+y^{3}+y^{4}+y^{5}),\\\ {\bf
X}_{4}^{k}&(x,y)\mapsto(x,y^{k},xy^{2}+y^{3}+x^{3}y+y^{4}),\\\ {\bf
Y}^{k}_{4}&(x,y)\mapsto(x,y^{k},-xy^{2}+x^{2}y+y^{4}+y^{5}),\end{array}$

where the constants $a_{j,j^{\prime}},b_{j,j^{\prime}}$ (resp. $c_{j}$) are as
in Proposition 4.23 (resp. 4.25) and $j,j^{\prime}=1,\ldots,k-1$, with $j\neq
j^{\prime}$.

It is worth noting that $3$-folding map-germs can have $\mathcal{A}$-simple
singularities and their corresponding strata in $\mathcal{S}_{k}$ are
$\mathcal{A}$-constant. For $k\geq 4$, none of the $k$-folding map-germs are
$\mathcal{A}$-simple, except for immersions and for the $C_{3}$-singularity of
$F_{4}$, and the strata of $\mathcal{S}_{k}$ give rise to moduli of finitely
determined map-germs with constant invariants $C,T,\mu(\mathcal{D})$ and
$r(\mathcal{D})$.

The robust features captured by $k$-folding map-germs on a generic surface are
sketched in Figure 1. An interesting finding is that, having studied
symmetries of infinitely many orders (for any $k\geq 2$), we obtain a finite
collection of robust features that occur along curves and a finite collection
of special points on these curves if we discard the ${\bf V}^{k,j,j^{\prime}}$
and ${\bf W}^{k,j}$-points.

Figure 1: Robust features captured by $k$-folding map-germs away from umbilic
points for $k\geq 4$, even and divisible by $3$ (see Theorem 6.5 and Remarks
6.6 for the robust curves at umbilic points).

##  2 Preliminaries

We introduce here $k$-folding map-germs and notation from singularity theory
that are needed in the paper. We start with the singularity theory notation,
see for example [25, 31] for more details.

###  2.1 Singularities of map-germs

We deal with germs $F\colon(M,p)\to(N,F(p))$ of holomorphic maps between
complex manifolds. Taking coordinate chartes, this is the same as dealing with
map-germs $(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$.

Let ${\mathcal{O}}_{n}$ be the local ring of germs of holomorphic functions
$(\mathbb{C}^{n},0)\to\mathbb{C}$ and ${\mathfrak{m}}_{n}$ its maximal ideal
(which is the subset of germs that vanish at the origin). Denote by
$\mathcal{O}(n,p)$ the $\mathcal{O}_{n}$-module of holomorphic map-germs
$(\mathbb{C}^{n},0)\to\mathbb{C}^{p}$, so
$\mathcal{O}(n,p)=\bigoplus_{p}\mathcal{O}_{n}$.

Let ${\mathcal{R}}$ (resp. ${\mathcal{L}}$) be the group of bi-holomorphic
germs $(\mathbb{C}^{n},0)\to(\mathbb{C}^{n},0)$ (resp.
$(\mathbb{C}^{p},0)\to(\mathbb{C}^{p},0)$). The group
${\mathcal{A}}={\mathcal{R}}\times{\mathcal{L}}$ of right-left equivalence
acts on ${\mathfrak{m}}_{n}.{\mathcal{O}}(n,p)$ by $(h_{1},h_{2}).G=h_{2}\circ
G\circ h_{1}^{-1}$. Two germs $H,G$ are said to be $\mathcal{A}$-equivalent,
and write $H\sim_{\mathcal{A}}G$, if $H=(h_{1},h_{2}).G$ for some
$(h_{1},h_{2})\in\mathcal{A}$.

The $l$-jet space of map-germs in $\mathfrak{m}_{n}\cdot\mathcal{O}(n,p)$ is
by definition

$J^{l}(n,p)={\mathfrak{m}_{n}\cdot\mathcal{O}}(n,p)/{\mathfrak{m}}_{n}^{l+1}\cdot{}{\mathcal{O}}(n,p).$

Given a germ $G\in\mathfrak{m}_{n}\cdot{\mathcal{O}}(n,p)$, we identified its
$l$-jet $j^{l}G$ with its Taylor polynomial of degree $l$ at the origin. Let
$\mathcal{A}_{l}$ be the subgroup of $\mathcal{A}$ whose elements have
$l$-jets the germ of the identity. The group $\mathcal{A}_{l}$ is a normal
subgroup of $\mathcal{A}$. Define
$\mathcal{A}^{(l)}=\mathcal{A}/\mathcal{A}_{l}$. The elements of
$\mathcal{A}^{(l)}$ are the $l$-jets of the elements of $\mathcal{A}$. The
action of $\mathcal{A}$ on ${\mathfrak{m}}_{n}.{\mathcal{O}}(n,p)$ induces an
action of the jet group $\mathcal{A}^{(l)}$ on $J^{l}(n,p)$ as follows. For
$j^{l}G\in J^{l}(n,p)$ and $j^{l}(h_{1},h_{2})\in\mathcal{A}^{(l)}$,
$j^{l}(h_{1},h_{2}).j^{r}G=j^{l}((h_{1},h_{2}).G).$

A germ $G$ is said to be finitely $\mathcal{A}$-determined if there exist an
integer $l$ such that $G\sim_{\mathcal{A}}H$ for any $H$ with $j^{l}H=j^{l}G$;
$j^{l}G$ is then said to be a sufficient jet of $G$. The germ $G$ is then said
to be $l$-$\mathcal{A}$-determined. The least $l$ satisfying this property is
called the degree of determinacy of $G$.

There are classifications of finitely determined map-germs for various pairs
$(n,p)$. When $p=1$, there is Arnold’s extensive list of the
$\mathcal{R}$-classification of germs of functions ([1]). For $(n,p)=(2,2)$,
classifications were carried out by several authors, the most extensive ones
are given in [17, 29]. Here we need only the singularities of
$\mathcal{A}_{e}$-codimension $\leq 2$, which we reproduce in Table 9. For
$(n,p)=(2,3)$, Mond [23] produced an extensive list of finitely
$\mathcal{A}$-determined map-germs. We use in this paper the following
singularities from [23]:

Immersion | $(x,y,0)$
---|---
Cross-cap | $(x,y^{2},xy)$
$S_{k}$ |  $(x,y^{2},y^{3}+x^{k+1}y)$, $k\geq 1$
$B_{k}$ |  $(x,y^{2},x^{2}y+y^{2k+1})$, $k\geq 2$
$C_{3}$ | $(x,y^{2},xy^{3}+x^{3}y)$
$H_{k}$ |  $(x,xy+y^{3k-1},y^{3})$, $k\geq 2$
$X_{4}$ | $(x,y^{3},x^{2}y+xy^{2}+y^{4})$

The notion of a simple germ is defined in [1] as follows. Let $X$ be a
manifold and $\mathcal{G}$ a Lie group acting on $X$. The modality of a point
$g\in X$ under the action of $\mathcal{G}$ on $X$ is the least number $m$ such
that a sufficiently small neighbourhood of $g$ may be covered by a finite
number of $m$-parameter families of orbits. The point $g$ is said to be simple
if its modality is $0$, that is, a sufficiently small neighbourhood intersects
only a finite number of orbits. The modality of a finitely
$\mathcal{A}$-determined map-germ is the modality of a sufficient jet in the
jet-space under the action of the jet-group.

We also need the notion of topological equivalence. We say that two germs
$H,G\in{\mathfrak{m}}_{n}\cdot{\mathcal{O}(n,p)}$ are topologically equivalent
if $H=h_{2}\circ G\circ h_{1}^{-1}$ for some germs of homeomorphisms $h_{1}$
and $h_{2}$ of, respectively, the source and target.

###  2.2 Reflections and $k$-folding maps

In all this paper, we fix the inner product $\langle
a,b\rangle=\sum_{i}a_{i}\overline{b_{i}}$ in $\mathbb{C}^{3}$.

Let $\pi$ be an element of the affine Grassmannian ${\rm Graff}(2,3)$ of
planes in $\mathbb{C}^{3}$. A plane $\pi$ has equation $\left<q,v\right>=d$,
where $v$ is a fixed non-zero vector orthogonal to $\pi$ and $d$ is a fixed
scalar. However, any non-zero scalar multiple of $(d,v)$ gives an equation of
$\pi$, so $\pi$ is identified with the class
$\overline{(d,v)}\in\mathbb{C}P^{3}$ of $(d,v)\in\mathbb{C}^{4}$.

Let $\pi:\left<q,v\right>=d$ be a plane in $\mathbb{C}^{3}$. The orthogonal
projection of a point $p\in\mathbb{C}^{3}$ to $\pi$ along the vector $v$ is
the point $q=p+\lambda v\in\pi$ with
$\lambda=(d-\left<p,v\right>)/\left<v,v\right>$.

Consider the map $\omega_{k}^{\pi}\colon\mathbb{C}^{3}\to\mathbb{C}^{3}$ given
by

$\omega_{k}^{\pi}(p)=q+\lambda^{k}v=p+\lambda v+\lambda^{k}v.$

If we take $(d^{\prime},v^{\prime})=(\alpha d,\alpha v)$,
$\alpha\in\mathbb{C}\setminus 0$, as another representative of
$\pi=\overline{(d,v)}$, then

$\begin{array}[]{rcl}p+\lambda^{\prime}v^{\prime}+\lambda^{\prime
k}v^{\prime}&=&p+\frac{\alpha(d-\left<p,v\right>))}{\alpha^{2}\left<v,v\right>}\alpha
v+\frac{\alpha^{k}(d-\left<p,v\right>)^{k})}{\alpha^{2k}\left<v,v\right>}\alpha
v\\\ &=&p+\lambda v+\lambda^{k}\alpha^{1-k}v\\\
&=&q+\lambda^{k}\alpha^{1-k}v.\end{array}$

Clearly, the map $\omega_{k}^{\pi}$ depends on the points on the line $(\alpha
d,\alpha v)\in\mathbb{C}^{4}$ and not merely on the class of the line
$\overline{(d,v)}\in\mathbb{C}P^{3}$. However, all these maps are
$\mathcal{L}$-equivalent: the bi-holomorphic map $q-\lambda v\mapsto
q-\lambda\alpha^{\frac{1-k}{k}}v$ composed (on the left) with the map
$\omega_{k}^{\pi}$ with $\pi$ represented by $(d,v)$ gives the map
$\omega_{k}^{\pi}$ with $\pi$ represented by $(\alpha d,\alpha v)$. Therefore,
the $\mathcal{L}$-class of $\omega_{k}^{\pi}$ depends only on $\pi$.

###### Definition 2.1

The _Whitney fold of order $k$_ ($k$-fold for short) with respect to a plane
$\pi\in{\rm Graff}(2,3)$ is the $\mathcal{L}$-class of the map
$\omega^{k}_{\pi}$. We still denote by $\omega_{k}^{\pi}$ any representative
of $\omega_{k}^{\pi}$ obtained by choosing a representative $(p,v)$ of
$\pi=\overline{(d,v)}\in\mathbb{C}P^{3}$.

A $k$-fold may be viewed as generalisation of the Whitney fold
$(x,y^{2},z)\mapsto(x,y^{2},z)$. While the Whitney fold folds the space along
the plane $\\{y=0\\}$ and identifies the points $(x,y,z)$ and $(x,-y,z)$, the
Whitney fold of order $k$ with respect to a plane $\pi$ represented by $(d,v)$
is a generically a $k$-to-one branched cover, ramified along $\pi$, and
identifies $k$-tuples of points $q-\lambda v,\ q-\xi\lambda v,\ \dots\ ,\
q-\xi^{k-1}\lambda v,$ where $\xi=e^{2\pi i/k}$ is a primitive $k^{th}$-root
of unity and $q\in\pi$.

The map $\omega_{k}^{\pi}$ can also be viewed as the quotient map associated
to the action of the cyclic group $\mathbb{Z}/k\mathbb{Z}$, regarded as a
complex reflection group whose hyperplane arrangement consist of the single
plane $\pi$. We regard $\mathbb{Z}/k\mathbb{Z}$ as the group generated by the
order $k$ complex reflection $q-\lambda v\mapsto q-\xi\lambda v$. Observe
that, even though the plane $\pi$ does not determine $\omega_{k}^{\pi}$
uniquely (it depends on the choice of a representative of
$\pi=\overline{(d,v)}\in\mathbb{C}P^{3}$), the action of
$\mathbb{Z}/k\mathbb{Z}$ on $\mathbb{C}^{3}$ is determined uniquely by $\pi$.

Given any subset $X\subseteq\mathbb{C}^{3}$, $\omega_{k}^{\pi}(X)$ encodes the
order $k$ reflectional symmetries of $X$ with respect to $\pi$. See [18] for a
recent work on singular maps related to reflection groups.

###### Definition 2.2

Let $M\subset\mathbb{C}^{3}$ be a complex surface, $p$ a point on $M$ and
$k\geq 2$ an integer. Given $\pi\in{\rm Graff}(2,3)$, the $k$-folding map-germ
on $M$ at $p$ with respect to $\pi$ is the $\mathcal{A}$-class of the
restriction of $\omega_{k}^{\pi}$ to $M$ at $p$. We denote any representative
of the class by
$F_{k}^{\pi}\colon(M,p)\to(\mathbb{C}^{3},\omega_{k}^{\pi}(p)).$

###### Remarks 2.3

1\. All the map-germs $F_{k}^{\pi}$ with $\pi$ represented by $(\alpha
d,\alpha v)$, $\alpha\in\mathbb{C}$, are $\mathcal{A}$-equivalent as the maps
$\omega_{k}^{\pi}$ are $\mathcal{L}$-equivalent. Thus, the $\mathcal{A}$-class
of $F_{k}^{\pi}$ depends only on $\pi$ and not on the choice of a
representative $(p,v)$ of $\pi=\overline{(d,v)}\in\mathbb{C}P^{3}$. In all the
paper, we work with a representative of the $\mathcal{A}$-class of
$F_{k}^{\pi}$.

2\. If $p\notin\pi$, then $F_{k}^{\pi}$ is the germ of an immersion. Thus, to
obtain any meaningful local geometric information about the surface $M$ we
should take the plane $\pi$ passing through the point $p\in M$.

3\. The image of $F_{k}^{\pi}$ is the image by $\omega_{k}^{\pi}$ of the germ
$(M,p)$, so for $p\in\pi\cap M$, $F_{k}^{\pi}$ captures order $k$ local
symmetries of $M$ with respect to $\pi$. The aim of this paper is to
understand how these local symmetries are captured by the
$\mathcal{A}$-singularities of $F_{k}^{\pi}$.

4\. Let $p_{0}\in\pi\cap M$ and $(d,v)$ a representative of $\pi$. If $v\notin
T_{p_{0}}M$, then $F^{\pi}_{k}$ is a germ of an immersion and is
$\mathcal{A}$-equivalent to $(x,y)\mapsto(x,y^{k},y)$. Suppose that $v\in
T_{p_{0}}M$. We choose a coordinates system in $\mathbb{C}^{3}$ so that
$p_{0}$ is the origin, the $z$-axis is along a normal vector to $M$ at
$p_{0}$, the $y$-axis along $v$ and the $x$-axis orthogonal to the previous
two axes. Then we can take $M$ locally at $p_{0}$ as the graph $z=f(x,y)$ of
some holomorphic map $f$ in a neighbourhood $U$ of the origin. In this
coordinate system, we have $\pi=\pi_{0}:y=0$. Consequently, the $k$-folding
map-germ on $M$ at $p_{0}$ is the germ
$F_{k}=F_{k}^{\pi_{0}}:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$, given in
standard form $F_{k}(x,y)=(x,y^{k},f(x,y))$. In view of this, we shall always
take a given $k$-folding map-germ in standard form (1).

5\. Definition 2.2 is adapted as follows for the real case. When $M$ is an
analytic surface in $\mathbb{R}^{3}$, denote by $M_{\mathbb{C},p}$ its local
complexification at $p$ and by $\pi_{\mathbb{C}}$ the complexification of
$\pi$. The $k$-folding map-germ on $M$ at $p$ with respect to $\pi$ is then
defined as the $\mathcal{A}$-class of the restriction of
$\omega_{k}^{\pi_{\mathbb{C}}}$ to $M_{\mathbb{C},p}$ at $p$. When $M$ is a
smooth surface, we consider the $k$-folding map-germ of a given jet of (a
parametrisation of) $M$ at $p$.

##  3 Topological invariants

We recall the definitions of some key $\mathcal{A}$-invariants of map-germs
$(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$. These are the Milnor number of the
double point curve $\mu(\mathcal{D})$, the number of cross-caps $C$ and the
number of triple points $T$. We give formulae for computing these invariants
for $k$-folding map-germs, and use the invariants to study the finite
$\mathcal{A}$-determinacy and topological equivalence of these germs.

###  3.1 The double point curve

We start by recalling the definition of the double and triple point spaces of
a corank one map-germ $F\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$ from
[20]. Any such germ can be written in a suitable coordinate system in the form
$F(x,y)=(x,f_{n}(x,y),f_{n+1}(x,y)),$ with
$x=(x_{1},\dots,x_{n-1})\in(\mathbb{C}^{n-1},0)$ and $y\in(\mathbb{C},0)$.

Given $h\in\mathcal{O}_{n}$, the iterated divided differences of $h$ are
defined as

$\begin{array}[]{rcl}h[x,y,y^{\prime}]&=&\dfrac{h(x,y^{\prime})-h(x,y)}{y^{\prime}-y}\in\mathcal{O}_{n+1},\\\
h[x,y,y^{\prime},y^{\prime\prime}]&=&\dfrac{h[x,y,y^{\prime\prime}]-h[x,y,y^{\prime}]}{y^{\prime\prime}-y^{\prime}}\in\mathcal{O}_{n+2}.\end{array}$

The multiple point ideals of a map-germ $F$ as above are defined as

$\begin{array}[]{rcl}I^{2}(F)&=&\langle
f_{n}[x,y,y^{\prime}],f_{n+1}[x,y,y^{\prime}]\rangle\subseteq\mathcal{O}_{n+1},\\\
I^{3}(F)&=&\langle
f_{n}[x,y,y^{\prime}],f_{n}[x,y,y^{\prime},y^{\prime\prime}],f_{n+1}[x,y,y^{\prime}],f_{n+1}[x,y,y^{\prime},y^{\prime\prime}]\rangle\subseteq\mathcal{O}_{n+2}.\end{array}$

The double and triple point spaces of $F$ are, respectively,

$\begin{array}[]{rcl}{\mathcal{D}}^{2}(F)&=&V(I^{2}(F)),\\\
{\mathcal{D}}^{3}(F)&=&V(I^{3}(F)).\end{array}$

By counting variables and generators, it follows that ${\mathcal{D}}^{2}(F)$
(resp. ${\mathcal{D}}^{3}(F)$) is a complete intersection whenever it has
dimension $n-1$ (resp. $n-2$).

The double point space ${\mathcal{D}}^{2}(F)$, as a subset of
$(\mathbb{C}^{n-1}\times\mathbb{C}\times\mathbb{C},0)$, consists of points
$(x,y,y^{\prime})$ such that either $y\neq y^{\prime}$ and
$F(x,y)=F(x,y^{\prime})$ or $y^{\prime}=y$ and $F$ is singular at $(x,y)$.

To define the source double point space, we assume that $F$ is finite. Then,
the projection
$\pi\colon{\mathcal{D}}^{2}(F)\to\mathbb{C}^{n-1}\times\mathbb{C}$ given by
$(x,y,y^{\prime})\mapsto(x,y)$ is also finite. As a consequence, the image of
$\pi$ can be given a complex structure as the $0$-th Fitting ideal
$\mathcal{F}_{0}(\pi_{*}\mathcal{O}_{{\mathcal{D}}^{2}(F)})$ of the push
forward module $\pi_{*}\mathcal{O}_{{\mathcal{D}}^{2}(F)}$ (see [26] for
details). The source double point space ${\mathcal{D}}(F)$ is defined as the
projection $\pi({\mathcal{D}}^{2}(F))$ endowed with this complex space
structure, that is,

${\mathcal{D}}(F)=V(\mathcal{F}_{0}(\pi_{*}\mathcal{O}_{{\mathcal{D}}^{2}(F)}))\subseteq\mathbb{C}^{n}.$

To compute the source double point space of a $k$-folding map-germ, we need
the following result.

###### Lemma 3.1

Let $Z$ be a germ of an $n$-dimensional Cohen-Macaulay space and let
$h_{1},\dots,h_{r}$ in $\mathcal{O}_{Z}$ be regular elements. Write
$X_{j}=V(h_{j})$ and $X=V(h_{1}\dots h_{r})$. Let $\phi\colon
Z\to(\mathbb{C}^{n},0)$ be a germ of a morphism of complex spaces such that
the restrictions $\phi|_{X_{j}}\colon X_{j}\to(\mathbb{C}^{n},0)$ are finite.
Then
$\mathcal{F}_{0}((\phi|_{X})_{*}\mathcal{O}_{X})=\prod_{j=1}^{r}\mathcal{F}_{0}((\phi|_{X_{j}})_{*}\mathcal{O}_{X_{j}}).$

  * Proof

It is enough to prove the statement for the case $r=2$. We can assume that
$X_{1}$ and $X_{2}$ have no common irreducible component as topological
spaces. Indeed, consider the two subspaces $\mathcal{X}_{1}=V(h_{1}-t)$ and
$\mathcal{X}_{2}=V(h_{2})$ of $Z\times(\mathbb{C},0)$ and the map
$\phi\times\operatorname{Id}\colon
Z\times(\mathbb{C},0)\to(\mathbb{C}^{n+1},0)$. The spaces $\mathcal{X}_{1}$
and $\mathcal{X}_{2}$ have no common irreducible component. Moreover, if the
statement holds for $\phi\times\operatorname{Id}$, then it holds for $\phi$.
This is a consequence of the fact that Fitting ideals commute with base change
(see Lemma 1.2 in [26]). Now consider the disjoint union $X_{1}\sqcup
X_{2}\subseteq Z\sqcup Z$ and the commutative diagram

${X_{1}\sqcup
X_{2}}$${X}$${(\mathbb{C}^{n},0)}$$\scriptstyle{\alpha}$$\scriptstyle{\psi}$$\scriptstyle{\phi|_{X}}$

The map $\alpha$ is generically a local isomorphism (that is, a local
isomorphism on a Zariski open and dense subset) because $X_{1}$ and $X_{2}$
are assumed to have no common component. Moreover, both $X$ and $X_{1}\sqcup
X_{2}$ are Cohen Macaulay spaces, which implies that the ideals
$\mathcal{F}_{0}(\psi_{*}\mathcal{O}_{X_{1}\sqcup X_{2}})$ and
$\mathcal{F}_{0}((\phi|_{X})_{*}\mathcal{O}_{X})$ are principal. Since
$\alpha$ is generically a local isomorphism, they are necessarily equal. The
statement then follows from the equalities

$\mathcal{F}_{0}(\psi_{*}\mathcal{O}_{X_{1}\sqcup
X_{2}})=\mathcal{F}_{0}((\phi|_{X_{1}})_{*}\mathcal{O}_{X_{1}}\oplus(\phi|_{X_{2}})_{*}\mathcal{O}_{X_{2}}))=\mathcal{F}_{0}((\phi|_{X_{1}})_{*}\mathcal{O}_{X_{1}})\cdot\mathcal{F}_{0}((\phi|_{X_{2}})_{*}\mathcal{O}_{X_{2}}).$

$\Box$

###### Theorem 3.2

For a $k$-folding map-germ $F_{k}(x,y)=(x,y^{k},f(x,y))$, the double point
space ${\mathcal{D}}$ is the zero locus $V(\lambda)$, where
$\lambda=\prod_{j=1}^{k-1}\lambda_{j}$ and

$\lambda_{j}=\frac{f(x,y)-f(x,\xi^{j}y)}{(1-\xi^{j})y},$

for $1\leq j\leq k-1.$ We have thus a decomposition
${\mathcal{D}}=\bigcup_{j=1}^{k-1}{\mathcal{D}}_{j}$, with
${\mathcal{D}}_{j}=V(\lambda_{j})$.

  * Proof

The double point space ${\mathcal{D}}^{2}(F_{k})$ is the intersection of the
zero loci of the divided differences

$\frac{((y^{\prime})^{k}-y^{k})}{y^{\prime}-y}\quad{\rm
and}\quad\frac{f(x,y)-f(x,y^{\prime})}{y-y^{\prime}}.$

Since
$((y^{\prime})^{k}-y^{k})/{(y^{\prime}-y)}=\prod_{j=1}^{k-1}(y^{\prime}-\xi^{j}y)$,
we conclude that, as a set, the space ${\mathcal{D}}^{2}(F_{k})$ is the union
of the spaces

${\mathcal{D}}^{2}_{j}=\left\\{(x,y,\xi^{j}y)\in(\mathbb{C}^{n-1}\times\mathbb{C}\times\mathbb{C},0)\mid\frac{f(x,y)-f(x,\xi^{j}y)}{(1-\xi^{j})y}=0\right\\},$

for $j=1,\ldots,k-1$. Each of the sets ${\mathcal{D}}^{2}_{j}$ projects to
$V(\lambda_{j})$, which shows that ${\mathcal{D}}(F_{k})=V(\lambda)$ as sets.

To show the equality as complex spaces, observe that the possible dimension of
${\mathcal{D}}^{2}(F_{k})$ is one or two. If ${\mathcal{D}}^{2}(F_{k})$ has
dimension two, then some branch ${\mathcal{D}}^{2}_{j}$ has dimension two.
Therefore, the corresponding function $\lambda_{j}$ is identically zero, which
in turn implies $\lambda=0$. Since $F_{k}$ is finite by construction, the
projection ${\mathcal{D}}^{2}(F_{k})$ is finite, hence the image of
${\mathcal{D}}^{2}_{j}$ is a germ of a two dimensional analytic closed subset
of $(\mathbb{C}^{2},0)$, so is equal to $(\mathbb{C}^{2},0)$. This implies
that ${\mathcal{D}}(F_{k})=(\mathbb{C}^{2},0)=V(0)$.

Suppose now that ${\mathcal{D}}^{2}(F_{k})$ has dimension one. This implies
that the functions $\prod_{j=1}^{k-1}(y^{\prime}-\xi^{j}y)$ and
$f[x,y,y^{\prime}]$ form a regular sequence. Applying Lemma 3.1 with
$Z=V(f[x,y,y^{\prime}])$ gives
${\mathcal{D}}(F_{k})=V\left(\prod_{j=1}^{k-1}\mathcal{F}_{0}((\pi|_{{\mathcal{D}}^{2}_{j}})_{*}\mathcal{O}_{{\mathcal{D}}^{2}_{j}}))\right)$,
where the ${\mathcal{D}}^{2}_{j}$, $j=1,\ldots,k-1$, are given the natural
complex space structure. Each of the morphisms
$\pi|_{{\mathcal{D}}^{2}_{j}}\colon{\mathcal{D}}^{2}_{j}\to\mathbb{C}^{2}$
consists of forgetting the third coordinate of the tuple $(x,y,\xi^{j}y)$, and
this implies
$\mathcal{F}_{0}((\pi|_{{\mathcal{D}}^{2}_{j}})_{*}\mathcal{O}_{{\mathcal{D}}^{2}_{j}}))=\langle\lambda_{j}\rangle$.
$\Box$

Now we introduce some results to we use to check finite
$\mathcal{A}$-determinacy of a $k$-folding map-germ $F_{k}$ and topological
triviality in families of such germs. The first of these results was proven in
[20] for corank one map-germs, then extended to arbitrary corank in [21].

###### Theorem 3.3

A finite map-germ $F\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ is finitely
$\mathcal{A}$-determined if, and only if, its double point curve
${\mathcal{D}}$ is reduced.

The decomposition ${\mathcal{D}}=\bigcup_{j=1}^{k-1}{\mathcal{D}}_{j}$ in
Theorem 3.2 can be used to compute $\mu({\mathcal{D}})$, making it easier to
apply Theorem 3.3 (and Theorem 3.10 below). We denote by
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$ the intersection
multiplicity of two distinct branches of the double point curve. Clearly,
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}={\mathcal{D}}_{j^{\prime}}\cdot{\mathcal{D}}_{j}$,
and

${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{\langle\lambda_{j},\lambda_{j^{\prime}}\rangle}.$

###### Proposition 3.4

A $k$-folding map-germ $F_{k}$ is finitely $\mathcal{A}$-determined if, and
only if, the Milnor numbers $\mu({\mathcal{D}}_{j})$, $j=1,\ldots,k-1$, and
the intersection multiplicities
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$ of all pairs
${\mathcal{D}}_{j}$ and ${\mathcal{D}}_{j}^{\prime}$, with $j^{\prime}\neq j$,
are finite. In that case,

$\mu({\mathcal{D}}(F_{k}))=\sum_{j=1}^{k-1}\mu({\mathcal{D}}_{j})+2\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\
j<j^{\prime}\end{subarray}}^{k-1}{\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}-k+2.$

  * Proof

By Theorem 3.3, $F_{k}$ is finitely $\mathcal{A}$-determined if, and only if,
$\mu({\mathcal{D}}(F_{k}))$ is finite, equivalently, ${\mathcal{D}}(F_{k})$
has an isolated singularity. This occurs if, and only if, every branch
${\mathcal{D}}_{j}$ has an isolated singularity and no pair of branches
${\mathcal{D}}_{j}$ and ${\mathcal{D}}_{j^{\prime}}$, with $j\neq j^{\prime}$,
have a common component. Using the formula $\mu=2\delta-r+1$ for plane curves
(see [22]) and the property $\delta(X\cup Y)=\delta(X)+\delta(Y)+X\cdot Y$, we
get

$\displaystyle\mu({\mathcal{D}}(F_{k}))$
$\displaystyle=2\delta({\mathcal{D}}(F_{k}))-r({\mathcal{D}}(F_{k}))+1$
$\displaystyle=\sum_{j=1}^{k-1}(2\delta({\mathcal{D}_{j}})-r(D_{j})+1)-k+2+2\sum_{j,j^{\prime}=1,j<j^{\prime}}^{k-1}{\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$
$\displaystyle=\sum_{j=1}^{k-1}\mu({\mathcal{D}}_{j})+2\sum_{j,j^{\prime}=1,j<j^{\prime}}^{k-1}{\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}-k+2.$

$\Box$

###### Remarks 3.5

1\. Suppose that ${\mathcal{D}}_{j}$ is a germ of a regular curve parametrised
by a regular map-germ $\alpha\colon(\mathbb{C},0)\to(\mathbb{C}^{2},0)$. Then,
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=\mathrm{ord}(h_{j^{\prime}}\circ\alpha),$
which is the degree of the first non zero term in the Taylor expansion of
$\lambda_{j^{\prime}}(\alpha(t))$.

2\. If both ${\mathcal{D}}_{j}$ and ${\mathcal{D}}_{j^{\prime}}$ are regular
curves, we refer to ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$ as the
_order of contact between ${\mathcal{D}}_{j}$ and
${\mathcal{D}}_{j^{\prime}}$_. We have
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=1$ if, and only if, the two
curves intersect transversally. Suppose they are tangential and parametrised,
respectively, by $t\mapsto(t,\gamma_{j}(t))$ and
$t\mapsto(t,\gamma_{j^{\prime}}(t))$. Then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=\mathrm{ord}(\gamma_{j}-\gamma_{j^{\prime}}).$

3\. Let $F_{k}$ be a finitely $\mathcal{A}$-determined $k$-folding map-germ.
Then any pair of branches ${\mathcal{D}}_{j}$ and
${\mathcal{D}}_{j}^{\prime}$, with $j\neq j^{\prime}$, cannot have any common
irreducible component, otherwise ${\mathcal{D}}(F_{k})$ would fail to be
reduced. Hence, we have
$r({\mathcal{D}})=\sum_{j=1}^{k-1}r({\mathcal{D}}_{j}).$

The defining functions $\lambda_{j}$ of the branches ${\mathcal{D}}_{j}$ of
the double point curve play a major role in our study of finite
$\mathcal{A}$-determinacy and topological equivalence of $k$-folding map-
germs. We take $F_{k}(x,y)=(x,y^{k},f(x,y))$ and write, for any given integer
$p\geq 1$,

$j^{p}f(x,y)=\sum_{q=1}^{p}\sum_{s=0}^{q}a_{qs}x^{q-s}y^{s}.$ (2)

Then,

$j^{p-1}\lambda_{j}=\sum_{q=1}^{p}\sum_{s=1}^{q}\vartheta_{sj}a_{qs}x^{q-s}y^{s-1},$
(3)

with

$\vartheta_{sj}=\frac{1-\xi^{sj}}{1-\xi^{j}}=1+\xi^{j}+\dots+\xi^{(s-1)j}.$

The constants $\vartheta_{sj}$ play a significant role in determining the
singularity type of the germs $\lambda_{j}$ and in computing
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$. The following properties
are needed in §4.

###### Lemma 3.6

The numbers $\vartheta_{sj}$ satisfy the following properties:

  * (1)

$\vartheta_{sj}=0$ if, and only if, $k\mid sj$.

  * (2)

$\vartheta_{sj}=1$ if, and only if, $k\mid(s-1)j$.

  * (3)

If $\vartheta_{sj}=\vartheta_{sj^{\prime}}$, then $\vartheta_{sj}$ is either
$0$ or $1$.

  * Proof

We observe that $\vartheta_{0j}=0$, $\vartheta_{1j}=1$ and, for any integers
$m,n$, we have $\vartheta_{mj}=\vartheta_{(m+n)j}$ if, and only if, $k\mid
nj$. For (1) we take $m=0$ and $n=s$, and for (2) we take $m=1$ and $n=s-1$.

For (3), we observe that the constants $\vartheta_{sj}$ lie in the images of
the curves $\gamma_{n}:S^{1}\subset\mathbb{C}\to\mathbb{C}$ given by
$\gamma_{n}(z)={(1-z^{n+1})}/{(1-z)}=1+z+\ldots+z^{n}.$ We show that the self
intersection points of the curves $\gamma_{n}$ are $0$ and $1$ (for $n\geq
3$).

Write $z=e^{i\theta}$, with $\theta\in[0,2\pi)$. Then
$\gamma_{n}(\theta)=x+iy$, with $(x,y)\in\mathbb{R}^{2}$, gives
$1-e^{i(n+1)\theta}=(x+iy)(1-e^{i\theta}).$ Therefore,

$\begin{array}[]{rcl}\cos((n+1)\theta)&=&1-x+x\cos(\theta)-y\sin(\theta),\\\
\sin((n+1)\theta)&=&-y+y\cos(\theta)+x\sin(\theta).\end{array}$

Now the identity $\cos((n+1)\theta)^{2}+\sin((n+1)\theta)^{2}=1$ gives

$(1-\cos(\theta))(x^{2}+y^{2}-x)-y\sin(\theta)=0.$ (4)

Suppose that $y=0$. Then equation (4) becomes $(1-\cos(\theta))x(x-1)=0$, so
$x=0$ or $1$ or $\theta=0$.

When $x=0$, we have $1-e^{i\theta}\neq 0$, so $1-e^{i(n+1)\theta}=0$. That
gives $\theta=\frac{2\pi j}{n+1},j=1,\ldots,n$. Therefore, $\gamma_{n}$ passes
$n$-times through the origin.

When $x=1$, we get $e^{in\theta}=1$, so $\theta=\frac{2\pi
j}{n},j=1,\ldots,n-1$. Therefore, $\gamma_{n}$ passes $(n-1)$-times through
the point $1$.

When $\theta=0$, we have $\gamma_{n}(0)=n+1$ and the curve has no self-
intersections at that point. See Figure 2 for the cases $n=5,6,7.$

Suppose now that $y\neq 0$. Then equation (4) can be written as
$\cot({\theta}/{2})={(x^{2}+y^{2}-x)}/{y}.$ This shows that, for any
$(x,y)\in\mathbb{R}^{2}$ with $y\neq 0$, there is at most one
$\theta\in[0,2\pi)$ satisfying $\gamma_{n}(\theta)=x+iy$. Therefore, the only
self-intersection points of $\gamma_{n}$ are $0$ and $1$. $\Box$

Figure 2: Maple plots of the curves $\gamma_{n}$ for $n=5,6,7$ (from left to
right).

###### Remark 3.7

As $k$ cannot divide $j$, the condition $k\mid sj$ in Lemma 3.6 (1) can also
be written as $d=\gcd(k,s)\neq 1\text{ and
}j\in\\{\frac{k}{d},\dots,\frac{(d-1)k}{d}\\}.$ Of course, the same applies to
the condition $k\mid(s-1)j$ in Lemma 3.6 (2).

###  3.2 Cross-caps and triple points

The number of cross-caps and triple points are invariants of finitely
$\mathcal{A}$-determined map-germs
$F\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ that can be described using
stable deformations. A stable mapping $U\to(\mathbb{C}^{3},0)$ with $U$ an
open neighbourhood of the origin in $\mathbb{C}^{2}$ exhibits only regular
points, transverse double points along curves, cross-caps and isolated
transverse triple points. Every stable deformation $F_{t}$ of $F$ exhibits the
same number $C$ of cross-caps and $T$ of triple points [24]. For a corank one
map-germ, these are given by the formulae

$C=\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{JF},\quad
T=\frac{1}{6}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{4}}{I^{3}(F)},$

where $JF$ is the ideal generated by the $2\times 2$ minors of the
differential matrix of $F$ [24] (the formula for $C$ holds without the corank
one assumption).

For $j,j^{\prime}\in\\{1,\dots,k-1\\}$, with $j\neq j^{\prime}$, we set

$\lambda_{j,j^{\prime}}=\frac{\lambda_{j}-\lambda_{j^{\prime}}}{y}$

and define

$T_{j,j^{\prime}}=\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{\langle\lambda_{j},\lambda_{j,j^{\prime}}\rangle}.$

Observe that $T_{j,j^{\prime}}=T_{j^{\prime},j}$ for all $j\neq j^{\prime}$.

###### Proposition 3.8

The number of cross-caps and of triple points of a finitely
$\mathcal{A}$-determined $k$-folding map-germ $F_{k}$ are given by

$C=\operatorname{dim}_{\mathbb{C}}\dfrac{\mathcal{O}_{2}}{\langle
y^{k-1},\frac{\partial{f}}{\partial y}(x,y)\rangle}\quad\mbox{and}\quad
T=\frac{1}{3}\sum_{1\leq j<j^{\prime}\leq k-1}T_{j,j^{\prime}}.$

  * Proof

The formula for $C$ is trivial. For $T$, we claim that

$J=\langle\frac{y^{\prime k}-y^{k}}{y^{\prime}-y},\frac{\frac{y^{\prime\prime
k}-y^{k}}{y^{\prime\prime}-y}-\frac{y^{\prime
k}-y^{k}}{y^{\prime}-y}}{y^{\prime\prime}-y}\rangle\subseteq I^{3}(F_{k})$

is a radical ideal of $\mathcal{O}_{4}$ whose zero locus is

$Z=\\{(x,y,y^{\prime},y^{\prime\prime})\mid
y^{\prime}=\xi^{j}y,y^{\prime\prime}=\xi^{j^{\prime}}\text{for some
}j,j^{\prime}\in\\{1,\dots,k-1\\}\text{ with }j\neq j^{\prime}\\}.$

Clearly, $Z$ is contained in the zero locus $V(J)$. The converse inclusion
follows by considering the degree of the generators of $J$ and by the fact
that $V(J)$ is generically reduced. Since $V(J)$ is also a complete
intersection, it is reduced. Thus $J$ is a radical ideal.

The irreducible decomposition of $Z$ consists of the branches

$Z_{j,j^{\prime}}=\\{(x,y,\xi^{j}y,\xi^{j^{\prime}}y)\mid
x,y\in\mathbb{C}\\}.$

Thus, we may decompose ${\mathcal{D}}^{3}(F_{k})$, as a set, as the union of
the (not necessarily irreducible) branches

${\mathcal{D}}^{3}_{j,j^{\prime}}(F_{k})=V(y^{\prime}-\xi^{j}y,y^{\prime\prime}-\xi^{j^{\prime}}y,f[x,y,y^{\prime}],f[x,y,y^{\prime},y^{\prime\prime}]).$

Eliminating $y^{\prime}$ and $y^{\prime\prime}$, gives
${\mathcal{D}}^{3}_{j,j^{\prime}}(F_{k})=\langle\lambda_{j},\lambda_{j,j^{\prime}}\rangle$.
Therefore, we only need to show that
$\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{4}}{I^{3}(F_{k})}=\sum_{j\neq
j^{\prime}}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{\langle\lambda_{j},\lambda_{j,j^{\prime}}\rangle}$.
It is clear that the same decomposition of ${\mathcal{D}}^{3}(F_{k})$ applies
to an unfolding $\tilde{F}_{k}(x,y,t)=(x,y^{k},f_{t}(x,y),t)$ of $F_{k}$, and
fixing a representative and a nonzero parameter $\delta$, the same holds for
$(\tilde{F}_{k})_{\delta}$. This gives a decomposition of
${\mathcal{D}}^{3}(({\tilde{F}_{k}})_{\delta})$ into the branches
${\mathcal{D}}^{3}_{j,j^{\prime}}(({\tilde{F}_{k}})_{\delta})$.

It is possible to choose an unfolding where the branches
${\mathcal{D}}^{3}_{j,j^{\prime}}(({\tilde{F}_{k}})_{\delta})$ are pairwise
disjoint. Indeed, it follows from their defining equations that two spaces
${\mathcal{D}}^{3}_{j,j^{\prime}}(({\tilde{F}_{k}})_{\delta})$ and
${\mathcal{D}}^{3}_{s,s^{\prime}}(({\tilde{F}_{k}})_{\delta})$ can only
intersect on $\\{y=0\\}$, so it is enough to find an unfolding where
${\mathcal{D}}^{3}_{j,j^{\prime}}(({\tilde{F}_{k}})_{\delta})\cap\\{y=0\\}$ is
empty. Again, it follows from the defining equations that a point
$(x,0,0,0)\in{\mathcal{D}}^{3}_{j,j^{\prime}}(({\tilde{F}_{k}})_{\delta})$ if
$\frac{\partial f_{\delta}}{\partial x}(x,0)=\frac{\partial
f^{2}_{\delta}}{\partial x^{2}}(x,0)=0$, a condition that can be avoided by
choosing a suitable deformation $f_{t}$ of $f$.

We take now an unfolding as above. Since $Z$ is reduced,
${\mathcal{D}}^{3}(({\tilde{F}_{k}})_{\delta})$ is isomorphic to the union of
the ${\mathcal{D}}^{3}_{j,j^{\prime}}(({\tilde{F}_{k}})_{\delta})$ as complex
spaces. The equality we need to show follows now from the constancy of the
numbers involved under continuous deformations. Defining
$\lambda_{j}^{\delta}$ and $\lambda_{j,j^{\prime}}^{\delta}$ in the obvious
way, we obtain

$\displaystyle\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{4}}{I^{3}(F_{k})}$
$\displaystyle=\sum_{(x,y,y^{\prime},y^{\prime\prime})}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{4}}{I^{3}(({\tilde{F}_{k}})_{\delta})_{(x,y,y^{\prime},y^{\prime\prime})}}$
$\displaystyle=\sum_{(x,y)}\sum_{j\neq
j^{\prime}}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{\langle\lambda_{j}^{\delta},\lambda_{j,j^{\prime}}^{\delta}\rangle}=\sum_{j\neq
j^{\prime}}\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{\langle\lambda_{j},\lambda_{j,j^{\prime}}\rangle}.$

Since $T_{j,j^{\prime}}=T_{j^{\prime},j}$, we add only the numbers
$T_{j,j^{\prime}}$, with $j<j^{\prime}$, and replace $1/6$ by $1/3$. $\Box$

###  3.3 Topological triviality

We say that two subspaces $S$ and $S^{\prime}$ of a topological space $X$ have
the _same topological type_ if there is a homeomorphism $X\to X$ restricting
to an homeomorphism $S\to S^{\prime}$. Milnor showed that two isolated
hypersurface singularities with same topological type have the same Milnor
number. In the case of our invariants, a similar result can be obtained using
the results in [28] and [30] (see [4] for details).

###### Proposition 3.9

Let $F,G:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be finitely
$\mathcal{A}$-determined map-germs. If $F$ and $G$ are topologically
equivalent, then $\mu({\mathcal{D}}(F))=\mu({\mathcal{D}}(G))$,
$r({\mathcal{D}}(F))=r({\mathcal{D}}(G))$, $C(F)=C(G)$ and $T(F)=T(G)$.

The Milnor number of the double point curve is enough to determine the
topological triviality of families of finitely $\mathcal{A}$-determined map-
germs.

######  Proposition 3.10 (Corollary 40 [3])

A family of finitely $\mathcal{A}$-determined map-germs
$G_{t}\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ is topologically trivial
if, and only if, $\mu({\mathcal{D}}(G_{t}))$ is constant along the parameter
$t$.

Using the upper semi-continuity of the numbers involved, Proposition 3.10 can
be combined with Proposition 3.4 to yield the following result.

###### Corollary 3.11

Let $F_{k}^{t}=(x,y^{k},f_{t}(x,y))$ be a family of finitely
$\mathcal{A}$-determined $k$-folding map-germs. The following statements are
equivalent:

  * (1)

The family $F_{k}^{t}$ is topologically trivial.

  * (2)

The numbers $\mu({\mathcal{D}}_{j})$ and
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$ are constant along the
family $F_{k}^{t}$.

  * (3)

The numbers
$C,\mu({\mathcal{D}}_{j}),{\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}$
and $T_{j,j^{\prime}}$ are constant along the family $F_{k}^{t}$.

##  4 The jet space stratification

In this section, we study the singularities of $k$-folding map-germs which we
take in standard form $F_{k}(x,y)=(x,y^{k},f(x,y))$ (see Remarks 2.3(4)). We
identify the set of such germs with the set $\mathcal{O}_{2}$ of germs, at the
origin, of holomorphic functions $f$. For each $k$, we obtain a stratification
$\mathcal{S}_{k}$ of $J^{11}(2,1)$ (and hence of $J^{l}(2,1)$ for $l\geq 11$).
The stratification consists of the strata of codimension $\leq 4$ stated in
Theorem 1.1 together with the complement of their union (i.e., the union of
strata of codimension $\geq 5$). Every stratum of codimension $\leq 4$ of
$\mathcal{S}_{k}$ consists of finitely $\mathcal{A}$-determined and pairwise
topologically equivalent $k$-folding map-germs. The jet level $l=11$ is
determined by the conditions defining the strata of $\mathcal{S}_{k}$ which
involve the coefficients of $f$ in (2) up to degree $11$ (see Tables 1, 2, 4,
6).

As pointed out in the introduction, the case $k=2$ was studied in [6, 10, 32].
The stratification $\mathcal{S}_{2}$ can be recovered from the results in this
paper. The different $\mathcal{A}$-classes obtain in [6] correspond to
different topological classes. This follows by analysing the invariants
$C,T,\mu(\mathcal{D})$ and $r(\mathcal{D})$. We shall suppose here that $k\geq
3$ and write the jets of $f$ as in (2).

It is clear that $F_{k}$ is an immersion if and only if $a_{11}\neq 0$. All
immersions are $\mathcal{A}$-finitely determined and pairwise topologically
equivalent. Moreover, only immersions have $\mathcal{D}=\emptyset$, hence
$r(\mathcal{D})=0$. We define $a_{11}\neq 0$ as the open stratum of
$\mathcal{S}_{k}$ (corresponding to the jets of all immersions) and choose

${\bf M}^{k}_{0}\colon(x,y)\mapsto(x,y^{k},y)$

as a normal form for topological equivalence of germs in this stratum.

The strata corresponding to singular germs are organized into four branches
according to the following result.

###### Lemma 4.1

For $k\geq 3$, every singular $k$-folding map-germ is $\mathcal{A}$-equivalent
to a $k$-folding map-germ whose $2$-jet is equal to
$(x,0,xy+y^{2}),(x,0,y^{2}),(x,0,xy)$, or $(x,0,0).$

  * Proof

As we are assuming $F_{k}$ to be singular at the origin, we have $a_{11}=0$.
Then $F_{k}$ is $\mathcal{A}$-equivalent to a germ whose $2$-jet is
$(x,0,a_{21}xy+a_{22}y^{2})$. Depending on the coefficients $a_{21}$ and
$a_{22}$, the $2$-jet can be taken to one of the following forms:

$\begin{array}[]{lcll}(x,0,xy+y^{2})&\iff&a_{11}=0,a_{21}a_{22}\neq
0&\mbox{\rm(Branch 1)}\\\ (x,0,y^{2})&\iff&a_{11}=a_{21}=0,\,a_{22}\neq
0&\mbox{\rm(Branch 2)}\\\ (x,0,xy)&\iff&a_{11}=a_{22}=0,\,a_{21}\neq
0&\mbox{\rm(Branch 3)}\\\ (x,0,0)&\iff&a_{11}=a_{21}=a_{22}=0&\mbox{\rm(Branch
4)}\end{array}$

$\Box$

We have the following about $\mathcal{A}$-simplicity of germs of $k$-folding
maps.

###### Proposition 4.2

There are no $\mathcal{A}$-simple $k$-folding map-germs for $k\geq 5$.

  * Proof

It is enough to show that the orbit of a map-germ $F_{k}$ with a 2-jet
$(x,0,xy+y^{2})$ is not simple as the orbits of germs in the remaining
branches in Lemma 4.1 are adjacent to it. For such a germ, we have
$j^{k}F_{k}\sim_{\mathcal{A}^{(k)}}(x,(y-\frac{1}{2}x)^{k},y^{2})$. The result
follows by Theorem 1:1 in [23] as there are no $\mathcal{A}$-simple germs of
the form $(x,y^{2},f(x,y))$ with $j^{4}f\equiv 0$.

When $k=4$ and for $F_{4}$ in Branch 1, we have
$j^{4}F_{4}\sim_{\mathcal{A}^{(4)}}(x,xy^{3}+x^{3}y,y^{2})$. This is a
$C_{3}$-singularity and is $\mathcal{A}$-simple [23]. For $F_{4}$ in Branch 2,
we have $j^{4}F_{4}\sim_{\mathcal{A}^{(4)}}(x,0,y^{2})$ so it leads to non
$\mathcal{A}$-simple germs. By adjacency, the germs in Branches 3 and 4 also
lead to non $\mathcal{A}$-simple germs. Therefore, the $C_{3}$-singularity is
the only $\mathcal{A}$-simple singularity of $4$-folding map-germs.

The case $k=3$ is treated in §4.1 where there are several $\mathcal{A}$-simple
singularities of $3$-folding map-germs. $\Box$

###### Remarks 4.3

1\. The degree of $\mathcal{A}$-determinacy of a singular germ $F_{k}$ is
greater or equal to $k$ (the germ $(x,y)\mapsto(x,0,f(x,y))$ is not finitely
$\mathcal{A}$-determined for any $f$).

2\. For germs in Branch 1 or Branch 2, we have
$F_{k}\sim_{\mathcal{A}}(x,yp(x,y^{2}),y^{2})$ for some germ $p$. One can
study the germ $p$, as was done in [23], instead of $F_{k}$, but this blurs
the order of the original $k$-folding map-germ. Also, the approach in [23] of
reducing the action of $\mathcal{A}$ on the set of $2$-folding map-germs to
the action of a subgroup of $\mathcal{K}$ on $\mathfrak{m}_{2}$ does not
extend to germs of $k$-folding maps for $k\geq 3$.

###  4.1 The case $k=3$

When $k=3$ we get several $\mathcal{A}$-simple map-germs. For this reason, we
treat this case separately.

###### Theorem 4.4

The only $\mathcal{A}$-simple singularities a $3$-folding map-germ $F_{3}$ can
have are those of type $S_{2l-1}$, $l\geq 2$, or $H_{s}$, $s\geq 3$. In the
real case, the $S_{2l-1}$-singularities are of type $S_{2l-1}^{-}$. The strata
of $\mathcal{S}_{3}$ of codimension $\leq 4$ are given in Table 1.

Table 1: The strata of $\mathcal{S}_{3}$ of codimension $\leq 4$.

Normal form | Defining equations and open conditions | Codim
---|---|---
Immersion | $a_{11}\neq 0$ | 0
$S_{1}$ | $a_{11}=0,a_{22}a_{21}\neq 0$ | 1
$S_{3}$ | $a_{11}=a_{21}=0,a_{22}a_{31}\neq 0$ | 2
$S_{5}$ | $a_{11}=a_{21}=a_{31}=0,a_{22}a_{41}\neq 0$ | 3
$S_{7}$ | $a_{11}=a_{21}=a_{31}=a_{41}=0,a_{22}a_{51}\neq 0$ | 4
$H_{2}$ | $a_{11}=a_{22}=0,a_{21}\neq 0,CndH_{2}\neq 0$ | 2
$H_{3}$ | $a_{11}=a_{22}=CndH_{2}=0,a_{21}\neq 0,CndH_{3}\neq 0$ | 3
$H_{4}$ | $a_{11}=a_{22}=CndH_{2}=CndH_{3}=0,a_{21}\neq 0,CndH_{4}\neq 0$ | 4
$X_{4}$ | $a_{11}=a_{21}=a_{22}=0,a_{31}a_{32}a_{44}\neq 0$ | 4
${\bf U}_{4}^{3}$ | $a_{11}=a_{21}=a_{22}=a_{44}=0,a_{31}a_{32}\neq 0,CndUm_{8}\neq 0$ | 4
${\bf X}_{4}^{3}$ | $a_{11}=a_{21}=a_{22}=a_{31}=0,a_{32}a_{41}a_{44}\neq 0$ | 4
${\bf W}_{4}^{3,1}$ | $a_{11}=a_{21}=a_{22}=a_{32}=0,a_{31}a_{44}\neq 0,a_{31}a_{55}-a_{42}a_{44}\neq 0$ | 4

$\begin{array}[]{rcl}CndH_{2}&=&a_{32}a_{44}-a_{55}a_{21}\\\
CndH_{3}&=&a_{88}a_{21}^{3}-(a_{77}a_{32}+a_{44}a_{65})a_{21}^{2}+a_{44}(a_{42}a_{44}+a_{32}a_{54})a_{21}-a_{31}a_{44}^{2}a_{32}.\\\
CndH_{4}&=&a_{11,11}a_{21}^{5}-(a_{44}a_{98}+a_{10,10}a_{32}+a_{77}a_{65})a_{21}^{4}\\\
&&+(a_{44}a_{87}a_{32}+a_{32}a_{54}a_{77}+a_{44}a_{54}a_{65}+2a_{44}a_{77}a_{42}+a_{44}^{2}a_{75})a_{21}^{3}\\\
&&-(a_{52}a_{44}^{2}+a_{44}a_{31}a_{65}+2a_{42}a_{44}a_{54}+a_{44}a_{64}a_{32}+2a_{77}a_{32}a_{31}+a_{54}^{2}a_{32})a_{44}a_{21}^{2}\\\
&&+(2a_{44}a_{31}a_{42}+a_{44}a_{41}a_{32}+3a_{32}a_{54}a_{31})a_{44}^{2}a_{21}-2a_{31}^{2}a_{32}a_{44}^{3}\\\
CndUm_{8}&=&a_{55}(a_{31}a_{55}-a_{54}a_{32})+a_{77}a_{32}^{2}\,\,\mbox{\rm(see
Table \ref{tab:ClassifPar})}\end{array}$

  * Proof

We can write $f(x,y)=f_{0}(x,y^{3})+yf_{1}(x,y^{3})+y^{2}f_{2}(x,y^{3})$, for
some germs of holomorphic functions $f_{i}$, $i=0,1,2$. Then
$F_{3}\sim_{\mathcal{A}}(x,y^{3},yf_{1}(x,y^{3})+y^{2}f_{2}(x,y^{3})).$

Suppose that $a_{10}=0$ ($F_{3}$ is singular) and $a_{22}\neq 0$. Then

$F_{3}\sim_{\mathcal{A}}(x,y^{3},y(g(x)+y^{3}h(x,y^{3}))+y^{2}(a_{22}+k(x,y^{3})))$

for some germs of holomorphic functions $g\in\mathfrak{m}_{1}$,
$h\in\mathcal{O}_{2}$ and $k\in\mathfrak{m}_{2}$. We can make successive
changes of coordinates in the target so that
$j^{p}F_{3}\sim_{\mathcal{A}^{(p)}}(x,y^{3},yL(x)+a_{22}y^{2})$ for any $p\geq
3$. It is not difficult to show that $F_{3}$ is finitely
$\mathcal{A}$-determined if, and only if, $ord(L)=ord(g)=ord(f_{y}(x,0))$ is
finite. Suppose that this is the case and denote by $l$ that order. Then
$j^{l}L(x)=a_{l1}x^{l}$, $a_{l1}\neq 0$, and the change of coordinates
$y\mapsto y-(a_{l1}/2a_{22})x^{l}$ in the source yields
$j^{2l+1}F_{3}\sim_{\mathcal{A}^{(2l+1)}}(x,y^{3}-3(a_{l1}/2a_{22})^{2}yx^{2l},a_{22}y^{2})$.
This is an $S_{2l-1}$-singularity, and since it is
$(2l+1)$-$\mathcal{A}$-determined, we have
$F_{3}\sim_{\mathcal{A}}(x,y^{3}-yx^{2l},y^{2})$. In the real case, this is an
$S_{2l-1}^{-}$-singularity.

The above calculations show, in particular, that we do not get the simple
singularities $B_{l}^{\pm}$, $C_{l}^{\pm}$ and $F_{4}$ whose 2-jets are
$\mathcal{A}^{(2)}$-equivalent to $(x,0,y^{2})$.

Similar calculations show that when $a_{11}=a_{22}=0$ and $a_{21}\neq 0$, we
get an $H_{l}$-singularity when the singularity of $F_{3}$ is finitely
$\mathcal{A}$-determined.

The remaining cases are studied in the same way as in the case $k\geq 4$
(Table 6). We get three strata with topological normal forms ${\bf
U}_{4}^{3},{\bf X}_{4}^{3}$ and ${\bf W}_{4}^{3,1}$ (the germs in these strata
are not $\mathcal{A}$-simple) together with the stratum represented by the
topological normal form ${\bf U}^{3}_{3}$, which is topologically equivalent
to Mond’s singularity $X_{4}$ (see §2 and [23]). $\Box$

###### Remark 4.5

The invariants associated to the simple singularities can be found in [24];
those associated to ${\bf U}_{4}^{3},{\bf X}_{4}^{3},{\bf W}_{4}^{3,1}$ are
given in Table 7.

###  4.2 The case $k\geq 4$

We consider here the case when $k\geq 4$, which we divide into the four
branches according to the $\mathcal{A}^{(2)}$-orbits in Lemma 4.1. In all that
follows, $j\in\\{1,\ldots,k-1\\}$, and subindices of singularities indicate
the codimension of the stratum.

####  4.2.1 Branch 1: $a_{11}=0$, $a_{21}a_{22}\neq 0$

###### Theorem 4.6

Any germ $F_{k}$ of a $k$-folding map satisfying $a_{11}=0$ and
$a_{21}a_{22}\neq 0$ is finitely $\mathcal{A}$-determined and is topologically
equivalent to

${\bf M}^{k}_{1}\colon(x,y)\mapsto(x,y^{k},xy+y^{2}).$

The invariants take the values $\mu({\mathcal{D}})=(k-2)^{2},\,C=k-1,\,T=0$
and $r({\mathcal{D}})=k-1$ and the double point curve of $F_{k}$ is the union
of $k-1$ regular curves intersecting transversally.

  * Proof

The functions defining the branches ${\mathcal{D}}_{j}$ of the double point
curve (see Theorem 3.2) are given by
$\lambda_{j}=a_{21}x+(1+\xi^{j})a_{22}y+O(2),$ where $O(l)$ denotes a
remainder of order $l$. Clearly, all of the branches ${\mathcal{D}}_{j}$ are
regular curves. As the scalars $1+\xi^{j}$ are pairwise distinct, the space
${\mathcal{D}}={\mathcal{D}}(F_{k})=\bigcup_{j}{\mathcal{D}}_{j}$ consists of
$k-1$ regular curves intersecting transversally at the origin.

We have $\mu({\mathcal{D}}_{j})=0$ and
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=1$, and from Proposition 3.4
we obtain $\mu({\mathcal{D}})=(k-2)^{2}$. By Theorem 3.3, any germ $F_{k}$
satisfying the conditions in the statement of the theorem is finitely
$\mathcal{A}$-determined. These germs form a stratum of codimension 1 defined
by $\\{a_{11}=0,a_{21}a_{22}\neq 0\\}$. Since this stratum is path connected,
we conclude by Theorem 3.10 that it consists of topologically equivalent
germs. We choose for a topological model the germ ${\bf M}^{k}_{1}$ given in
the statement of the theorem.

By Proposition 3.11, it is enough to compute $C$ and $T_{j,j^{\prime}}$ for
${\bf M}^{k}_{1}$ as these invariants are constant along the stratum. We have
$C=\operatorname{dim}_{\mathbb{C}}{\mathcal{O}_{2}}/{\langle
y^{k-1},x+2y\rangle}=k-1$ and
$T_{j,j^{\prime}}=\operatorname{dim}_{\mathbb{C}}{\mathcal{O}_{2}}/{\langle
x+(1+\xi^{j})y,1\rangle}=0.$ $\Box$

###### Remark 4.7

The singularity ${\bf M}^{4}_{1}$ is the $\mathcal{A}$-simple singularity
$C_{3}$ (see the proof of Proposition 4.2).

The proofs for the cases in the remaining branches follow by similar arguments
used in the proof of Theorem 4.6 (except for the calculations of $T$). To
avoid repetition, we highlight only key differences in each case. The notation
for the conditions that define the strata are those indicated in the tables.

####  4.2.2 Branch 2: $a_{11}=a_{21}=0$, $a_{22}\neq 0$

###### Theorem 4.8

The strata of codimension $\leq 4$ of finitely $\mathcal{A}$-determined
$k$-folding map-germs in the branch $a_{11}=a_{21}=0$, $a_{22}\neq 0$ are
those given in Table 2. The invariants associated to the germs in each stratum
are given in Table 3.

  * Proof

The result follows from Propositions 4.9, 4.11 and 4.12. $\Box$

Table 2: Strata of codimension $\leq 4$ in Branch 2.

Name | Defining equations and open condition | Codim
---|---|---
|  together with $a_{11}=a_{21}=0$, $a_{22}\neq 0$ |
${\bf M}^{k}_{2}$, $2\nmid k$ | $a_{31}\neq 0$ | 2
${\bf M}^{k}_{2}$, $2\mid k$ | $a_{33}a_{31}\neq 0$ | 2
${\bf M}^{k}_{3}$, $2\nmid k$ |  $a_{31}=0$, $a_{41}\neq 0$ | 3
${\bf M}^{k}_{3}$, $2\mid k$ |  $a_{31}=0$, $a_{33}a_{41}\neq 0$ | 3
${\bf M}^{k}_{4}$, $2\nmid k$ |  $a_{31}=a_{41}=0$, $a_{51}\neq 0$ | 4
${\bf M}^{k}_{4}$, $2\mid k$ |  $a_{31}=a_{41}=0$, $a_{33}a_{51}\neq 0$ | 4
${\bf N}^{k}_{3}$, $2\mid k$ |  $a_{33}=0$, $a_{31}\neq 0$, ${CndNA}_{3}\neq 0$ | 3
${\bf N}^{k}_{4}$, $2\mid k$ |  $a_{33}={CndNA}_{3}=0$, $a_{31}\neq 0$, ${CndNA}_{5}\neq 0$ | 4
${\bf O}^{k}_{4}$, $2\mid k$ |  $a_{31}=a_{33}=0$, $a_{41}a_{43}\neq 0$ | 4

$\begin{array}[]{l}{CndNA}_{3}=a_{43}^{2}-4a_{31}a_{55}\\\
{CndNA}_{5}=8a_{31}^{3}a_{77}-4a_{65}a_{43}a_{31}^{2}+2a_{53}a_{43}^{2}a_{31}-a_{41}^{2}a_{43}^{3}\end{array}$

For germs in this branch, the map-germ $(x,y)\mapsto(x,f(x,y))$ is finite and
generically two-to-one. Therefore, $T=0$ for any finitely
$\mathcal{A}$-determined map-germ in this branch.

The germs of the functions defining the double point branch
${\mathcal{D}}_{j}$ is given by

$\lambda_{j}={\vartheta_{2j}a_{22}y}+a_{31}x^{2}+\vartheta_{2j}a_{32}xy+\vartheta_{3j}a_{33}y^{2}+O(3).$

The branch ${\mathcal{D}}_{j}$ is thus regular if, and only if,
$\vartheta_{2j}\neq 0$. By Lemma 3.6, $\vartheta_{2j}=0$ when $k$ is even and
$j=k/2$.

Table 3: Topological invariants of germs in the strata in Table 2. Name | $C$ | $T$ | $\mu({\mathcal{D}})$ | $r({\mathcal{D}})$
---|---|---|---|---
${\bf M}^{k}_{2}$, $2\nmid k$ | $2k-2$ | 0 | $2(k-1)(k-2)-k+2$ | $k-1$
${\bf M}^{k}_{2}$, $2\mid k$ | $2k-2$ | 0 | $(k-1)(k-2)+3-k$ | $k$
${\bf M}^{k}_{3}$, $2\nmid k$ | $3k-3$ | 0 | $3(k-1)(k-2)-k+2$ | $k-1$
${\bf M}^{k}_{3}$, $2\mid k$ | $3k-3$ | 0 | $(k-1)(k-2)+4-k$ | $k-1$
${\bf M}^{k}_{4}$, $2\nmid k$ | $4k-4$ | 0 | $4(k-1)(k-2)-k+2$ | $k-1$
${\bf M}^{k}_{4}$, $2\mid k$ | $4k-4$ | 0 | $(k-1)(k-2)+5-k$ | $k$
${\bf N}^{k}_{3}$, $2\mid k$ | $2k-2$ | $0$ | $(2k-3)(k-2)+3$ | $k$
${\bf N}^{k}_{4}$, $2\mid k$ | $2k-2$ | $0$ | $(2k-3)(k-2)+5$ | $k$
${\bf O}^{k}_{4}$, $2\mid k$ | $3k-3$ | $0$ | $(3k-4)(k-2)+4$ | $k+1$

###### Proposition 4.9

Any $k$-folding map-germ $F_{k}$ satisfying $a_{11}=a_{21}=\ldots=a_{l1}=0$,
$a_{22}a_{(l+1)1}\neq 0$, for some $l\geq 2$, and $a_{33}\neq 0$ when $k=2p$,
is finitely $\mathcal{A}$-determined and is topologically equivalent to

${\bf M}^{k}_{l}\colon(x,y)\mapsto(x,y^{k},y^{2}+y^{3}+x^{l}y).$

The invariants $C$, $T$, $\mu(\mathcal{D})$ and $r(\mathcal{D})$ are as in
Table 2. All the double point branches are regular curves except for the
branch $\mathcal{D}_{p}$, when $k=2p$, which has an $A_{l-1}$-singularity. We
have ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=l$, for all $j\neq
j^{\prime}$.

  * Proof

Fix an index $j$ and assume that $2\nmid k$ or that $k=2p$ but $j\neq p$. Then
${\mathcal{D}}_{j}$ is a regular curve and can be parametrised by
$t\mapsto(t,\gamma_{j}(t))$, with
$\gamma_{j}(t)=-\frac{a_{l+1,1}}{(1+\xi^{j})a_{22}}t^{l}+O(l+1).$ Clearly, any
two distinct branches have order of contact equal to $l$.

Suppose now that $k=2p$. As $\vartheta^{2}_{p}=0$, the coefficients of
$x^{s}y$ in $\lambda_{p}$ vanish for all $s\geq 1$. Moreover, since
$\vartheta^{3}_{p}=1$, the function $\lambda_{p}$ is of the form
$\lambda_{p}=a_{l+1,1}x^{l}+a_{33}y^{2}+y^{3}h(y).$ This implies that
$\lambda_{p}$ is $\mathcal{R}$-equivalent to $y^{2}+x^{l}$ if and only if
$a_{33}\neq 0$, in which case it has an $A_{l-1}$-singularity. As for the
contact between branches,
$ord(\lambda_{p}(t,\frac{-a_{l+1,1}}{(1+\xi^{j})a_{22}}t^{l}+O(l+1))=l,$ hence
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{p}=l$. This determines
$\mu(\mathcal{D})$, and hence the topological triviality and the constancy of
the invariants along the stratum. $\Box$

###### Remarks 4.10

1\. Branch 1 can be considered as a particular case of the strata ${\bf
M}^{k}_{l}$ in Proposition 4.9 (the condition $a_{33}\neq 0$ is not needed
when $l=1$).

2\. When $2\nmid k$, the term $y^{3}$ in ${\bf M}^{k}_{l}$ is irrelevant for
topological equivalence. We include it to represent both $k$ even and $k$ odd
by the same map-germ. We do this for all subsequent topological normal forms.

###### Proposition 4.11

Suppose that $k=2p$. Any $k$-folding map-germ $F_{k}$ satisfying
$a_{11}=a_{21}=a_{33}=0$, $a_{22}a_{31}\neq 0$ and the additional conditions
in (a) or (b) below is finitely $\mathcal{A}$-determined and is topologically
equivalent to one of the germs

${\bf N}^{k}_{l}\colon(x,y)\mapsto(x,y^{k},y^{2}+x^{2}y+y^{2l-1}),\,l=3,4.$

The invariants $\mu({\mathcal{D}(F_{k})}),C,T,r({\mathcal{D}}(F_{k}))$ are as
in Table 3. The branches ${\mathcal{D}}_{j}$ are regular curves for all $j\neq
p$ and ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=2$ for all $j\neq
j^{\prime}$.

(a) If ${CndNA}_{3}\neq 0$, then the branch ${\mathcal{D}}_{p}$ has an
$A_{3}$-singularity and the map-germ is topologically equivalent to ${\bf
N}^{k}_{3}$.

(b) If ${CndNA}_{3}=0$ and ${CndNA}_{5}\neq 0$, then ${\mathcal{D}}_{p}$ has
an $A_{5}$-singularity and the map-germ is topologically equivalent to ${\bf
N}^{k}_{4}$.

  * Proof

We have
$\lambda_{p}=a_{31}x^{2}+a_{41}x^{3}+a_{43}xy^{2}+a_{51}x^{4}+a_{53}x^{2}y^{2}+a_{55}y^{4}+O(5).$
It has an $A_{3}$-singularity if, and only if, ${CndNA}_{3}\neq 0$. When
${CndNA}_{3}=0$, we need to consider the $7$-jet of $\lambda_{p}$. A
calculation shows that $\lambda_{p}$ has an $A_{5}$-singularity if, and only
if, ${CndNA}_{5}\neq 0$. In both cases, we have
$ord(\lambda_{p}(t,-{a_{31}}/{((1+\xi^{j})a_{22})}t^{2}+O(3)))=2$ for all
$j\neq p$. $\Box$

###### Proposition 4.12

Suppose that $k=2p$. Any $k$-folding map-germ $F_{k}$ satisfying
$a_{11}=a_{21}=a_{31}=a_{33}=0$ and $a_{22}a_{41}a_{43}\neq 0$ is finitely
$\mathcal{A}$-determined and is topologically equivalent to

${\bf O}^{k}_{4}\colon(x,y)\mapsto(x,y^{k},y^{2}+x^{3}y+xy^{3}).$

The codimension of the stratum is $4$ and the invariants
$\mu({\mathcal{D}}(F_{k})),C,T,$ $r({\mathcal{D}}(F_{k}))$ are as in Table 3.
The branches ${\mathcal{D}}_{j},$ $j\neq p$, are regular curves and
${\mathcal{D}}_{p}$ has a $D_{4}$-singularity. We have
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=2$ for the distinct regular
branches and ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{p}=3$, for $j\neq p$.

  * Proof

The result follows from the fact that
$\lambda_{p}=a_{41}x^{3}+a_{43}xy^{2}+O(4).$ $\Box$

####  4.2.3 Branch 3: $a_{11}=a_{22}=0$, $a_{21}\neq 0$

###### Theorem 4.13

The strata of codimension $\leq 4$ of finitely $\mathcal{A}$-determined
$k$-folding map-germs in the branch $a_{11}=a_{22}=0$, $a_{21}\neq 0$ are
those given in Table 4. The invariants of the germs in each stratum are given
in Table 5.

  * Proof

The result follows from Propositions 4.14, 4.16 and 4.18. $\Box$

Table 4: Strata of codimension $\leq 4$ in Branch 3.

Name | Defining equations and open conditions | Codim
---|---|---
|  together with $a_{11}=a_{22}=0$, $a_{21}\neq 0$ |
${\bf P}^{k}_{2}$, $3\nmid k$ | $a_{33}\neq 0$ | 2
${\bf P}^{k}_{2}$, $3\mid k$ |  $a_{33}\neq 0$, $CndH_{2}\neq 0$ | 2
${\bf P}^{k}_{3}$, $3\mid k$ |  $a_{33}\neq 0$, $CndH_{2}=0$, $CndH_{3}\neq 0$ | 3
${\bf P}^{k}_{4}$, $3\mid k$ |  $a_{33}\neq 0$, $CndH_{2}=CndH_{3}=0$, $CndH_{4}\neq 0$ | 4
${\bf Q}^{k}_{3}$, $3\nmid k$, $4\nmid k$ |  $a_{33}=0$, $a_{44}\neq 0$ | 3
${\bf Q}^{k}_{3}$, $3\mid k$, $4\nmid k$ |  $a_{33}=0$, $a_{44}\neq 0,CndH_{2}\neq 0$ | 3
${\bf Q}^{k}_{3}$, $3\nmid k$, $4\mid k$ |  $a_{33}=0$, $a_{44}\neq 0$, $CndQm_{5}\neq 0$ | 3
${\bf Q}^{k}_{3}$, $12\mid k$ |  $a_{33}=0$, $a_{44}\neq 0$, $CndH_{2}\neq 0$, $CndQm_{5}\neq 0$ | 3
${\bf Q}^{k}_{4}$, $3\mid k$, $4\nmid k$ |  $a_{33}=CndH_{2}=0$, $a_{44}\neq 0$, $CndH_{3}\neq 0$ | 4
${\bf Q}^{k}_{4}$, $12\mid k$ |  $a_{33}=CndH_{2}=0$, $CndH_{3}\neq 0$, $CndQm_{5}\neq 0$ | 4
$\widetilde{\bf Q}^{k}_{4}$, $3\nmid k$, $4\mid k$ |  $a_{33}=CndQm_{5}=0$, $a_{44}\neq 0$, $CndQm_{6}\neq 0$ | 4
$\widetilde{\bf Q}^{k}_{4}$, $12\mid k$ |  $a_{33}=CndQm_{5}=0$, $CndH_{2}\neq 0$, $CndQm_{6}\neq 0$ | 4
${\bf R}^{k}_{4}$, $4\nmid k$, $5\nmid k$ |  $a_{33}=a_{44}=0$, $a_{55}\neq 0$ | 4
${\bf R}^{k}_{4}$, $4\mid k$, $5\nmid k$ |  $a_{33}=a_{44}=0$, $a_{55}\neq 0$, $CndQm_{5}\neq 0$ | 4
${\bf R}^{k}_{4}$, $4\nmid k$, $5\mid k$ |  $a_{33}=a_{44}=0$, $a_{55}\neq 0$, $CndRm_{5}\neq 0$ | 4
${\bf R}^{k}_{4}$, $20\mid k$ |  $a_{33}=a_{44}=0$, $a_{55}\neq 0$, $CndQm_{5}\neq 0$, $CndRm_{5}\neq 0$ | 4

$\begin{array}[]{rcl}CndQm_{5}&=&a_{32}a_{55}-a_{21}a_{66}\\\
CndQm_{6}&=&a_{43}a_{55}-a_{21}a_{77}\\\
CndRm_{5}&=&a_{32}a_{66}-a_{21}a_{77}\end{array}$

Table 5: Topological invariants of germs in the strata in Table 4. Name | $C$ | $T$ | $\mu(\mathcal{D})$ | $r(\mathcal{D})$
---|---|---|---|---
${\bf P}^{k}_{2}$, $3\nmid k$ | $k-1$ | $\frac{(k-1)(k-2)}{6}$ | $(2k-3)(k-2)$ | $k-1$
${\bf P}^{k}_{2}$, $3\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+4}{6}$ | $(2k-3)(k-2)+4$ | $k-1$
${\bf P}^{k}_{3}$, $3\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+10}{6}$ | $(2k-3)(k-2)+10$ | $k-1$
${\bf P}^{k}_{4}$, $3\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+16}{6}$ | $(2k-3)(k-2)+16$ | $k-1$
${\bf Q}^{k}_{3}$, $3\nmid k,$ $4\nmid k$ | $k-1$ | $\frac{(k-1)(k-2)}{3}$ | $(3k-4)(k-2)$ | $k-1$
${\bf Q}^{k}_{3}$, $3\mid k,$ $4\nmid k$ | $k-1$ | $\frac{(k-1)(k-2)+1}{3}$ | $(3k-4)(k-2)+2$ | $k-1$
${\bf Q}^{k}_{3}$, $3\nmid k,$ $4\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+6}{3}$ | $(3k-4)(k-2)+12$ | $k-1$
${\bf Q}^{k}_{3}$, $12\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+7}{3}$ | $(3k-4)(k-2)+14$ | $k-1$
${\bf Q}^{k}_{4}$, $3\mid k$, $4\nmid k$ | $k-1$ | $\frac{(k-1)(k-2)+4}{3}$ | $(3k-4)(k-2)+8$ | $k-1$
${\bf Q}^{k}_{4}$, $12\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+10}{3}$ | $(3k-4)(k-2)+20$ | $k-1$
$\widetilde{\bf Q}^{k}_{4}$, $3\nmid k,4\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+9}{3}$ | $(3k-4)(k-2)+18$ | $k-1$
$\widetilde{{\bf Q}}^{k}_{4}$, $12\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+10}{3}$ | $(3k-4)(k-2)+20$ | $k-1$
${\bf R}^{k}_{4}$, $4\nmid k,$ $5\nmid k$ | $k-1$ | $\frac{(k-1)(k-2)}{2}$ | $(4k-5)(k-2)$ | $k-1$
${\bf R}^{k}_{4}$, $4\mid k,$ $5\nmid k$ | $k-1$ | $\frac{(k-1)(k-2)+2}{2}$ | $(4k-5)(k-2)+6$ | $k-1$
${\bf R}^{k}_{4}$, $4\nmid k,$ $5\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+8}{2}$ | $(4k-5)(k-2)+24$ | $k-1$
${\bf R}^{k}_{4}$, $20\mid k$ | $k-1$ | $\frac{(k-1)(k-2)+10}{2}$ | $(4k-5)(k-2)+30$ | $k-1$

For any finitely $\mathcal{A}$-determined $k$-folding map-germ in this branch,
we have $C=\operatorname{dim}_{\mathbb{C}}{\mathcal{O}_{2}}/{\langle
y^{k-1},a_{21}x+O(2)\rangle}=k-1.$

The double point branches are regular curves that can be parametrised by
$t\mapsto(\gamma_{j}(t),t)$, with

$\begin{array}[]{rcl}\gamma_{j}(t)&=&-\frac{1}{a_{21}}\vartheta_{3j}a_{33}t^{2}-\frac{1}{a_{21}^{2}}(\vartheta_{2j}\vartheta_{3j}a_{32}a_{33}-\vartheta_{4j}a_{21}a_{44})t^{3}\\\
&&-\frac{1}{a_{21}^{3}}(\vartheta_{2j}^{2}\vartheta_{3j}a_{32}^{2}a_{33}-\vartheta_{2j}\vartheta_{4j}a_{32}a_{44}a_{21}+\vartheta_{3j}^{2}a_{33}(a_{31}a_{33}-a_{43}a_{21})+\vartheta_{5j}a_{55}a_{21}^{2})t^{4}\\\
&&+O(5).\end{array}$

The strata are determined by the contact between the branches of the double
point curve which depend on $\vartheta_{sj}$ as well as on the coefficients
$a_{pq}$. We start with the case $a_{33}\neq 0$, where the strata depend on
the divisibility of $k$ by $3$.

###### Proposition 4.14

Suppose that $a_{11}=a_{22}=0$ and $a_{21}a_{33}\neq 0$. Any $k$-folding map-
germ $F_{k}$ in case (a) or satisfying the additional conditions in (b) is
finitely $\mathcal{A}$-determined and is topologically equivalent to one of
the map-germs

${\bf P}^{k}_{l}\colon(x,y)\mapsto(x,y^{k},xy+y^{3}+y^{3l-1}),\text{ for
}l=2,3,4.$

The invariants $\mu({\mathcal{D}}),C,T,r(\mathcal{D})$ are as in Table 5. We
have contact ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=2$, except for
${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}$ when $k=3p$ which is given in (b).

  * (a)

If $3\nmid k$, then $F_{k}$ is topologically equivalent to ${\bf P}^{k}_{2}$.

  * (b)

If $k=3p$, then the strata are as follows:

    * (b1)

If $CndH_{2}\neq 0$, then ${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=4$ and
$F_{k}$ is topologically equivalent to ${\bf P}^{k}_{2}$.

    * (b2)

If $CndH_{2}=0$ and $CndH_{3}\neq 0$, then
${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=7$ and $F_{k}$ is topologically
equivalent to ${\bf P}^{k}_{3}$.

    * (b2)

If $CndH_{2}=CndH_{3}=0$ and $CndH_{4}\neq 0$, then
${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=10$ and $F_{k}$ is topologically
equivalent to ${\bf P}^{k}_{4}$.

  * Proof

If $3\nmid k$, then by Lemma 3.6 we have $\vartheta_{3j}\neq 0$ and
$\vartheta_{3j}\neq\vartheta_{3j^{\prime}}$ for all $j\neq j^{\prime}$. This
implies ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=2$ for all $j\neq
j^{\prime}$.

If $k=3p$, then by Lemma 3.6 the equality
$\vartheta_{3j}=\vartheta_{3j^{\prime}}$ holds only when
$\\{j,j^{\prime}\\}=\\{p,2p\\}$. Again, we obtain
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=2$ for all $j\neq
j^{\prime}$ with $\\{j,j^{\prime}\\}\neq\\{p,2p\\}$.

We have $\vartheta_{3p}=\vartheta_{3(2p)}=0$,
$\vartheta_{4p}=\vartheta_{4(2p)}=1$ and $\vartheta_{5p}=\vartheta_{2(2p)}$
(Lemma 3.6). Using the parametrisations of ${\mathcal{D}}_{p}$ and
${\mathcal{D}}_{2p}$, we get ${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=4$ if,
and only if, $a_{32}a_{44}-a_{21}a_{55}\neq 0$, equivalently, $CndH_{2}\neq
0$.

When $CndH_{2}=0$, the exceptional branches ${\mathcal{D}}_{p}$ and
${\mathcal{D}}_{2p}$ are parametrised by
$x=-\frac{a_{44}}{a_{21}}y^{3}+\beta_{s}y^{7}+O(9)$, $s=1,2$, with
$\beta_{1}-\beta_{2}\neq 0$ if, and only if, $CndH_{3}\neq 0$. Then,
${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=7$.

When $CndH_{2}=CndH_{3}=0$, the exceptional branches are parametrised by
$x=-\frac{a_{44}}{a_{21}}y^{3}+\beta_{s}y^{10}+O(11)$, $s=1,2$, with
$\beta_{1}-\beta_{2}\neq 0$ if, and only if, $CndH_{4}\neq 0$. Then,
${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=10$.

The values of $T$ can be computed using the models ${\bf P}^{k}_{l}$. We have
$\lambda_{j}=x+\vartheta_{3j}y^{2}+\vartheta_{(3l-1)j}y^{3l-2}$ and
$\lambda_{j,j^{\prime}}=(\vartheta_{3j}-\vartheta_{3j^{\prime}})y+(\vartheta_{3l-1,j}-\vartheta_{3l-1,j^{\prime}})y^{3l-3}.$
By Lemma 3.6,
$T_{j,j^{\prime}}=\operatorname{dim}_{\mathbb{C}}\frac{\mathcal{O}_{2}}{\langle
x,y\rangle}=1$ when $3\nmid k$ or when $k=3p$ but
$\\{j,j^{\prime}\\}\neq\\{p,2p\\}$. For $k=3p$, we show that
$\vartheta_{3l-1,p}\neq\vartheta_{3l-1,2p}$ so $T_{p,2p}=3l-3$. $\Box$

###### Remark 4.15

The singularity ${\bf P}^{4}_{2}\colon(x,y)\mapsto(x,y^{4},xy+y^{3}+y^{5})$ is
topologically equivalent to the singularity
$T_{4}\colon(x,y)\mapsto(x,y^{4},xy+y^{3})$ in [23]. Observe that according to
Proposition 4.14, if $k$ is not divisible by $3$, then the $y^{3l-1}$ term can
be removed from the expression of ${\bf P}^{k}_{l}$, without changing the
topological class of the germ.

###### Proposition 4.16

Suppose that $a_{11}=a_{22}=a_{33}=0$ and $a_{21}a_{44}\neq 0$. Any
$k$-folding map-germ $F_{k}$ in case (a) or satisfying the additional
conditions in (b), (c) or (d) is finitely $\mathcal{A}$-determined and is
topologically equivalent to one of the following map-germs:

${\bf Q}^{k}_{3}\colon(x,y)\mapsto(x,y^{k},xy+y^{4}+y^{5}+y^{6})$ ${\bf
Q}^{k}_{4}\colon(x,y)\mapsto(x,y^{k},xy+y^{4}+y^{6}+y^{8})$ $\widetilde{{\bf
Q}}^{k}_{4}\colon(x,y)\mapsto(x,y^{k},xy+y^{4}+y^{5}+y^{7})$

The invariants are as in Table 5. We have
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=3$ except when $j$ and
$j^{\prime}$ are in the sets $J$ or $J^{\prime}$ below.

  * (a)

If $3\nmid k$ and $4\nmid k$, then there are no exceptional branches and the
germs in this stratum are topologically equivalent to ${\bf Q}_{3}^{k}$.

  * (b)

If $k=3p$ and $4\nmid k$, then $J=\\{p,2p\\}$.

    * (b1)

If $CndH_{2}\neq 0$, then ${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=4$ and
$F_{k}$ is topologically equivalent to ${\bf Q}_{3}^{k}$.

    * (b2)

If $CndH_{2}=0$ and $CndH_{3}\neq 0$, then
${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=7$ and $F_{k}$ is topologically
equivalent to ${\bf Q}_{4}^{k}$.

  * (c)

If $k=4p$, $3\nmid k$, then $J=\\{p,2p,3p\\}$.

    * (c1)

If $CndQm_{5}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=5$, for all $j,j^{\prime}\in
J,j\neq j^{\prime}$, and $F_{k}$ is topologically equivalent to ${\bf
Q}_{3}^{k}$.

    * (c2)

If $CndQm_{5}=0$ and $CndQm_{6}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=6$, for all $j,j^{\prime}\in
J,j\neq j^{\prime}$, and $F_{k}$ is topologically equivalent to
$\widetilde{\bf Q}_{4}^{k}$.

  * (d)

If $k=12p$, the exceptional contact between double point branches occurs when
the indices are in $J=\\{4p,8p\\}$ or $J^{\prime}=\\{3p,6p,9p\\}$. There are
three strata:

    * (d1)

If $CndH_{2}\neq 0$ and $CndQm_{5}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=4$ (resp.
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=5$) for all distinct pairs
with $j,j^{\prime}$ in $J$ (resp. $J^{\prime}$), and $F_{k}$ is topologically
equivalent to ${\bf Q}_{3}^{k}$.

    * (d2)

If $CndH_{2}=0,CndH_{3}\neq 0$ and $CndQm_{5}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=7$ (resp.
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=5$) for all distinct pairs
with $j,j^{\prime}$ in $J$ (resp. $J^{\prime}$), and $F_{k}$ is topologically
equivalent to ${\bf Q}_{4}^{k}$.

    * (d3)

If $CndQm_{5}=0,CndQm_{6}\neq 0,$ and $CndH_{2}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=4$ (resp.
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=6$) for all distinct pairs
with $j,j^{\prime}$ in $J$ (resp. $J^{\prime}$), and $F_{k}$ is topologically
equivalent to $\widetilde{\bf Q}_{4}^{k}$.

  * Proof

The branches $\mathcal{D}_{j}$ can be parametrised by
$t\mapsto(\gamma_{j}(t),t)$ with

$\begin{array}[]{rcl}\gamma_{j}(t)&=&-\frac{1}{a_{21}}\vartheta_{4j}a_{44}t^{3}-\frac{1}{a_{21}^{2}}(\vartheta_{2j}\vartheta_{4j}a_{32}a_{44}-\vartheta_{5j}a_{21}a_{55})t^{4}\\\
&&-\frac{1}{a_{21}^{3}}(\vartheta_{2j}^{2}\vartheta_{4j}a_{32}^{2}a_{44}-\vartheta_{3j}\vartheta_{4j}a_{43}a_{44}a_{21}-\vartheta_{2j}\vartheta_{5j}a_{21}a_{32}a_{55}+\vartheta_{6j}a_{66}a_{21}^{2})t^{5}+O(6).\end{array}$

(a) If $3\nmid k$, $4\nmid k$, then
$\vartheta_{4j}\neq\vartheta_{4j^{\prime}}$ for all $j,j^{\prime}$ with $j\neq
j^{\prime}$ (Lemma 3.6). Therefore,
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=3$ for all distinct pairs.

(b) If $k=3p$ and $4\nmid p$, then $\vartheta_{4j}\neq\vartheta_{4j^{\prime}}$
for all distinct pairs with $j$ or $j^{\prime}$ not in $J=\\{p,2p\\}$. For
such pairs, ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=3$.

Using the fact that
$\vartheta_{6p}=\vartheta_{6(2p)}=\vartheta_{3p}=\vartheta_{3(2p)}=0$,
$\vartheta_{4p}=\vartheta_{4(2p)}=1$ and
$\vartheta_{5p}=\vartheta_{2p}\neq\vartheta_{5(2p)}=\vartheta_{2(2p)}$, the
parametrisation of the exceptional branch $\mathcal{D}_{p}$ becomes

$\gamma_{p}(t)=-\frac{a_{44}}{a_{21}}t^{3}-\frac{\vartheta_{2p}(a_{32}a_{44}-a_{21}a_{55})}{a_{21}^{2}}t^{4}-\frac{\vartheta_{2p}^{2}a_{32}(a_{32}a_{44}-a_{21}a_{55})}{a_{21}^{3}}t^{5}+O(6).$

A parametrisation of $\mathcal{D}_{2p}$ is obtained by replacing
$\vartheta_{2p}$ by $\vartheta_{2(2p)}$ in the expression of $\gamma_{p}$.
Therefore, ${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=4$ if, and only if,
$a_{32}a_{44}-a_{21}a_{55}\neq 0$, i.e., $CndH_{2}\neq 0$.

When $CndH_{2}=0$, we have

$\gamma^{1}_{p}=-\frac{a_{44}}{a_{21}}t^{3}-\frac{a_{31}a_{44}^{2}-a_{21}a_{44}a_{54}+a_{21}^{2}a_{77}}{a_{21}^{3}}t^{6}+\\\
\frac{\vartheta^{2}_{p}CndH_{3}}{a^{4}_{21}}t^{7}+O(8).$

For $\gamma_{2p}$, we replace $\vartheta^{2}_{p}$ by $\vartheta^{2}_{2p}$ in
$\gamma_{p}$. It follows that ${\mathcal{D}}_{p}\cdot{\mathcal{D}}_{2p}=7$ if,
and only if, $CndH_{3}\neq 0$.

(c) If $k=4p$ and $3\nmid p$, we have
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=3$ for $j$ or $j^{\prime}$
not in $\\{p,2p,3p\\}$.

The parametrisation of $\mathcal{D}_{p}$ becomes
$\gamma_{p}(t)={a_{55}}/{a_{21}}t^{4}+{\vartheta^{2}_{p}(a_{32}a_{55}-a_{21}a_{66})}/{a_{21}^{2}}t^{5}+O(6),$
and similarly for $\gamma_{2p}$ and $\gamma_{3p}$ replacing $p$ by $2p$ and
$3p$ respectively. Consequently, the branches $\mathcal{D}_{p}$,
$\mathcal{D}_{2p}$, $\mathcal{D}_{3p}$ have pairwise order of contact $5$ if,
and only if, $a_{32}a_{55}-a_{21}a_{66}\neq 0$, i.e., $CndQm_{5}\neq 0.$

When $CndQm_{5}=0$,
$\gamma_{p}=-{a_{55}}/{a_{21}}t^{4}-\vartheta_{2p}\vartheta_{3p}({a_{21}a_{77}-a_{43}a_{55}})/{a_{21}^{2}}t^{6}+O(7)$,
with similar ajustements as above for $\gamma_{2p}$ and $\gamma_{3p}$.
Therefore, the three exceptional branches have pairwise order of contact $6$
if, and only if, $a_{21}a_{77}-a_{43}a_{55}\neq 0$, i.e., $CndQm_{6}\neq 0$.

(d) This follows by Lemma 3.6 and (b) and (c) above.

The contact between the branches determines $\mu(\mathcal{D})$, the
topological types and their associated strata. It remains to compute $T$ for
each normal form.

For ${\bf Q}^{k}_{3}$ we have
$\lambda_{j}=x+\vartheta_{4j}y^{3}+\vartheta_{5j}y^{5}+\vartheta_{6j}y^{5},$
and
$\lambda_{j,j^{\prime}}=(\vartheta_{4j}-\vartheta_{4j^{\prime}})y^{2}+(\vartheta_{5j}-\vartheta_{5j^{\prime}})y^{3}+(\vartheta_{6j}-\vartheta_{6j^{\prime}})y^{4}.$
Using the properties of $\vartheta_{sj}$ in Lemma 3.6, we have
$T_{j,j^{\prime}}=2$ unless $k=3p$ and $j,j^{\prime}\in\\{p,2p\\}$, or $k=4p$
and $j,j^{\prime}\in\\{p,2p,3p\\}$. In the first case we get
$T_{j,j^{\prime}}=3$ and in the second $T_{j,j^{\prime}}=4$.

The invariant for the germ ${\bf Q}^{k}_{4}$ differs from ${\bf Q}^{k}_{3}$
only when $k$ is divisible by $3$. For $k=3p$, we have $T_{p,2p}=6$ and
$T_{j,j^{\prime}}=4$ if $k=4q$ and $j,j^{\prime}\in\\{q,2q,3q\\}$. All other
indices $j$ and $j^{\prime}$ give $T_{j,j^{\prime}}=2$.

Similarly, for the germ $\widetilde{\bf Q}^{k}_{4}$ and for $k=4p$, we get
$T_{j,j^{\prime}}=5$ if $j,j^{\prime}\in\\{p,2p,3p\\}$. If $k=3q$, then
$T_{q,2q}=3$. All other indices $j$ and $j^{\prime}$ give
$T_{j,j^{\prime}}=2$. $\Box$

###### Remark 4.17

When $k$ is divisible by $12$, the germs ${\bf Q}^{k}_{4}$ and $\widetilde{\bf
Q}^{k}_{4}$ have the same invariants $C,T,\mu(D)$ and $r(D)$ but they are not
topologically equivalent as their associated sets of contacts between double
points branches are distinct (see [33]).

###### Proposition 4.18

Suppose that $a_{11}=a_{22}=a_{33}=a_{44}=0$ and $a_{21}a_{55}\neq 0$. Any
$k$-folding map-germ $F_{k}$ in case (a) or satisfying the additional
conditions in (b) or (c) is finitely $\mathcal{A}$-determined and is
topologically equivalent to

${\bf R}^{k}_{4}\colon(x,y)\mapsto(x,y^{k},xy+y^{5}+y^{6}+y^{7}).$

The invariants associated to the germs in the stratum are as in Table 5. We
have ${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=4$ except for the
exceptional pairs of branches below.

  * (a)

If $4\nmid k$ and $5\nmid k$, there are no additional conditions and no
exceptional branches.

  * (b)

If $k=4p$, $5\nmid k$ and $CndQm_{5}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=5$ for all distinct pairs
with $j,j^{\prime}$ in $J=\\{p,2p,3p\\}$.

  * (c)

If $k=5p$, $4\nmid k$ and $CndRm_{5}\neq 0$,
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=5$ for all distinct pairs
with $j,j^{\prime}$ in $J=\\{p,2p,3p,4p\\}$.

  * (d)

If $k=20p$, $CndQm_{5}\neq 0$ and $CndRm_{5}\neq 0$, then
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=5$ for all distinct pairs
with $j,j^{\prime}$ in $J=\\{5p,10p,15p\\}$ or in
$J^{\prime}=\\{4p,8p,12p,16p\\}$.

  * Proof

The branches $\mathcal{D}_{j}$ can be parametrised by
$t\mapsto(\gamma_{j}(t),t)$ with

$\begin{array}[]{rcl}\gamma_{j}(t)&=&-\frac{1}{a_{21}}\vartheta_{5j}a_{55}t^{4}+\frac{1}{a_{21}^{2}}(\vartheta_{2j}\vartheta_{5j}a_{32}a_{55}-\vartheta_{6j}a_{21}a_{66})t^{5}+\\\
&&\frac{1}{a_{21}^{3}}(a_{21}(\vartheta_{2j}\vartheta_{6j}a_{32}a_{66}-\vartheta_{7j}a_{21}a_{77})-((\vartheta_{2j})^{2}a_{32}^{2}-\vartheta_{3j}a_{21}a_{43})\vartheta_{5j}a_{55})t^{6}+O(7).\end{array}$

(a) If $4\nmid k$ and $5\nmid k$, then
$\vartheta_{5j}\neq\vartheta_{5j^{\prime}}$ for all $j,j^{\prime}$ with $j\neq
j^{\prime}$ (by Lemma 3.6). Therefore,
${\mathcal{D}}_{j}\cdot{\mathcal{D}}_{j^{\prime}}=4$ for all distinct pairs.

(b) If $k=4p$ and $5\nmid k$, then $\vartheta_{5p}=1$,
$\vartheta_{6p}=\vartheta_{2p}$, and we can write
$\gamma_{p}(t)=-a_{55}/a_{21}t^{4}+\vartheta_{2p}CndQm_{5}/a_{21}^{2}t^{5}+O(6)$.
We get similarly $\gamma_{2p}(t)$ and $\gamma_{3p}(t)$ by substituting
$\vartheta_{2p}$ by, respectively, $\vartheta_{2(2p)}$ and $\vartheta_{2(3p)}$
in $\gamma_{p}$. The result follows as
$\vartheta_{2s}\neq\vartheta_{2s^{\prime}}$ for
$s,s^{\prime}\in\\{p,2p,3p\\}$, $s\neq s^{\prime}$.

(c) If $k=5p$ and $4\nmid k$, then $\vartheta_{5p}=0$, $\vartheta_{6p}=1$,
$\vartheta_{7p}=\vartheta_{2p}$, so
$\gamma_{p}(t)=-a_{66}/a_{21}t^{5}+\vartheta_{2p}CndRm_{5}/a_{21}^{2}t^{6}+O(7)$.
The expressions $\gamma_{sp}$, $s=2,3,4,$ are obtaining by substituting
$\vartheta_{2p}$ by $\vartheta_{2(sp)}$ in $\gamma_{p}$.

(d) The case $k=20p$ follows in a similarly way to that of Proposition
4.16(d).

The calculations for $T$ are similar to those in the proof of Proposition
4.16. We obtain $T_{j,j^{\prime}}=4$ if $k=4p$ and
$j,j^{\prime}\in\\{p,2p,3p\\}$, and $T_{j,j^{\prime}}=5$ if $k=5q$ and
$j,j^{\prime}\in\\{q,2q,3q,4q\\}$. All remaining indices $j$ and $j^{\prime}$
give $T_{j,j^{\prime}}=3$. $\Box$

###### Remark 4.19

The singularities ${\bf Q}^{4}_{3}$ and ${\bf R}^{4}_{4}$ have the same
invariants, but the contact of their double point branches shows that they are
not topologically equivalent.

####  4.2.4 Branch 4: $a_{11}=a_{21}=a_{22}=0$

###### Theorem 4.20

The strata of codimension $\leq 4$ of finitely $\mathcal{A}$-determined
$k$-folding map-germs in the branch $a_{11}=a_{22}=a_{21}=0$ are those given
in Table 6. The invariants of the germs in each stratum are given in Table 7.

  * Proof

The result follows from Propositions 4.22, 4.23, 4.25, 4.26, 4.27. $\Box$

Table 6: strata of codimension $\leq 4$ in Branch 4.

Name | Defining equations and open conditions | Codim
---|---|---
|  together with $a_{11}=a_{21}=a_{22}=0$ |
${\bf U}^{k}_{3}$, $3\nmid k$ |  $a_{31}a_{33}\neq 0$, $\Delta_{j}^{k}\neq 0$, $\Omega_{j,k-j}\neq 0$ | 3
${\bf U}^{k}_{3}$, $3\mid k$ |  $a_{31}a_{33}\neq 0$, $a_{44}\neq 0$, $\Delta_{j}^{k}\neq 0$, $\Omega_{j,j^{\prime}}^{k}\neq 0$ | 3
${\bf U}^{k}_{4}$, $3\mid k$ |  $a_{31}a_{33}\neq 0$, $a_{44}=0$, $\Delta_{j}^{k}\neq 0$, $\Omega_{j,j^{\prime}}^{k}\neq 0$, $CndUm_{8}\neq 0$ | 4
${\bf V}_{4}^{k,j,j^{\prime}}$, (*) |  $a_{31}a_{33}\neq 0$, $\Omega_{j,j^{\prime}}^{k}=0$, $a_{32}\neq 0$, $a_{44}\neq 0$, ${CndVm_{5}}_{j,j^{\prime}}\neq 0$ | 4
${\bf W}_{4}^{k,j}$, $3\nmid k$, (**) |  $a_{31}a_{33}\neq 0$, $\Delta_{j}^{k}=0$, $CndWA_{2}\neq 0$ | 4
${\bf W}_{4}^{k,j}$, $3\mid k$, (**), (***) |  $a_{31}a_{33}\neq 0$, $\Delta_{j}^{k}=0$, $CndWA_{2}\neq 0$, $a_{44}\neq 0$ | 4
${\bf W}_{4}^{3p,p}$, $k=3p$ |  $a_{31}a_{33}\neq 0$, $a_{32}=0$, $a_{44}\neq 0$, ${Cnd\widetilde{W}m}_{8}\neq 0$ | 4
${\bf X}_{4}^{k}$, $2\nmid k$, $3\nmid k$ |  $a_{32}a_{33}\neq 0$, $a_{31}=0$, $a_{41}\neq 0$ | 4
${\bf X}_{4}^{k}$, $2\mid k$, $3\nmid k$ |  $a_{32}a_{33}\neq 0$, $a_{31}=0$, $a_{41}\neq 0$ | 4
${\bf X}_{4}^{k}$, $2\nmid k$, $3\mid k$ |  $a_{32}a_{33}\neq 0$, $a_{31}=0$, $a_{41}\neq 0$, $a_{44}\neq 0$ | 4
${\bf X}_{4}^{k}$, $6\mid k$ |  $a_{32}a_{33}\neq 0$, $a_{31}=0$, $a_{41}\neq 0$, $a_{44}\neq 0$ | 4
${\bf Y}_{4}^{k}$, $2\nmid k$ |  $a_{32}a_{31}\neq 0$, $a_{33}=0$, $a_{44}\neq 0$ | 4
${\bf Y}_{4}^{k}$, $2\mid k$ |  $a_{32}a_{31}\neq 0$, $a_{33}=0$, ${CndYA}_{3}\neq 0$, ${CndYm_{6}}_{j}\neq 0$, $a_{44}\neq 0$ | 4

$\begin{array}[]{rcl}\Delta_{j}^{k}&=&(a_{32}^{2}-4a_{31}a_{33})\xi^{2j}+2(a_{32}^{2}-2a_{31}a_{33})\xi^{j}+a_{32}^{2}-4a_{31}a_{33}\\\
\Omega_{j,j^{\prime}}^{k}&=&a_{31}a_{33}(1+\xi^{j}+\xi^{j^{\prime}})^{2}-a_{32}^{2}(\xi^{j}+\xi^{j^{\prime}}+\xi^{j+j^{\prime}})\\\
CndUm_{8}&=&a_{55}(a_{31}a_{55}-a_{54}a_{32})+a_{77}a_{32}^{2}\\\
{CndVm_{5}}_{j,j^{\prime}}&=&a_{32}^{2}a_{44}\beta_{j,j^{\prime}}+a_{32}a_{33}(2a_{33}a_{42}-a_{32}a_{43})\alpha_{j,j^{\prime}}+a_{41}a_{33}^{3},\,\mbox{\rm
with}\\\
&&\alpha_{j,j^{\prime}}=\frac{\xi^{j}+\xi^{j^{\prime}}+\xi^{j+j^{\prime}}}{(1+\xi^{j}+\xi^{j^{\prime}})^{2}},\
\beta_{j,j^{\prime}}=\frac{\xi^{j+3j^{\prime}}+\xi^{3j+j^{\prime}}+\xi^{3j^{\prime}}+2\xi^{2j+j^{\prime}}+2\xi^{j+2j^{\prime}}+\xi^{3j}+2\xi^{j+j^{\prime}}+\xi^{j}+\xi^{j^{\prime}}}{(1+\xi^{j}+\xi^{j^{\prime}})^{4}}.\\\
CndWA_{2}&=&(a_{41}a_{33}+a_{31}a_{43})a_{32}^{3}-2a_{31}(a_{42}a_{33}+2a_{31}a_{44})a_{32}^{2}+8a_{31}^{3}a_{33}a_{44}\\\
{Cnd\widetilde{W}m}_{8}&=&a_{31}a_{55}-a_{42}a_{44}\\\
{CndYA}_{3}&=&a_{43}^{2}-4a_{31}a_{55}\\\
{CndYm_{6}}_{j}&=&(a_{31}a_{44}^{2}+a_{55}a_{32}^{2}-a_{43}a_{44}a_{32})\xi^{4j}+a_{44}(2a_{31}a_{44}-a_{32}a_{43})\xi^{2j}+a_{31}a_{44}^{2}\\\
(*)&&\mbox{$j<j^{\prime}$; $(j,j^{\prime})\neq(p,2p)$ when $k=3p$}\\\
(**)&&\mbox{If $2\mid k$ then $j\neq k/2$}\\\ (***)&&\mbox{\rm If $k=3p$, then
$j\neq p,2p$}\end{array}$

Table 7: Topological invariants of germs in strata in Table 6. Name | $C$ | $T$ | $\mu(\mathcal{D})$ | $r(\mathcal{D})$
---|---|---|---|---
${\bf U}^{k}_{3}$, $3\nmid k$ | $2k-2$ | $\frac{(k-1)(k-2)}{3}$ | $4(k-1)(k-2)+1$ | $2k-2$
${\bf U}^{k}_{3}$, $3\mid k$ | $2k-2$ | $\frac{(k-1)(k-2)+1}{3}$ | $4(k-1)(k-2)+3$ | $2k-2$
${\bf U}^{k}_{4}$, $3\mid k$ | $2k-2$ | $\frac{(k-1)(k-2)+4}{3}$ | $4(k-1)(k-2)+9$ | $2k-2$
${\bf V}^{k,j,j^{\prime}}_{4},3\nmid k$ | $2k-2$ | $\frac{(k-1)(k-2)+3}{3}$ | $4(k-1)(k-2)+7$ | $2k-2$
${\bf V}^{k,j,j^{\prime}}_{4},3\mid k$ | $2k-2$ | $\frac{(k-1)(k-2)+4}{3}$ | $4(k-1)(k-2)+9$ | $2k-2$
${\bf W}^{k,j}_{4},3\nmid k$ | $2k-2$ | $\frac{(k-1)(k-2)}{3}$ | $4(k-1)(k-2)+3$ | $2k-4$
${\bf W}^{k,j}_{4},3\mid k$ | $2k-2$ | $\frac{(k-1)(k-2)+1}{3}$ | $4(k-1)(k-2)+5$ | $2k-4$
${\bf W}_{4}^{3p,p}$, $k=3p$ | $2k-2$ | $\frac{(k-1)(k-2)+4}{3}$ | $4(k-1)(k-2)+11$ | $2k-4$
${\bf X}^{k}_{4}$, $2\nmid k,3\nmid k$ | $3k-3$ | $\frac{(k-1)(k-2)}{3}$ | $5(k-1)(k-2)+1$ | $2k-2$
${\bf X}^{k}_{4}$, $2\mid k,3\nmid k$ | $3k-3$ | $\frac{(k-1)(k-2)}{3}$ | $5(k-1)(k-2)+2$ | $2k-3$
${\bf X}^{k}_{4}$, $2\nmid k,3\mid k$ | $3k-3$ | $\frac{(k-1)(k-2)+1}{3}$ | $5(k-1)(k-2)+3$ | $2k-2$
${\bf X}^{k}_{4}$, $6\mid k$ | $3k-3$ | $\frac{(k-1)(k-2)+1}{3}$ | $5(k-1)(k-2)+4$ | $2k-3$
${\bf Y}^{k}_{4}$, $2\nmid k$ | $2k-2$ | $\frac{(k-1)(k-2)}{2}$ | $5(k-1)(k-2)+1$ | $2k-2$
${\bf Y}^{k}_{4}$, $2\mid k$ | $2k-2$ | $\frac{k(k-2)}{2}$ | $5(k-1)(k-2)+3(k-1)$ | $2k-2$

The functions defining the double point branches $\mathcal{D}_{j}$ have the
following initial terms:

$\lambda_{j}(x,y)=a_{31}x^{2}+a_{32}\vartheta_{2j}xy+a_{33}\vartheta_{3j}y^{2}+O(3).$

Consequently, the branches $\mathcal{D}_{j}$ are singular. A branch
$\mathcal{D}_{j}$ has an $A_{1}$-singularity unless the discriminant
$\Delta_{j}^{k}$ of the quadratic part of $\lambda_{j}$ vanishes. We have

$\Delta_{j}^{k}=(a_{32}^{2}-4a_{31}a_{33})\xi^{2j}+2(a_{32}^{2}-2a_{31}a_{33})\xi^{j}+a_{32}^{2}-4a_{31}a_{33}.$
(5)

An $A_{1}$-singularity is a transverse intersection of two regular curves. Two
branches $\mathcal{D}_{j}$ and $\mathcal{D}_{j^{\prime}}$ with an
$A_{1}$-singularity may have one or both of their components being tangential
(that is, the tangent cones of the two branches have a non-trivial
intersection). Taking the resultant of $j^{2}\lambda_{j}$ and
$j^{2}\lambda_{j^{\prime}}$ with respect to one of the variables, we find that
this happens if, and only if, $a_{31}a_{33}=0$ or

$\Omega^{k}_{j,j^{\prime}}=a_{31}a_{33}(1+\xi^{j}+\xi^{j^{\prime}})^{2}-a_{32}^{2}(\xi^{j}+\xi^{j^{\prime}}+\xi^{j+j^{\prime}})=0.$
(6)

We have $1+\xi^{j}+\xi^{j^{\prime}}=0$ or
$\xi^{j}+\xi^{j^{\prime}}+\xi^{j+j^{\prime}}=0$ if, and only if, $k=3p$ and
$j,j^{\prime}\in\\{p,2p\\}$. Therefore, if $3\nmid k$ or if $k=3p$ and
$j,j^{\prime}\notin\\{p,2p\\}$, $V(\Omega^{k}_{j,j^{\prime}})$ is a
codimension 1 algebraic variety in $J^{l}(2,1)$, for $l\geq 3$. For such
pairs, we set

$\alpha_{j,j^{\prime}}=\frac{\xi^{j}+\xi^{j^{\prime}}+\xi^{j+j^{\prime}}}{(1+\xi^{j}+\xi^{j^{\prime}})^{2}}\quad\mbox{\rm
and}\quad\alpha=\frac{a_{32}^{2}}{a_{31}a_{33}}.$ (7)

We have the following properties of $\Delta_{j}^{k}$ and
$\Omega_{j,j^{\prime}}^{k}$ when $a_{31}a_{33}\neq 0$; the case
$a_{31}a_{33}=0$ is dealt with in Propositions 4.26 and 4.27.

###### Proposition 4.21

Suppose that $a_{31}a_{33}\neq 0$ and that $3\nmid k$ or $k=3p$ and
$j,j^{\prime}\notin\\{p,2p\\}$. Then:

(1) $\Delta_{j}^{k}=\xi^{2j}\Delta_{k-j}^{k}$, and if $\Delta_{j}^{k}=0$ then
$\Delta_{s}^{k}\neq 0$ for $s\notin\\{j,k-j\\}$, so the solutions of
$\Delta_{j}^{k}=0$ in the $k^{th}$-roots of unity come in pairs.

(2) For $\Delta_{j}^{k}$ to vanish requires $\alpha$ in (7) to belong to the
real semi-line $(-\infty,3)$.

(3) $\alpha_{j,j^{\prime}}=\alpha_{j^{\prime},j}$ for all pairs
$(j,j^{\prime})$. We have
$\alpha_{j,j^{\prime}}=\alpha_{k-j,j^{\prime}-j}=\alpha_{k-j^{\prime},j-j^{\prime}}$
and the pairs $(j,j^{\prime}),(k-j,j^{\prime}-j),(k-j^{\prime},j-j^{\prime})$
are pairwise distinct. Furthermore,
$\alpha^{k}_{l,q}=\alpha^{k}_{j,j^{\prime}}$ if, and only if, $(l,q)$ or
$(q,l)$ is one of those $3$ pairs.

(4) $\alpha_{j,j^{\prime}}$ is real if, and only if, $j^{\prime}=k-j$ or
$j^{\prime}=2j$. In that case, by (3), $\alpha_{k-j,2j}$ is also real. Then,
$\alpha_{j,k-j}=\alpha_{j,2j}=\alpha_{k-j,2j}=(1+\xi^{j}+\xi^{k-j})^{-1}$.

(5) If $\alpha$ is real then $\Omega^{k}_{j,j^{\prime}}=0$ if, and only if,
$j^{\prime}=k-j$ or $j^{\prime}=2j$. Then by (3), we also have
$\Omega^{k}_{k-j,2j}=0$.

(6) If $\alpha$ is real, then $\Omega^{k}_{j_{0},k-j_{0}}=0$ implies
$\Delta_{j}^{k}\neq 0$ for all $j$. Conversely, if $\Delta_{j_{0}}^{k}=0$,
then $\Omega^{k}_{j,k-j}\neq 0$ for all $j$.

(7) If $\Delta_{j}^{k}\Delta_{j^{\prime}}^{k}\neq 0$ and
$\Omega^{k}_{j,j^{\prime}}\neq 0$ for all $j,j^{\prime}$ with $j^{\prime}\neq
j$, then $\mathcal{D}_{j}\cdot\mathcal{D}_{j^{\prime}}=4$.

  * Proof

(1) As $\xi^{-j}=\xi^{k-j}$, factoring out $\xi^{2j}$ in (5) gives
$\Delta_{j}^{k}=\xi^{2j}\Delta_{k-j}^{k}$.

If $a_{32}^{2}-4a_{31}a_{33}=0$, then for $\Delta_{j}^{k}$ to vanish requires
$a_{32}^{2}-2a_{31}a_{33}=0$. This would imply $a_{31}a_{33}=0$. Therefore,
under the hypothesis of the proposition, we can assume that
$a_{32}^{2}-4a_{31}a_{33}\neq 0$. Then

$\Delta_{j}^{k}=0\iff\xi^{2j}+\frac{2(\alpha-2)}{\alpha-4}\xi^{j}+1=0$

with $\alpha$ as in (7). If $\xi^{j}$ is a solution of the above quadratic
equation, then so is $\xi^{k-j}=\xi^{-j}$. Therefore, for $\alpha$ fixed, if
$\Delta_{j}^{k}=0$, then $\Delta_{s}^{k}\neq 0$ for $s\notin\\{j,k-j\\}$.

(2) When $a_{31}a_{33}\neq 0$, we can write
$\Delta_{j}^{k}=a_{31}a_{33}((1+\xi^{j})^{2}\alpha-4(1+\xi^{j}+\xi^{2j})).$
Clearly, $\Delta_{j}^{k}\neq 0$ when $\xi^{j}=-1$. Thus, $\Delta_{j}^{k}=0$
if, and only if, $\alpha=4(1+\xi^{j}+\xi^{2j})/(1+\xi^{j})^{2}$, which shows
that if $\Delta_{j}^{k}=0$ then $\alpha$ must be real. The discriminant of the
quadratic equation $(\xi^{j}+1)^{2}\alpha-4(\xi^{2j}+\xi^{j}+1)=0$ in
$\xi^{j}$ is $4(\alpha-3)$, so $\alpha<3$ as the solutions $\xi^{j}$ are not
real.

(3) Clearly as
$\alpha_{j,j^{\prime}}=\frac{\xi^{j}+\xi^{j^{\prime}}+\xi^{j+j^{\prime}}}{(1+\xi^{j}+\xi^{j^{\prime}})^{2}}$,
we have $\alpha_{j,j^{\prime}}=\alpha_{j^{\prime},j}$. Factoring our
$\xi^{2j}$ (resp. $\xi^{2j^{\prime}}$) from the numerator and denominator
gives
$\alpha_{k-j,j^{\prime}-j}=\alpha_{k-j,j^{\prime}-j}=\alpha_{k-j^{\prime},j-j^{\prime}}$.

We now seek pairs $(l,q)$ for which $\alpha_{l,q}=\alpha_{j,j^{\prime}}$. We
know from the above that we have at least 3 such pairs. To show that these are
the only ones, we write $\alpha_{j,j^{\prime}}=c+id$ (so $(c,d)\neq(0,0)$) and
represent points on the unit circle, with $-1$ removed, in the form
$z=\frac{1-t^{2}}{1+t^{2}}+i\frac{2t}{1+t^{2}}$ and
$w=\frac{1-s^{2}}{1+s^{2}}+i\frac{2s}{1+s^{2}}$, with $t,s\in\mathbb{R}$. We
set

$\alpha_{z,w}=\frac{z+w+zw}{(1+z+w)^{2}}.$

The real and imaginary parts of $\alpha_{z,w}-(c+id)$ vanish if, and only if,
$P_{(c,d)}(s,t)=Q_{(c,d)}(s,t)=0$, where

${\small\begin{array}[]{rcl}P_{(c,d)}(s,t)&=&(cs^{4}+4ds^{3}+s^{4}-6cs^{2}-4ds+2s^{2}+c+1)t^{4}+\\\
&&4(s^{2}+1)(ds^{2}-2cs-d+s)t^{3}-2(3cs^{4}-s^{4}+10cs^{2}+8ds-c+1)t^{2}-\\\
&&4(s^{2}+1)(ds^{2}+2cs+3d-s)t+\\\
&&cs^{4}-4ds^{3}+s^{4}+2cs^{2}-12ds-2s^{2}+9c-3,\\\
Q_{(c,d)}(s,t)&=&-(ds^{4}-4cs^{3}-6ds^{2}+4cs+d)t^{4}+4(s^{2}+1)(cs^{2}+2ds-c+1)t^{3}+\\\
&&2(3ds^{4}+10ds^{2}+2s^{3}-8cs-d+2s)t^{2}-\\\
&&4(s^{2}+1)(cs^{2}-2ds+3c-1)t-ds^{4}-4cs^{3}-2ds^{2}+4s^{3}-12cs-9d+4s.\end{array}}$

Observe that $P_{(c,d)}$ and $Q_{(c,d)}$ are symmetric polynomials. Their
resultant with respect to $t$ vanishes if, and only if, $s^{2}=3$ or
$R_{(c,d)}(s)=0$, with

${\small\begin{array}[]{rcl}R_{(c,d)}(s)&=&(c^{2}+d^{2}-1)^{2}s^{6}-(9c^{4}+18c^{2}d^{2}+9d^{4}+2c^{2}+2d^{2}+8c-3)s^{4}+\\\
&&(27c^{4}+54c^{2}d^{2}+27d^{4}+18c^{2}+18d^{2}-16c+3)s^{2}-\\\
&&27c^{4}-54c^{2}d^{2}-27d^{4}+18c^{2}+18d^{2}-8c+1.\end{array}}$

We have $s^{2}=3$ if, and only if, $k=3p$ and $w=\xi^{p}$ or $w=\xi^{2p}$, and
this is excluded from the hypotheses. The component $R_{(c,d)}$ of the
resultant is a cubic polynomial in $s^{2}$ provided $c^{2}+d^{2}-1\neq 0$,
i.e., $|\alpha_{j,j^{\prime}}|\neq 1$. Suppose that
$|\alpha_{j,j^{\prime}}|\neq 1$. Then the discriminant of $R_{(c,d)}$ vanishes
if, and only if, $d=0$ (i.e., $\alpha_{j,j^{\prime}}$ is real and this is
treated in (4) below) or

$\delta_{R}=27c^{4}+54c^{2}d^{2}+27d^{4}-18c^{2}-18d^{2}+8c-1=0.$

For $(c,d)$ in the interior region bounded by the curve $\delta_{R}=0$ (see
Figure 3), $R_{(c,d)}$ has a unique solution in $s^{2}$. As we know that there
are at least three distinct solution pairs to the problem, it follows that
$(c,d)$ must be in the exterior region $(R^{+})$ bounded by the curve
$\delta_{R}=0$.

Observe that $s=0$ is a root of $R_{(c,d)}$ if, and only if, $\delta_{R}=0$.
Therefore, the roots of $R_{(c,d)}$ in $s^{2}$ do not change sign in
$(R^{+})$. Choosing any point in that region, we find that they are all
positive. It follows that $R_{(c,d)}$ has six roots $\pm s_{1},\pm s_{2},\pm
s_{3}$. These correspond to six points on the unit circle $w_{1},w_{2},w_{3}$
and $\overline{w}_{1},\overline{w}_{2},\overline{w}_{3}.$

For each root of $R_{(c,d)}$ we show, by considering the subresultant (see for
example [19]) of $P_{(c,d)}$ and $Q_{(c,d)}$ that $P_{(c,d)}(\pm s_{i},t)$ and
$Q_{(c,d)}(\pm s_{i},t)$ have only one common root. As $P_{(c,d)}$ and
$Q_{(c,d)}$ are symmetric polynomials, that common root is a root of
$R_{(c,d)}$. Interchanging $w_{i}$ with $\overline{w}_{i}$ if necessary, we
can set $w_{1}=\xi^{j}$, $w_{1}=\xi^{k-j}$, $w_{1}=\xi^{k-j^{\prime}}$, Then
the solutions of $\alpha_{z,w}-c-id=0$ are exactly, up to permutation of $z$
and $w$, $(\xi^{j},\xi^{j^{\prime}})$, $(\xi^{k-j},\xi^{j^{\prime}-j})$,
$(\xi^{k-j^{\prime}},\xi^{j-j^{\prime}})$.

We turn now to the case when $|\alpha_{j,j^{\prime}}|=1$, i.e.,
$c^{2}+d^{2}-1=0$. This occurs if, and only if, $2\mid k$ and
$j^{\prime}=j+\frac{k}{2}$ or $j^{\prime}=\frac{k}{2}$. Suppose that $2\mid k$
and $j^{\prime}=j+\frac{k}{2}$. Then $\alpha_{j,j+\frac{k}{2}}=-\xi^{2j}$, and
$\alpha_{l,l+\frac{k}{2}}=\alpha_{j,j+\frac{k}{2}}$ if, and only if, $l=j$ or
$l=j+\frac{k}{2}$. In both cases, we get only the pair $(j,j+\frac{k}{2})$.
Now $\alpha_{\frac{k}{2},l}=-\xi^{-2l}$, so
$\alpha_{\frac{k}{2},l}=\alpha_{j,j+\frac{k}{2}}$ if, and only if, $l=k-j$ or
$l=\frac{k}{2}-j$. This shows that
$\alpha_{j,j+\frac{k}{2}}=\alpha_{k-j,\frac{k}{2}}=\alpha_{\frac{k}{2}-j,\frac{k}{2}}$
and the equality $\alpha_{j,j+\frac{k}{2}}=\alpha_{l,q}$ holds only for these
three pairs.

Figure 3: The red curve is the discriminant of $R_{(c,d)}$, the blue curve is
the unit circle and the green line are the values of $\alpha_{p,w}$ and
$\alpha_{2p,w}$ when $k=3p$.

(4) We can write $\alpha_{j,j^{\prime}}$ in the form

$\alpha_{j,j^{\prime}}=\frac{6(1+\Re(\xi^{j})+\Re(\xi^{j^{\prime}})+\Re(\xi^{j^{\prime}-j}))+\xi^{j-2j^{\prime}}+\xi^{j+j^{\prime}}+\xi^{j^{\prime}-2j}}{|1+\xi^{j}+\xi^{j^{\prime}}|^{2}}.$

Therefore, $\alpha_{j,j^{\prime}}$ is real if, and only if,
$\zeta=\xi^{j-2j^{\prime}}+\xi^{j+j^{\prime}}+\xi^{j^{\prime}-2j}$ is real.
Setting $\theta=j-2j^{\prime}$ and $\phi=j^{\prime}-2j$, we get
$\Im(\zeta)=\sin(\frac{2\pi\theta}{k})+\sin(\frac{2\pi\phi}{k})-\sin(\frac{2\pi(\theta+\phi)}{k})$.
Then

$\begin{array}[]{rcl}\Im(\zeta)=0&\iff&\sin(\frac{2\pi\theta}{k})+\sin(\frac{2\pi\phi}{k})=\sin(\frac{2\pi(\theta+\phi)}{k})\\\
&\iff&2\sin(\frac{\pi(\theta+\phi)}{k})\cos(\frac{\pi(\theta-\phi)}{k})=2\sin(\frac{\pi(\theta+\phi)}{k})\cos(\frac{\pi(\theta+\phi)}{k}).\end{array}$

Now, $\sin(\frac{\pi(\theta+\phi)}{k})=0$ when $j^{\prime}=k-j$ and
$\cos(\frac{\pi(\theta-\phi)}{k})=\cos(\frac{\pi(\theta+\phi)}{k})$ when
$j^{\prime}=2j$ or $j=2j^{\prime}$. Clearly,
$\alpha_{j,k-j}=1/(1+\xi^{j}+\xi^{k-j})=\alpha_{j,2j}.$

(5) As $a_{31}a_{33}\neq 0$, we can write
$\Omega_{j,j^{\prime}}^{k}=a_{31}a_{33}(1-\alpha_{j,j^{\prime}}\alpha)$, so
$\Omega_{j,j^{\prime}}^{k}=0$ if, and only if,
$\alpha_{j,j^{\prime}}=1/\alpha$ and the statement follows by (4).

(6) When $\alpha$ is real, by (4), we have $\Omega_{j_{0},k-j_{0}}=0$ when
$\xi^{j}_{0}+\xi^{k-j_{0}}+1=\alpha$. It follows that $\xi^{j}_{0}=\eta_{1}$
with $\eta_{1}=\frac{\alpha-1}{2}\pm
i\sqrt{1-\left(\frac{\alpha-1}{2}\right)^{2}}\,.$ Suppose that there exist a
$j$ for which $\Delta_{j}^{k}=0$, equivalently,
$\xi^{2j}+\frac{2(\alpha-2)}{(\alpha-4)}\xi^{j}+1=0$. Then
$\xi^{j}+\xi^{k-j}=-\frac{2(\alpha-2)}{\alpha-4}$ which gives
$\xi_{j}=\eta_{2}$ with $\eta_{2}=-\frac{\alpha-2}{\alpha-4}\pm
i\sqrt{1-\left(\frac{\alpha-2}{\alpha-4}\right)^{2}}\,.$ We have
$\eta_{1}=\eta_{2}$ if, and only if, $\alpha=0$ or $3$. Then $k=3p$ and
$j_{0}=p$ or $2p$, which is excluded from our hypotheses, so
$\eta_{1}\neq\eta_{2}$.

The complex number $\eta_{1}$ (resp. $\eta_{2}$) is a $k^{th}$-root of unity
if, and only if, $\alpha$ is a root of the polynomial $P_{1}(\alpha)$ (resp.
$P_{2}(\alpha)$) of degree $k$ obtained by taking the numerator of
$\Re({\eta_{1}}^{k})-1$ (resp. $\Re({\eta_{2}}^{k})-1$). But if $P_{1}$ and
$P_{2}$ have one common root, all the other roots must also be common. (This
follows from the fact that the map $\cos\theta=-(\alpha-2)/(\alpha-4)$ is a
bijection for $\alpha\in(-\infty,3]$ and $\cos\theta=(\alpha-1)/2$ is also a
bijection for $\alpha\in(-1,3]$.) As $P_{1}$ and $P_{2}$ are distinct
polynomials, it follows that they have no common roots. Consequently,
$\Delta_{j}^{k}\neq 0$ for all $j$.

The argument for showing that $\Omega^{k}_{j,k-j}\neq 0$ when
$\Delta_{j_{0}}^{k}=0$ is the same as above.

(7) We have, with the hypothesis,
$\mathcal{D}_{j}\cdot\mathcal{D}_{j^{\prime}}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{2}\right>=4$.
$\Box$

In view of Proposition 4.21 (7), we give in the rest of this section
$\mathcal{D}_{j}\cdot\mathcal{D}_{j^{\prime}}$ for the exceptional branches
only, i.e, when one or both branches have a singularity more degenerate than
$A_{1}$ or one or both of their components are tangential (which is equivalent
to $\mathcal{D}_{j}\cdot\mathcal{D}_{j^{\prime}}>4$). We start with the case
when all the branches have an $A_{1}$-singularity.

###### Proposition 4.22

Suppose that $a_{11}=a_{21}=a_{22}=0$, $a_{31}a_{33}\neq 0$ and
$\Delta_{j}^{k}\neq 0$ for all $j$. Any $k$-folding map-germ $F_{k}$
satisfying the additional conditions in (a) or (b) is finitely
$\mathcal{A}$-determined and is topologically equivalent to one of the
following germs:

${\bf
U}^{k}_{l}\colon(x,y)\mapsto(x,y^{k},x^{2}y+2xy^{2}+y^{3}+y^{3l-5}),\,l=3,4.\\\
$

Every branch of the double point curve has an $A_{1}$-singularity and the
invariants associated to the germs in these strata are as in Table 7.

  * (a)

If $3\nmid k$ and $\Omega^{k}_{j,j^{\prime}}\neq 0$ for all distinct pairs,
then $F_{k}$ is topologically equivalent to ${\bf U}^{k}_{3}$.

  * (b)

Suppose that $k=3p$ and $\Omega^{k}_{j,j^{\prime}}\neq 0$ for all distinct
pairs with $j,j^{\prime}\neq\\{p,2p\\}$.

    * (b1)

If $a_{44}\neq 0$, then ${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=5$ and $F_{k}$
is topologically equivalent to ${\bf U}^{k}_{3}$.

    * (b2)

If $a_{44}=0$ and $CndUm_{8}\neq 0$, then
${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=8$ and $F_{k}$ is topologically
equivalent to ${\bf U}^{k}_{4}$.

  * Proof

Each branch of the double point curve consists of a transverse intersection of
two regular curves. In (a) all of these curves are pairwise transverse.

In (b), $j^{2}\lambda_{s}(x,y)=x(a_{31}x+a_{32}\vartheta_{2s}y)$ for $s=p,2p$.
Observe that $a_{32}\neq 0$ as we supposed $\Delta^{k}_{s}\neq 0$, so the
branches $\mathcal{D}_{p}$ and $\mathcal{D}_{2p}$ have one common line $x=0$
in their tangent cone and their associated curves tangent to this line are
parametrised by $t\mapsto(\gamma_{s}(t),t)$ with
$\gamma_{s}(t)=-(a_{44}/({\vartheta_{2s}a_{32}}))t^{2}+O(3)$, so they have
order of contact 2 when $a_{44}\neq 0$. Then
$\mathcal{D}_{p}\cdot\mathcal{D}_{2p}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{3}\right>=5$.

When $a_{44}=0$, the two curves are parametrised by
$t\mapsto(\gamma_{s}(t),t)$ with

$\gamma_{s}(t)=-\frac{a_{55}}{a_{32}}t^{3}+\frac{2}{\vartheta_{2s}a_{32}^{3}}(a_{55}(a_{31}a_{55}-a_{54}a_{32})+a_{77}a_{32}^{2})t^{5}+O(6),$

and have contact order 5 when
$(a_{55}(a_{31}a_{55}-a_{54}a_{32})+a_{77}a_{32}^{2}\neq 0$, i.e.,
$CndUm_{8}\neq 0$. Then
$\mathcal{D}_{p}\cdot\mathcal{D}_{2p}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{6}\right>=8.$

For the topological normal forms, we choose $a_{31},a_{32},a_{33}$ real with
$\alpha>3$. By Proposition 4.21 (2), all $\Delta_{j}^{k}$ are non-zero. Also,
by Proposition 4.21 (4) and (5) $\Omega_{j,k-j}=0$ when
$\xi^{j}+\xi^{k-j}+1=\alpha$, which cannot happen when $\alpha>3$, so the
$\Omega_{j,j^{\prime}}^{k}$ in (a) and (b) are non-zero. As the strata are
connected sets, we can choose $a_{31}=a_{32}=1,a_{32}=2$ so that the
conditions on $\Delta_{j}^{k}$ and $\Omega_{j,j^{\prime}}^{k}$ are satisfied.

The number of triple points is calculated as in Proposition 4.16. If $k=3p$,
the germ ${\bf U}^{k}_{3}$ (resp. ${\bf U}^{k}_{4}$) has $T_{p,2p}=3$ (resp.
$T_{p,2p}=6$), while all other indices $j$ and $j^{\prime}$ give
$T_{j,j^{\prime}}=2$. $\Box$

###### Proposition 4.23

Suppose that $a_{11}=a_{21}=a_{22}=0$, $a_{31}a_{33}\neq 0$, $a_{32}\neq 0$
and $\Omega^{k}_{j_{0},j_{0}^{\prime}}=0$ for some pair
$(j_{0},j^{\prime}_{0})$, with $j_{0},j^{\prime}_{0}\notin\\{p,2p\\}$ when
$k=3p$. Then ${\mathcal{D}}_{s}\cdot\mathcal{D}_{q}=5$ for
$(s,q)\in\\{(j_{0},j^{\prime}_{0}),(k-j_{0},j^{\prime}_{0}-j_{0}),(k-j^{\prime}_{0},j_{0}-j^{\prime}_{0})\\}$
if, and only if, ${CndVm_{5}}_{j_{0},j_{0}^{\prime}}\neq 0$. For $k=3p$, we
have ${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=5$ if, and only if, $a_{44}\neq
0$. Any $k$-folding map-germ $F_{k}$ satisfying the above conditions is
finitely $\mathcal{A}$-determined and is topologically equivalent to

${\bf
V}^{k,j_{0},j_{0}^{\prime}}_{4}\colon(x,y)\mapsto(x,y^{k},x^{2}y+xy^{2}+\alpha_{j_{0},j^{\prime}_{0}}y^{3}+\big{(}1-\frac{\beta_{j_{0},j^{\prime}_{0}}}{\alpha^{3}_{j_{0},j^{\prime}_{0}}}\big{)}x^{3}y+y^{4}).$

The invariants associated to germs in these strata are as in Table 7.

  * Proof

When $a_{32}=0$ and $k=3p$, we have $\Delta_{p}^{k}=\Delta_{2p}^{k}=0$. This
case is dealt with in Proposition 4.25. With the hypothesis and Proposition
4.21(5), we have
$\Omega_{j_{0},j^{\prime}_{0}}=\Omega_{k-j_{0},j^{\prime}_{0}-j_{0}}=\Omega_{k-j^{\prime}_{0},j_{0}-j^{\prime}_{0}}=0.$

We need to consider the order of contact between the two tangential components
of the pairs $(\mathcal{D}_{s},\mathcal{D}_{q})$ with
$(s,q)\in\\{(j_{0},j^{\prime}_{0}),(k-j_{0},j^{\prime}_{0}-j_{0}),(k-j^{\prime}_{0},j_{0}-j^{\prime}_{0})\\}$.
Using the fact that if two quadratic equations $x^{2}+a_{i}x+b_{i}=0$,
$i=1,2$, have one root in common, then the root is given by
$x=-(b_{1}-b_{2})/(a_{1}-a_{2})$, we can get the initial terms of
parametrisations of the tangential components of the above pairs. These are
given by $t\mapsto(t,\gamma_{l}(t))$, $l=s,q$, with
$\gamma_{l}(t)=-\frac{a_{33}}{a_{32}}\big{(}1+\xi^{s}+\xi^{q}\big{)}t+\lambda_{l}t^{2}+O(3),l=s,q.$
A calculation shows that $\lambda_{s}-\lambda_{q}=0$ if, and only if,

${CndVm_{5}}_{s,q}=a_{32}^{2}a_{44}\beta_{s,q}+a_{32}a_{33}(2a_{33}a_{42}-a_{32}a_{43})\alpha_{s,q}+a_{41}a_{33}^{3}=0,$

with

$\beta_{j,j^{\prime}}=\frac{\xi^{j+3j^{\prime}}+\xi^{3j+j^{\prime}}+\xi^{3j^{\prime}}+2\xi^{2j+j^{\prime}}+2\xi^{j+2j^{\prime}}+\xi^{3j}+2\xi^{j+j^{\prime}}+\xi^{j}+\xi^{j^{\prime}}}{(1+\xi^{j}+\xi^{j^{\prime}})^{4}}.$
(8)

Observe that
$\beta_{j,j^{\prime}}=\beta_{k-j,j^{\prime}-j}=\beta_{k-j^{\prime},j-j^{\prime}}$,
so ${CndVm_{5}}_{s,q}$ has the same value for
$(s,q)\in\\{(j_{0},j^{\prime}_{0}),(k-j_{0},j^{\prime}_{0}-j_{0}),(k-j^{\prime}_{0},j_{0}-j^{\prime}_{0})\\}.$
It follows that
${\mathcal{D}}_{s}\cdot\mathcal{D}_{q}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{3}\right>=5$
if, and only if, ${CndVm_{5}}_{j_{0},j_{0}^{\prime}}\neq 0$.

For the topological model, we take $a_{31}=a_{32}=1$ and
$a_{33}=\alpha_{j_{0},j^{\prime}_{0}}$ so $\Omega_{j_{0},j^{\prime}_{0}}=0$.
We also set $a_{44}=1$, $a_{42}=a_{43}=0$, then
${CndVm_{5}}_{j_{0},j^{\prime}_{0}}=\beta_{j_{0},j^{\prime}_{0}}+a_{41}\alpha_{j_{0},j^{\prime}_{0}}^{3}$.
We set
$a_{41}=1-\beta_{j_{0},j^{\prime}_{0}}/\alpha_{j_{0},j^{\prime}_{0}}^{3}$, so
that ${CndVm_{5}}_{j_{0},j^{\prime}_{0}}\neq 0$. For calculating the triple
points, we find that $T_{j,j^{\prime}}=3$ for all $j$ and $j^{\prime}$ with
$j\neq j^{\prime}$. $\Box$

We deal now with the case when one pair of branches of the double point curve
has a singularity more degenerate than $A_{1}$ (see Proposition 4.21(1)).

###### Remark 4.24

For $k=3p$, the singularities ${\bf U}^{k}_{4}$ and ${\bf
V}^{k,j,j^{\prime}}_{4}$ have the same invariants, but the intersection
numbers between their double point branches shows that they are not
topologically equivalent.

###### Proposition 4.25

Suppose that $a_{11}=a_{21}=a_{22}=0$, $a_{31}a_{33}\neq 0$. The strata below
are of codimension $4$, and the invariants associated to germs in the strata
are as in Table 7.

(1) Suppose that $\Delta_{j_{0}}^{k}=\Delta_{k-j_{0}}^{k}=0$, with
$j_{0}\notin\\{p,2p\\}$ when $k=3p$ (so $a_{32}\neq 0{\rm)}$. The branches
$\mathcal{D}_{j_{0}}$ and $\mathcal{D}_{k-j_{0}}$ have an $A_{2}$-singularity
if, and only if, $CndWA_{2}\neq 0$. We have
${\mathcal{D}}_{j_{0}}\cdot\mathcal{D}_{j}={\mathcal{D}}_{k-j_{0}}\cdot\mathcal{D}_{j}=4$
for all distinct pairs. When $k=3p$ and $a_{44}\neq 0$, we have
${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=5$. Any $k$-folding map-germ $F_{k}$
satisfying the above conditions is $\mathcal{A}$-finitely determined and is
topologically equivalent to

${\bf
W}^{k,j_{0}}_{4}\colon(x,y)\mapsto(x,y^{k},x^{2}y+xy^{2}+\frac{(\xi^{j_{0}}+1)^{2}}{4(\xi^{2j_{0}}+\xi^{j_{0}}+1)}y^{3}+4x^{2}y^{2}+y^{4}).$

(2) If $k=3p$ and $j_{0}=p$, then ${\mathcal{D}}_{p}$ and $\mathcal{D}_{2p}$
have an $A_{2}$-singularity if, and only if, $a_{44}\neq 0$. We have
${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=8$ if, and only if,
${Cnd\widetilde{W}m}_{8}\neq 0$, and ${\mathcal{D}}_{s}\cdot\mathcal{D}_{j}=5$
for $s=p,2p$ and $j\neq s$. Any $k$-folding map-germ $F_{k}$ satisfying the
above conditions is $\mathcal{A}$-finitely determined and is topologically
equivalent to

${\bf W}^{3p,p}_{4}\colon(x,y)\mapsto(x,y^{k},x^{2}y+y^{3}+y^{4}+y^{5}).$

  * Proof

The condition for an $A_{2}$-singularity of $\mathcal{D}_{j_{0}}$ and
$\mathcal{D}_{k-j_{0}}$ follows by analysing the 3-jets of, respectively,
$\lambda_{j_{0}}$ and $\lambda_{k-j_{0}}$. Observe that when $2\mid k$,
$\xi^{j_{0}}=-1$ is a solution of $\Delta_{j_{0}}^{k}=\Delta_{k-j_{0}}^{k}=0$
if, and only if, $a_{31}a_{33}=0$. As we are assuming $a_{31}a_{33}\neq 0$, we
have $j_{0}\neq k/2$ in (1) when $2\mid k$.

With the hypothesis in (1),
${\mathcal{D}}_{j_{0}}\cdot\mathcal{D}_{j}={\mathcal{D}}_{k-j_{0}}\cdot\mathcal{D}_{j}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<x^{2}+y^{3},x^{2}-y^{2}\right>=4$
for any distinct pairs.

For (2), we have
${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<x^{2}+y^{3},y^{4}\right>=8$
when ${Cnd\widetilde{W}m}_{8}\neq 0$.

For the topological normal forms, we use the fact that $\Delta_{j_{0}}^{k}=0$
if and only if
$\alpha={a_{32}^{2}}/({a_{31}a_{33}})=4(\xi^{2j_{0}}+\xi^{j_{0}}+1)/{(\xi^{j_{0}}+1)^{2}}$
(see the proof of Proposition 4.21) and set $a_{31}=a_{32}=1$, so
$a_{33}={(\xi^{j_{0}}+1)^{2}}/{4(\xi^{2j_{0}}+\xi^{j_{0}}+1)}$. We have
$T_{j,j^{\prime}}=2$ for all $j$ and $j^{\prime}$ with $j\neq j^{\prime}$.
$\Box$

###### Proposition 4.26

Suppose that $a_{11}=a_{21}=a_{22}=0$, $a_{31}=0$, $a_{33}a_{32}\neq 0$. All
the branches of the double point curve share only one line in their tangent
cones except when $k=3p$ where the branches $\mathcal{D}_{p}$ and
$\mathcal{D}_{2p}$ have the same tangent cone. Any $k$-folding map-germ
$F_{k}$ satisfying the additional conditions in (a), (b), (c) or (d) is
finitely $\mathcal{A}$-determined and is topologically equivalent to

${\bf X}_{4}^{k}\colon(x,y)\mapsto(x,y^{k},xy^{2}+y^{3}+x^{3}y+y^{4}).$

The stratum is of codimension $4$, and the invariants associated to the germs
in the strata above are as in Table 7.

  * (a)

If $2\nmid k$, $3\nmid k$, all of the branches $\mathcal{D}_{j}$ of the double
point curve have an $A_{1}$-singularity and
${\mathcal{D}}_{j}\cdot\mathcal{D}_{j^{\prime}}=5$, $j\neq j^{\prime}$, when
$a_{41}\neq 0$.

  * (b)

If $k=2p$, $3\nmid k$, the branches $\mathcal{D}_{j}$, $j\neq p$, behave as in
(a). The branch $\mathcal{D}_{p}$ has an $A_{2}$-singularity if, and only if,
$a_{41}\neq 0$. We also have ${\mathcal{D}}_{j}\cdot\mathcal{D}_{p}=5$, $j\neq
p$.

  * (c)

If $k=3p$, $2\nmid k$, the branches $\mathcal{D}_{j}$ behave as in (a) except
for $\mathcal{D}_{p}$ and $\mathcal{D}_{2p}$ where
${\mathcal{D}}_{p}\cdot\mathcal{D}_{2p}=6$ when $a_{41}a_{44}\neq 0$.

  * (d)

If $k=6p$, when $a_{41}a_{44}\neq 0$, the exceptional branches of the double
point curve behave as in (b) and (c), the remaining branches behave as in (a).

  * Proof

We have
$j^{2}\lambda_{j}(x,y)=y(\vartheta_{2j}a_{32}x+\vartheta_{3j}a_{33}y)$. For
(a), ${\vartheta_{2j}}\neq 0$ for all $j$ and
$\frac{\vartheta_{2j}}{\vartheta_{2j}}\neq\frac{\vartheta_{3j^{\prime}}}{\vartheta_{2j^{\prime}}}$
for $j\neq j^{\prime}$, so all of the branches of the double point curve have
an $A_{1}$-singularity and one common line $y=0$ in their tangent cone. The
component of ${\mathcal{D}}_{j}$ with tangent $y=0$ can be parametrised by
$t\mapsto(t,\gamma_{j}(t))$ with by
$\gamma_{j}(t)=-{a_{41}}/({(1+\xi^{j})a_{32}})t^{2}+O(3)$. Therefore, when
$a_{41}\neq 0$, we have
${\mathcal{D}}_{j}\cdot\mathcal{D}_{j^{\prime}}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,y^{2}+x^{3}\right>=5$.

The remaining parts of the proof follow similarly and are omitted. $\Box$

###### Proposition 4.27

Suppose that $a_{11}=a_{21}=a_{22}=0$, $a_{33}=0$, $a_{31}a_{32}\neq 0$. All
the branches of the double point curve share only one line in their tangent
cones. Any $k$-folding map-germ $F_{k}$ satisfying the additional conditions
in (a) or (b) is finitely $\mathcal{A}$-determined and is topologically
equivalent to

${\bf Y}^{k}_{4}\colon(x,y)\mapsto(x,y^{k},-xy^{2}+x^{2}y+y^{4}+y^{5}).$

The stratum is of codimension $4$, and the invariants associated to germs in
the stratum are as in Table 7.

  * (a)

If $2\nmid k$, all of the branches of the double point curve have an
$A_{1}$-singularity and ${\mathcal{D}}_{j}\cdot\mathcal{D}_{j^{\prime}}=5$,
$j\neq j^{\prime}$, when $a_{44}\neq 0$.

  * (b)

If $k=2p$, the branch ${\mathcal{D}}_{p}$ has an $A_{3}$-singularity if, and
only if, ${CndYA}_{3}\neq 0$. Then ${\mathcal{D}}_{p}\cdot\mathcal{D}_{j}=6$,
$j\neq p$, and ${\mathcal{D}}_{j}\cdot\mathcal{D}_{p+j}=6$, $1\leq j<p$, if
and only if ${CndYm_{6}}_{j}\neq 0$. The other pairs of branches behave as in
(a).

  * Proof

We have $j^{2}\lambda_{j}(x,y)=x(a_{31}x+a_{32}\vartheta_{2j}y)$ so when
$2\nmid k$, all of the branches of the double point curve have an
$A_{1}$-singularity as $\vartheta_{2j}\neq 0$ for all $j$. All the branches
have only $x=0$ as a common line in their tangent cone. The components of the
branches which are tangent to $x=0$ are parametrised by
$t\mapsto(\gamma_{j}(t),t)$, with
$\gamma_{j}(t)=-{a_{44}(1+\xi^{2j})}/{a_{32}}t^{2}+O(3)$. When $a_{44}\neq 0$,
${\mathcal{D}}_{j}\cdot\mathcal{D}_{j^{\prime}}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{3}\right>=5$.

When $k=2p$, we have $\vartheta_{2p}=\vartheta_{4p}=0$, so considering the
4-jet of $\lambda_{p}$, we find that the branch $\mathcal{D}_{p}$ has an
$A_{3}$-singularity if, and only if, ${CndYA}_{3}\neq 0$. When this is the
case,
${\mathcal{D}}_{p}\cdot\mathcal{D}_{j}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{4}\right>=6$,
for $j\neq p$.

All pairs of branches $\mathcal{D}_{j}$, $\mathcal{D}_{p+j}$, $1\leq j<p$ have
tangential component parametrised by $t\mapsto(\gamma_{s}(t),t)$, with
$\gamma_{s}(t)=-{a_{44}(1+\xi^{2}_{j})}/{a_{32}}t^{2}+\alpha_{s}t^{3}+O(4)$,
$s=j,p+j$, with $\alpha_{1}-\alpha_{2}\neq 0$ if and only if
${CndYm_{6}}_{j}\neq 0$. When this is the case,
${\mathcal{D}}_{j}\cdot\mathcal{D}_{p+j}=\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{2}/\left<xy,x^{2}+y^{4}\right>=6$,
for $1\leq j<p$. The other pairs of branches behave as when $2\nmid k$. $\Box$

###### Remark 4.28

The stratification $\mathcal{S}_{k}$ is determined, for each $k\geq 4$, by the
conditions in Tables 2, 4 and 6 ($\mathcal{S}_{2}$ follows from the results in
[10] and $\mathcal{S}_{3}$ is indicated in Table 1). Each stratum corresponds
to a single topological class and all the topological classes listed in Tables
2, 4 and 6 are topologically distinct. Indeed, as pointed out in Remark 4.17
(resp. Remark 4.19 and resp. Remark 4.24) ${\bf Q}^{k}_{4}$ and
$\widetilde{\bf Q}^{k}_{4}$ (resp. ${\bf Q}^{4}_{3}$ and ${\bf R}^{4}_{4}$ and
resp. ${\bf U}^{k}_{4}$ and ${\bf V}^{k,j,j^{\prime}}_{4}$) are not
topologically equivalent. For the remaing classes, it follows by comparing the
invariants in Tables 3, 5 and 7, that they are all topologically distinct.

##  5 Generic singularities of $k$-folding map-germs

We define, by varying the plane $\pi\in{\rm Graff}(2,3)$, the family of
Whitney $k$-folds $\Omega^{k}:\mathbb{C}^{3}\times{\rm
Graff}(2,3)\to\mathbb{C}^{3},$ given by
$\Omega^{k}(p,\pi)=\omega^{k}_{\pi}(p)$, with $\omega^{k}_{\pi}$ as in
Definition 2.2.

Given a complex surface $M$ in $\mathbb{C}^{3}$, we call the restriction of
$\Omega^{k}$ to $M$ the family of $k$-folding maps on $M$ and denote it by
${\mathcal{F}}_{k}$. We have
${\mathcal{F}}_{k}(p,\pi)=F^{\pi}_{k}(p)=\omega^{k}_{\pi}(p)$ for all $p\in M$
and $\pi\in{\rm Graff}(2,3)$.

Recall that a property of surfaces is said to be generic if it is satisfied in
a residual set of embeddings of the surfaces to $\mathbb{C}^{3}$. The image of
a surface $M$ by an embedding in the residual set is then called generic, or
simply that $M$ is generic. When $k$ is large, the $\mathcal{A}$-singularities
of $F^{\pi}_{k}$ may have high $\mathcal{A}_{e}$-codimensions (for the cases
in this paper, this means high modality). However, they do occur on generic
surfaces. To make sense of this, we follow a similar approach to that in [6]
and proceed as follows.

As we are interested here in the local singularities of $k$-folding maps, we
consider the setting in Remarks 2.3(4) at a point $p_{0}\in M$ and choose a
suitable system of coordinates so that $\pi_{0}:y=0$ and
$F_{k}=F^{\pi_{0}}_{k}(x,y)=(x,y^{k},f(x,y))$ for $(x,y)$ in a small enough
neighbourhood $U$ of the origin. A plane $\pi=\overline{(d,v)}$ near $\pi_{0}$
is obtained by applying a translation $T_{\pi}$ followed by an orthogonal
transformation $R_{\pi}\in U(3)$ to $\pi_{0}$. We choose $T_{\pi}$ and
$R_{\pi}$ as follows. The translation $T_{\pi}$ takes the origin to the point
of intersection of $\pi$ with the $y$-axis (the point exists because $\pi_{0}$
is orthogonal to the $y$-axis and $\pi$ is close to $\pi_{0}$). The
transformation $R_{\pi}$ near the identity (and is taken to be the identity if
$v$ is parallel to $v_{0}$) takes $v/||v||$ to $v_{0}=(0,1,0)$ and fixes the
line through the origin orthogonal to $v_{0}$ and $v$. By varying the planes
$\pi$ in a neighbourhood $V$ of $\pi_{0}$ in ${\rm Graff}(2,3)$, we get the
family of $k$-folding maps given by ${\mathcal{F}}_{k}:U\times
V\to\mathbb{C}^{3}$, with

${\mathcal{F}}_{k}((x,y),\pi)=(R_{\pi}\circ
T_{\pi})\circ\omega_{k}\circ(R_{\pi}\circ T_{\pi})^{-1}(\phi(x,y)),$

with $\phi(x,y)=(x,y,f(x,y))$. (We choose $R_{\pi}$ and $T_{\pi}$ to depend
analytically on $\pi$.) The $\mathcal{A}$-type of the singularity of
$F^{\pi}_{k}$ at a given point in $U$ is the same as that of the germ of

$\tilde{\mathcal{F}}_{k}((x,y),\pi)=\tilde{F}_{\pi}^{k}(x,y)=(R_{\pi}\circ
T_{\pi})^{-1}\circ{\mathcal{F}}_{k}((x,y),\pi)=\omega_{k}\circ(R_{\pi}\circ
T_{\pi})^{-1}(\phi(x,y))$

at that point. At any point $p=(x,y)\in U$, there exist a bi-holomorphic map-
germ $K:(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},(x,y))$ such that

$(R_{\pi}\circ
T_{\pi})^{-1}(\phi(K(X,Y))=(\bar{x}+X,\bar{y}+Y,\bar{z}+g_{p}(X,Y)),$

for some germ of a holomorphic function $g_{p}$, where
$(\bar{x},\bar{y},\bar{z})=(R_{\pi}\circ T_{\pi})^{-1}(\phi(x,y)).$ Composing
$\tilde{F}_{\pi}^{k}$ with $K$ gives the germ $(\tilde{F}_{\pi}^{k})_{p}$ of
$\tilde{F}_{\pi}^{k}$ at $p$

$\begin{array}[]{rll}(\tilde{F}_{\pi}^{k})_{p}(X,Y)&=&(\bar{x}+X,(\bar{y}+Y)^{k},\bar{z}+g_{p}(X,Y))\\\
&\sim_{\mathcal{A}}&(X,(\bar{y}+Y)^{k},g_{p}(X,Y)).\end{array}$

Observe that $g_{p}$ depends on the choice of $R_{\pi}$, $T_{\pi}$ and of the
coordinates system, but the $\mathcal{A}$-class of the resulting germs
$(\tilde{F}_{\pi}^{k})_{p}$ is independent of these choices.

Clearly, a necessary condition for $(\tilde{F}^{\pi}_{k})_{p}$ to be singular
at the origin is $\bar{y}=0$, equivalently, $p\in\pi$. We define the family of
maps $\Phi:U\times V\to\mathbb{C}\times J^{l}(2,1)$ given by

$\Phi(p,\pi)=(\langle v,p\rangle-d,j^{l}g_{p}(0,0)).$

The map $\Phi$ plays a similar role to the Monge-Taylor map in [6]. Here we
include the first component to capture the planes through a given point $p\in
M$ which can give rise to singular $k$-folding map-germs at $p$.

We stratify $\mathbb{C}\times J^{l}(2,1)$ by $0\times\mathcal{S}_{k}$,
together with $\mathbb{C}\times J^{l}(2,1)\setminus 0\times J^{l}(2,1)$.
Following standard transversality arguments (see for example [6]), one can
show that for a residual set of local embeddings of $M$ in $\mathbb{C}^{3}$,
the map $\Phi$ is transverse to the strata in $\mathbb{C}\times J^{l}(2,1)$.
As the domain of the family $\Phi$ is of dimension 5, for a generic local
embedding of $M$ in $\mathbb{C}^{3}$, $\Phi$ intersects a stratum $0\times X$
only when $X$ has codimension $\leq 4$ in $J^{l}(2,1)$. This means that the
only singularities of $k$-folding maps that can occur on a generic surface are
those belonging to the strata listed in Tables 1, 2, 4, 6. Furthermore, $\Phi$
is transverse to these strata. Therefore, the locus of points where the
$l$-jets of the $k$-folding map-germs belong to a stratum $X$ of codimension
$3$ (resp. $4$) is a regular curve on $M$ (resp. isolated points on this
curve). The above discussion also clarifies why the singularities of
$F^{\pi}_{k}$ occur generically even though they may have high
$\mathcal{A}_{e}$-codimensions: they belong to strata of low codimension. (The
strata can be viewed as a gluing of non $\mathcal{A}$-equivalent orbits which
depends on a finite set of moduli.)

It is worth observing that the above discussion is valid for smooth surfaces
in $\mathbb{R}^{3}$.

##  6 $k$-folding map-germs on surfaces in $\mathbb{R}^{3}$

We consider in this section the geometry of $k$-folding maps-germs on smooth
(i.e., regular and of class $C^{\infty}$) surfaces in $\mathbb{R}^{3}$.

###  6.1 Robust features on surfaces

Robust features on a surface in $\mathbb{R}^{3}$ is a terminology introduced
by Ian Porteous to indicate special characteristic features that can be traced
when the surface evolves. What is sought after in applications are robust
features which are represented by curves or points on the surface as these
form a “skeletal structure” of the surface (open regions bounded by robust
curves are also robust features). They play an important role in computer
vision and shape recognition (see for example [27]) as they can be used to
distinguish two shapes (surfaces) from each other and, in some cases,
reconstruct the surface.

We consider here the parabolic, ridge, sub-parabolic and flecnodal curves on a
smooth surface $M$ in $\mathbb{R}^{3}$ and special points on these curves (see
[15] for references on work on these curves from a singularity theory point of
view). These are robust features on $M$. We recall briefly what they are.

The parabolic curve is the locus of points where the Gaussian curvature
vanishes. It is captured by the contact of the surface $M$ with planes: it is
the locus of points where the height function along a normal direction to $M$
has an $A_{\geq 2}$-singularity. The parabolic curve is regular on a generic
surface, and the height function has an $A_{2}$-singularity at its points
except at isolated ones, called Cusps of Gauss, where it has an
$A_{3}$-singularity.

The ridge is the locus of points on $M$ where a principal curvature is
extremal along its associated lines of principal curvature. It is also the
locus of points on $M$ which correspond to singular points on its focal set.
The ridge is captured by the contact of $M$ with spheres: it is the locus of
points on $M$ where the distance squared function from a given point in
$\mathbb{R}^{3}$ (the point belong to the focal set) has an $A_{\geq
3}$-singularity. Away from umbilic points (i.e., points where the principal
curvatures coincide), the ridge is a regular curve on a generic surface and
the distance squared function has an $A_{3}$-singularity at its points except
at isolated points where it has an $A_{4}$-singularity. At umbilics, the ridge
consist of one regular curve or a transverse intersection of three regular
curves. (Umbilics and $A_{4}$-points are used as seed points for drawing
ridges on a given shape, see [27].)

The sub-parabolic curve is the locus of points on $M$ corresponding to
parabolic points of its focal set. It is the locus of geodesic inflections of
the lines of principal curvature; it is also the locus of points along which a
principal curvature is extremal along the other lines of curvature. It is
captured by the singularities of the 2-folding map on $M$: it is the locus of
points where some map $F_{2}^{\pi}$ has an $S_{\geq 2}$-singularity (or one
which is adjacent to an $S_{2}$-singularity). The singularities of $F_{2}$ and
their geometric characterisation on a generic surface parametrised in Monge
form $z=f(x,y)$ at a given non-umbilic point, with $f$ as in (2), are as in
Table 8. At umbilics, the sub-parabolic curve consist of one regular curve or
a transverse intersection of three regular curves. Observe that the ridge is
also captured by the singularities of $F_{2}^{\pi}$ (see Table 8).

Table 8: Geometric characterisation of the singularities of the folding-map $F_{2}$. Name |  Algebraic conditions and geometric meaning
---|---
Crosscap |  $a_{21}\neq 0$
$B_{1}=S_{1}$ |  $a_{21}=0,\ a_{31}\neq 0,\ a_{33}\neq 0$
|  General smooth point of focal set
$B_{2}$ |  $a_{21}=0,a_{31}\neq 0,a_{33}=0,CndNA_{3}=4a_{31}a_{55}-a_{43}^{2}\neq 0$
|  General cusp point of focal set corresponding to a point on the ridge curve
$B_{3}$ |  $a_{21}=0,a_{31}\neq 0,a_{33}=0,CndNA_{3}=0$,
|
${CndNA}_{5}=8a_{31}^{3}a_{77}-4a_{65}a_{43}a_{31}^{2}+2a_{53}a_{43}^{2}a_{31}-a_{41}^{2}a_{43}^{3}\neq
0$
|  (Cusp) point of focal set in closure of parabolic curve on symmetry set
$S_{2}$ |  $a_{21}=0,\ a_{31}=0,\ a_{33}\neq 0,\ a_{41}\neq 0$
|  Parabolic smooth point of focal set corresponding to a point on the sub-
parabolic curve
$S_{3}$ |  $a_{21}=0,\ a_{31}=0,\ a_{33}\neq 0,\ a_{41}=0,\ a_{51}\neq 0$
|  Cusp of Gauss at smooth point of focal set
$C_{3}$ |  $a_{21}=0,\ a_{31}=0,\ a_{33}=0,\ a_{41}\neq 0,\ a_{43}\neq 0$
|  Intersection point of cuspidal-edge and parabolic curve on focal set

The flecnodal curve is the locus of geodesic inflections of the asymptotic
curves. It is captured by the contact of the surface $M$ with lines: it is the
locus of points where the orthogonal projection of the surface has a
singularity of type swallowtail or worse. We recall briefly some results on
these projections as they are needed for interpreting the singularities of
$k$-folding maps.

The orthogonal projection $P_{v}$ of $M$ along the direction $v\in S^{2}$ to
the plane $T_{v}S^{2}$ is given by $P_{v}(p)=p-(p\cdot v)v$, with $p\in M$.
This can be represented locally by a map-germ from the plane to the plane.
Varying $v$ yields the family of orthogonal projections $P:M\times S^{2}\to
TS^{2}$ given by $P(p,v)=(v,P_{v}(p)).$

A transversality theorem asserts that for an open and dense (i.e., generic)
set of embeddings $\phi:U\to\mathbb{R}^{3}$, the surface $M=\phi(U)$ has the
following property: for any $v\in S^{2}$ the map-germ $P_{v}$ has only local
singularities $\mathcal{A}$-equivalent to one in Table 9 at any point on $M$.
By translating $p_{0}\in M$ to the origin and taking $M$ locally at $p_{0}$ in
Monge form, an open subset of $M$ is parametrised by $\phi(x,y)=(x,y,f(x,y))$,
with $f$ having no constant nor linear terms. Projecting along the direction
$v=(0,1,0)$ gives $P_{v}(x,y)=(x,f(x,y))$, which is singular at the origin.
Table 9 shows the conditions on the coefficients of $f$ for the map-germ
$P_{v}$ to have a singularity at the origin of $\mathcal{A}_{e}$-codimension
$\leq 2$, where $d_{e}(G,\mathcal{A})$ denotes the
$\mathcal{A}_{e}$-codimension of $G$ (see for example [15]),

Table 9: $\mathcal{A}_{e}$-codimension $\leq 2$ singularities of map-germs $G:(\mathbb{R}^{2},0)\to(\mathbb{R}^{2},0)$. Name | Normal form | $d_{e}(G,{\mathcal{A}})$ |  Algebraic conditions on $f$ in (2)
---|---|---|---
|  |  |  for the singularities of $G(x,y)=(x,f(x,y))$
Fold | $(x,y^{2})$ | 0 | $a_{22}\neq 0$
Cusp | $(x,xy+y^{3})$ | 0 |  $a_{22}=0$, $a_{21}\neq 0$, $a_{33}\neq 0$
Swallowtail | $(x,xy+y^{4})$ | 1 |  $a_{22}=0$, $a_{33}=0$, $a_{21}\neq 0$, $a_{44}\neq 0$
Lips/beaks | $(x,y^{3}\pm x^{2}y)$ | 1 |  $a_{22}=0$, $a_{21}=0$, $a_{33}\neq 0$, $a_{32}^{2}-3a_{31}a_{33}\neq 0$
Goose | $(x,y^{3}+x^{3}y)$ | 2 |  $a_{22}=0$, $a_{21}=0$, $a_{32}^{2}-3a_{31}a_{33}=0$, $a_{33}\neq 0$,
|  |  | $27a_{41}a_{33}^{3}-18a_{42}a_{32}a_{33}^{2}+9a_{43}a_{32}^{2}a_{33}-4a_{44}a_{32}^{3}\neq 0$
Butterfly | $(x,xy+y^{5}\pm y^{7})$ | 2 |  $a_{22}=0$, $a_{33}=0$, $a_{44}=0$, $a_{21}\neq 0$, $a_{55}\neq 0$,
|  |  | $(8a_{55}a_{77}-5a_{66}^{2})a_{21}^{2}+2a_{55}(a_{32}a_{66}-20a_{43}a_{55})a_{21}+$
|  |  | $35a_{32}^{2}a_{55}^{2}\neq 0$
Gulls | $(x,xy^{2}+y^{4}+y^{5})$ | 2 |  $a_{22}=0$, $a_{21}=0$, $a_{33}=0$, $a_{32}\neq 0$, $a_{44}\neq 0$,
|  |  | $a_{55}a_{32}^{2}-2a_{43}a_{44}a_{32}+4a_{31}a_{44}^{2}\neq 0$

The projection $P_{v}$ has a fold singularity at $p$ if, and and only if, $v$
is a non-asymptotic tangent direction to $M$ at $p$. The singularity at $p$ is
of type cusp or worse if, and only if, $v$ is an asymptotic direction at $p$.
For a generic surface $M$, the closure of the set of points where $P_{v}$ has
a swallowtail (resp. lips/beaks) singularity is a precisely the flecnodal
(resp. parabolic) curve. The flecnodal curve meets tangentially the parabolic
curve at the cusps of the Gauss map, which are the gulls singularities of