Cross-cap singularities counted with sign
A method for computing the algebraic number of cross-cap singularities for mapping from m-dimensional compact manifold with boundary M ⊂ Rᵐ into R²ᵐ⁻¹, m is odd, is presented. As an application, the intersection number of an immersion g : Sᵐ⁻¹ (r) → R²ᵐ⁻² is described as the algebraic number of cros...
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irk-123456789-1883612023-02-27T01:27:46Z Cross-cap singularities counted with sign Krzyzanowska, I. A method for computing the algebraic number of cross-cap singularities for mapping from m-dimensional compact manifold with boundary M ⊂ Rᵐ into R²ᵐ⁻¹, m is odd, is presented. As an application, the intersection number of an immersion g : Sᵐ⁻¹ (r) → R²ᵐ⁻² is described as the algebraic number of cross-caps of a mapping naturally associated with g. 2018 Article Cross-cap singularities counted with sign / I. Krzyzanowska // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 257–268. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC: 14P25, 57R45, 57R42, 12Y05 http://dspace.nbuv.gov.ua/handle/123456789/188361 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A method for computing the algebraic number of cross-cap singularities for mapping from m-dimensional compact manifold with boundary M ⊂ Rᵐ into R²ᵐ⁻¹, m is odd, is presented. As an application, the intersection number of an immersion g : Sᵐ⁻¹ (r) → R²ᵐ⁻² is described as the algebraic number of cross-caps of a mapping naturally associated with g. |
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Krzyzanowska, I. Cross-cap singularities counted with sign Algebra and Discrete Mathematics |
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Krzyzanowska, I. |
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Cross-cap singularities counted with sign |
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Cross-cap singularities counted with sign |
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Cross-cap singularities counted with sign |
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Cross-cap singularities counted with sign |
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cross-cap singularities counted with sign |
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Cross-cap singularities counted with sign / I. Krzyzanowska // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 257–268. — Бібліогр.: 11 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT krzyzanowskai crosscapsingularitiescountedwithsign |
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2025-07-16T10:23:14Z |
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“adm-n2” — 2018/7/24 — 22:32 — page 257 — #95
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 2, pp. 257–268
c© Journal “Algebra and Discrete Mathematics”
Cross-cap singularities counted with sign
Iwona Krzyżanowska
Communicated by I. Protasov
Abstract. A method for computing the algebraic number
of cross-cap singularities for mapping from m-dimensional com-
pact manifold with boundary M ⊂ R
m into R
2m−1, m is odd, is
presented. As an application, the intersection number of an immer-
sion g : Sm−1(r) → R
2m−2 is described as the algebraic number of
cross-caps of a mapping naturally associated with g.
Introduction
Mappings from the m-dimensional, smooth, orientable manifold M
into R
2m−1 are natural object of study. In [9],Whitney described typical
mappings from M into R
2m−1. Those mappings have only isolated critical
points, called cross-caps (or Whintey umbrellas).
According to [1, Theorem 4.6], [11, Lemma 2], a mapping M → R
2m−1
has a cross-cap at p ∈ M , if and only if in the local coordinate system
near p this mapping has the form
(x1, . . . , xm) 7→ (x21, x2, . . . , xm, x1x2, . . . , x1xm).
In [11], for m odd, Whitney presented a method to associate a sign
with a cross-cap. Put ζ(f) to be an algebraic sum of cross-caps of f : M →
R
2m−1, where M is m-dimensional compact orientable manifold. Then
according to Whitney, [11, Theorem 3], ζ(f) = 0, if M is closed. If M
2010 MSC: 14P25, 57R45, 57R42, 12Y05.
Key words and phrases: cross-cap, immersion, Stiefel manifold, intersection
number, signature.
“adm-n2” — 2018/7/24 — 22:32 — page 258 — #96
258 Cross-cap singularities counted with sign
has a boundary, then following Whitney, [11, Theorem 4], for a homotopy
ft : M → R
2m−1 regular in some open neighbourhood of ∂M , if the only
singular points of f0 and f1 are cross-caps, then ζ(f0) = ζ(f1). Moreover
arbitrarily close to any mapping h : M → R
2m−1, there is mapping regular
near boundary, with only cross-caps as singular points (see [11]). In the
case where m even, it is impossible to associate sign with cross-cap in the
same way as in the odd case, but if m is even, it is enough to consider
number of cross-caps mod2, to get similar results (see [11]).
In [6], the authors studied a mapping α from a compact and oriented
(n− k)-manifold M into the Stiefel manifold Ṽk(R
n), for n− k even. They
constructed a mapping α̃ : Sk−1 ×M → R
n \ {0} associated with α, and
defined Λ(α) as half of topological degree of α̃. In case M = Sn−k, they
showed that Λ(α) corresponds with the class of α in πn−kṼk(R
n) ≃ Z.
According to [6], in the case where M ⊂ R
n−k+1 is an algebraic hypersur-
face and α is polynomial, with some additional assumptions concerning
M and α, Λ(α) can be presented as a sum of signatures of two quadratic
forms defined on R[x1, . . . , xn−k+1]. And so, easily computed.
In this paper we prove that in the case where m is odd, for f :
(M,∂M) → R
2m−1, where M ⊂ R
m, ζ(f) can be expressed as Λ(α), for
some α associated with f . And so, with some additional assumptions con-
cerning M and f , ζ(f) can be easily computed for polynomial mapping f .
Moreover we present a method that can be used to check effectively that
f has only cross-caps as singular points. In case when m is even, the
effective method to compute number of cross-caps modulo 2 is presented
in [5].
Take a smooth map g : R
m → R
2m−2, let us assume that g|Sm−1
is an immersion. In [10], Whitney introduced the intersection number
I(g|Sm−1) of immersion g|Sm−1 . In this paper we show that I(g|Sm−1), can
be presented as an algebraic sum of cross-caps of the mapping (ω, g)|B̄m,
where ω is sum of squares of coordinates.
Take f : (Rm, 0) → R
2m−1 with cross-cap at 0. In [3], Ikegami and
Saeki defined the sign of a cross-cap singularity for mapping f as the
intersection number of immersion f |S : S = f−1(S2m−2(ǫ)) → S2m−2(ǫ),
for ǫ small enough. It is easy to see that this definition complies with
Whitney definition from [11]. In [3], the authors showed that for generic
map (in sense of [3]) g : (Rm, 0) → R
2m−1, the number of cross-caps
appearing in a C∞ stable perturbation of g, counted with signs, is an
invariant of the topological A+-equivalence class of g, and is equal to the
intersection number of g|S : S = g−1(S2m−2(ǫ)) → S2m−2(ǫ). Using our
methods, this number can be easily computed for polynomial mappings.
“adm-n2” — 2018/7/24 — 22:32 — page 259 — #97
I . Krzyżanowska 259
We use notation Sn(r), Bn(r), B̄n(r) for sphere, open ball, closed
ball (resp.) centred at the origin of radios r and dimension n. If we omit
symbol r, we assume that r = 1.
1. Cross-cap singularities
Let M,N be smooth manifolds. Take a smooth mapping f : M → N .
Lemma 1. Let W be a submanifold of N . Take p ∈ M such that f(p) ∈
W . Let us assume that there is a neighbourhood U of f(p) in N and a
smooth mapping φ : U → R
s such that rankDφ(f(p)) = k = codimW and
W ∩ U = φ−1(0). Then f ⋔ W at p if and only if rankD(φ ◦ f)(p) = k.
Proof. Of course KerDφ(f(p)) = Tf(p)W , and so we get dimTf(p)N =
dimKerDφ(f(p)) + k. Then:
f ⋔ W at p ⇐⇒ Tf(p)N = Tf(p)W +Df(p)TpM
⇐⇒ Tf(p)N = KerDφ(f(p)) +Df(p)TpM.
The above equality holds if and only if there exist vectors v1, . . . , vk in
Df(p)TpM , such that any nontrivial combination of v1, . . . , vk is outside
the KerDφ(f(p)) and so rankDφ(f(p)) [v1 . . . vk] = k. We get that f ⋔ W
at p if and only if rankD(φ ◦ f)(p) = k.
By j1f we mean the canonical mapping associated with f , from M
into the spaces of 1-jets J1(M,N). We say that f : M → N is 1-generic,
if j1f ⋔ Sr, for r > 0, where Sr = {σ ∈ J1(M,N) | corankσ = r}. Put
Sr(f) = {x ∈ M | corankDf(p) = r} = (j1f)−1(Sr).
Let us assume that M and N are manifolds of dimension m and
2m−1 respectively. In this case (see [1]) codimSr = r2+ r(m−1), and so
codimS1 = m and codimSr > m, for r > 2. So f is 1-generic if and only
if f ⋔ S1 and Sr(f) = ∅ for r > 2. The typical singularity for mapping
f : M → N is a cross-cap singularity. Following [9], [11], [1] we present
equivalent definitions of a cross-cap.
Definition 1. A point p is a cross-cap of a mapping f : M → N if the
following equivalent conditions are fulfilled:
1) p ∈ S1(f) and j1f ⋔ S1 at p;
2) there are coordinate systems near p and f(p), such that
∂f
∂x1
(p) = 0 (1)
“adm-n2” — 2018/7/24 — 22:32 — page 260 — #98
260 Cross-cap singularities counted with sign
and vectors
∂2f
∂x21
(p),
∂f
∂x2
(p), . . . ,
∂f
∂xm
(p),
∂2f
∂x1∂x2
(p), . . . ,
∂2f
∂x1∂xm
(p) (2)
are linearly independent;
3) there are coordinate systems near p and f(p) such that the mapping
f has the form
(x1, . . . , xm) 7→ (x21, x2, . . . , xm, x1x2, . . . , x1xm).
According to [9, Section 2], if p is a cross-cap singularity and (1) holds,
then vectors (2) are linearly independent.
Take f = (f1, . . . , f2m−1) : R
m → R
2m−1. Put µ : Rm → R
s such that
µ(x) is given by all the m-minors of Df(x). Of course s =
(
2m−1
m
)
.
Lemma 2. A point p ∈ R
m is a cross-cap singularity of f if and only if
rankDf(p) = m− 1 and rankDµ(p) = m.
Proof. A point p is a cross-cap singularity if and only if p ∈ S1(f) and
j1f ⋔ S1 at p. Note that p ∈ S1(f) if and only if rankDf(p) = m− 1.
Of course J1(Rm,R2m−1) ∼= R
m × R
2m−1 × M(2m − 1,m), where
M(2m− 1,m) is a space of real matrices of dimension (2m− 1)×m. Take
an open neighbourhood U of j1f(p) in J1(Rm,R2m−1), and a mapping
φ : U → R
s,
where φ(x, y, [aij ]) is given by all m-minors of [aij ]. We may assume that
det
∂(f1, . . . , fm−1)
∂(x1, . . . , xm−1)
(p) 6= 0.
Put A = [aij ]16i,j6m−1 the submatrix of [aij ], then for U small enough,
detA 6= 0. Let Mi be the determinant of submatrix of [aij ] composed of
first m− 1 rows and row number (m+ i− 1), for i = 1, . . . ,m. Then
Mi = (−1)2m−1+i detA · am+i−1,m + bi,
for i = 1, . . . ,m and bi does not depend on amm, . . . , a2m−1,m, and so
rank
∂(M1, . . . ,Mm)
∂(am,m, . . . , a2m−1,m)
= m.
We get that
rankDφ(j1f(p)) > m.
“adm-n2” — 2018/7/24 — 22:32 — page 261 — #99
I . Krzyżanowska 261
Let us recall that codimS1 = m. We can choose U small enough such that
φ−1(0) = U ∩ S1.
So we get that rankDφ(j1f(p)) = codimS1 = m. Of course φ ◦ j1f = µ
in the small neighbourhood of p. According to Lemma 1, j1f ⋔ S1 at p if
and only if rankDµ(p) = m.
2. Algebraic sum of cross-cap singularities
First we want to recall some well-known facts concerning the topological
degree. Let (N, ∂N) be n-dimensional compact oriented manifold with
boundary. For smooth mapping f : N → R
n such that f |∂N : ∂N →
R
n \ {0}, by deg f |∂N or deg(f,N, 0) we denote the topological degree of
mapping f/|f | : ∂N → Sn−1. Note that if f−1(0) is a finite set then
deg f |∂N =
∑
p∈f−1(0)
degp f,
where degp f stands for the local topological degree of f at p (see [8]).
Let M be a m-dimensional manifold and m be odd. Take a smooth
mapping f : M → R
2m−1 and let p ∈ M be a cross-cap of f . According
to [11], p is called positive (negative) if the vectors (2) determine the
negative (positive) orientation of R2m−1. According to [11, Lemma 3], this
definition does not depend on choosing the coordinate system on M .
Let us assume, that f : Rm → R
2m−1 is a smooth mapping such that
0 is a cross-cap of f . Of course it is an isolated critical point of f . Denote
by vi the ith column of Df , for i = 1, . . . ,m. There exists r > 0 such that
v1(x), . . . , vm(x) are linearly independent for x ∈ B̄m(r) \ {0}. Following
[6] we can define
α̃(β, x) = β1v1(x) + . . .+ βmvm(x)
= Df(x)(β) : Sm−1 × B̄m(r) → R
2m−1.
Then the topological degree of the mapping
α̃|Sm−1×Sm−1(r) : S
m−1 × Sm−1(r) → R
2m−1 \ {0}
is well defined. By [6, Proposition 2.4], deg(α̃|Sm−1×Sm−1(r)) is even.
“adm-n2” — 2018/7/24 — 22:32 — page 262 — #100
262 Cross-cap singularities counted with sign
Theorem 1. Let m be odd. If 0 is a cross-cap of a mapping f : Rm →
R
2m−1, then it is positive if and only if 1
2 deg(α̃|Sm−1×Sm−1(r)) = −1, and
so it is negative if and only if 1
2 deg(α̃|Sm−1×Sm−1(r)) = +1.
Proof. We can find linear coordinate system φ : Rm → R
m, such that
φ(0) = 0 and f ◦ φ fulfills condition (1) at 0. Denote by A the matrix of
φ. Let w1, . . . , wm denote columns of D(f ◦ φ). Then w1(0) = 0 and since
0 is a cross-cap then vectors
∂w1
∂x1
(0), w2(0), . . . , wm(0),
∂w1
∂x2
(0), . . . ,
∂w1
∂xm
(0) (3)
are linearly independent. Put γ̃(β, x) = (β1w1(x) + . . . + βmwm(x)) :
Sm−1 × B̄m(r) → R
2m−1. We can assume that r is such that γ̃ 6= 0 on
Sm−1 × B̄m(r) \ {0}. Let us see that
γ̃(β, x) = D(f ◦ φ)(x) ·
β1
...
βm
= Df(φ(x)) ·A ·
β1
...
βm
= Df(φ(x)) ·
φ1(β)
...
φm(β)
.
So γ̃ = α̃ ◦ (φ × φ). It is easy to see that φ × φ preserve the orienta-
tion of Sm−1 × Sm−1(r). We can assume that r > 0 is so small, that
deg(α̃|Sm−1×Sm−1(r)) = deg(α̃|φ(Sm−1)×φ(Sm−1(r))). So we get that
deg(γ̃|Sm−1×Sm−1(r)) = deg(α̃|φ(Sm−1)×φ(Sm−1(r))) deg(φ× φ) =
= deg(α̃|Sm−1×Sm−1(r)).
Since f ◦ φ fulfils (1), vectors w2, . . . , wm are independent on B̄m(r).
Let us see that γ̃(β, x) = 0 on Sm−1 × B̄m(r) if and only if x = 0 and
β = (±1, 0, . . . , 0). So deg(γ̃|Sm−1×Sm−1(r)) is a sum of local topological
degrees of γ̃ at (1, 0, . . . , 0; 0, . . . , 0) and at (−1, 0, . . . , 0; 0, . . . , 0).
Near the point (1, 0, . . . , 0; 0, . . . , 0) the well-oriented parametrisation
of Sm−1 × B̄m(r) is given by
(β2, . . . , βm;x) = (
√
1− β2
2 − . . .− β2
m, β2, . . . , βm;x).
And then the derivative matrix of γ̃ at (1, 0, . . . , 0; 0, . . . , 0) has a form
A1 =
[
w2(0) . . . wm(0) ∂w1
∂x1
(0) . . . ∂w1
∂xm
(0)
]
.
“adm-n2” — 2018/7/24 — 22:32 — page 263 — #101
I . Krzyżanowska 263
Near (−1, 0, . . . , 0; 0, . . . , 0) the well-oriented parametrisation of Sm−1 ×
B̄m(r) is given by
(β2, . . . , βm;x) = (−
√
1− β2
2 − . . .− β2
m,−β2, . . . , βm;x).
And then the derivative matrix of γ̃ at (−1, 0, . . . , 0; 0, . . . , 0) has a form
A2 =
[
−w2(0) . . . wm(0) −∂w1
∂x1
(0) . . . − ∂w1
∂xm
(0)
]
.
Let us recall that m is odd. System of vectors (3) is independent, so 0
is a regular value of γ̃, and
1
2
deg(γ̃|Sm−1×Sm−1(r)) =
1
2
(sgn detA1 + sgn detA2) = sgn detA1.
Moreover 0 is a positive cross-cap if and only if vectors (3) determine
negative orientation of a R
2m−1, i. e. if and only if 1
2 deg(α̃|Sm−1×Sm−1(r)) =
−1.
Let U ⊂ R
m be an open bounded set and f : U → R
2m−1 be smooth.
We say that f is generic if only critical points of f are cross-caps and f is
regular in the neighborhood of ∂U . Let us denote by ζ(f) the algebraic
sum of cross-caps of f . Then using Theorem 1 we get the following.
Proposition 1. Let U ⊂ R
m, (m is odd), be a bounded m-dimensional
manifold such that U is an m-dimensional manifold with a boundary.
For f : U ⊂ R
m → R
2m−1 generic, ζ(f) = −1
2 deg(α̃), where α̃(β, x) =
Df(x)(β) : Sm−1 × ∂U → R
2m−1 \ {0}.
Proposition 2. Let U ⊂ R
m, (m is odd), be a bounded m-dimensional
manifold such that U is an m-dimensional manifold with a boundary. Take
h : U ⊂ R
m → R
2m−1 a smooth mapping such that h is regular in a
neighborhood of ∂U . Then for every generic f : U ⊂ R
m → R
2m−1 close
enough to h in C1-topology we have, ζ(f) = −1
2 deg(α̃), where α̃(β, x) =
Dh(x)(β) : Sm−1 × ∂U → R
2m−1 \ {0}.
3. Examples
To compute some examples we want first to recall the theory presented
in [6].
Take α = (α1, . . . , αk) : R
n−k+1 → M(n, k) a polynomial mapping,
n− k even, where M(n, k) is a space of real matrices of dimension n× k.
“adm-n2” — 2018/7/24 — 22:32 — page 264 — #102
264 Cross-cap singularities counted with sign
By [aij(x)], 1 6 i 6 n, 1 6 j 6 k, we denote the matrix given by α(x) (i.e.
αj(x) stands in the jth column). Then one can define α̃ : Rk ×R
n−k+1 →
R
n as
α̃(β, x) = β1α1(x) + . . .+ βkαk(x) = [aij(x)]
β1
...
βk
.
Let I be the ideal in R[x1, . . . , xn−k+1] generated by all k × k minors of
[aij(x)], and V (I) = {x ∈ R
n−k+1 | h(x) = 0 for all h ∈ I}.
Take
m(x) = det
a12(x) . . . a1k(x)
ak−1,2(x) . . . ak−1,k(x)
.
For k 6 i 6 n, we define
∆i(x) = det
a11(x) . . . a1k(x)
. . .
ak−1,1(x) . . . ak−1,k(x)
ai1(x) . . . aik(x)
.
Put A = R[x1, . . . , xn−k+1]/I. Let us assume that dimA < ∞, so
that V (I) is finite. For h ∈ A, we denote by T (h) the trace of the linear
endomorphism A ∋ a 7→ h · a ∈ A. Then T : A → R is a linear functional.
Let u ∈ R[x1, . . . , xn−k+1]. Assume that U = {x | u(x) > 0} is
bounded and ∇u(x) 6= 0 at each x ∈ u−1(0) = ∂U . Then U is a compact
manifold with boundary, and dimU = n− k + 1.
Put δ = ∂(∆k, . . . ,∆n)/∂(x1, . . . , xn−k+1). With u and δ we associate
quadratic forms Θδ, Θu·δ : A → R given by Θδ(a) = T (δ · a2) and
Θu·δ(a) = T (u · δ · a2).
Theorem 2. [6, Theorem 3.3] If n − k is even, α = (α1, . . . , αk) :
R
n−k+1 → M(n, k) is a polynomial mapping such that dimA < ∞,
I + 〈m〉 = R[x1, . . . , xn−k+1] and quadratic forms Θδ, Θu·δ : A → R
are non-degenerate, then the restricted mapping α|∂U goes into Ṽk(R
n)
and
Λ(α|∂U ) =
1
2
deg(α̃|Sk−1×∂U ) =
1
2
(signatureΘδ + signatureΘu·δ),
where α̃(β, x) = β1α1(x) + . . .+ βkαk(x).
Using the theory presented in [6], particularly [6, Theorem 3.3], and
computer system Singular ([2]), one can apply the results from Sections
1 and 2 to compute algebraic sum of cross-caps for polynomial mappings.
“adm-n2” — 2018/7/24 — 22:32 — page 265 — #103
I . Krzyżanowska 265
Example 1. Let us take f : R3 → R
5 given by
f(x, y, z) = (12y2 + z, 6x2 + y2 + 6y, 18xy + 13y2 + 9x,
8x2z + 10xz2 + 5x2 + 3xz, x2y + 4xyz + yz + 4z2).
Applying Lemma 2 and using Singular one can check that f is
1-generic. Moreover, according to Proposition 1 and [6], one can check
that
ζ(f |B̄3(
√
3)) = 2, ζ(f |B̄3(10)) = 1.
We can also check that f has 11 cross-caps in R
3, 6 of them are positive,
5 negative.
Example 2. Take f : R5 → R
9 given by
f(s, t, x, y, z) = (y, z, t, 20x2+17sz+x, 13sy+13sz+5t, 25st+4x2+28z,
3x2 + 19yz + 22s, 11ts2 + 8t2z + xz, 27txy + 9sxz + 20st).
One may check that f is 1-generic, has 3 cross-caps in R
5 and
ζ(f |B̄3(1/10)) = 0, ζ(f |B̄3(2)) = −1, ζ(f |B̄3(1000)) = 1.
4. Intersection number of immersions
Take n-dimensional, compact, oriented manifold N and immersion
g : N → R
2n. As in [10] we say that an immersion g : N → R
2n has a
regular self-intersection at the point g(p) = g(q) if
Dg(p)TpN +Dg(q)TqN = R
2n.
An immersion g : N → R
2n is called completely regular if it has only
regular self-intersections and no triple points.
Assume that n is even. Let g : N → R
2n be a completely regular
immersion having a regular self-intersection at the point g(p) = g(q).
Let u1, . . . , un ∈ TpN , v1, . . . , vn ∈ TqN be sets of well-oriented, inde-
pendent vectors in respective tangent spaces of N . Then the vectors
Dg(p)u1, . . . , Dg(p)un, Dg(q)v1, . . . , Dg(q)vn form a basis in R
2n. As in
[10] we will say that the self-intersection at the point g(p) = g(q) is positive
or negative according to whether this basis determines the positive or
negative orientation of R2n.
Following [10], the intersection number I(g) of a completely regular
immersion g is the algebraic sum of its self-intersections. For any immersion
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266 Cross-cap singularities counted with sign
g : N → R
2n the intersection number I(g) is defined as the intersection
number of a completely regular immersion g̃, regularly homotopic to g
(homotopy by immersions). For other equivalent description of I(g) see
[7], [4].
As in previous Sections we assume that m is odd. Take a smooth map
g = (g1, . . . , g2m−2) : R
m → R
2m−2. Denote by ω = x21 + . . .+ x2m. Then
Sm−1(r) = {x| ω(x) = r2}. According to [4, Lemma 18], g|Sm−1(r) is an
immersion if and only if
rank
2x1 . . . 2xm
∂g1
∂x1
(x) . . . ∂g1
∂xm
(x)
. . .
∂g2m−2
∂x1
(x) . . . ∂g2m−2
∂xm
(x)
= m,
for x ∈ Sm−1(r).
Take 0 < r1 < r2, such that g|Sm−1(r1) and g|Sm−1(r2) are immersions.
Denote by P = {x| r21 6 w(x) 6 r22}. Then P is an m-dimensional oriented
manifold with boundary. Then (ω, g) : Rm → R
2m−1 is a regular map in
the neighbourhood of ∂P . Let us define α̃ : Sm−1 × P → R
2m−1 as
α̃(β, x) =
2x1 . . . 2xm
∂g1
∂x1
(x) . . . ∂g1
∂xm
(x)
. . .
∂g2m−2
∂x1
(x) . . . ∂g2m−2
∂xm
(x)
β1
...
βm
Proposition 3. Let us assume that g|Sm−1(r1) and g|Sm−1(r2) are immer-
sions, then
I(g|Sm−1(r2))− I(g|Sm−1(r1)) = ζ((ω, g)|P ).
Proof. Let us recall that m is odd. Then
deg(α̃|Sm−1×∂P ) = deg(α̃|Sm−1×Sm−1(r2))− deg(α̃|Sm−1×Sm−1(r1)).
According to [6, Theorem 4.2], we get that
deg(α̃|Sm−1×Sm−1(ri)) = I(g|Sm−1(ri)),
for i = 1, 2. Then applying Proposition 2 we get that ζ((ω, g)|P ) =
−1
2 deg(α̃|Sm−1×∂P ). And so
ζ((ω, g)|P ) = I(g|Sm−1(r2))− I(g|Sm−1(r1)).
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I . Krzyżanowska 267
Corollary 1. If g|Sm−1(r) is an immersion, then
I(g|Sm−1(r)) = ζ((ω, g)|B̄m(r)).
Remark 1. If the only singular points of (ω, g)|B̄m(r) are cross-caps, then
the intersection number of an immersion g|Sm−1(r) is equal to the alge-
braic sum of cross-caps of (ω, g)|B̄m(r). Also, in generic case,the difference
between intersection numbers of immersions g|Sm−1(r1) and g|Sm−1(r2), is
equal to the algebraic sum of cross-caps of (ω, g) appearing in P .
Remark 2. If (ω, g) has finite number of singular points, and all of them
are cross-caps, then for any R > 0 big enough, g|Sm−1(R) is an immersion
with the same intersection number equal to the algebraic sum of cross-caps
of (ω, g).
Example 3. Take g : R3 → R
4 given by
g = (−3y2 + 5yz − x+ 2,−4x2 + z2 + 9y − 6z + 5,
4x2z − 2x2 + 2xy − y − 3, 3y2z + xy − 4yz + 4x− 5y − 5),
and ω = x2 + y2 + z2. In the same way as in Section 3, one may check
that the only singular points of (ω, g) are cros-s-caps, moreover (ω, g) has
8 cross-caps, 5 of them are positive and 3 negative. According to previous
results g|S2(r) is an immersion for all r > 0, except at most 8 values of r.
And if g|S2(r) is an immersion, then
−3 6 I(g|S2(r)) 6 5.
Moreover for R > 0 big enough g|S2(R) is an immersion with
I(g|S2(R)) = 2.
References
[1] M. Golubitsky, V. Guillemin, Stable mappings and their singularities, 1973 by
Springer-Verlag New York.
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Algebra System for Polynomial Computations.
[3] K. Ikegami, O. Saeki, Cobordism of Morse maps and its applications to map germs,
Math. Proc. Cambridge Philos. Soc. 147, no. 1, 2009, pp. 235–254.
[4] I. Karolkiewicz, A. Nowel, Z. Szafraniec, An algebraic formula for the intersection
number of a polynomial immersion, J. Pure Appl. Algebra vol. 214, no. 3, 2010,
pp. 269–280.
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268 Cross-cap singularities counted with sign
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Contact information
I. Krzyżanowska Institute of Mathematics, University of Gdańsk,
80-952 Gdańsk, Wita Stwosza 57, Poland
E-Mail(s): iwona.krzyzanowska@mat.ug.edu.pl
Received by the editors: 22.09.2015
and in final form 02.03.2018.
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