# Cross-caps, triple points and a linking invariant for finitely determined
germs

Gergő Pintér  Department of Theoretical Physics, Institute of Physics,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111
Budapest, Hungary<EMAIL_ADDRESS>and  András Sándor  Alfréd Rényi
Institute of Mathematics, Reáltanoda u. 13-15, Budapest, 1053, Hungary
Central European University, Dept. of Mathematics, Nádor u. 9., Budapest,
1051, Hungary<EMAIL_ADDRESS>
###### Abstract.

It was recently proved that for finitely determined germs
$\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ the number $C(\Phi)$ of Whitney
umbrella points and the number $T(\Phi)$ of triple values of a stable
deformation are topological invariants. The proof uses the fact that the
combination $C(\Phi)-3T(\Phi)$ is topological since it equals the linking
invariant of the associated immersion $S^{3}\looparrowright S^{5}$ introduced
by Ekholm and Szűcs. We provide a new, direct proof for this equality. We also
clarify the relation between various definitions of the linking invariant.

###### Key words and phrases:

hypersurface singularities, non-isolated singularities, links of
singularities, finite determinacy, immersion theory, stable immersions, cross
caps, topological invariants

##  1\. Introduction

### 1.1.

Let $\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a finitely determined
(also called $\mathcal{A}$-finite) holomorphic germ. In this case
$\mathcal{A}$-finiteness means that $\Phi$ is a stable immersion off the
origin [22, 15]. For these germs the number of the complex Whitney umbrella
(cross cap) points $C(\Phi)$ and the triple values $T(\Phi)$ of a stable
holomorphic deformation are well-defined analytic invariants [12, 13].
Recently in [1] J. Fernández de Bobadilla, G. Peñafort, and J. E. Sampaio
proved that these invariants are topological, moreover they are determined by
the embedded topological type of the image of $\Phi$. One of the main
ingredients of their proof is the formula

(1.1.1) $L(\Phi|_{\mathfrak{S}})=C(\Phi)-3T(\Phi)$

from [17], which expresses the naturally topological Ekholm–Szűcs invariant
(also called triple point invariant or linking invariant)
$L(\Phi|_{\mathfrak{S}})$ of the associated stable immersion
$\Phi|_{\mathfrak{S}}:\mathfrak{S}\simeq S^{3}\looparrowright S^{5}$ in terms
of $C$ and $T$. However, the formula (1.1.1) is proved in [17] in a rather
complicated way, by using two Smale invariant formulas. The main purpose of
this article is to provide a new direct proof for this formula.

The Ekholm–Szűcs invariant $L(f)$ of a stable immersion
$f:S^{3}\looparrowright\mathbb{R}^{5}$ measures the linking of the image with
a copy of the double values, shifted slightly along a suitable chosen normal
vector field. In the literature different versions of the definition of $L$
can be found (see [2, 3, 4, 20]), whose relation is not completely clarified.
We verify their equivalence, i.e. $L_{1}(f)=-L_{2}(f)$, based on their
opposite behavior through regular homotopies.

Although our proof of the main theorem (1.1.1) is self-contained, an
independent secondary goal of this article is to clarify the enigmatic
relation between several variants of the linking invariant $L$ and other
related invariants, used in the study of generic $\mathcal{C}^{\infty}$ real
maps and immersions.

###  1.2. Structure of the article

In the Preliminaries (Section 2), we summarise the properties of finitely
determined holomorphic germs we will use. We outline the definitions of $C$
and $T$ and their invariance for analytic, $\mathcal{C}^{\infty}$ and
topological left-right equivalence. We introduce the associated immersion and
we describe the double point structure of $\Phi$.

In Section 3, we collect the different definitions of the Ekholm–Szűcs
invariant $L$ of stable immersions $S^{3}\looparrowright S^{5}$ from the
literature. We show that they agree up to sign and we clarify that sign. Then
we define an invariant for finitely determined germs by applying $L$ to the
associated immersions, and we prove its topological left-right invariance.

In Section 4, we provide a new, direct proof for the correspondence $L=C-3T$.
We use local calculations near complex cross cap points and triple values.

Finally, Appendix A is a brief summary of the applications of $L$ and another
similar linking invariant in the study of generic real maps and immersion
theory. We collect the most relevant results and clear up the context of this
article, including the main steps of the original proof of (1.1.1). Then we
compare the new local calculation for the complex cross cap points with an
older one in [17], and clarify its consequences for the Ekholm–Szűcs Smale
invariant formula.

### Acknowledgements

We are very grateful to László Fehér, András Szűcs, Tamás Terpai, András
Némethi, József Bodnár and David Mond for several very helpful conversations
regarding this topic. We also thank the reviewer for their insightful
suggestions.

GP thanks his physicist colleagues András Pályi, György Frank, Dániel Varjas
and János Asbóth for the new inspiration to the singularity theory research.

##  2\. Preliminaries

###  2.1. Invariants of a stabilization

A holomorphic germ $\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ is _finitely
$\mathcal{A}$-determined_ (briefly, _finitely determined_), if there is an
integer $k$ such that the $k$-th Taylor polynomial of $\Phi$ determines it up
to left-right equivalence, or equivalently, the $\mathcal{A}$-codimension of
$\Phi$ is finite. By Mather–Gaffney criterion [22, 15], $\Phi$ is finitely
determined if and only if its restriction
$\Phi|_{\mathbb{C}^{2}\setminus\\{0\\}}$ is stable. This means that a
sufficiently small representative of $\Phi|_{\mathbb{C}^{2}\setminus\\{0\\}}$
has only (1) regular simple points and (2) double values with transverse
intersection of the regular branches.

The only possible multigerms of a stabilization (stable deformation) of a
holomorphic germ $\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ are (1)
regular simple points, (2) double values with transverse intersection of the
regular branches, (3) triple values with regular intersection of the regular
branches and (4) simple _Whitney umbrella (cross cap)_ points. The Whitney
umbrellas and the triple values are isolated points, up to analytic
$\mathcal{A}$-equivalence they have local normal forms

(2.1.1) $\mbox{Whitney umbrella (cross cap): }\ (s,t)\mapsto(s^{2},st,t)$
(2.1.2) $\mbox{Triple
value:}\left\\{\begin{array}[]{ccc}(s_{1},t_{1})&\mapsto&(0,s_{1},t_{1})\\\
(s_{2},t_{2})&\mapsto&(t_{2},0,s_{2})\\\
(s_{3},t_{3})&\mapsto&(s_{3},t_{3},0)\\\ \end{array}\right.$

The numbers $C(\Phi)$ of the cross caps and $T(\Phi)$ of the triple values are
independent of the stabilization, they are analytic invariants of the finitely
determined germs $\Phi$. Both invariants were introduced by Mond [12, 13],
they can be defined in algebraic way as well, without referring to a
stabilization, as follows.

Let $C_{alg}(\Phi)$ be the codimension of the _ramification ideal_ , which is
the ideal in the local ring $\mathcal{O}_{(\mathbb{C}^{2},0)}$ generated by
the determinants of the $2\times 2$ minors of the Jacobian matrix of
$\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$. $T_{alg}(\Phi)$ is the
codimension of the second Fitting ideal associated with $\Phi$ in
$\mathcal{O}_{(\mathbb{C}^{3},0)}$ [16]. If $\Phi$ is finitely determined,
then both $C_{alg}(\Phi)$ and $T_{alg}(\Phi)$ are finite, and any
stabilization of $\Phi$ has $C(\Phi)=C_{alg}(\Phi)$ number of cross caps and
$T(\Phi)=T_{alg}(\Phi)$ number of triple values. The invariants $T$ and $C$
appear in several different contexts, see for example [14, 9, 11, 10, 15, 19].

The analytic invariance of $C$ and $T$ means the following. Let $\Phi_{1}$ and
$\Phi_{2}$ be finitely determined germs, analytic $\mathcal{A}$-equivalent to
each other. That is, there exist germs of biholomorphisms
$\phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ and
$\psi:(\mathbb{C}^{3},0)\to(\mathbb{C}^{3},0)$ such that

(2.1.3) $\Phi_{2}=\psi\circ\Phi_{1}\circ\phi$

holds, i.e. the diagram below commutes.

(2.1.4)
${(\mathbb{C}^{2},0)}$${(\mathbb{C}^{3},0)}$${(\mathbb{C}^{2},0)}$${(\mathbb{C}^{3},0)}$$\scriptstyle{\Phi_{1}}$$\scriptstyle{\psi}$$\scriptstyle{\Phi_{2}}$$\scriptstyle{\phi}$

Then

(2.1.5) $C(\Phi_{1})=C(\Phi_{2})\mbox{ and }T(\Phi_{1})=T(\Phi_{2}).$

In [17] it is proved that $C$ and $T$ are $\mathcal{C}^{\infty}$-invariants as
well. That is, (2.1.5) holds also for $\mathcal{C}^{\infty}$ left-right
equivalent germs, i.e. for two holomorphic finitely determined germs for which
(2.1.4) holds with some germs of $\mathcal{C}^{\infty}$-diffeomorphisms
$\phi:(\mathbb{R}^{4},0)\to(\mathbb{R}^{4},0)$ and
$\psi:(\mathbb{R}^{6},0)\to(\mathbb{R}^{6},0)$. (Here, $\mathbb{C}^{n}$ and
$\mathbb{R}^{2n}$ are naturally identified.)

The topological invariance of $C$ and $T$ would mean that (2.1.5) holds also
for topologically left-right equivalent germs, that is when we only require
$\phi$ and $\psi$ to be germs of homeomorphisms. This invariance was an open
question for a long time. In [17] A. Némethi and the first author proved that
the linear combination $C-3T$ is a topological invariant. This follows from
$L=C-3T$ (formula (1.1.1)) which expresses a topological invariant (the
Ekholm–Szűcs invariant) of the associated immersion, see the next sections. In
this article, we present a new direct proof of formula $L=C-3T$. (We also
prove the topological invariance of the Ekholm–Szűcs invariant, see
Proposition 3.2.2. This fact is very natural and has been implicitly used
previously, but according to the authors’ knowledge, it has not been published
yet.)

In [1] J. Fernández de Bobadilla, G. Peñafort, and J. E. Sampaio proved that
$C$ and $T$ are topological invariants, moreover they are determined by the
embedded topological type of the image of $\Phi$. A key ingredient of their
proof is the topological invariance of $C-3T$, which follows from the formula
$L=C-3T$.

###  2.2. The associated immersion

Let $\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a finitely determined
germ. Such a germ, on the level of links of the spacegerms
$(\mathbb{C}^{2},0)$ and $(\mathbb{C}^{3},0)$, provides a stable immersion
$\Phi|_{S^{3}}:S^{3}\looparrowright S^{5}$ as follows. The preimage
$\mathfrak{S}:=\Phi^{-1}(S^{5}_{\epsilon})$ of the $5$-sphere
$S^{5}_{\epsilon}\subset\mathbb{C}^{3}$ around the origin, with a sufficiently
small radius $\epsilon$, is diffeomorphic to $S^{3}$. The restriction
$\Phi|_{\mathfrak{S}}:\mathfrak{S}\looparrowright S^{5}_{\epsilon}$ is the
immersion associated with $\Phi$. The regular homotopy class of
$\Phi|_{\mathfrak{S}}$ is independent of all the choices. The immersions
obtained by different choices are regular homotopic to each other through
stable immersions. See [17, 2.1.] or [19, Subsection 1.1.2.].

###  2.3. The image and the double points

Write $(X,0)$ for $({\rm im}(\Phi),0)$ and let
$f:(\mathbb{C}^{3},0)\to(\mathbb{C},0)$ be the reduced equation of $(X,0)$.
Note that $(X,0)$ is a non-isolated hypersurface singularity, except when
$\Phi$ is a regular map (see [17]). We denote by
$(\Sigma,0)=(\partial_{x_{1}}f,\partial_{x_{2}}f,\partial_{x_{3}}f)^{-1}(0)\subset(\mathbb{C}^{3},0)$
the reduced singular locus of $(X,0)$ – that is the closure of the set of
double values of $\Phi$. Also, we denote by $(D,0)$ the reduced double point
curve $\Phi^{-1}(\Sigma)\subset(\mathbb{C}^{2},0)$. The reduced equation of
$D$ is $d:(\mathbb{C}^{2},0)\to(\mathbb{C},0)$. (In fact, the finite
determinacy of the germ $\Phi$ is equivalent with the fact that the double
point curve $D$ is reduced; see e.g. [11].)

Let $\Upsilon\subset S^{5}_{\epsilon}$ be the link of $\Sigma$. It is exactly
the set of double values of $\Phi|_{\mathfrak{S}}$. Let
$\gamma=\Phi^{-1}(\Upsilon)\subset\mathfrak{S}^{3}$ denote the set of double
points of $\Phi|_{\mathfrak{S}}$, that is, $\gamma\subset\mathfrak{S}$ is the
link of $D$. All link components are considered with their natural
orientations.

(2.3.1) $\begin{array}[]{ccc}(\mathbb{C}^{2},0)&\to&(\mathbb{C}^{3},0)\\\
\cup&&\cup\\\ (D,0)&\to&(\Sigma,0)\\\ \cup&&\cup\\\
\gamma=D\cap\mathfrak{S}^{3}&\to&\Upsilon=\Sigma\cap
S_{\epsilon}^{5}\end{array}$

Figure 1. Notations of the various parts of the space germs.

##  3\. The Ekholm–Szűcs linking invariant

###  3.1. The Ekholm–Szűcs invariant of stable immersions

The invariant $L(f)$ of a stable immersion
$f:S^{3}\looparrowright\mathbb{R}^{5}$ measures the linking of a shifted copy
of the double values with the whole image of $f$. Different versions of the
definition can be found in the literature, for references see below. In this
paragraph, we review these definitions and prove their equivalence via their
behavior along regular homotopies. We present the whole argument in the
simplest case, for immersions $S^{3}\looparrowright\mathbb{R}^{5}$, although
originally they were introduced for different levels of generality (for other
manifolds, higher dimensions) in [2, 3, 4, 20]. This discussion is an extended
version of the summery in [19, 2.2.2.].

A stable immersion $f:S^{3}\looparrowright\mathbb{R}^{5}$ has only simple
values and double values with transverse intersection of the two branches. Let
$\gamma\subset S^{3}$ be the double point locus of $f$, that is
$\gamma=\\{p\in S^{3}\ |\ \exists p^{\prime}\in S^{3}:\ p\neq p^{\prime}\mbox{
and }f(p)=f(p^{\prime})\\}$. The locus $\gamma$ is a closed $1$-manifold, i.e.
a link in $S^{3}$ with possibly more components. The map
$f|_{\gamma}:\gamma\to f(\gamma)$ is a $2$-fold covering. $\gamma$ is endowed
with an involution $\iota:\gamma\to\gamma$ such that $\iota(p)\neq p$ and
$f(p)=f(\iota(p))$ hold for all $p\in\gamma$.

The first definition of $L(f)$ is from [2, 6.2.]. Let $v$ be a vector field
along $\gamma$ tangent to $S^{3}$ and nowhere tangent to $\gamma$, i.e. $v$
represents a section of the normal bundle $TS^{3}|_{\gamma}/T\gamma$ of
$\gamma\subset S^{3}$. We also require that $[\tilde{\gamma}]$ is $0$ in
$H_{1}(S^{3}\setminus\gamma,\mathbb{Z})$, where $\tilde{\gamma}\subset S^{3}$
is the result of pushing $\gamma$ slightly along $v$. Such a vector field $v$
is unique up to homotopy, and for instance each of the two vectors of a
Seifert framing provides such a vector field. If $v$ is such a vector field,
then the linking number $\mbox{lk}_{S^{3}}(\gamma,\tilde{\gamma})$ equals to
$0$, but the reverse is not true, since
$\mbox{lk}_{S^{3}}(\gamma,\tilde{\gamma})$ is the sum of the components of
$[\tilde{\gamma}]\in H_{1}(S^{3}\setminus\gamma,\mathbb{Z})$. (All the linking
numbers appearing are considered with respect to the natural orientation of
the curves and submanifolds involved.) Let $q=f(p)=f(\iota(p))$ be a double
value of $f$. Then $w(q)=df_{p}(v(p))+df_{\iota(p)}(v(\iota(p))$ defines a
vector field $w$ along $f(\gamma)$ that is nowhere tangent to the branches of
$f$. In this sense $w$ is a normal vector field of $f$ along $f(\gamma)$. Let
$\widetilde{f(\gamma)}\subset\mathbb{R}^{5}$ be the result of pushing
$f(\gamma)$ slightly along $w$, then $\widetilde{f(\gamma)}$ and $f(S^{3})$
are disjoint. The first invariant is the linking number

(3.1.1) $L_{1}(f):=\mbox{lk}_{\mathbb{R}^{5}}(\widetilde{f(\gamma)},f(S^{3}))$

(or equivalently, $L_{1}(f)=[\widetilde{f(\gamma)}]\in
H_{1}(\mathbb{R}^{5}\setminus f(S^{3}),\mathbb{Z})\cong\mathbb{Z}$). Note that
Ekholm used an other notation: in [2, 2.2., 6.2.] our $L_{1}(f)$ is denoted by
$\mbox{lk}(f)$, and $L(f)$ is defined as $\lfloor\mbox{lk}(f)/3\rfloor$.

The second definition is [4, Definition 11.], [20, Definition 2.2.]. It works
only with further assumptions, see Remark 3.1.3 below. The normal bundle
$\nu(f)$ of $f$ is trivial, since the oriented rank–$2$ vector bundles over
$S^{3}$ are classified by $\pi_{2}(SO(2))=0$. Any two trivializations are
homotopic, since their difference represents an element in $\pi_{3}(SO(2))=0$.
Let $(v_{1},v_{2})$ be the homotopically unique normal framing of $f$, and at
a double value $q=f(p)=f(\iota(p))$ define $u(q)=v_{1}(p)+v_{1}(\iota(p))$.
$u$ is a normal vector field along $f(\gamma)$, and let
$\overline{f(\gamma)}\subset\mathbb{R}^{5}$ be the result of pushing
$f(\gamma)$ slightly along $u$. Then $\overline{f(\gamma)}$ and $f(S^{3})$ are
disjoint. The invariant is the linking number (or equivalently, the homology
class)

(3.1.2)
$L_{2}(f):=\mbox{lk}_{\mathbb{R}^{5}}(\overline{f(\gamma)},f(S^{3}))=[\overline{f(\gamma)}]\in
H_{1}(\mathbb{R}^{5}\setminus f(S^{3}),\mathbb{Z})\cong\mathbb{Z}\mbox{.}$

Note that the framing $(v_{1},v_{2})$ can be replaced by an arbitrary nonzero
normal vector field $v$ of $f$, since it can be extended to a framing whose
first component is $v$.

######  Remark 3.1.3.

Without further assumptions it is possible that $u(q)$ is tangent to one of
the branches of $f$, hence it can happen that $\overline{f(\gamma)}\cap
f(S^{3})\neq\emptyset$. To avoid this problem one has to choose a unit normal
vector field $v$ or has to assume that the intersection of the branches is
orthogonal, which can be reached by a regular homotopy through stable
immersions. In this paper all the calculations uses $L_{1}$ and not $L_{2}$.

The third definition is in [4, Definition 4.], see also [3, 4.5., 4.6.]. Let
$v$ be a nonzero normal vector field of $f$ along $\gamma$, that is, a nowhere
zero section of $\nu(f)|_{\gamma}$. Let $[v]$ be the homology class
represented by $v$ in $H_{1}(E_{0}(\nu(f)),\mathbb{Z})\cong\mathbb{Z}$, where
$E_{0}(\nu(f))$ denotes the total space of the bundle of nonzero normal
vectors of $f$. Let $u_{v}(q)=v(p)+v(\iota(p))$ be the value of the vector
field $u_{v}$ along $f(\gamma)$ at the point $q=f(p)=f(\iota(p))$. Let
$\overline{f(\gamma)}^{(v)}$ be the result of pushing $f(\gamma)$ slightly
along $u_{v}$, then $\overline{f(\gamma)}^{(v)}$ and $f(S^{3})$ are disjoint.
The invariant is

(3.1.4)
$L_{v}(f):=\mbox{lk}_{\mathbb{R}^{5}}(\overline{f(\gamma)}^{(v)},f(S^{3}))-[v]=[\overline{f(\gamma)}^{(v)}]-[v]\mbox{,}$

where $[\overline{f(\gamma)}^{(v)}]\in H_{1}(\mathbb{R}^{5}\setminus
f(S^{3}),\mathbb{Z})\cong\mathbb{Z}$.

By [3, Lemma 4.15.] $L_{v}(f)$ is well-defined, that is, $L_{v}(f)$ does not
depend on the choice of the normal field $v$. Moreover, if $v$ is the
restriction of a (global) normal vector field of $f$ to $\gamma$, then
$[v]=0$. Indeed, the restriction of the normal field of $f$ to a Seifert
surface $H$ of $\gamma$ results a surface $\overline{H}\subset E_{0}(\nu(f))$,
whose boundary is the image of $v:\gamma\to E_{0}(\nu(f))$. Hence
$L_{v}(f)=L_{2}(f)$.

The invariants $L_{1}$, $L_{2}$ are equal to each other with opposite sign.
This follows from the fact that they behave in an inverse way along regular
homotopies, i.e. they change with the same number with opposite sign when a
stable regular homotopy steps through first order instabilities: immersions
with (1) one triple value (“triple point moves”) or (2) a self-tangency
(“self-tangency moves”). For definitions we refer to [2, 3]. The proof of
Proposition 3.1.5 is a result of a discussion with Tamás Terpai and András
Szűcs.

######  Proposition 3.1.5.

(a) $L_{1}(f)$ and $L_{2}(f)$ are invariants of stable immersions. They change
by $\pm 3$ under triple point moves and do not change under self tangency
moves. In other words: if $f$ and $g$ are regular homotopic stable immersions,
$h:S^{3}\times[0,1]\to\mathbb{R}^{5}$ is a stable regular homotopy between
them, then $\pm(L_{i}(f)-L_{i}(g))$ is equal to three times the algebraic
number of the triple values of the map
$H:S^{3}\times[0,1]\to\mathbb{R}^{5}\times[0,1]$, $H(x,t)=(h(x,t),t)$.

(b) In the above setup $L_{1}(f)-L_{1}(g)=-(L_{2}(f)-L_{2}(g))$.

(c) The three definitions are equivalent:

$L_{1}(f)=-L_{2}(f)=-L_{v}(f).$

###### Proof.

Part (a) is proved for $L_{1}$ in [2, Lemma 6.2.1.] and for $L_{2}=L_{v}$ in
[3, Theorem 1.].

For part (b), we compare the change of $L_{1}$ and $L_{2}$ through a triple
point move. In the proof of [2, Lemma 6.2.1.] Ekholm defines a local model of
the triple point move where $L_{1}$ increases by $3$. On the other hand, in
the discussion preceding [3, Definition 6.3] he provides a convention to
measure the change of $L_{2}$. If we check this convention on the previous
local model, we obtain that $L_{2}$ decreases by $3$ through that triple point
move. Hence $L_{1}$ and $L_{2}$ changes in opposite ways at each triple point
move.

Using part (a) and part (b), we prove part (c) as follows. Since $L_{1}$ and
$L_{2}$ changes in opposite way along a regular homotopy, $L_{1}+L_{2}$ is a
regular homotopy invariant. Moreover $L_{1}$ and $L_{2}$ are additive under
connected sum, see [3, Lemma 5.2., Proposition 5.4.], [2, 6.5.]. It follows
that $L_{1}+L_{2}$ defines a homomorphism from
$\mbox{Imm}(S^{3},\mathbb{R}^{5})$ to $\mathbb{Z}$. If
$f:S^{3}\hookrightarrow\mathbb{R}^{5}$ is an embedding, then
$L_{1}(f)=L_{2}(f)=0$, hence $L_{1}+L_{2}$ is $0$ on the $24$-index subgroup
$\mbox{Emb}(S^{3},\mathbb{R}^{5})$ of
$\mbox{Imm}(S^{3},\mathbb{R}^{5})\cong\mathbb{Z}$. It follows that
$L_{1}+L_{2}$ is $0$ for every stable immersion, hence $L_{1}=-L_{2}$. ∎

We fix the following convention.

######  Notation 3.1.6.

$L(f):=L_{1}(f)$.

In the continuation of this article, $L$ will be studied thoroughly, and a new
direct proof of $L_{2}(f)=-L_{1}(f)$ will be provided.

###  3.2. Ekholm–Szűcs invariant for finitely determined germs

The definition of $L_{1}(f)$ and $L_{2}(f)$ of immersions
$f:S^{3}\looparrowright\mathbb{R}^{5}$ cannot be applied directly for
$\Phi|_{\mathfrak{S}}:\mathfrak{S}\looparrowright S^{5}$. In fact, the shifted
copy of $\upsilon\in S^{5}$ by a normal vector field is a curve in
$\mathbb{C}^{3}=\mathbb{R}^{6}$, but not exatly in $S^{5}$. To solve this
technical difficulty we recall one of the definitions of the linking number.

######  Definition 3.2.1.

Let $N^{n},M^{m}\subset S^{k}=\partial B^{k+1}$ be two closed oriented
submanifolds with dimensions $n+m+1=k$. Choose any oriented homological
membranes $\widetilde{M},\widetilde{N}\subset B^{k+1}$ for them, that is,
$\widetilde{M}$ and $\widetilde{N}$ are singular chains in $B^{k+1}$ of
dimensions $n+1$, respectively $m+1$, with coefficients in $\mathbb{Z}$, whose
boundaries are $\partial\widetilde{N}=N$, $\partial\widetilde{M}=M$. Then the
linking number $\mbox{lk}_{S^{k}}(N,M)$ of $N$ and $M$ in $S^{k}$ is defined
as the intersection number $\mbox{int}_{B^{k+1}}(\widetilde{M},\widetilde{N})$
of $\widetilde{M}$ and $\widetilde{N}$ in $B^{k+1}$.

For the definition of $L_{1}(\Phi|_{\mathfrak{S}})$ consider
$\overline{\mbox{grad}(d)}$, the conjugate of the gradient vector field of $d$
defined on $D$. Its restriction to $\gamma\subset\mathfrak{S}$ is a
representative of the homotopically unique Seifert framing of $\gamma$. Then
the sum of the two copies of $d\Phi(\overline{\mbox{grad}(d)})$ is a nonzero
normal vector field along $\Sigma\setminus\\{0\\}$, which extends to the
origin with $0$. Let $\widetilde{\Sigma}$ be a copy of $\Sigma$ shifted along
this vector field. Define $\widetilde{\Upsilon}:=\widetilde{\Sigma}\cap S^{5}$
and
$L_{1}(\Phi|_{\mathfrak{S}})=\mbox{lk}_{S^{5}}(\widetilde{\Upsilon},\Phi(\mathfrak{S}))$.
The invariant $L_{1}(\Phi|_{\mathfrak{S}})$ is equal to the intersection
number of any pair of membranes in $B^{6}$ with boundaries
$\widetilde{\Upsilon}$ and $\Phi|_{\mathfrak{S}}$. Especially,
$L_{1}(\Phi|_{\mathfrak{S}})$ is the intersection number of
$\widetilde{\Sigma}$ and $X$. Unfortunately, however, they intersect each
other only at the origin, which is a singular point of possibly both
membranes, hence the intersection number cannot be calculated directly.
Instead, we will repeat the whole procedure with the _analytic stabilization_
of $\Phi$, and that will lead to the formula
$L_{1}(\Phi|_{\mathfrak{S}})=C(\Phi)-3T(\Phi)$.

$L_{2}(\Phi|_{\mathfrak{S}})$ can be defined in a similar way, by using
$\overline{\partial_{s}\Phi\times\partial_{t}\Phi}$ as a representative of the
homotopically unique global normal field of $\Phi|_{\mathfrak{S}}$. We can
define the shifted copy $\widetilde{\Sigma}^{(2)}$ of $\Sigma$, and
$\widetilde{\Upsilon}^{(2)}:=\widetilde{\Sigma}^{(2)}\cap S^{5}$. However, by
Remark 3.1.3, we cannot guarantee that $\widetilde{\Upsilon}^{(2)}$ and
$\Phi(\mathfrak{S})$ are disjoint. Although the formula
$L_{2}(\Phi|_{\mathfrak{S}})=3T(\Phi)-C(\Phi)$ can be supported by local
calculation, the precise proof in this way is technically complicated. On the
other hand, $L_{2}$ can be computed directly for the Whitney umbrella to
support that $L_{1}=-L_{2}$ holds, see Appendix A.

The topological invariance of
$L(\Phi|_{\mathfrak{S}})=L_{1}(\Phi|_{\mathfrak{S}})$ is almost trivial, since
the linking number is a topological (homological) invariant. However, its
proof has been nowhere explained in detail.

######  Proposition 3.2.2.

$L(\Phi|_{\mathfrak{S}})$ is a topological invariant of $\Phi$. That is if
$\Phi_{1}$ and $\Phi_{2}$ are finitely determined germs topologically
$\mathcal{A}$-equivalent to each other (see Subsection 2.1), then

(3.2.3) $L(\Phi_{1}|_{\mathfrak{S_{1}}})=L(\Phi_{2}|_{\mathfrak{S_{2}}}).$

###### Proof.

The topological equivalence of the germs means that there exist germs of
homeomorphisms $\phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ and
$\psi:(\mathbb{C}^{3},0)\to(\mathbb{C}^{3},0)$ such that
$\Phi_{2}=\psi\circ\Phi_{1}\circ\phi$ holds. The double point curves
$(D_{1},0)=(d_{1}^{-1}(0),0)$ of $\Phi_{1}$ and $(D_{2},0)=(d_{2}^{-1}(0),0)$
of $\Phi_{2}$ are topologically equivalent germs of curves, in fact,
$D_{1}=\phi(D_{2})$. Their links $\gamma_{1}$, $\gamma_{2}$ are of the same
type as links in $\mathfrak{S}_{1}\cong\mathfrak{S}_{2}\cong S^{3}$.

Although the normal vector field $\mbox{grad}(d_{2})$ along $\gamma_{2}$
cannot be pushed forward by $\phi$ since it is not necessarily differentiable,
the slightly pushed out copy $\widetilde{\gamma_{2}}$ can be. The image
$\phi(\widetilde{\gamma_{2}})$ determines a normal vector field denoted by
$\phi_{*}(\mbox{grad}(d_{2}))$ along $\gamma_{1}$, which is homotopic to
$\mbox{grad}(d_{1})$ since both vector field represent the Seifert framing.
Hence the sum of the two copies of $d\Phi_{1}(\mbox{grad}(d_{1}))$ and
$d\Phi_{1}(\phi_{*}(\mbox{grad}(d_{2})))$ are homotopic normal fields along
$\Upsilon_{1}$, thus the pushed out copies $\widetilde{\Upsilon}_{1}$ and
$\widetilde{\Upsilon}_{1}^{(2)}$ of $\Upsilon_{1}$ along these vector fields
are homotopic in
$S^{5}\setminus\Phi_{1}(\mathfrak{S}_{1})=S^{5}\setminus\Phi_{1}(\phi(\mathfrak{S}_{2}))$.
Therefore,
$\mbox{lk}_{S^{5}}(\Phi_{1}(\mathfrak{S}_{1}),\widetilde{\Upsilon}_{1})=\mbox{lk}_{S^{5}}(\Phi_{1}(\mathfrak{S}_{1}),\widetilde{\Upsilon}_{1}^{(2)})$.
Finally, applying $\psi$ to the whole configuration does not change the
linking numbers, and
$\psi(\Phi_{1}(\mathfrak{S}_{1}))=\Phi_{2}(\mathfrak{S}_{2})$,
$\psi(\Upsilon_{1})=\Upsilon_{2}$,
$\psi(\widetilde{\Upsilon}_{1}^{(2)})=\widetilde{\Upsilon}_{2}$. ∎

######  Remark 3.2.4.

$L(f)$ can be defined for stable immersions
$f:M^{3}\looparrowright\mathbb{R}^{5}$ of closed oriented $3$-manifolds
$M^{3}$, with trivial normal bundle, see [20, Definition 2.5.]. Especially
$M^{3}$ can be a disjoint union of some copies of $S^{3}$. In this way for
multigerms $\Phi=(\Phi_{i}):\sqcup(\mathbb{C}^{2},0)_{i}\to(\mathbb{C}^{3},0)$
the invariant $L(\Phi|_{M^{3}})$ is defined, where
$M^{3}=\sqcup\mathfrak{S}_{i}$ with
$\mathfrak{S}_{i}=\Phi_{i}^{-1}(S^{5}_{\epsilon})$. We will use this extension
of $L$ for ordinary triple values.

######  Remark 3.2.5.

Recall Remark 2.2.7 from [19]. $L$ can be defined also for nonstable
immersions which do not have triple values, by the following argument. Any
immersion $f$ admits a small perturbation by regular homotopy to a stable
immersion $\widetilde{f}$, and if $f$ does not have triple values, then any
two stable perturbations can be joined with a regular homotopy without
stepping through a triple point. Thus $L(f)$ can be defined as
$L(\widetilde{f})$ of any small stable perturbation $\widetilde{f}$ of $f$.

Consequently $L(\Phi|_{\mathfrak{S}})$ can be defined not only for finitely
determined germs but for germs with finite $C$ and $T$, since for these germs
$\Phi|_{\mathfrak{S}}$ is not a stable immersion, but it does not have triple
points. Moreover the equation (1.1.1) holds for these germs too, since the
proof uses an analytic stabilization of $\Phi$, not $\Phi$ itself. See also
Corollary 3.6.3., Remark 3.6.4. in [19] or [17]. An interesting example is
$\Phi(s,t)=(s^{2},t^{2},st)$, which is the double cover of the $A_{1}$
singularity. See Subsection 3.7.2. of [19].

However it is not clear for these germs, how can $L(\Phi|_{\mathfrak{S}})$ be
computed directly from the topology of $\Phi$, without stabilizing it.

##  4\. Main theorem

###  4.1. Proof of the main theorem

######  Theorem 4.1.1.

For a finitely determined holomorphic germ
$\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$

$L(\Phi|_{\mathfrak{S}})=C(\Phi)-3T(\Phi).$

###### Proof.

Let $\Phi_{\lambda}:\mathfrak{B}_{\lambda}\to B^{6}_{\epsilon}$ be an analytic
stabilization of $\Phi_{0}=\Phi$. Here,
$\mathfrak{B}_{\lambda}=\Phi_{\lambda}^{-1}(B^{6}_{\epsilon})$ with boundary
$\partial\mathfrak{B}_{\lambda}=\mathfrak{S}_{\lambda}=\Phi_{\lambda}^{-1}(S^{5}_{\epsilon})$.

Decreasing $\lambda$ to $0$ induces a diffeomorphism
$\mathfrak{B}_{\lambda}\simeq\mathfrak{B}$, respectively
$\mathfrak{S}_{\lambda}\simeq\mathfrak{S}$, and a regular homotopy through
stable immersions between $\Phi_{\lambda}|_{\mathfrak{S}_{\lambda}}$ and
$\Phi|_{\mathfrak{S}}$. It implies – as recognised in [17, Section 9] – that

(4.1.2) $L(\Phi|_{\mathfrak{S}})=L(\Phi_{\lambda}|_{\mathfrak{S}_{\lambda}}).$

We denote the corresponding double point sets respectively by $D_{\lambda}$,
$\Sigma_{\lambda}$, $\gamma_{\lambda}$, $\Upsilon_{\lambda}$, as defined in
Subsection 3.2 . The reduced equation of $D_{\lambda}$ is
$d_{\lambda}:\mathfrak{B}_{\lambda}\to\mathbb{C}$.

We are going to count $L(\Phi_{\lambda})=L_{1}(\Phi_{\lambda})$. According to
definitions 3.1.1 and 3.2.1, we want to construct membranes bounding
$\Phi_{\lambda}(\mathfrak{S}_{\lambda})$ and $\widetilde{\Upsilon}_{\lambda}$,
and count their intersection number. The first membrane is simply the whole
image $\Phi_{\lambda}(\mathfrak{B}_{\lambda})$.

For the second membrane, consider the normal vector field
$w=\overline{\mbox{grad}(d_{\lambda})}$ on
$D_{\lambda}\subset\mathfrak{B_{\lambda}}$, its restriction represents the
Seifert framing on $\gamma_{\lambda}\subset\mathfrak{S}_{\lambda}$. The
pushforward $\mathrm{d}\Phi_{\lambda}(w)$ gives a double valued vector field
at each point of $\Sigma_{\lambda}$. We add up the two vectors pointwise and
pushout $\Sigma_{\lambda}$ slightly along the obtained vector field $v$ to get
$\widetilde{\Sigma}_{\lambda}$. (Notice that at triple points the vector field
$v$ has three values, but they are all zeroes.)

By the construction in Subsection 3.2, the boundary is
$\partial\widetilde{\Sigma}_{\lambda}=\widetilde{\Upsilon}_{\lambda}$ and

(4.1.3)
$L(\Phi_{\lambda}|_{\mathfrak{S}_{\lambda}})=\mathrm{int}(\Phi_{\lambda}(\mathfrak{B}_{\lambda}),\widetilde{\Sigma}_{\lambda}).$

As the two components of $v$ are tangent to the two branches of the image at a
double point, the pushout $\widetilde{\Sigma}_{\lambda}$ has no intersection
point with the whole image near an ordinary double point.

Besides double points, the only two types of singular points that may occur in
the stabilized map $\Phi_{\lambda}$ are Whitney umbrella points and triple
points. With the above remark, it means that we only have to count the
intersection points near these points.

Umbrella points and triple points are left-right equivalent to the standard
copies of them, see (2.1.1) and (2.1.2). In the following two lemmas, we
calculate the intersection numbers for these normal forms – which are, in
fact, the Ekholm–Szűcs invariants of these (multi)-germs. After stating the
lemmas we will deduce the global invariant by gluing these pieces together to
complete the proof.

######  Lemma 4.1.4.

The Ekholm–Szűcs invariant of the standard Whitney umbrella

$\Phi(s,t)=(s^{2},st,t)$

is

$L(\Phi|_{\mathfrak{S}})=1$

where $\mathfrak{S}=\Phi^{-1}(S^{5}_{\epsilon})$.

######  Lemma 4.1.5.

The standard triple value is the regular intersection of three branches. We
parametrize it the following way

$\Phi\left\\{\begin{array}[]{ccc}\Phi_{1}:(s_{1},t_{1})&\mapsto&(0,s_{1},t_{1})\\\
\Phi_{2}:(s_{2},t_{2})&\mapsto&(t_{2},0,s_{2})\\\
\Phi_{3}:(s_{3},t_{3})&\mapsto&(s_{3},t_{3},0)\\\ \end{array}\right.$

(The pairs $(s_{i},t_{i})$ are local coordinates around the three preimages of
the triple point.) The Ekholm–Szűcs invariant of this multigerm is

$L(\Phi|_{\mathfrak{S}})=-3.$

The above results suggest that each umbrella point and triple point
contributes $1$ and respectively $-3$ to the global Ekholm–Szűcs invariant.
This is, in fact, the case and the brief argument is the following. The
(multi)germs at the umbrella and triple points of $\Phi_{\lambda}$ are left-
right equivalent of their standard form, hence, by the left-right invariance
of $L$ (see 3.2.2), the membrane of $\Phi_{\lambda}$ shall be replaced locally
by the one coming from the standard forms.

More precisely, let us take an umbrella point or a triple value $p_{i}$ in
$\mathbb{C}^{3}$ and take a small balls $U_{i}\subset\mathbb{C}^{3}$ around
$p_{i}$ and $V_{i}\subset\mathbb{C}^{2}$ around $\Phi_{\lambda}^{-1}(p_{i})$
and biholomorphisms $\phi_{i}:(U_{i},p_{i})\to(\mathbb{C}^{3},0)$ and
$\psi_{i}:(\mathbb{C}^{2}_{r},\underline{0}_{r})\to(V_{i},\Phi_{\lambda}^{-1}(p_{i}))$
so that

$\phi_{i}\circ\Phi_{\lambda}\circ\psi_{i}:(\mathbb{C}^{2}_{r},\underline{0}_{r})\to(\mathbb{C}^{3},0)$

is a standard umbrella (respectively triple point) at $\underline{0}_{r}$.
(Here we use the notation for multi germs:
$(\mathbb{C}^{2}_{r},\underline{0}_{r})=\bigsqcup_{i=1}^{r}(\mathbb{C}^{2},0)$
with $r=1$ for a Whitney umbrella point and $r=3$ for a triple value $p_{i}$.)

We pull back the two membranes of the standard Whitney umbrella (resp. triple
value) via $\phi_{i}$ to define new membranes inside $U_{i}$. On one hand we
obtain another pushout of $\Sigma_{\lambda}$ instead of
$\widetilde{\Sigma}_{\lambda}$, let us denote it by $M_{i}\subset U_{i}$. On
the other hand, we get back a piece of the other original membrane,
$\Phi_{\lambda}(\mathfrak{B}_{\lambda})\cap U_{i}$.

Taking a look at the boundary of $U_{i}$, we find that both
$\widetilde{\Sigma}_{\lambda}\cap\partial U_{i}$ and $M_{i}\cap\partial U_{i}$
have the same linking number with
$\Phi_{\lambda}(\mathfrak{B}_{\lambda})\cap\partial U_{i}$: that is the
$L_{1}$ invariant of the umbrella point or the triple point. Therefore we can
construct a collar $N_{i}$ that connects
$\widetilde{\Sigma}_{\lambda}\cap\partial U_{i}$ and $M_{i}\cap\partial U_{i}$
in $\partial U_{i}$, in a way that $N_{i}$ has an intersection number 0 with
$\Phi_{\lambda}(\mathfrak{B}_{\lambda})\cap\partial U_{i}$.

Gluing all these pieces together, we obtain a membrane replacing
$\widetilde{\Sigma}_{\lambda}$:

(4.1.6)
$\mathcal{M}=(\widetilde{\Sigma}_{\lambda}\setminus\bigcup_{i}U_{i})\cup\bigcup_{i}(N_{i}\cup
M_{i}).$

The intersection number
$\mathrm{int}(\Phi_{\lambda}(\mathfrak{B}_{\lambda}),M_{i})$ equals $1$ for an
umbrella point and $-3$ for a triple value,
$\mathrm{int}(\Phi_{\lambda}(\mathfrak{B}_{\lambda}),N_{i})=0$ and
$\mathrm{int}(\Phi_{\lambda}(\mathfrak{B}_{\lambda}),(\widetilde{\Sigma}_{\lambda}\setminus\bigcup_{i}U_{i}))=0$,
hence

(4.1.7)
$L(\Phi|_{\mathfrak{S}})=\mathrm{int}(\Phi_{\lambda}(\mathfrak{B}_{\lambda}),\mathcal{M})=C(\Phi)-3T(\Phi).$

∎

###  4.2. Proofs of the lemmas

###### Proof of Lemma 4.1.4.

Consider the standard Whitney umbrella $\Phi(s,t)=(s^{2},st,t)$. The closure
of the set of double values of $\Phi$ is

$\Sigma=\\{y=z=0\\}=\\{(x,0,0):x\in\mathbb{C}\\}.$

This is the image of the double point curve
$D=\\{t=0\\}=\\{(s,0):s\in\mathbb{C}\\}$. The link of $D$ is $\gamma$ and
$\Phi(\gamma)=\Upsilon$. We compute the linking number
$\mbox{lk}_{S^{5}}(\widetilde{\Upsilon},\Phi(\mathfrak{S}))$ by defining
membranes bounded by $\widetilde{\Upsilon}$ and $\Phi(\mathfrak{S})$ and
taking their intersection multiplicity.

Let the membrane of $\widetilde{\Upsilon}$ be the shifted copy of the curve of
double values $\widetilde{\Sigma}$. More precisely, we push $\Sigma$ out from
$X=\mbox{im}(\Phi)$ along the pushforward $\mathrm{d}\Phi(v)$ of the vector
field $v(s,0)=\overline{\mbox{grad}(t)}(s,0)=(0,1)$ that is normal to $D$. The
differential of our germ is

$\mathrm{d}\Phi(s,t)=\begin{pmatrix}2s&0\\\ t&s\\\ 0&1\end{pmatrix}$

making the pushforward of the normal vector field

$\mathrm{d}\Phi(v(s,0))=\begin{pmatrix}2s&0\\\ 0&s\\\
0&1\end{pmatrix}\cdot\begin{pmatrix}0\\\ 1\end{pmatrix}=\begin{pmatrix}0\\\
s\\\ 1\end{pmatrix}.$

At any double point $(x,0,0)\in\Sigma$, we have two preimages

$\Phi^{-1}\\{(x,0,0)\\}=\\{(\sqrt{x},0),(-\sqrt{x},0)\\}.$

The pushforward of the normal vectors at these points are
$\mathrm{d}\Phi(v(\pm\sqrt{x},0))=(0,\pm\sqrt{x},1)$, hence the sum of the two
vectors provides the vector field

$w(x,0,0)=\begin{pmatrix}0\\\ \sqrt{x}\\\ 1\end{pmatrix}+\begin{pmatrix}0\\\
-\sqrt{x}\\\ 1\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 2\end{pmatrix}$

along $\Sigma\setminus\\{0\\}$. The vector field $w$ can be extended
continuously to the origin as it is constant. Therefore, when we push the
double point out by $w$, we obtain $(x,0,0)+\delta w(x,0,0)=(x,0,2\delta)$ for
a some $\delta\ll\epsilon$.

Thus the resulting membrane is

$\widetilde{\Sigma}=\\{(x,0,2\delta):x\in\mathbb{C}\\}\cap B_{\varepsilon}.$

On the other hand, let the membrane of $\Phi(\mathfrak{S})$ be simply the
image of the ball $\Phi(\mathfrak{B})=X\cap B_{\epsilon}$. That is

$\\{(x,y,z):xz^{2}=y^{2}\\}\cap B_{\varepsilon}.$

The two membranes $\widetilde{\Sigma}$ and $X\cap\mathfrak{B}$ intersect
transversely at $(0,0,2\delta)$. The sign of the intersection is positive as
the two membranes have the complex orientations. ∎

Figure 2. Pushing out $\Sigma$ with the sum of the two pushforward vector
fields.

###### Proof of Lemma 4.1.5.

Consider the standard triple value

$\Phi\left\\{\begin{array}[]{ccc}\Phi_{1}:(s_{1},t_{1})&\mapsto&(0,s_{1},t_{1})\\\
\Phi_{2}:(s_{2},t_{2})&\mapsto&(t_{2},0,s_{2})\\\
\Phi_{3}:(s_{3},t_{3})&\mapsto&(s_{3},t_{3},0)\\\ \end{array}\right.$

In this case, the set of double values is

$\Sigma=\\{(x,0,0)\\}\cup\\{(0,y,0)\\}\cup\\{(0,0,z)\\}$

with $x,y,z\in\mathbb{C}$. The curve $\Sigma$ has three components meeting at
the origin. Also, $\Sigma$ has preimages in each two-dimensional chart:

$D_{i}=\\{(s_{i},0)\\}\cup\\{(0,t_{i})\\}=\\{s_{i}t_{i}=0\\}$

for $i\in\\{1,2,3\\}$.

The membrane we pull over $X\cap S^{5}$ is again the whole of the image $X\cap
B^{6}$. Note that $X\cap S^{5}$ is diffeomorphic to the disjoint union of
three copies of $S^{3}$. Thus the membrane consists of three components
$X_{x}=\\{x=0\\}\cap B,X_{y}=\\{y=0\\}\cap B$, and $X_{z}=\\{z=0\\}\cap B$,
meeting at the origin.

Now, we describe the membrane for $\widetilde{\Upsilon}$. Let us see what
happens if we push out the double values using the sum of the normal vector
fields in the preimage – as before.

The normal vector fields corresponding to $D_{i}=\\{s_{i}t_{i}=0\\}$ are
$v_{i}(s_{i},t_{i})=\overline{\mbox{grad}(s_{i}t_{i})}=(\overline{t_{i}},\overline{s_{i}})$.
The differentials of the three map germs are

$\mathrm{d}\Phi_{1}(s_{1},t_{1})=\begin{pmatrix}0&0\\\ 1&0\\\
0&1\end{pmatrix},\quad\mathrm{d}\Phi_{2}(s_{2},t_{2})=\begin{pmatrix}0&1\\\
0&0\\\
1&0\end{pmatrix},\quad\mathrm{d}\Phi_{3}(s_{3},t_{3})=\begin{pmatrix}1&0\\\
0&1\\\ 0&0\end{pmatrix}.$

Let us show how our construction works on one component of $\Sigma$. We denote
$X_{y}\cap X_{z}=\\{(x,0,0):x\in\mathbb{C}\\}$ by $\Sigma_{x}$. A point of
$\Sigma_{x}$ has, again, two preimages:
$\Phi_{2}^{-1}(x,0,0)=(0,x)\in\mathbb{C}_{s_{2},t_{2}}$ and
$\Phi_{3}^{-1}(x,0,0)=(x,0)\in\mathbb{C}_{s_{3},t_{3}}$. The corresponding
normal vectors are $v_{2}(0,x)=(\overline{x},0)$ and
$v_{3}(x,0)=(0,\overline{x})$. When we push these vectors forward with the
respective differentials, we obtain

$\mathrm{d}\Phi_{2}(v_{2}(0,x))=\begin{pmatrix}0&1\\\ 0&0\\\
1&0\end{pmatrix}\cdot\begin{pmatrix}\overline{x}\\\
0\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ \overline{x}\end{pmatrix}$

and similarly $\mathrm{d}\Phi_{3}(v_{3}(0,x))=(0,\overline{x},0)^{T}$. Hence,
by pushing the initial point $(x,0,0)$ out with the sum of these, we reach
$(x,0,0)+\delta(0,0,\overline{x})+\delta(0,\overline{x},0)=(x,\delta\overline{x},\delta\overline{x})\in\widetilde{\Sigma}_{x}$.

Because of the cyclic symmetry of the presentation, the other two components
behave similarly, resulting in the membrane

$\widetilde{\Sigma}=\\{(x,\delta\overline{x},\delta\overline{x})\\}\cup\\{(\delta\overline{y},y,\delta\overline{y})\\}\cup\\{(\delta\overline{z},\delta\overline{z},z)\\}=:\widetilde{\Sigma}_{x}\cup\widetilde{\Sigma}_{y}\cup\widetilde{\Sigma}_{z}$

for some $x,y,z\in\mathbb{C}$ with $\widetilde{\Sigma}$ being in $B$. One
problem with this membrane is that each vector field vanishes at the origin
hence in the end we have not moved the point of $\Sigma$ at the origin. Thus
$\widetilde{\Sigma}$ meets $X$ only at the origin but with some multiplicity
that is somewhat difficult to count. Fortunately, each pair of components
$(X_{\alpha},\widetilde{\Sigma}_{\beta})$ intersect transversally. We only
need to compute the sign of each such intersection and sum them up.

Take $\widetilde{\Sigma}_{x}=\\{(x,\delta\overline{x},\delta\overline{x})\\}$
first. It intersects $X_{x}=\\{x=0\\}$ with positive sign, and the other two
with negative – as the corresponding coordinate functions are antiholomorphic.
The membranes $\widetilde{\Sigma}_{y}$ and $\widetilde{\Sigma}_{z}$ behave
similarly. We can summerize this in the formula

$\mathrm{int}_{0}\big{(}\widetilde{\Sigma}_{\alpha},X_{\beta}\big{)}=\left\\{\begin{array}[]{ll}+1&\text{if
}\alpha=\beta\\\ -1&\text{if }\alpha\neq\beta.\end{array}\right.$

Therefore the total intersection number is

$\mathrm{int}_{0}\big{(}\widetilde{\Sigma},X\big{)}=\sum_{\alpha,\beta\in\\{x,y,z\\}}\mathrm{int}_{0}\big{(}\widetilde{\Sigma}_{\alpha},X_{\beta}\big{)}=3\cdot
1+6\cdot(-1)=-3.$

∎

######  Remark 4.2.1.

Note that we could also move the components of $\widetilde{\Sigma}$ away from
the origin in order to see the nine points of intersection apart. A
perturbation of the form

$\Sigma^{\prime}=\\{(x-\varepsilon_{1},\delta(\overline{x-\varepsilon_{2}}),\delta(\overline{x-\varepsilon_{3}}))\\}\cup...$

with $|\varepsilon_{i}|\ll\varepsilon$ would do so. In turn, these
modifications would not change the topology of the membrane on the boundary of
$B$.

##  5\. Final remark about future plans

This article has a continuation in progress, in which the relation of $L(f)$
with the surgery coefficients of the Milnor fiber boundary of $({\rm
Im}(\Phi),0)\subset(\mathbb{C}^{3},0)$ – described in [18] – will be
explained. In that paper we will also provide a new direct proof for part (c)
of Proposition 3.1.5.

##  Appendix A Outline of related results

The aim of this section is to clarify the role of $L$ and another related
linking invariant in the study of generic $\mathcal{C}^{\infty}$ real maps and
immersions, and clear up the context of our result. We also clear up some
sensitive sign ambiguities related to the Ekholm–Szűcs Smale invariant
formula.

The other linking invariant $l$ is defined for real generic maps. While $L$
measures the linking of the double values of an immersion with the image of
it, $l$ measures the linking of the set of singular points in the target of a
generic map with the image of the map.

The Ekholm–Szűcs formula for the Smale invariant of an immersion uses both
linking invariant, $L$ of the immersion and $l$ of a singular Seifert surface
of the immersion. The original proof of our main formula (1.1.1) is based on
the Ekholm–Szűcs Smale invariant formula and the ‘holomorphic Smale invariant
formula’ of Némethi and the first author.

Invariants of generic maps Smale invariant formula using singular Seifert
surface $L$ of stable immersions $L=C-3T$ Holomorphic Smale invariant formula
$[$Ekholm–Szűcs$]$$[$Némethi, Pintér$]$ Figure 3. Mindmap for the original
proof of (1.1.1).

###  A.1. A linking invariant of real generic maps

The $\mathbb{Z}_{2}$ or integer valued invariant $l(f)$ is defined for real
generic maps $f:M^{2k}\to\mathbb{R}^{3k}$ of closed smooth manifolds $M^{2k}$.
It measures the linking of a pushout copy of the singular values with the
image of the map as follows. (See, for reference, [5, 4, 20].)

Such a map $f$ has (1) regular simple points, (2) double values with
transverse intersection of the regular branches, (3) triple values with
regular intersection of the regular branches and (4) singular values. The
dimension of the set of double values is $k$, and the triple values are
isolated. The set of singular values is a $k-1$ dimensional family of
generalized real Whitney umbrella points, whose local form is

(A.1.1)
$f_{\mbox{wh}}:(\mathbb{R}\times\mathbb{R}^{k},0)\to(\mathbb{R}\times\mathbb{R}^{k}\times\mathbb{R}^{k},0),$
(A.1.2) $f_{\mbox{wh}}(s,\underline{t})=(s^{2},s\underline{t},\underline{t}).$

The closure $\Delta(f)$ of the set of double values of $f$ is an immersed
manifold with boundary. $\Delta(f)$ has triple self intersection at the triple
values of $f$ and the boundary of $\Delta(f)$ is the set of the Whitney
umbrella points (singular values) $\Sigma(f)=\partial\Delta(f)$.

The invariant $l(f)$ is defined as the linking number

(A.1.3) $l(f)=\mbox{lk}_{\mathbb{R}^{3k}}(\Sigma^{\prime}(f),f(M^{3k}))$

of the copy $\Sigma^{\prime}(f)$ of $\Sigma(f)$ shifted slightly along the
outward normal field of $\Sigma(f)\subset\Delta(f)$ and the image $f(M^{3k})$
of $f$.

In general $l(f)$ and the number of triple values is defined only modulo 2 –
because the lack of orientation on $\Delta(f)$ – and these $\mathbb{Z}_{2}$
versions are denoted by $l_{2}(f)$ and $t_{2}(f)$ respectively. If $k$ is even
and $M^{2k}$ is oriented, then $l(f)$ is a well defined as an integer, [5]. In
these cases each triple value can be given a sign, and the sum of these signs
is the integer $t(f)$.

Ekholm and Szűcs expressed some characteristic numbers of $M^{2k}$ in terms of
$l$ and $t$. Namely, in [5] they proved the equality

(A.1.4)
$l_{2}(f)+t_{2}(f)=\overline{w}^{2}_{k}[M]+\overline{w}_{k-1}\overline{w}_{k+1}[M]$

in $\mathbb{Z}_{2}$, where the terms on the right hand side are products of
the normal Stiefel-Whitney classes of $M^{2k}$ evaluated on the fundamental
class $[M]$ of $M^{2k}$.

For $k=2n$ and $M^{2k}$ oriented, the equation of integers

(A.1.5) $3t(f)-3l(f)=\overline{p}_{n}[M]$

is proved in [4], where $\overline{p}_{n}[M]$ is the $n$-th normal Pontryagin
number of $M^{4n}$. By using Hirzebruch signature theorem, for $k=2$ one can
rewrite the formula (A.1.5) as

(A.1.6) $l(f)-t(f)=\sigma(M^{4}),$

where $\sigma(M^{4})$ is the signature of $M^{4}$, see [4].

The proofs of these formulas use methods similar to that of our proof. Namely,
each of them considers a set of certain type singularities of a map, and deals
with the pushout copy of it along a suitably defined normal vector field, then
counts the intersection point, see for example [4, Lemma 3]. When proving the
formula (A.1.4) in [5, Theorem 1], the set of double values of the map
$f:M^{2k}\to\mathbb{R}^{3k}$ is shifted slightly along a vector field, which
is defined as the sum of the two vectors coming from a suitable normal vector
field of the double point set in the source. At this rate it is even more
similar to the method we use to prove equation (1.1.1).

Despite the similarity in methods, none of the equations (A.1.4), (A.1.5) and
(A.1.6) can be directly applied for the setup of this article, that is for
holomorphic stabilizations $\Phi_{\lambda}$ of holomorphic germs
$\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$, for the following reasons.
First, the domain of $\Phi_{\lambda}$ is a 4-ball, which is not a closed
manifold, moreover it is topologically trivial. Second, the stabilization
$\Phi_{\lambda}$ is stable as a holomorphic map, but it is not stable (not
generic) in the $\mathcal{C}^{\infty}$-sense, considered as a map from
$\mathbb{R}^{4}$ to $\mathbb{R}^{6}$. Indeed, each isolated complex cross cap
point can be further deformed to obtain a stable real $\mathcal{C}^{\infty}$
map with a circle of generalized real cross cap points, see [17]. Also, in
contrast to the real case, the complex cross cap points are not boundary
points of the set of double values. Third, one could try to relate the above
results to the immersion on the boundary in our case. Then, however, the
dimensions do not match and these immersions do not have triple points or
singular points whatsoever. What is more, the Smale invariant formula (A.2.3)
hints that $t(f)$ and $l(f)$ should really be considered for the membranes and
not the boundary.

###  A.2. Smale invariant formulas

If $M^{4}$ is an oriented 4-manifold with boundary, the ‘defect’ of the
equation (A.1.6) provides information about the restriction of the map to the
boundary. In the simplest case, the manifold $M^{4}$, with boundary $\partial
M^{4}$ diffeomorphic with $S^{3}$, is mapped to the upper half space
$\mathbb{R}^{6}_{+}=\\{(x_{1},\dots,x_{6})\in\mathbb{R}^{6}\ |\ x_{6}>0\\}$
with a generic map

(A.2.1) $\hat{f}:M^{4}\to\mathbb{R}^{6}_{+},$

whose restriction is assumed to be a stable immersion

(A.2.2) $f:=\hat{f}|_{\partial M^{4}}:S^{3}\looparrowright\mathbb{R}^{5}.$

In this case $\hat{f}$ is referred as a singular Seifert surface of the
immersion $f$.

Recall that the immersions of $S^{3}$ to $\mathbb{R}^{5}$ are classified up to
regular homotopy by an integer valued invariant called Smale invariant and
denoted by $\Omega$. That is, two immersions
$f_{1},f_{2}:S^{3}\looparrowright\mathbb{R}^{5}$ are regular homotopic if and
only if $\Omega(f_{1})=\Omega(f_{2})$, and for every integer $n\in\mathbb{Z}$
there is an immersion $g:S^{3}\looparrowright\mathbb{R}^{5}$ with Smale
invariant $\Omega(g)=n$. The Smale invariant can be constructed in many
different ways, see for example [21, 7, 17, 19]. Eventually the Smale
invariant $\Omega(f)$ is constructed as an element of the homotopy group
$\pi_{3}(SO(5))$, which is isomorphic with the infinite cyclic group
$(\mathbb{Z},+)$.

Then by [4] the Smale invariant $\Omega(f)$ of a stable immersion
$f:S^{3}\looparrowright\mathbb{R}^{5}$ can be expressed with the invariants of
a singular Seifert surface $\hat{f}:M^{4}\to\mathbb{R}^{6}_{+}$ and $L$ as

(A.2.3) $\Omega(f)=\frac{1}{2}(3\sigma(M^{4})+3t(\hat{f})-3l(\hat{f})+L(f)).$

Several variants and generalizations of the Ekholm–Szűcs formula (A.2.3)
appeared in the literature, see [7, 20, 8, 6] or the brief summary of these
results in [19, Ch.2].

For immersions $\Phi|_{\mathfrak{S}}:\mathfrak{S}\cong S^{3}\looparrowright
S^{5}$ associated with finitely determined holomorphic germs
$\Phi:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ Némethi and the first author
[17] proved the ‘holomorphic Smale invariant formula’

(A.2.4) $\Omega(\Phi|_{\mathfrak{S}})=-C(\Phi).$

The proof of this formula is self-contained in the sense that it is
independent of the above results.

A singular Seifert surface for $\Phi|_{\mathfrak{S}}$ can be constructed from
a holomorphic stabilization $\Phi_{\lambda}$ of $\Phi$ by a canonical
$\mathcal{C}^{\infty}$ stabilization of the complex Whitney umbrella points.
In this way the Ekholm–Szűcs formula (A.2.3) can be applied. By comparing it
with the equation (A.2.4) and using calculations on concrete examples, [17]
proves the main theorem (1.1.1) of this article, namely

(A.2.5) $L(\Phi|_{\mathfrak{S}})=C(\Phi)-3T(\Phi).$

The evolution of these results is summed up by Figure 3.

However, the proof of each Smale invariant formula is rather complicated, and
the identification of the signs of the terms are widely nontrivial (see the
next paragraph). Furthermore the correspondence (1.1.1) becomes important in
the proof of the topological invariance of $C$ and $T$. This was the
motivation to publish a new direct proof for (1.1.1), which does not use any
of the above results – although the techniques are similar to those ones used
in their proofs. An additional benefit of our proof is the simple
identification of the sign: (1.1.1) is sign correct with the $L_{1}$ version
of $L$. This fact has further consequences for the singular Seifert surface
formula (A.2.3) as explained in the next paragraph.

###  A.3. Remarks on sign and orientation

This paragraph is a brief summary of the issues related to the signs in the
Smale invariant formulas. We unravel an imprecision in [4] and [20]: although
the linking invariant $L$ is defined in these articles using the construction
denoted by $L_{2}$ in Subsection 3.1, the Smale invariant formula (A.2.3) is
satisfied by using $L=L_{1}$.

The Smale invariant does not have a canonical sign, since by default it is an
element of the group $\pi_{3}(SO(5))\cong(\mathbb{Z},+)$. To identify this
group with $\mathbb{Z}$, one has to fix a generator in $\pi_{3}(SO(5))$ and
declare it to be $+1$. That was done in [17], and the formula (A.2.4) is sign-
correct with that fixed generator. In other words it is proved that the Smale
invariant of the immersion associated with the complex Whitney umbrella is
$-1$ times the fixed generator.

The formula (A.2.3) is proved in [4] without considering the sign of the Smale
invariant. More precisely, they proved that the right hand side of the formula
is a complete regular homotopy invariant, therefore it must agree with the
Smale invariant up to sign. Nevertheless it is shown in [17] that the foruma
(A.2.3) is correct with the fixed generator of $\pi_{3}(SO(5))$.

However the sign of $L$ is not specified directly in [17, 19]. It is chosen to
satisfy the Ekholm–Szűcs formula (A.2.3) with this choice. For example, the
invariant $L(\Phi|_{\mathfrak{S}})$ of the complex Whitney umbrella
$\Phi(s,t)=(s^{2},st,t)$ is computed in [17, 10.1.] up to sign by using the
‘$L_{2}$’ construction (see also [19, 3.7.1]), resulting
$L(\Phi|_{\mathfrak{S}})=\pm 1$. Using the sign convention adapted to the
formula (A.2.3), $L(\Phi|_{\mathfrak{S}})$ of the complex Whitney umbrella is
declared to be $+1$, and $L=C-3T$ is concluded with this sign convention.

Now, from the proof of Lemma 4.1.4 it is clear that
$L_{1}(\Phi|_{\mathfrak{S}})=+1$ for the Whitney umbrella, hence
$L_{2}(\Phi|_{\mathfrak{S}})=-1$ by part (c) of Proposition 3.1.5. Therefore
to make the Ekholm–Szűcs Smale invariant formula (A.2.3) correct, $L$ has to
defined to be $L_{1}$, in contrast to the definitions given in [4] and [20].
Note that by changing the sign of $L$ in the formula not only the sign of the
right hand side changes, but the absolute value changes as well.

On the other hand, in the calculation of $L_{2}(\Phi|_{\mathfrak{S}})$ of the
complex Whitney umbrella in [17, 10.1.] the sign of the intersection point can
be determined directly. Both membranes ($\Phi(\mathfrak{B})$ and $H$ in [17]
and [19]) has complex (but not holomorphic) parametrization. These
parametrizations induce the correct orientations in the sense that the induced
orientation on the boundary agrees with the original orientation of the
boundary. A direct calculation of the determinant shows that the intersection
point has negative sign. Hence $L_{2}(\Phi|_{\mathfrak{S}})=-1$ can be
discovered directly, which is equal to $-L_{1}(\Phi|_{\mathfrak{S}})$
according to Proposition 3.1.5.

######  Remark A.3.1.

By default, the orientation induced on the boundary of an oriented manifold
depends on a choice of a convention, called ‘boundary convention’, for example
‘outward normal first’. Although, at first sight, the boundary convention
seems to play a key role in the identification of the signs of the Smale
invariant formulas and $L$, this is not the case.

The correct sign of the formulas (A.2.3) and (A.2.4) are independent of the
choice of the boundary convention. Briefly speaking its reason is that in the
construction of the Smale invariant $S^{3}$ is considered as the boundary of
the 4-ball in $\mathbb{R}^{4}$. By changing the boundary convention, the
orientation of the boundary of the singular Seifert surface changes, as well
as the orientation of $S^{3}=\partial B^{4}$ in the construction of the Smale
invariant, but the value of the Smale invariant and the right hand side of the
formulas remain the same. See [17] or [19, Ch.3] for details.

The invariant $L$ of finitely determined holomorphic germs $\Phi$ is also
independent of the choice of the boundary convention. Recall that
$L(\Phi|_{\mathfrak{S}})$ is defined as the intersection number of two
oriented ‘membranes’ in $B^{6}$ whose boundaries are $\Phi(\mathfrak{S})$ and
$\widetilde{\Upsilon}=\partial\widetilde{\Sigma}$ respectively. Although
$\Phi(\mathfrak{S})$ and $\widetilde{\Upsilon}=\partial\widetilde{\Sigma}$ are
originally oriented as the boundaries of $\Phi(\mathfrak{B})$ and
$\widetilde{\Sigma}$ after choosing a boundary convention, all in all, the
correct orientations of the membranes do not depend on the choice of the
boundary convention. Indeed, the correct orientation means that the membrane
induces the same orientation on the boundary as the original membrane,
whichever boundary convention is used. Cf. [17, 19].

## Funding and competing interests

### Declarations

GP has been supported by the National Research Development and Innovation
Office (NKFIH) through the OTKA Grant FK 132146.

AS was partially supported by the Élvonal (Frontier) grant KKP126683 of the
NKFIH, and Central European University.

The authors declare they have no financial interests.

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