SURFACES WITH ASSIGNED APPARENT CONTOUR

DOMENICO LUMINATI

Introduction

In all the paper we shall denote by S^{1} the unitary circle in Euclidean plane ℝ^{2} and by ∐_{h=1}^{k} S^{1} the disjoint union of k identical copies of the circle; all surfaces and maps will be supposed to be smooth (i.e. of class C^{∞}). Furthermore we shall often use “cut and paste” techniques, that work nicely in C^{0} or PL category. Since in low dimension these categories are the same as the C^{∞} one, we shall not care to give technical details.

Let S and N be smooth, compact surfaces, p ∈ S.

Definition. A map F : S → N is said to be excellent at p if its germ at p is equivalent (left-right) to one of the following three normal forms:

i) (x, y) ↦ (x, y),  ii) (x, y) ↦ (x, y^{2}),  iii) (x, y) ↦ (x, y^{3} − xy).

Clearly, if F is excellent at all p, then the set Σ_{F} of its critical points is a smooth curve in S (i.e. Σ_{F} is a finite union of disjoint circles).

Definition. F is said to be excellent if the following two hold:

(1) F is excellent at all p ∈ S;

(2) the apparent contour of F (i.e. the set Γ_{F} = F (Σ_{F}) ⊂ N) is a smooth curve except for a finite number of singularities of the following two local kinds:

Crossing: {(x, y) ∈ ℝ^{2} | xy = 0}; Cusp: {(x, y) ∈ ℝ^{2} | x^{3} − y^{2} = 0}.

A classical theorem of Whitney [22] asserts that excellent maps are stable and constitute an open, dense subset of the set of C^{∞} maps between two surfaces. This is the reason why excellent maps are often called generic.

Given a curve Γ ⊂ N satisfying condition (2) of the previous definition, we ask whether a surface S and an excellent map F : S → N exist such that Γ_{F} = Γ. Moreover we should like to know how many, up to right equivalence, such maps exist (recall that F_{1}, F_{2} : S → N are said to be right equivalent if there exists a diffeomorphism φ : S → S such that F_{2} = F_{1} ∘ φ). This problem can be restated as follows: given a curve, say f : ∐_{h=1}^{k} S^{1} → N, with only cusp and normal crossings (Definition 1.1) find out all surfaces S ⊃ ∐_{h=1}^{k} S^{1} and a maps F : S → N such that Σ_{F} = ∐_{h=1}^{k} S^{1} and F _{∐_{h=1}^{k} S^{1}} = f.

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The basic idea, that will be used to solve this problem, is not dissimilar from the one used by Francis and Troyer [8], [9] to solve the problem for plane curves and arises from a very simple remark by Haefliger [11]: let F be an excellent extension of f and U a tubular neighborhood of Σ_{F}, then F restricted to S −U is an immersion. Furthermore, one can suppose that F restricted to ∂U is a curve with only normal crossings. Hence the problem reduces to the following two sub–problems: i) find out local, excellent extensions of f to a union of cylinders and Möbius bands, ii) find out immersions that extend the curves resulting as “boundary of the local extensions”, and the last problem can be solved using Blank’s methods [1], [19].

In §1 we define an extension of Blank’s word for curves with cusps and normal crossings, from which, by a purely combinatorial algorithm, we construct a set (the set of minimal assemblages), which is in one to one correspondence with the set of excellent extensions of the curve, up to right equivalence (Theorem 1.34).

In §0 we sketch, without any proof, the methods introduced by Blank [1], [19], and subsequently developed by other authors ([14], [4], [5], [10], [3], [2]), to find all (up to right equivalence) the extensions, that are immersions, of a curve with only normal crossings. We include this section because our notations and statements differs slightly from those in the original papers. For a complete survey on this matter see also the first chapter of [13].

Finally, given a line bundle π : E → N and a curve with cusps and normal crossings Γ ⊂ N, we ask whether a generic surface (i.e. π _{S} is excellent) S ⊂ E, having Γ as apparent contour, exists. Since we can construct all excellent extensions of the given curve, the problem reduce to find a factorization F = π ∘ F_{1} of a given excellent map F, asking for F_{1} : S → E to be an embedding. In §2 we use the methods developed in §1 to find combinatorial, necessary and sufficient conditions in order that F possesses such a factorization (Theorem 2.32).

0. Immersions with assigned boundary

Definition 0.1. Let k be a positive integer, we call generic k-curve an immersion f : ∐_{h=1}^{k} S^{1} → N, whose image Γ = ∐_{h=1}^{k} Γ_{h} is smooth up to a finite number of normal crossings.

Definition 0.2. Let S be a surface (not necessarily connected) with boundary and f a generic k-curve, we say that F : S → N extends f if and only if F is an immersion, ∂S = ∐_{h=1}^{k} S^{1} and F _{∂S} = f. We denote by E(f) the set of all extensions of f. Fixed a point ∞ ∈ N − Γ, we denote by E_{∞}^{β}(f) the set of all extensions such that #F^{−1}(∞) = β.

If f is a generic k-curve, then the bundle f^{∗}T N → ∐_{h=1}^{k} S^{1} (i.e. the pull-back via f of the tangent bundle to N) has a canonical trivial sub-bundle, τf, spanned by the never zero cross section f′. Let νf = f^{∗}T N/τf. The two sub-bundle τf and νf are called the tangent and normal bundles to f.

Definition 0.3. We say that f is sided if ν_{f} is the trivial bundle. A side for f is a never zero cross section of νf, up to multiplication by a positive function.

Remark 0.4. Since νf can be realized as a sub-bundle of f^{∗}T N, in such a way that f^{∗}T N = τf ⊕ νf, f is sided if and only if there exists a vector field along f, which is transversal to f.

A notion of rotation number can be given for a sided curve in a surface, with respect to a fixed vector field X on N, not vanishing on Γ. We denote by R_{X}(f) the rotation number of the sided curve f with respect to the vector field X. We do not give its definition here (for a definition see [20], [4], [13]), we only remark that this number essentially counts how many times a vector specifying the side turns with respect to X and that it coincide with the usual rotation number for plane curves, endowing such a curve with the side induced by the orientations of ℝ^{2} and taking X to be a constant (never zero) vector field.

Let S be a compact surface with boundary and let F : S → N be a generic immersion (i.e. F _{∂S} is a generic curve), then F _{∂S} has a canonical side defined by the image of an inward pointing vector field on ∂S. The following fact holds (see [4], [13]):

Proposition 0.5. Let F : S → N be a generic immersion and let X be a vector field having at most one isolated zero at the point ∞ ∉ Γ. Then

(0-1)  R_{X}(F _{∂S}) = χ(S) − βχ(N).

Here χ denotes Euler’s characteristic and β = #F^{−1}(∞).

Let f : ∐_{h=1}^{k} S^{1} → N be a sided curve and ∞ ∉ Γ a fixed point.

Definition 0.6. A set of segments for the curve f is a set R = {r_{i}} = A ∪ Ω, A = {α_{i}} and Ω = {ω_{j}}, such that:

(1) each α_{i} is an oriented, smooth arc, diffeomorphic to a segment of the real line, starting at a point in some component of N − Γ and ending to ∞;

(2) the ω_{j}’s are smooth (diffeomorphic to S^{1}) representatives of a minimal system of generators of π_{1}(N, ∞);

(3) each r_{i} is in general position with respect to Γ (i.e. misses all crossings and is transversal to Γ);

(4) for all i ≠ j r_{i} ∩ r_{j} = ∞. We say that R is a system of segments if and only if the following holds too:

(5) if C is a component of N − Γ and ∞ ∉ C, then some α_{i} starts from C.

Given a set of segments for the k-curve f, fix a neighborhood, U_{∞}, of the point ∞, such that U_{∞} ∩ Γ = ∅. For each r ∈ R fix an orientable neighborhood U_{r_{i}} of r − U_{∞} and an orientation on it. Label each point in R ∩ Γ by a letter x_{i,j,μ}^{e}, being i, j integers and e, μ = ±1, according to the following rules (see Figure 1):

(1) x_{i,j,μ}^{e} ∈ r_{i};

(2) e = 1 if the segment r_{i} crosses Γ from left to right (positive point), with respect to the side of f; e = −1 otherwise (negative point);

(3) μ = 1 if Γ crosses r_{i} from left to right, with respect to the orientation of U_{r_{i}}; μ = −1 otherwise;

(4) for all points x_{i,j,μ}^{e}, x_{i,j′,μ′}^{e} ∈ r_{i}, j < j′ if and only if x_{i,j,μ}^{e} < x_{i,j′,μ′}^{e} with respect to the ordering induced by the orientation on r_{i}.

Figure 1^{\begin{aligned}^{\begin{matrix}e=−1 \\ μ=− \end{matrix} e=1} μ=+ & e=1 \\ & μ=− \end{aligned} ; e=−1 ; μ=+ ∞ ∞}

Finally, define a set of words w = {w_{1}, …, w_{k}} by the following construction: fix points p_{h} ∈ Γ_{h}, being neither crossings of Γ nor points in R ∩ Γ; for each h, walk along Γ_{h}, starting from p_{h} and following its orientation, and write down all the labels you meat, until you come back to p_{h}. Call w_{h} the word obtained this way.

Definition 0.7. The set w = {w_{1}, …, w_{k}} is called the k-word of the generic k-curve f, with respect to the set of segments R.

Remark 0.8. Clearly, w depends on the choice of the base points p_{h}. Indeed, if we change the base point p_{h}, then the word w_{h} changes by the action of a cyclic permutation. We shall always consider words up to cyclic permutations.

Definition 0.9. We call 0-assemblage for the k-word w, a set 𝔄, whose elements are unordered pairs {x_{i,j,μ}^{−1}, x_{i,j′,μ′}} of letters in w, such that:

(1) if {x_{i,j,μ}^{−1}, x_{i,j′,μ′}} ∈ 𝔄, then j < j′;

(2) each letter, corresponding to a point in Ω ∩ Γ, appears in some pair of 𝔄;

(3) each negative letter (i.e. e = −1) appears in some pair of 𝔄;

(4) each letter appears at most in one pair of 𝔄.

Given f and R as above, fix a neighborhood V_{∞} of ∞ and a diffeomorphism Φ_{∞} : V_{∞} → B^{2} (here B^{2} denotes the unitary open ball in ℝ^{2}), such that:

(1) Γ ∩ V_{∞} = ∅;

(2) Φ_{∞}(R ∩ V_{∞}) is a set of rays starting at the origin;

and for each integer l ≥ 1, consider the curve g_{l} : S^{1} → N, g_{l}(t) = Φ_{∞}^{−1}((\frac{1}{3} + \frac{1}{3l})e^{it}), endowed with the side pointing outside V_{∞}.

Definition 0.10. We call β-expansion of the k-curve f, the (k + β)-curve f^{β}, obtained adding g_{1}, …, g_{β} to f. If w is a word for f, we call β-expansion of the k-word w, the (k + β)-word w^{β} for f^{β} defined by the same set of segments as w. Finally, we call β-assemblage for the word w a 0-assemblage for w^{β}.

Remark 0.11. w^{β} = w ∪ {u_{1}, …, u_{β}}, u_{l} being the word for g_{l}. Observe that the words u_{1}, …, u_{β} are equals up to shift the index j of all letters. We define an action of the symmetric group 𝔖_{β} on A^{β}(w), σ𝔄 being the set obtained from 𝔄 exchanging each letter in u_{l} with the corresponding one in u_{σl}. Clearly σ𝔄 fulfills conditions (1)–(4) in Definition 0.9, hence it is actually a β-assemblage for w.

A β-assemblage, 𝔄, defines a graph, G(𝔄), as follows: take k + β disjoint, oriented circles C_{1}, …, C_{k} and C_{k+1}, …, C_{k+β}, on each circle C_{h}, h = 1, …, k C_{k+l}, l = 1, …, β] choose points corresponding to the letters of the word w_{h} [u_{l}], ordered as in w_{h} [u_{l}], and join by an extra edge (we call such an edge a proper edge) all pairs of points corresponding to pairs of letters in 𝔄. Finally weight each proper edge by −1 or 1 according as the μ signs of its vertices agree or not.

A graph as just described (i.e. all vertices lay on k disjoint, oriented circles and all proper edges are weighted by ±1) will be called a weighted k-circular graph. We denote by l(G) the number of connected components of G. If 𝔄 is a β-assemblage, then l(𝔄) will denote the number of components of its associated graph.

Definition 0.12. Let G be a connected weighted k-circular graph and S a surface with k boundary component. We say that G is embeddable in S if and only if there exists a map φ : G → S such that the following three hold:

(1) φ is an homeomorphisms onto its image;

(2) φ(⋃_{h} C_{h}) = ∂S; for each proper edge ℓ of G, let U_{ℓ} be a neighborhood of φ(ℓ) homeomorphic to ℓ × [−1, 1]. Two semiorientations are naturally defined on ∂S ∩ U_{ℓ}: the one induced as boundary of U_{ℓ} and the one induced from the orientation of the circles C_{h}.

(3) the just described semiorientations on ∂S ∩U_{ℓ} agree if and only if the weight of the edge ℓ is equal to 1.

Obviously, every weighted k-circular graph is embeddable in some surface.

Definition 0.13. If G is a connected, weighted k-circular graph we call genus of G the number g(G) = min{g(S) | G is embeddable in S}.

It is easy to prove the following (see [13]):

Proposition 0.14. Let G′ be obtained from G by reversing the orientation of a circle and changing the weight of all proper edges having exactly one vertex on that circle, then g(G′) = g(G). If G is embeddable in S and g(G) = g(S), then S is orientable if and only if there exists a finite sequence G = G_{0}, …, G_{n} of graphs such that for all i, G_{i+1} is obtained from G_{i} as above, and all proper edges of G_{n} are positively weighted, in this case we say G is positive.

The second part of the previous statement allows us to define another noteworthy number associated to the graph:

Definition 0.15. We call weighted genus of the connected k-circular graph G the number h(G) = 2g(G) or g(G) according as G is positive or not. If G is not connected and G_{i} are its connected component, we call weighted genus of G the number h(G) = ∑_{i} h(G_{i}). We call weighted genus of a β-assemblage, the weighted genus of its associated graph and we denote it by h(𝔄).

The following proposition holds:

Proposition 0.16. Let f : ∐_{h=1}^{k} S^{1} → N be a generic curve, and w be the word for f defined by a system of segments, then for all 𝔄 ∈ A^{β}(w)

(0-2)  R_{X}(f) ≥ 2l(𝔄) − k − h(𝔄) − βχ(N).

Definition 0.17. Let w be defined by a system of segments. A β-assemblage 𝔄 ∈ A^{β}(w) is said to be minimal if and only if in (0-2) equality holds. We denote by A_{m}^{β}(w) the set of minimal β-assemblages of w.

It is not hard to see that if σ ∈ 𝔖_{β}, 𝔄 ∈ A_{m}^{β}(w), then σ𝔄 ∈ A_{m}^{β}(w), hence 𝔖_{β} acts on A_{m}^{β}(w). Denote right equivalence by ∼^{r}, the following theorem holds:

Theorem 0.18. Let f : ∐_{h=1}^{k} S^{1} → N be a generic k-curve and let w be the k-word for f defined by a system of segments. Then there exists a bijection between the set E_{∞}^{β}(f)/∼^{r} and the set A_{m}^{β}(w)/𝔖_{β}.

Sketch of proof. We only show how to define such a bijection. First suppose β = 0. Let F ∈ E_{∞}^{0}(w) be fixed. The set R̃ = F^{−1}(R) is made up by smooth arcs in S, which are oriented by the immersion. Label each point of R̃ ∩ ∂S by the same label as the corresponding point in Γ, and observe that each positive labeled point (i.e. e = 1) is the ending point of some arc, while each negative labeled point (i.e. e = −1) is the starting point of some arc. Furthermore, since F^{−1}(∞) = ∅, every arc must end to ∂S. Pairing the letters corresponding to the vertices of those arcs in R̃, that have both vertices in ∂S, we actually obtain a minimal 0-assemblage, 𝔄(F), and the mapping F ↦ 𝔄(F) is a bijection E_{∞}^{0}(f)/∼^{r} → A_{m}^{0}(w).

If F ∈ E_{∞}^{β}(f), let D_{1}, …, D_{β} ⊂ N be the disks bounding the curves g_{l} that define the β-expansion of f (see Definition 0.10), call {p_{1}, …, p_{β}} = F^{−1}(∞) and U_{i}^{j} the interior of the connected component of F^{−1}(D_{i}) containing p_{j}. For each σ ∈ 𝔖_{β}, let S_{σ} = S − ⋃_{i} U_{i}^{σi}. Clearly F_{σ} = F _{S_{σ}} ∈ E_{∞}^{0}(f^{β}), hence, by the case β = 0, we obtain an assemblage 𝔄(F_{σ}) ∈ A_{m}^{0}(w^{β}) = A_{m}^{β}(w). Such a construction actually defines a bijection between E_{∞}^{β}(f)/∼^{r} and A_{m}^{β}(w)/𝔖_{β}. □

The statement of Theorem 0.18 can be a little improved.

Definition 0.19. A connected component, say C, of N − Γ is said to be negative if the side of the curve points outward C at all points in ∂C; otherwise it is said to be positive. A set of segments is called a reduced system of segment if:

(5′) at least one of the α_{i}’s starts from each negative component.

Remark 0.20. Proposition 0.16, Definition 0.17 and finally Theorem 0.18 can be restated and proved (see [13]), replacing the words “system of segments” by “reduced system of segments”. These improved forms of such statements will be used in §1.

Now we prove a lemma, that we shall use in §1. We need one more definition:

Definition 0.21. We say that a generic k-curve has a curl if there exist t_{0}, t_{1} ∈ S^{1} such that f(t_{0}) = f(t_{1}) and f _{[t_{0},t_{1}]} bounds a disk, D ⊂ N, such that D ∩ Γ = ∂D. We say that the curl is positive or negative according as D − ∂D is a positive or negative component.

Lemma 0.22. Suppose that the k-curve f has a positive curl, then for every word w defined by a reduced system of segments, A_{m}^{β}(w) = ∅.

Proof. It is enough to prove that A_{m}^{β}(w) = ∅ for the word defined by some reduced system of segments. Let D be the disk bounded by the curl and observe

\begin{matrix}^{f f^{∗}} & D \\ Figure 2 \end{matrix}

negative positive

Figure 3

Figure 4

that we can construct a reduced system of segment R such that R∩D = ∅. Indeed, fix ∞ ∉ D, since the curl is positive, there is no need to draw segments starting from D and all other segments can be drawn far from D. Let f^{∗} be the curve obtained removing the curl as suggested in Figure 2 and let w and w^{∗} the words defined by R for f and f^{∗}. Clearly, R is a reduced system of segments for f^{∗} too and since R ∩ D = ∅, w = w^{∗}. Applying (0-2) to f^{∗} and using the fact that R_{X}(f) = R_{X}(f^{∗}) + 1, we obtain: R_{X}(f) > 2l(𝔄) − k − h(𝔄) − βχ(N) for all 𝔄 ∈ A^{β}(w). □

1. Extending curves with cusps and normal crossings

Let I be an open interval of the real line ℝ, and f : I → N be a curve.

Definition 1.1. A point t_{0} ∈ I is called a cusp point if and only if there exist germs of diffeomorphism γ : (I, t_{0}) → (ℝ, 0) and φ : (N, f(t_{0}) → (ℝ^{2}, 0) such that φ ∘ f ∘ γ^{−1}(s) = (s^{2}, s^{3}). We say that f : ∐_{h=1}^{k} S^{1} → N is a k-curve with cusp and normal crossing (briefly a CN k-curve) if f′(t) ≠ 0, except for a finite number of cusp points, and f is injective except for a finite number of normal crossings.

Let C be the set of cusp points of the CN k-curve f : ∐_{h=1}^{k} S^{1} → N.

Definition 1.2. A side for f : ∐_{h=1}^{k} S^{1} → N is a side for f _{∐_{h=1}^{k} S^{1}−C}, that, in a neighborhood of each cusp, is directed either inward or outward the cusp on both branches of the cusp itself. A cusp is said to be positive or negative according as the side is outward or inward pointing (see Figure 3). Finally, we say that a side for f is coherent if and only if all cusps are negative.

Let F : S → N be an excellent mapping, by definition F _{Σ_{F}} is a CN curve.

Proposition 1.3. The curve F _{Σ_{F}} has a coherent side.

Proof. Take the folding side of the map (see Figure 4 and [18]). □

Local extensions. Let f : S^{1} → N be a sided CN curve and let M be either a cylinder or a Möbius band (i.e. either M = S^{1} × [−1, 1] or M = ℝ × [−1, 1]/_{(x,t)=(x+1,−t)}) and identify S^{1} with the central circle in M (i.e. either S^{1} × {0} or S^{1} = ℝ × {0}/_{(x,0)=(x+1,0)}). By a local excellent extension of f we shall mean an excellent mapping F : M → N such that Σ_{F} = S^{1}, F _{S^{1}} = f and inducing the given side on f. If f : ∐_{h=1}^{k} S^{1} → N is a sided CN k-curve a local

if q ∈ D ; \begin{matrix}f_{D,1}^{∗} & f_{D,2}^{∗} \\ q \end{matrix}  if q ∉ D ; \begin{matrix}f_{D,2}^{∗} & f_{D,1}^{∗} \\ q \end{matrix}

Figure 5

\begin{matrix}\begin{matrix}p \\ p \end{matrix} ; f_{D,2}^{∗} ; f_{D,1}^{∗} & f_{D}^{∗∗} \\ Figure 6 \end{matrix}

excellent of f will mean a mapping F : ∐_{h=1}^{k} M_{h} → N such that F _{M_{h}} is a local excellent extension of f restricted to the h^{th} circle.

Let f : S^{1} → N be a sided CN curve and let D ⊆ C.

Definition 1.4. We call D-deformation of the first kind of f the generic 2curve f_{D}^{∗} = f_{D,1}^{∗} ⨿ f_{D,2}^{∗}, that one obtains by doubling f and deforming its cusps as suggested in Figure 5. Fix a point p ∈ Γ being neither a crossing nor a cusp, we call D-deformation of the second kind the generic 1-curve f_{D}^{∗∗} obtained modifying f_{D}^{∗} in a neighborhood of p as suggested in Figure 6.

Remark 1.5. It is clear that f_{D}^{∗} = f_{C−D}^{∗} and f_{D}^{∗∗} = f_{C−D}^{∗∗}. By local arguments, it is easy to see that if F : M → N is a local excellent extension of f, then there exists a tubular neighborhood U of S^{1} such that either f _{∂U} = f_{D}^{∗} or F _{∂U} = f_{D}^{∗∗} (according as M is either a cylinder or a Möbius band) for some D ⊆ C. This set D is unique up to complementation. We call such a D, up to complementation, the deformation set induced by F.

Let f : ∐_{h=1}^{k} S^{1} → N be a sided CN k-curve, D as above and let H ⊆ {1, …, k}; a pair (D, H) will be called a deformation pair. Denote by H̄ the complement of H and let k_{H} be the number k_{H} = 2#H + #H̄.

Definition 1.6. We call (D, H)-deformation of f the generic k_{H}-curve defined by f_{(D,H)} = {(f_{h})_{D}^{∗} | h ∈ H} ⨿ {(f_{h})_{D}^{∗∗} | h ∈ H̄, }.

Remark 1.7. Let C_{h} be the set of cusp points in the h^{th} circle and D_{h} = D∩C_{h}, if D′ is such that either D′_{h} = D_{h} or D′_{h} = C_{h} − D_{h} for all h, then f_{(D,H)} = f_{(D′,H)}. If F is a local excellent extension of f, then there exists a tubular neighborhood U of ∐_{h=1}^{k} S^{1} such that F _{∂U} = f_{(D,H)} for some pair (D, H). Furthermore if (D, H) and (D′, H′) are two different such pairs, then H = H′ and D, D′ are as above. We call such a pair (D, H), up to this relation, the deformation pair induced by F.

Remark 1.8. Let F : M → N be a local excellent extension of f : S^{1} → N and let t_{0} be a cusp point. If v ∈ T_{t_{0}}M − ker(dF (t_{0})), then dF (t_{0})[v] is a vector tangent to the cusp. Identify M with the quotient space of ℝ × [−1, 1] by the usual relations in such a way that π^{−1}(p) = ℤ, where p is the fixed point in Γ and π is the quotient map. Let F̃ = F ∘ π. It can be easily shown that the deformation set induced by F is given, up to complementation, by D = {t ∈ (0, 1) | (t, 0) is a cusp point and dF̃(t, 0)[(0, 1)] points inward the cusp}.

Using this characterization, it is not hard to prove the following proposition:

Figure 7  Figure 8  Figure 9

Proposition 1.9. Let f : ∐_{h=1}^{k} S^{1} → N be a CN k-curve endowed with e coherent side, then for all deformation pairs (D, H), there exists a local excellent extension of f inducing (D, H).

Let F, G : (ℝ^{2}, 0) → (ℝ^{2}, 0) be two generic, singular germs having the same apparent contour (i.e. F (Σ_{F}) = G(Σ_{G}) = Γ), then they have the same normal form with respect to left-right equivalence, say ii) or iii) in introduction. Standard arguments in singularity theory, prove the following lemma.

Lemma 1.10. Let F, G be as above and suppose they induce the same side, then there exists a germ of diffeomorphism Φ such that F = G ∘ Φ. If their normal form is iii) such a germ Φ is uniquely determined. If their normal form is ii) there are exactly two such germs, one of which is orientation preserving and the other one is orientation reversing.

Proposition 1.11. Let f : ∐_{h=1}^{k} S^{1} → N be a CN k-curve and F, G two local, excellent extensions of f. F and G are locally (around ∐_{h=1}^{k} S^{1}) right equivalent if and only if they induce the same deformation pair.

Proof. If F ∼^{r} G, they obviously induce the same deformation pair. Conversely, suppose that F and G induce the same pair. Clearly, it is enough to prove the thesis for a CN 1-curve. Let M be the domain of F and G. By the previous lemma we can find finite open covers {U_{i}}_{i=1,...,n}, {V_{i}}_{i=1,...,n} of S^{1} and diffeomorphisms Φ_{i} : U_{i} → V_{i} such that F _{U_{i}} = G _{V_{i}} ∘ Φ_{i}. Lifting all maps to the universal covering of M, we see that, since F and G induce the same deformation set, all the uniquely determined diffeomorphisms are orientation preserving [reversing]. To end the proof it is enough to choose all the other ones to be orientation preserving [reversing] and paste them together. □

Rotation number. Let f : ∐_{h=1}^{k} S^{1} → N be a sided CN k-curve, and let X be a vector field on N with no zeros on Γ. Denote by f^{∗} the generic k-curve obtained by deforming all cusps in the way suggested in Figure 7. With the notations introduced above, f^{∗} = ∐_{h=1}^{k}(f_{h})_{C,1}^{∗}. Since a deformation of f has a side, canonically induced by the side of f (Figure 8), we can give the following definition:

Definition 1.12. We call rotation number of f with respect to X the number R_{X}(f) = R_{X}(f^{∗}), where f^{∗} is endowed with the side induced by f.

Lemma 1.13. Let f : S^{1} → N be a sided, CN 1-curve and let D ⊆ C, then R_{X}(f_{D}^{∗∗}) = R_{X}(f_{D}^{∗}) = 2R_{X}(f) − c^{−} + c^{+}, where c^{+} and c^{−} denote respectively the number of positive and negative cusps.

Proof. Since rotation number does not change modifying a curve as suggested in Figure 9, the left hand side equality holds. Denote by c_{D}^{±} the number of positive and negative cusps in D. f_{D,1}^{∗} [f_{D,2}^{∗}] is obtained adding c_{C−D}^{+} [c_{D}^{+}] positive curls and c_{C−D}^{−} [c_{D}^{−}] negative curls to f^{∗}, hence R_{X}(f_{D,1}^{∗}) = R_{X}(f^{∗}) + c_{C−D}^{+} − c_{C−D}^{−} and R_{X}(f_{D,2}^{∗}) = R_{X}(f^{∗})+c_{D}^{+} −c_{D}^{−}. Sum these two equalities and obtain the thesis. □

Lemma 1.14. Let f : ∐_{h=1}^{k} S^{1} → N be a CN k-curve and (D, H) a deformation pair, then R_{X}(f_{(D,H)}) = 2R_{X}(f) − c^{−} + c^{+}.

Proof. Follows immediately by the previous lemma. □

From now on, we shall suppose to have fixed a point ∞ ∉ Γ and a vector field X vanishing at most at ∞.

Proposition 1.15. Let F : S → N be an excellent map, then:

(1-1)  2R_{X}(F _{Σ_{F}} ) − c = χ(S) − βχ(N).

Where c denotes the number of cusps, and β = #F^{−1}(∞).

Proof. Put f = F _{Σ_{F}}. By Proposition 1.3, all cusps are negative, hence, by Lemma 1.14, R_{X}(f_{(D,H)}) = 2R(f) − c, for all deformation pairs (D, H). Let U be a tubular neighborhood of Σ_{F}, such that F _{∂U} = f_{(D,H)}. The two sides on f_{(D,H)}, induced respectively by the immersion F _{S−U} and by the side of f, are clearly the same, hence, by (0-1), R_{X}(f_{(D,H)}) = χ(S − U) − βχ(N). To end the proof, observe that χ(S) = χ(S − U). □

Remark 1.16. The previous proposition gives one more condition in order that a curve may be the apparent contour of an excellent map. Nevertheless it is not hard to construct examples of curves fulfilling (1-1) for some choice of a surface S, but being the apparent contour of no excellent map S → N. We remark also that (1-1) implies a generalization of a classical theorem of Thom [21], asserting that χ(S) ≡ c mod 2 for all excellent maps F : S → ℝ^{2}. Indeed, denoting by deg_{2} the modulo two degree, we have:

Theorem 1.17. Let F : S → N be an excellent map, then χ(S) ≡ c mod 2, if and only if either χ(N) is even or deg_{2}(F) = 0.

Words. Let f : ∐_{h=1}^{k} S^{1} → N be a CN k-curve and ∞ ∈ N − Γ a fixed point.

Definition 1.18. A set of segments for f is a set R = A ∪ B ∪ Ω of smooth, oriented arcs in N, such that:

(1) each r ∈ A ∪ B is diffeomorphic to a closed interval of the real line. The ending point of such arcs being ∞ and the starting point being either a point in a connected component of N − Γ, if r ∈ A, or a cusp, if r ∈ B;

(2) Ω is a set of smooth (diffeomorphic to S^{1}) representatives of a minimal set of generators of π_{1}(N, ∞);

(3) r ∩ r′ = ∞ for all r, r′ ∈ R, r ≠ r′;

(4) each r ∈ R is in general position with respect to Γ, that is r contains neither crossings nor cusps (except at most the starting point), is transversal to Γ and if it starts from a cusp, it points outward the cusp.

Figure 10^{\begin{aligned}\begin{aligned}\begin{aligned}\begin{matrix}e=−1 \\ μ=−1 \end{matrix} & e=1 \\ & μ=−1 \end{aligned} & e=−1 \\ & μ=1 \end{aligned} & e=1 \\ & μ=1 \end{aligned}}

We say that R is a system of segments for f if the following holds too:

(5) at least one segment starts from each component of N − Γ not containing ∞ and from each cusp.

Definition 1.19. Suppose f is sided, and let r ∈ R. We say p ∈ R ∩ Γ is positive if and only if either p is not a cusp and r crosses Γ from the left to the right or p is a positive cusp; otherwise we say p is negative.

Fix a neighborhood U_{∞} of ∞ not intersecting Γ and oriented neighborhoods V_{r} of r − U_{∞} for all r ∈ R. Next put labels x_{i,j,μ}^{e} on each point in R ∩ Γ according to the same rules as in §1, except for cusp points where the index μ is +1 or −1 according as the curve crosses the segment from left to right or from right to left (see Figure 10). A set of k words can be associated to f as in §1.

Definition 1.20. We call the set w = {w_{1}, …, w_{k}} obtained this way, the k-word of f, with respect to the set of segments R.

Remark 1.21. As for generic curves (Remark 0.8), the word of a CN curve will be considered up to the action of cyclic permutations.

Let f : S^{1} → N be a CN curve, w a word for f and D ⊆ C.

Definition 1.22. We call D-deformations of the first and second kind of w respectively the words w_{D}^{∗} = {w_{D,1}^{∗}, w_{D,2}^{∗}} and w_{D}^{∗∗} = w_{D,1}^{∗}w_{D,2}^{∗}, where w_{D,1}^{∗} is obtained from w by erasing all letters corresponding to cusps in D and w_{D,2}^{∗} by erasing the ones corresponding to cusps in C − D.

Let f be a CN k-curve, w a k-word for f and (D, H) a deformation pair.

Definition 1.23. The (D, H)-deformation of w is the k_{H}-word w_{(D,H)} = {(w_{h})_{D}^{∗} | h ∈ H} ⨿ {(w_{h})_{D}^{∗∗} | h ∈ H̄}, where H̄ and k_{H} are as in Definition 1.6.

Definition 1.24. A β-assemblage for w is a pair A = ((D, H), 𝔄), where (D, H) is a deformation pair and 𝔄 is a β-assemblage for w_{(D,H)} (as defined in Definition 0.10). We denote by A^{β}(w) the set of β-assemblages of w. If A = ((D, H), 𝔄) ∈ A^{β}(w), we call number of components and weighted genus of A respectively the numbers l(A) = l(𝔄) and h(A) = h(𝔄).

Remark 1.25. All letters, except the ones corresponding to cusps, appear twice in the deformed word w_{(D,H)}. In the previous definition, the two copies of the same letter, must be considered as different ones.

Proposition 1.26. Let R be a system of segments for f, then R is a reduced system of segments for f_{(D,H)}.

Figure 11

Proof. Clearly, R is a set of segments for f_{(D,H)}, (the deformation can be done in such a way that R is a set of segments for f_{(D,H)}). Each connected components of N − Γ_{(D,H)} either is essentially a component of N − Γ or is generated by the deformation. At least one segment starts from each component of the first kind. The new ones are bounded by either the curl around a cusp or parallel branches of Γ_{(D,H)}. In the first case, the segment, starting from the corresponding cusp, starts from the new component while in the second case, it is easy to see that such components are positive (see Figure 11), hence condition (5′) of Definition 0.19 holds. □

Remark 1.27. Let w̃_{(D,H)} the word for f_{(D,H)} defined by R, clearly w̃_{(D,H)} is obtained renumbering the second index of all letters in w_{(D,H)} in such a way to preserve the previous ordering of letters belonging to the same segment; hence a β-assemblage A = ((D, H), 𝔄) for w defines naturally a β-assemblage 𝔄̃_{A} for w̃_{(D,H)}. Furthermore, since letters having the same indexes in w_{(D,H)} have also the same exponent, the corresponding letters in w̃_{(D,H)} cannot be paired by any β-assemblage. This argument proves the following:

Proposition 1.28. The mapping A ↦ 𝔄̃_{A} is a bijection between the set of βassemblages of w having (D, H) as deformation pair and A^{β}(w̃_{(D,H)}). Furthermore A and 𝔄̃_{A} have the same number of components and weighted genus.

Proposition 1.29. Let f be a CN k-curve and w the word defined by a system of segments. Then for all A ∈ A^{β}(w),

(1-2)  2R_{X}(f) − c_{−} + c_{+} ≥ 2l(A) − k_{H} − h(A) − βχ(N).

Proof. Use Proposition 1.28, (0-2) and Lemma 1.14. □

Definition 1.30. Let w be the k-word for the CN k-curve f, defined by a system of segments. A ∈ A^{β}(w) will be called minimal if in (1-2) equality holds. We shall denote by A_{m}^{β}(w) the set of minimal β-assemblages of w.

Remark 1.31. The mapping A ↦ 𝔄̃_{A} is a bijection between the set of minimal β-assemblages of w, having (D, H) as deformation pair, and A_{m}^{β}(w̃_{(D,H)}).

Let f be a CN k-curve, and suppose that the h^{th} component of f has no cusps. Let (D, H) be a deformation pair, then (w_{h})_{D,1}^{∗} = (w_{h})_{D,2}^{∗} = w_{h}. We define an involution ι_{h} on the set of letters of w_{(D,h)} by mapping each letter of (w_{h})_{D,1}^{∗} to the corresponding one of (w_{h})_{D,2}^{∗} and leaving fixed all other letters.

Let A = ((D, H), 𝔄) be a β-assemblage of w, we denote by ι_{h}A the β-assemblage ((D, H), ι_{h}𝔄), where ι_{h}𝔄 = {{ι_{h}(x_{i,j,μ}^{−1}), ι_{h}(x_{i,j′,μ′})} | {x_{i,j,μ}^{−1}, x_{i,j′,μ′}} ∈ 𝔄}.

Definition 1.32. We say that A is ι-equivalent to A′ (in symbols A ∼^{ι} A′), if and only if there exist involutions ι_{h_{1}}, …, ι_{h_{n}} such that A′ = ι_{h_{n}} … ι_{h_{1}}A. We say that A = ((D, H), 𝔄) and A′ = ((D′, H′), 𝔄′) are δ-equivalent (in symbols A ∼^{δ} A′) if the two deformation pairs are as in Remark 1.7, and 𝔄 = 𝔄′.

As in §1, we have an action of the symmetric group 𝔖_{β}, on the set A^{β}(w), given by σA = ((D, H), σ𝔄). We denote by ∼ the equivalence relation on A^{β}(w) generated by ∼^{ι}, ∼^{δ} and the action of 𝔖_{β} and simply call it equivalence.

Remark 1.33. If A ∼ A′, then A and A′ have the same number of components and the same weighted genus.

Let E_{∞}^{β}(f) denote the set of all excellent mappings defined on some surface, S ⊃ ∐_{h=1}^{k} S^{1}, such that Σ_{F} = ∐_{h=1}^{k} S^{1}, F _{∐_{h=1}^{k} S^{1}} = f and #F^{−1}(∞) = β. We shall prove the following theorem:

Theorem 1.34. Let w be the word for f defined by a system of segments. The set E_{∞}^{β}(f)/∼^{r} is in one to one correspondence with the set A_{m}^{β}(w)/∼.

Before proving the theorem, we give some elementary remarks.

Lemma 1.35. Let A, A′ ∈ A^{β}(w), σ ∈ 𝔖_{β}. If A ∼^{ι} A′, then σA ∼^{ι} σA′. If A ∼^{δ} A′, then σA ∼^{δ} σA′.

Remark 1.36. Denote by ∼^{ιδ} the equivalence relation generated by ∼^{ι} and ∼^{δ}. We can consider the quotient relation over A^{β}(w)/𝔖_{β}; we denote it by ∼^{ιδ}

again. Denote by [A] the class of A in A^{β}(w)/𝔖_{β}; by Lemma 1.35 we have that

[A] ∼^{ιδ} [A′] ⟺ ∃ σ ∈ 𝔖_{β} : σA ∼^{ιδ} A′; furthermore, A_{m}^{β}(w)/∼ = (A_{m}^{β}(w)/𝔖_{β} )/∼^{ιδ}.

Remark 1.37. Let (D, H) be a deformation pair and w̃_{(D,H)} be the word for f_{(D,H)} defined by the same segments as w. Let A = ((D, H), 𝔄) ∈ A^{β}(w) and let 𝔄̃_{A} ∈ A^{β}(w̃_{(D,H)}) the corresponding assemblage (see Proposition 1.28). By the very definitions, we have σ𝔄̃_{A} = 𝔄̃_{σA} for all σ and A. It follows that the mapping 𝔄̃ defines a map [𝔄̃] : [A] ↦ [𝔄̃]_{[A]} = [𝔄̃_{A}], which is a bijection between the set of β-assemblages of w having (D, H) as deformation pair, modulo the action of 𝔖_{β} and the set A^{β}(w̃_{(D,H)})/𝔖_{β}.

Proof of Theorem 1.34. We begin trying to define a map A_{1}^{β} : E_{∞}^{β}(f) → A_{m}^{β}(w)/𝔖_{β}. Fix F ∈ E_{∞}^{β}(f) and choose a tubular neighborhood, say U, of Σ_{F}, in such a way that F _{∂(S−U)} = f_{(D,H)}, for some choice of (D, H). Clearly F _{S−U} is an immersion that extends f_{(D,H)} and that takes exactly β times the value ∞. Let 𝔄^{β}(F) ∈ A_{m}^{β}(w̃_{(D,H)})/𝔖_{β} be the class of minimal assemblages associated to

F _{S−U} by Theorem 0.18. Put A_{1}^{β}(F) ∈ A_{m}^{β}(w)/𝔖_{β} to be the unique class such that

[𝔄̃]_{A_{1}^{β}(F)} = 𝔄^{β}(F) (Remark 1.37). Observe that A_{1}^{β}(F) is not uniquely determined, indeed its construction depends upon:

(a) the choice of the tubular neighborhood;

(b) the choice of the deformation set; if f_{h} is a component with no cusps and U_{h} is the correspondent component of U,

(c) the choice of the branch of ∂U, used as domain of f_{D,1}^{∗}.

As we shall see soon, different choices lead to ∼^{ιδ}-equivalent classes, thus defining a map A_{2}^{β} : E_{∞}^{β}(f) → A_{m}^{β}(w)/∼. To see this, we describe more explicitly the above construction.

Let {p_{1}, …, p_{β}} = F^{−1}(∞) and let D_{1}, …, D_{β} ⊂ S be disks such that p_{i} ∈ D_{i} and F (∂D_{i}) = g_{i}(S_{1}), being g_{1}, …, g_{β} as in Definition 0.10. Denote by Σ^{β} the set Σ_{F} ∪ ⋃_{i=1}^{β} ∂D_{i}. Since the segments of R are in general position with respect to Γ (condition (4) of Definition 1.18), we see that (F _{(S−⋃ D̊_{i})})^{−1}(R) is a set of smooth arcs with at least one ending point on Σ^{β}. Give to all this points the same label as the corresponding point on Γ^{β} = F (Σ^{β}).

Let γ be an arc of (F _{(S−⋃ D̊_{i})})^{−1}(R). Clearly, γ intersects ∂U in as many points as γ ∩ Σ_{F}. Label these points as the corresponding ending point of γ. Reading letters on ∂U, we get the word w_{(D,H)}, while reading letters on ∂D_{i} we get the words u_{i} (for all i = 1, …, k) of the β expansion (see §1). Pairing together letters corresponding to ending points of the same arc, we get a β-assemblage for w_{(D,H)}, hence a β-assemblage for w. By the way we associated an assemblage to an immersion (see the sketch of proof of Theorem 0.18), the just described two constructions lead to the same class in A_{m}^{β}(w)/𝔖_{β}.

From the last construction, it is clear that, leaving U and the deformation set D fixed, but exchanging the role of the two branches of ∂U_{h}, the assemblage changes by the action of the involution ι_{h}. It is also clear that choosing a different deformation set, the assemblage changes by a δ-equivalence. Finally if U, U′ are two different tubular neighborhood, we may define a map from the set of letters in w_{(D,H)} into itself, simply mapping each letter represented by an intersection of the arc γ with ∂U, to the letter represented by the corresponding intersection of γ with ∂U′. It is clear that the assemblages, defined by the two choices, differs up to the action of this map and that this map is a composition of ι-involutions, hence the two assemblages are ι-equivalent.

By the last construction one can easily prove the following:

Proposition 1.38. If F, F′ ∈ E_{∞}^{β}(f) are right equivalent extensions of f, then A_{2}^{β}(F) = A_{2}^{β}(F′).

By this proposition, A_{2}^{β} defines a map: A^{β} : E_{∞}^{β}(f)/∼^{r} → A_{m}^{β}(w)/∼. To end the proof of Theorem 1.34, we prove the following:

Proposition 1.39. The map A^{β}, is a bijection.

Proof. A^{β} is injective. Let F : S → N and F′ : S′ → N be such that A^{β}(F) = A^{β}(F′), then they induce the same deformation pair. By Proposition 1.11 there exist neighborhoods U and U′ of ∐_{h=1}^{k} S^{1} respectively in S and S′ and a diffeomorphism φ_{1} : U′ → U such that F′_{U′} = (F _{U}) ∘ φ_{1}. On the other hand, F _{(S−U)} and F′_{(S′−U′)} are two extension of f_{(D,H)} inducing the same class of assemblages in A_{m}^{β}(w̃_{(D,H)})/𝔖_{β}, hence, by Theorem 0.18, there exist a diffeo-

morphism φ_{2} : S′ − U′ → S − U such that F′_{(S′−U′)} = F _{(S−U)} ∘ φ_{2}. A local

analysis shows that the two diffeomorphisms φ_{1} and φ_{2} paste together, defining a diffeomorphism φ : S′ → S such that F′ = F ∘ φ.

To prove that A^{β} is surjective we need the following lemma:

Lemma 1.40. If w has minimal assemblages, then all cusps of f are negative.

Proof. By contradiction, suppose that f has a positive cusp, then every deformation of f has a positive curl. Hence, by Lemma 0.22, A_{m}^{β}(w̃_{(D,H)}) = ∅ for all deformation pair (D, H). By Remark 1.31, this means that A_{m}^{β}(w) = ∅. □

Finally we prove that A^{β} is surjective. Fix A = ((D, H), 𝔄) ∈ A_{m}^{β}(w), by the previous lemma, f has a coherent side, hence, by Proposition 1.11, there exists a local, excellent extension F_{1} : M → N of f such that F_{1}_{∂M} = f_{(D,H)}. By Proposition 1.28, 𝔄̃_{A} is a minimal β-assemblage of w̃_{(D,H)}, hence, by Theorem 0.18, there exist a surface S̃ and an immersion F_{2} : S̃ → N such that ∂S̃ = ∐_{h=1}^{k_{H}} S^{1} and F_{2}_{∂S̃} = f_{(D,H)}. Once again a local analysis shows that M and S̃ can be pasted along their boundary in such a way that the resulting map F, obtained by pasting F_{1} and F_{2}, is an excellent map extending f. By construction, A^{β}(F) = A. □

2. Factorization of excellent mappings

Classically the problem to find out a factorization F = π ∘ F_{1} for an excellent map F : S → N into a given line bundle π : E → N was posed asking for F_{1} to be an immersion. This problem was solved first by Haefliger [11] in the case N = ℝ^{2} and E = ℝ^{3}; successively Millet [15] remarked that Hafliger’s proof worked for an arbitrary surface N and the trivial bundle (E = N × ℝ) too. Finally, the author [12] used again Hafliger’s methods to solve the general case: let C be a component of Σ_{F}, and put c_{C} = #{cusps in C}, ν_{C} = ±1 according as C has a trivial normal bundle or not and ε_{C} = ±1 according as F _{C}^{∗}E → C is the trivial bundle or not. The following theorem holds:

Theorem. F factors through an immersion into E if and only if for all components C of Σ_{F}, (−1)^{c_{C}} ν_{C}ε_{C} = 1.

As announced in the introduction, in this section we deal with the problem of finding a factorization, F = π ∘ F_{1}, asking for F_{1} to be an embedding. We begin with some lemmas and propositions, that we shall use later.

From now on Σ and Γ will denote the set of critical points and the apparent contour of an excellent map.

Lemma 2.1. Let F : S → N be an excellent map and let D ⊂ N be an embedded disk such that: ∂D is in general position with respect to Γ, D contains no cusp and D ∩ Γ is a union of simple arcs. Then F restricted to any component of F^{−1}(D) − Σ is injective.

Proof. Fix a point ∞ ∉ D and a vector field X on N, vanishing at most at ∞. Let C be a connected component of F^{−1}(D) − Σ and let C_{ε} be obtained by removing a small neighborhood of Σ. The following facts hold: F _{C_{ε}} is an immersion; ∂C_{ε} is made up by branches of F^{−1}(∂D) and arcs parallel to Σ; if k is the number of boundary components of ∂C_{ε}, then F (∂C_{ε}) is made up by k loops obtained by arcs of ∂D and branches of Γ. It is easy to see that each of these loops, endowed with the side induced by the immersion, has rotation number greater or equal to 1, hence R_{X}(F _{∂C_{ε}}) ≥ k. By (0-1), R_{X}(F _{∂C_{ε}}) = χ(C_{ε}) ≤ 1, hence k = 1, C_{ε} is a disk and F _{∂C_{ε}} cannot have any crossing. Thus F _{C_{ε}} is 1 to 1. An exhaustion argument ends the proof. □

Let R be a system of segments for F _{Σ}. The set F^{−1}(R) is made up by smooth, oriented arcs having at least the ending point on Σ ∪ F^{−1}(∞). Denote by H(F) the set of all such arcs starting from Σ ∪ F^{−1}(∞).

Remark 2.2. Since ω ∈ Ω is a closed loop based at ∞, F^{−1}(Ω) ⊆ H(F).

Lemma 2.3. F restricted to each component of S − H(F) is injective.

Proof. By contradiction, suppose p, q ∈ S − H(F) be such that F (p) = F (q) and let γ : [0, 1] → S − H(F) be a path joining p and q. It is not hard to see that we can suppose that p, q ∈ S − F^{−1}(R), γ([0, 1]) ⊂ S − F^{−1}(R) and F ∘ γ is a simple curve in general position with respect to Γ (see [10]). Let D ⊂ N be the disk bounded by F ∘ γ. Since ∂D ∩ R = ∅, D ∩ R = ∅, and since R is a system of segments for F _{Σ}, D contains neither cusp nor curls of Γ. A contradiction follows by Lemma 2.1. □

We now state two topological lemmas, whose proof is an easy exercise.

Lemma 2.4. Let X be an arcwise connected topological space, U, V ⊂ X two open subset such that X = U ∪ V and U ∩ V is not arcwise connected. Then H_{1}(X) ≠ 0, where H_{1} denotes the first homology group.

Lemma 2.5. Let U_{1}, …, U_{n}, n ≥ 3, be open subsets of a topological space X. If U_{i} ∩ U_{j} ≠ ∅ ⟺ |i − j| ≡ 0, 1 mod n. Then either ⋂_{i=1}^{n} U_{i} ≠ ∅, and in this case n = 3, or U_{i} ∩ (⋃_{i≠j} U_{j}) is disconnected ∀i.

From the above two lemmas, we have the following:

Proposition 2.6. Let U_{1}, …, U_{n} (n ≥ 3) be arcwise connected, open subsets of the topological space X, fulfilling the hypothesis of Lemma 2.5. Then either n = 3 and ⋂_{i} U_{i} ≠ ∅ or H_{1}(⋃_{i} U_{i}) ≠ 0.

From now on, C will denote the set of connected components of S − H(F).

Proposition 2.7. Every C ∈ C is diffeomorphic to an open disk.

Proof. Let C ∈ C. As already remarked, F^{−1}(Ω) ⊆ H(F), hence F (C) ⊂ N − Ω. Since Ω is a minimal system of generators of π_{1}(N, ∞), N − Ω is diffeomorphic to an open disk, hence, by Lemma 2.3, we can say that C is a planar surface, that is an open disk with some holes. By contradiction, assume F (C) is not an open disk, hence F (C) has at least two boundary components, furthermore the boundary of F (C) is made up by branches of Γ and branches of segments of R. Let γ be one of the internal components of ∂F (C). If γ contains a branch of the segment r, since the segment ends at ∞, the segment itself cuts F (C) from boundary to boundary. But this contradicts the fact that C does not contain arcs in F^{−1}(R) starting and ending at its boundary. If γ is made up only by branches of Γ, then it bounds a connected component of N − Γ. The segment starting from this connected component, leads to a contradiction as in the previous case. □

Proposition 2.8. Let C_{1}, C_{2} ∈ C, then F (C_{1}) ∩ F (C_{2}) is connected.

Proof. By contradiction, suppose F (C_{1}) ∩ F (C_{2}) is disconnected. By Lemma 2.4, F (C_{1}) ∪ F (C_{2}) has at least two boundary components. A contradiction follows as in the proof of the previous proposition. □

Factorization. Let S and N be surfaces and π : E → N a line bundle.

Definition 2.9. We will say that a map F : S → N factors through π, if there exists an embedding F_{1} : S → E such that F = π ∘ F_{1}.

Remark 2.10. Finding a map F_{1} such that F = π ∘ F_{1} is the same as finding a section σ of the line bundle F^{∗}E, induced by F (see the diagram below). Hence F factors through π if and only if there exists a section σ : S → F^{∗}E of the induced bundle, such that π^{∗}F ∘ σ is an embedding. If E = N × ℝ is the trivial bundle, this condition reduce to the existence of a function h : S → ℝ such that (F, h) : S → N × ℝ is an embedding. We shall call such a function an height for F.

F^{∗}E ⟶^{π^{∗}F} E
σ ↑↓ F^{∗}π A A A A C A^{F_{1}} ↓π
S ⟶_{F} N

Remark 2.11. Suppose F : S → N and F′ : S′ → N are right equivalent maps, it is easy to see that F factors through π if and only if F′ do. For excellent maps this means that conditions in orders that a map factors through π “must” be contained in the assemblage associated to the map itself.

Let F^{−1}(∞) = {p_{1}, …, p_{β}} and let D_{1}, …, D_{β} disks in S such that p_{i} ∈ D_{i} and F _{∂D_{i}} = g_{i} (g_{i} as in Definition 0.10), and let S̃ = S − ⋃_{i=1}^{β} D̊_{i}, F̃ = F S̃.

Proposition 2.12. F factors through π, if and only if F̃ do.

Proof. Let F̃_{1} : S̃ → E be an embedding such that π ∘ F̃_{1} = F̃ and let U_{∞} be an open ball around ∞, such that U_{∞} ∩ Γ = ∅ and containing all curves g_{i}. Let U_{i} the connected component of F^{−1}(U_{∞}) containing p_{i}. Clearly, U_{i} ∩ U_{j} = ∅ for all i ≠ j, and F is a diffeomorphism between U_{i} and U_{∞}. Take a trivialization Φ : E _{U_{∞}} → U_{∞} ×ℝ, then Φ∘F̃_{1} ; ⋃_{i}(U_{i}−D̊_{i})(p) = (F (p), h(p)). Since F̃_{1} is injective, we may suppose that h assumes constant values, say c_{i}, on U_{i}−D_{i}, such that c_{i} ≠ c_{j} for all i ≠ j. Extend F̃_{1}, to the required embedding F_{1}, defining F_{1}(p) = Φ^{−1}(F (p), c_{i}) for all p ∈ D_{i}. □

The trivial bundle. First of all we consider the case E = N × ℝ. In Remark 2.10 we saw that F factors through E = N × ℝ if and only if there exists an height function h for F.

Proposition 2.13. Let h be an height function for F, then for all C_{1}, C_{2} ∈ C such that F (C_{1}) ∩ F (C_{2}) ≠ ∅, one (and only one) of the following two holds:

(1) ∀x ∈ C_{1}, y ∈ C_{2} F (x) = F (y) ⟹ h(x) < h(y);

(2) ∀x ∈ C_{1}, y ∈ C_{2} F (x) = F (y) ⟹ h(x) > h(y).

Proof. Clearly at most one of the two holds. By contradiction, suppose that both are false, then there exist x_{1}, x_{2} ∈ C_{1}, y_{1}, y_{2} ∈ C_{2} such that F (x_{1}) = F (y_{1}) = z_{1}, F (x_{2}) = F (y_{2}) = z_{2} and h(x_{1}) < h(y_{1}), h(x_{2}) > h(y_{2}). By Proposition 2.8 F (C_{1}) ∩ F (C_{2}) is connected, let γ : [0, 1] → F (C_{1}) ∩ F (C_{2}) be a path joining z_{1} and z_{2}, and let γ_{1} and γ_{2} its liftings to C_{1} and C_{2} respectively (such liftings exists by Lemma 2.3). Put h_{i} = h ∘ γ_{i}, then h_{1}(0) = h(x_{1}) < h(y_{1}) = h_{2}(0) and h_{1}(1) = h(x_{2}) > h(y_{2}) = h_{2}(1), therefore there exists t_{0} ∈ (0, 1) such that h_{1}(t_{0}) = h_{2}(t_{0}). Call p_{1} = γ_{1}(t_{0}) ∈ C_{1}, p_{2} = γ_{2}(t_{0}) ∈ C_{2}, then p_{1} ≠ p_{2} and F (p_{1}) = F (p_{2}), h(p_{1}) = h(p_{2}), that is a contradiction. □

If h is an height for F we can give a structure of oriented graph to the set C: C_{1}, C_{2} ∈ C are joined by an edge pointing to C_{2} (in symbols C_{1} → C_{2}) if and only if C_{1} and C_{2} verify (1) in Proposition 2.13. We denote by L(h) this oriented graph.

Proposition 2.14. The graph L(h) has no loops.

Proof. By contradiction, let C_{1} → C_{2} → ⋯ → C_{n} → C_{1} be a loop of minimal length. Observe that F (C_{i})∩F (C_{j}) ≠ ∅ ⟺ |i−j| ≡ 0, 1 mod n, on the contrary we could use Proposition 2.13 to find a shorter loop. By Proposition 2.6, either n = 3 and ⋂_{i} F (C_{i}) ≠ ∅ or H_{1}(⋃_{i} F (C_{i})) ≠ 0. In the first case, take points x_{i} ∈ C_{i} such that F (x_{1}) = F (x_{2}) = F (x_{3}), by definition of →, h(x_{1}) < h(x_{2}), h(x_{2}) < h(x_{3}) and h(x_{3}) < h(x_{1}), that is a contradiction. In the second case, a contradiction follows as in the proof of Proposition 2.7. □

As in the definition of the word of a curve, for all r ∈ R, fix an orientation of a small tubular neighborhood of r − ∞.

Proposition 2.15. Let C_{1}, C_{2}, C_{3}, C_{4} ∈ C and ℓ, ℓ_{1}, ℓ_{2}, be edges of the graph H(F). The following two holds:

(A) suppose that C_{1}, C_{2} paste along ℓ (i.e. C̄_{1} ∩ C̄_{2} ⊇ ℓ) and F (C_{3}) ∩ F (ℓ) ≠ ∅.

Then either C_{1} → C_{3} and C_{2} → C_{3} or C_{3} → C_{1} and C_{3} → C_{2}; (B) suppose that C_{1}, C_{2} paste along ℓ_{1}, C_{3}, C_{4} paste along ℓ_{2} and F (ℓ_{1}) ∩

F (ℓ_{2}) ≠ ∅. Let r ∈ R be the segment containing F (ℓ_{i}) and suppose that F (C_{1}), F (C_{3}) stand on the left side of r while F (C_{2}), F (C_{4}) stand on its right side. Then either C_{1} → C_{3} and C_{2} → C_{4} or C_{3} → C_{1} and C_{4} → C_{2}.

Proof. Let C_{1}, C_{2}, C_{3} be as in (A), then F (C_{1})∩F (C_{3}) and F (C_{2})∩F (C_{3}) are both not empty, hence some edge of L(h) joins C_{1}, C_{3} and C_{2}, C_{3}. By contradiction, suppose (A) is false, say C_{1} → C_{3} and C_{3} → C_{2}. Let z ∈ F (C_{3})∩F (ℓ) and let p ∈ ℓ such that F (p) = z. Take a small neighborhood U of p, such that U ∩ C_{3} = ∅ and F (U) ⊂ F (C_{3}), and a path γ in U joining two points p_{1} ∈ C_{1}, p_{2} ∈ C_{2}. Let γ_{3} be the lifting of F ∘γ to C_{3}. Clearly F ∘γ_{3} = F ∘γ, but C_{1} → C_{3} ⟹ h(γ(0)) < h(γ_{3}(0)) and C_{3} → C_{2} ⟹ h(γ(1)) > h(γ_{3}(1)). As in the previous proposition this fact leads to a contradiction. A completely analogous argument proves (B). □

Proposition 2.14 and Proposition 2.15 give necessary conditions in order that an excellent map F may factor through N×ℝ. We now prove that these conditions are sufficient too.

Theorem 2.16. Let F : S → N be an excellent map. F factors through N×ℝ if and only if the set C can be given a structure of oriented graph with no loops, fulfilling conditions (A) and (B) of Proposition 2.15.

Proof. Let S̃ be as in Proposition 2.12. We proceed in two steps: Step 1: definition of an height on S̃ minus a tubular neighborhood of Σ; Step 2: extension of such an height to the tube. The thesis will follow by Proposition 2.12.

Let L the graph in the assumption, since it has no loops, → extends to an ordering on C, call ≺ such an ordering. Clearly ≺ fulfills (A) and (B). Put C = {C_{1}, …, C_{n}}, being C_{s} ≺ C_{s+1} for all s. Step 1. Let U be a tubular neighborhood of Σ such that F _{∂U} is a generic curve and denote S′ = S̃−U, Γ′ = F (∂S′) and if C ∈ C denote C′ = C ∩S′. Let V_{∞} be a ball around ∞, contained in the internal side of all curves g_{i} (see Definition 0.10), that is to say F (S̃) ⊂ (N − V_{∞}). For all r ∈ R fix an oriented neighborhood V_{r} of r − V_{∞} in N − V_{∞} and a diffeomorphism φ_{r} : V_{r} → [0, t_{r}] × [−1, 1], such that:

(1) V_{r} ∩ V_{r′} = ∅ ∀r ≠ r′;

(2) ∀ r ∈ R, V_{r} contains no crossings of Γ′;

(3) ∀ r ∈ R, φ_{r}(r) ⊆ [0, t_{r}] × {0};

(4) ∀ r ∈ R, (r ∪ Γ′) ∩ V_{r} is connected (this fact and (2) say that V_{r} contains only branches of Γ′ intersecting r);

(5) let Γ_{r,j} be the component of Γ′ ∩V_{r} intersecting r at the j^{th} point, counting according the orientation of r, then φ_{r}(Γ_{r,j}) = {j} × [−1, 1];

(6) ∀ r ∈ R, φ_{r} and φ_{r}_{r} : r → ℝ are orientation preserving. Put V = ⋃_{r∈R} V_{r}, for any proper edge ℓ of H(F), denote by U_{ℓ} the connected component of F^{−1}(V) ∩ S′ containing ℓ ∩ S′ and by r(ℓ) the segment containing F (ℓ), let s(ℓ) and t(ℓ) be the two indexes such that C_{s(ℓ)}, C_{t(ℓ)} ∈ C are the two components pasting along ℓ, the former on the left of ℓ, the latter on the right (left and right with respect to the orientations induced by F). For all C ∈ C denote C^{∗} = (C ∩ S′) − ⋃_{ℓ} U_{ℓ}. Finally let ψ : ℝ × [−1, 1] → ℝ be a C^{∞} function such that: ψ(x, y) = 0 ∀y ∈ [−1, −2/3], ψ(x, y) = 1 ∀y ∈ [2/3, 1] and ψ(x, y) ∈ [0, 1] ∀x, y. We define the function h_{1} : S′ → ℝ,

(2-1)  h_{1}(x) = \{ \begin{matrix}s & if x ∈ C_{s}^{∗} \\ s(ℓ) + (t(ℓ) − s(ℓ))ψ(φ_{r(ℓ)}(F (x))) & if x ∈ U_{ℓ}. \end{matrix}

Clearly such a function is smooth. We now prove that (F _{S′}, h_{1}) : S′ → N × ℝ is injective. Let x ≠ y ∈ S′, three cases appear: (I) x ∈ C_{s}^{∗} and y ∈ C_{s′}^{∗}; (II) x ∈ C_{s}^{∗} and y ∈ U_{ℓ}; (III) x ∈ U_{ℓ} and y ∈ U_{ℓ′}. Case (I). If s = s′, by Lemma 2.3, F (x) ≠ F (y); if s ≠ s′, clearly h_{1}(x) ≠ h_{1}(y). Case (II). If F (x) ≠ F (y) there is nothing to prove. If F (x) = F (y), then F (C_{s}^{∗}) ∩ F (U_{ℓ}) ≠ ∅, therefore, by the way we chose the V_{r}’s, F (C_{s}) ∩ r(ℓ) ≠ ∅; by (A), this means either C_{s} ≺ C_{s(ℓ)} and C_{s} ≺ C_{t(ℓ)} or C_{s(ℓ)} ≺ C_{s} and C_{t(ℓ)} ≺ C_{s},

[figure: ^{D_{j_{2}(t̄)}  D_{j_{3}(t̄)}}; D_{j_{1}(t̄)}; Figure 12; ^{D_{j_{1}(t̄)}  D_{j_{2}(t̄)}}; ^{D_{j_{3}(t̄)}  D_{j_{4}(t̄)}}; Figure 13]

that is to say either s < s(ℓ) and s < t(ℓ) or s(ℓ) < s and t(ℓ) < s. By definition of h_{1}, min{s(ℓ), t(ℓ)} < h_{1}(y) < max{s(ℓ), t(ℓ)}, hence s = h_{1}(x) ≠ h_{1}(y). Case (III). Suppose F (x) = F (y) = p. The V_{r}’s are pairwise disjoint, hence r(ℓ) = r(ℓ′) and since F (U_{ℓ}) ∩ F (U_{ℓ′}) ≠ ∅, F (ℓ) ∩ F (ℓ′) ≠ ∅. By (B), either C_{s(ℓ)} ≺ C_{s(ℓ′)} and C_{t(ℓ)} ≺ C_{t(ℓ′)} or C_{s(ℓ′)} ≺ C_{s(ℓ)} and C_{t(ℓ′)} ≺ C_{t(ℓ)}, that is to say either s(ℓ) < s(ℓ′) and t(ℓ) < t(ℓ′) or s(ℓ′) < s(ℓ) and t(ℓ′) < t(ℓ). Using the definition of h_{1}, an easy computation shows that either h_{1}(x) < h_{2}(y) or h_{2}(y) < h_{1}(x). This ends the first step. Step 2. For the sake of simplicity, we suppose Σ is connected, it will be clear that the same proof works in the general case too. Let U be as above and put

M = \{ \begin{matrix}ℝ × [−1, 1]/_{(x,y)=(x+1,y)} & if U is orientable, \\ ℝ × [−1, 1]/_{(x,y)=(x+1,−y)} & if U is unorientable. \end{matrix}

Let φ : M → U be a diffeomorphism such that φ({y = 0}) = Σ and the inverse image of U ∩ H(F) is made up by vertical segments, that is if p : ℝ × [−1, 1] → M denotes the quotient map, we suppose that (φ ∘ p)^{−1}(H(F)) ∩ [0, 1] × [−1, 1] has the form: ⋃_{i}({t_{i}} × [0, 1]) ∪ ⋃_{j}({s_{j}} × [−1, 0]) ∪ ⋃_{l}({z_{l}} × [−1, 1]). Denote by D_{j} both the generic component of M − φ^{−1}(H(F)) and the corresponding component of U − H(F) and by C_{i(j)} ∈ C the component of S − H(F) containing D_{j}. The following two lemmas are an immediate consequence of (A) and (B) respectively.

Lemma 2.17. Let t̄ ∈ [0.1) a cusp point (with the just said notations, either t̄= t_{i} or t = s_{j}) and let D_{j_{1}(t̄)}, D_{j_{2}(t̄)}, D_{j_{3}(t̄)} the three components having (t̄, 0) in their closure. Let D_{j_{1}(t̄)} the one opposed to the edge starting at (t̄, 0) (see Figure 12); then either C_{i(j_{2}(t̄))} ≺ C_{i(j_{1}(t̄))} ≺ C_{i(j_{3}(t̄))} or C_{i(j_{3}(t̄))} ≺ C_{i(j_{1}(t̄))} ≺ C_{i(j_{2}(t̄))}.

Lemma 2.18. Let D_{j_{1}(z_{l})}, D_{j_{2}(z_{l})}, D_{j_{3}(z_{l})}, D_{j_{4}(z_{l})} be the four components having (z_{l}, 0) in their closure (numbered as in Figure 13); then either C_{i(j_{1}(z_{l}))} ≺ C_{i(j_{3}(z_{l}))} and C_{i(j_{2}(z_{l}))} ≺ C_{i(j_{4}(z_{l}))} or C_{i(j_{1}(z_{l}))} ≻ C_{i(j_{3}(z_{l}))} and C_{i(j_{2}(z_{l}))} ≻ C_{i(j_{4}(z_{l}))}.

Denote h_{1,−1}(t) = h_{1} ∘ p(t, −1), h_{1,1}(t) = h_{1} ∘ p(t, 1). By the previous two lemmas, and the way we constructed h_{1} in Step 1, we can suppose h_{1,1} and h_{1,−1} have the following properties:

(1) h_{1,1}(t + 1) = h_{1,1}(t) and h_{1,−1}(t + 1) = h_{1,−1}(t) if U is orientable, otherwise h_{1,1}(t + 1) = h_{1,−1}(t) and h_{1,−1}(t + 1) = h_{1,1}(t);

(2) there exists ε > 0 such that:

a) h_{1,1} is strictly monotonic on [t_{i} − ε, t_{i} + ε] and [z_{l} − ε, z_{l} + ε];

b) h_{1,−1} is strictly monotonic on [s_{j} − ε, s_{j} + ε] and [z_{l} − ε, z_{l} + ε];

c) h_{1,1} e h_{1,−1} assume constant values on the other intervals;

(3) ∀ t ∈ [0, 1) h_{1,1}(t) = h_{1,−1}(t) ⟺ either t = t_{i} or t = s_{j} for some i or j.

Figure 14  Figure 15  Figure 16

These conditions imply that we can extend h_{1} in a neighborhood [t̄−δ, t̄+δ]×[−1, 1] of each cusp point (t̄, 0), as suggested in Figure 14, in such a way that (F, h_{1}) : [t̄ − δ, t̄ + δ] × [−1, 1] → N × ℝ is an embedding. Moreover, if D_{j_{l}(t̄)} are the three components having (t̄, 0) in their closure and M(t̄) = max{i(j_{2}(t̄)), i(j_{3}(t̄))}, m(t̄) = min{i(j_{2}(t̄)), i(j_{3}(t̄))}, we can suppose that

(2-2)  m(t̄) ≤ h_{1}(t, y) ≤ M(t̄) ∀ (t, y) ∈ [t̄− δ, t̄+ δ] × [−1, 1];

and, up to choose U and δ small enough, that

(2-3)  F ([t̄_{1} − δ, t̄_{1} + δ] × [−1, 1]) ∩ F ([t̄_{2} − δ, t̄_{2} + δ] × [−1, 1]) = ∅

for all pairs of cusp points (t̄_{1}, 0), (t̄_{2}, 0).

Lemma 2.19. The mapping (F, h_{1}) : S′ ∪ ⋃t̄([t̄− δ, t̄+ δ] × [−1, 1]) → N × ℝ is an embedding.

Proof. By the way we extended h_{1}, the map is clearly an immersion. By contradiction, suppose that there exist x_{1}, x_{2} ∈ S′ ∪ ⋃t̄[t̄− δ, t̄+ δ] × [−1, 1] such that F (x_{1}) = F (x_{2}) and h_{1}(x_{1}) = h_{1}(x_{2}). Since (F, h_{1}) restricted to both S′ and [t̄− δ, t̄+ δ] × [−1, 1] is injective and by (2-3), we see that x_{1} ∈ [t̄− δ, t̄+ δ] × [−1, 1] and x_{2} ∈ S′. By construction, h_{1} assumes a constant value in a neighborhood of x_{1}, let C_{i} be the component of S′ − H(F) containing it. F (C_{i}) contains the cusp corresponding to (t̄, 0), hence F (C_{i}) ∩ F (D_{j_{l}(t̄)}) ≠ ∅ for all l = 1, 2, 3. Use twice (A) and get either i < m(t̄) = min_{l} i(j_{l}(t̄)) or i > M(t̄) = max_{l} i(j_{l}(t̄)). Using (2-2) and the fact that h_{1}(x_{2}) = i, we get a contradiction. □

Only remains to extend h_{1} over U − ⋃t̄([t̄− δ, t̄+ δ] × [−1, 1]). If both h_{1,1} and h_{1,−1} assume constant values, an extension can be easily found as suggested in Figure 15. Furthermore, conditions (2) and (3) ensure we can define local extensions of h_{1}, over the sets [z_{l} − ε, z_{l} + ε] × [−1, 1], in such a way that the value of such an extension at the point (t, y) is between h_{1,1}(t) and h_{1,−1}(t) (see Figure 16). Clearly all such extensions can be pasted to give a function h : S̃ → ℝ. An argument in all similar to the one used to prove the previous lemma, proves the following too:

Lemma 2.20. The map (F, h) : S → N × ℝ is injective.

Since (F, h) is an immersion, this ends the proof of Theorem 2.16. □

Now, we give a description of line bundles over surfaces, fitting in with our purpose to find conditions in order that an excellent mapping may factor through a line bundle.

Line bundles. Let N_{1} be a connected, compact surface with one boundary component and π_{1} : E_{1} → N_{1} a line bundle. Suppose E_{1}_{∂N_{1}} is the trivial bundle and let Φ : S^{1} × ℝ → E_{1}_{∂N_{1}} a trivialization. Denote by φ the restriction of Φ to the null section. We can consider the line bundle over N = N_{1} ∪_{φ} D^{2}, whose total space is E = E_{1} ∪_{Φ} (D^{2} × ℝ) and whose projection is the obvious one. Here we have denoted by X ∪_{f} Y the union of X and Y along the mapping f : A ⊂ X → Y.

Definition 2.21. The bundle constructed this way will be called the closure of E_{1} and denoted by Ê_{1}.

If N is a compact, connected surface without boundary and D ⊂ N is an embedded disk, then E _{D} is the trivial bundle, hence the following hold:

Proposition 2.22. Every line bundle over N is the closure of some bundle over a compact, connected surface with one boundary component.

Since the only line bundle over S^{2} is the trivial one, from now on we shall suppose that g(N) > 0. Put h(N) = 2g(N) if N is orientable, h(N) = g(N) otherwise. It is a classical fact in topology, that a surface N is the quotient of a 2h(N)-gon, by pairwise identification of its edges, and such a polygon can be found by cutting N along a minimal system of generators of its fundamental group. Fix the following data: a point ∞ ∈ N, a set Ω = {ω_{i}} of smooth curves, as in (2) of Definition 1.18 and a small open ball U_{∞} around the point ∞. Put N_{1} = N − U_{∞}, and let P_{N_{1}} be the 4h-gon resulting by cutting N_{1} along all the curves ω_{i} ∩ N_{1}. Call ℓ_{i}, ℓ′_{i} the two edges resulting by the cut along ω_{i}. For all i, let ψ_{i} : ℓ_{i} → ℓ′_{i} be the diffeomorphism giving the identification. Suppose to be given a distribution of weights, ρ_{i} = ±1, on the set Ω, and define E_{ρ} = P_{N_{1}} ×ℝ/∼, being (x_{1}, t_{1}) ∼ (x_{2}, t_{2}) if and only if either the two pairs are equal or x_{1} ∈ ℓ_{i}, x_{2} = ψ_{i}(x_{1}) and t_{2} = ρ_{i}t_{1} for some i. E_{ρ}, endowed with the obvious projection, is a line bundle over N_{1}.

Proposition 2.23. The line bundle E_{ρ}_{∂N_{1}} is the trivial bundle.

Proof. Such a bundle is obtained successively pasting 2h copies of [0, 1] × ℝ, by means of mappings of the following two kinds: (1, t) ↦ (0, t), (1, t) ↦ (0, −t), and both kinds of such identifications are even in number. □

By the previous proposition and Proposition 2.22 we see that every distribution of weights, ρ, over Ω generates a line bundle over N, simply taking the closure, Ê_{ρ}, of the just constructed bundle E_{ρ}. Simple technical arguments prove the following:

Proposition 2.24. If E is a line bundle over N, then there exist a distribution of weights ρ over Ω such that E is isomorphic to Ê_{ρ}.

The general case. We turn back to our problem. Let F : S → N be an excellent map and π : E → N a line bundle. Let R, U_{∞}, S̃ and N_{1} = N − U_{∞} be as above. By Proposition 2.24, we can suppose E = Ê_{ρ} for some distribution of weights ρ, hence by Proposition 2.12 and Remark 2.10, our problem reduces to find a section σ of F̃^{∗}E_{ρ} → S̃, making π_{ρ}^{∗}F̃ ∘σ into an embedding. First of all we try to understand what ”finding a section of the line bundle F̃^{∗}E_{ρ}” means. Let S′ be the surface obtained from S̃ cutting along F̃^{−1}(Ω), and let ξ : S′ → S̃ be the quotient map given by the cut. Let φ : P_{N_{1}} → N_{1} be the mapping generated by the cut of N_{1} along Ω, then there exists a unique map F′ : S′ → P_{N_{1}} such that φ∘F′ = F̃ ∘ξ. By this fact, by the way E_{ρ} was defined and by the following commutative diagram

P_{N_{1}} × ℝ ⟵ ξ^{∗}F̃^{∗}E_{ρ} ⟶ F̃^{∗}E_{ρ} ⟶ E_{ρ} ⟵ P_{N_{1}} × ℝ
↓  ↓  ↓  ↓^{π_{ρ}}  ↓
P_{N_{1}}  ⟵_{F′} S′  ⟶_{ξ} S̃ \overrightarrow{F̃} N_{1} ⟵_{φ} P_{N_{1}}

we see that finding a section of F̃^{∗}E_{ρ} is the same as finding a function h : S′ → ℝ, such that:

(2-4) h(x_{1}) = ρ_{i}h(x_{2}) if F (ξ(x_{1})) = F (ξ(x_{2})) ∈ ω_{i} ∀ x_{1}, x_{2} s.t. ξ(x_{1}) = ξ(x_{2})

and satisfying the analogous conditions on the differentials. We shall denote by σ_{h} the section of F̃^{∗}E_{ρ} defined by h. Clearly the following holds:

Proposition 2.25. The mapping π^{∗}F̃ ∘ σ_{h} : S̃ → E_{ρ} is an embedding, if and only if (F′, h) : S′ → P_{N_{1}} × ℝ is.

We shall call such a function h a strange height for the mapping F.

We introduce some further notations. Put H̃(F) = (H(F) ∩ S̃) ∪ ∂S̃ and let H′(F) ⊂ S′ be defined by H′(F) = ξ^{−1}(H̃(F)). Denote by C the set of connected components of S − H(F), by C̃ the set of components of S̃ − H̃(F) and finally by C′ the set of components of S′ −H′(F). It is immediate that inclusion S̃ ↪ S gives a bijection between C̃ and C, while ξ : S′ → S̃ gives a bijection between C′ and C̃. For all C ∈ C denote by C̃ ∈ C̃ and by C′ ∈ C′ the corresponding components (i.e. C̃ is the interior of C ∩ S̃ and C′ is the component such that ξ(C′) = C̃).

Proposition 2.26. Let C_{1}, C_{2} ∈ C. F (C_{1}) ∩ F (C_{2}) ≠ ∅ if and only if F (C̃_{1}) ∩ F (C̃_{2}) ≠ ∅ if and only if F (C′_{1}) ∩ F (C′_{2}) ≠ ∅.

Proof. The second equivalence is immediate. The first follows from the fact that if F (C_{1})∩F (C_{2}) ≠ ∅ and F (C̃_{1})∩F (C̃_{2}) = ∅, then F (C_{1})∩F (C_{2}) ⊂ U_{∞} and the fact that ∂(F (C_{1})∩F (C_{2})) contains some branches of Γ, while U_{∞} ∩Γ = ∅. □

Proposition 2.27. Let C_{1}, C_{2} ∈ C, then F (C̃_{1})∩F (C̃_{2}) and F′(C′_{1})∩F′(C′_{2}) are connected.

Proof. Exactly the same as Proposition 2.8. □

Let h be a strange height for the excellent map F. The same arguments used in the proof of Proposition 2.13 prove the following:

Proposition 2.28. Let C′_{1}, C′_{2} ∈ C′ be such that F′(C′_{1}) ∩ F′(C′_{2}) ≠ ∅. One of the following two holds:

(1) ∀x ∈ C′_{1}, y ∈ C′_{2} if F (x) = F (y) then h(x) < h(y);

(2) ∀x ∈ C′_{1}, y ∈ C′_{2} if F (x) = F (y) then h(x) > h(y).

As in the case of the trivial bundle we give a structure of oriented graph to the set C, saying C_{1} → C_{2} if and only if the corresponding components C′_{1}, C′_{2} ∈ C′ satisfy condition (1). We call such a graph the graph of h and denote it by L(h).

Remark 2.29. By Proposition 2.26 we see that two components C_{1}, C_{2} ∈ C are joined by an edge if and only if F (C_{1}) ∩ F (C_{2}) ≠ ∅, hence the situation is in all similar to the one we had dealing with the trivial bundle.

As in the case of the trivial bundle, the following is proved:

Proposition 2.30. The graph L(h) has no loop.

Put Ω^{−} = {ω ∈ Ω | ρ(ω) = −1}, an analogous of Proposition 2.15 holds in this case too.

Proposition 2.31. Let h be a strange height for F. If C_{1}, C_{2}, C_{3}, C_{4} ∈ C and ℓ, ℓ_{1}, ℓ_{2} are proper edges of H(F) the following two hold: (A1) suppose that C_{1}, C_{2} paste along ℓ (i.e. C̄_{1} ∩ C̄_{2} ⊇ ℓ) and F (C_{3}) ∩ F (ℓ) ≠ ∅.

Then either C_{1} → C_{3} and C_{2} → C_{3} or C_{3} → C_{1} and C_{3} → C_{2}; (B1) suppose that C_{1}, C_{2} paste along ℓ_{1}, C_{3}, C_{4} paste along ℓ_{2} and F (ℓ_{1}) ∩

F (ℓ_{2}) ≠ ∅. Let r ∈ R be the segment containing F (ℓ_{i}), and suppose that F (C_{1}), F (C_{3}) stand on the left side of r while F (C_{2}), F (C_{4}) stand on its right side. If r ∉ Ω^{−}, then either C_{1} → C_{3} and C_{2} → C_{4} or C_{3} → C_{1} and C_{4} → C_{2}. If r ∈ Ω^{−}, then either C_{1} → C_{3} and C_{4} → C_{2} or C_{3} → C_{1} and C_{2} → C_{4}.

Proof. Observe that a situation as in (A1) occurs only if F (ℓ) is contained either in Γ or in R − Ω, hence the same argument used to prove (A) of Proposition 2.15 proves (A1) too. We now prove (B1). If the segment r containing F (ℓ_{1}) and F (ℓ_{2}) is in R − Ω, the proof is again the same as the one of Proposition 2.15. Suppose F (ℓ_{1}), F (ℓ_{2}) ⊂ ω ∈ Ω and let x_{1} ∈ ℓ_{1}, x_{2} ∈ ℓ_{2} be such that F (x_{1}) = F (x_{2}). It is easy to see that we can suppose x_{1}, x_{2} ∈ S̃. Let ℓ′_{1}, ℓ′′_{1} be the edges of H′(F) such that ξ(ℓ′_{1}) = ξ(ℓ′′_{1}) = ℓ_{1} and let ℓ′_{2}, ℓ′′_{2} be such that ξ(ℓ′_{2}) = ξ(ℓ′′_{2}) = ℓ_{2}. The assumption in (B1) ensures that C′_{1}, C′_{2}, C′_{3} and C′_{4} verify the following:

∂C′_{1} ⊇ ℓ′_{1}, ∂C′_{2} ⊇ ℓ′′_{1}, ∂C′_{3} ⊇ ℓ′_{2}, ∂C′_{4} ⊇ ℓ′′_{2},
F′(C′_{1}) ∩ F′(C′_{3}) ≠ ∅, F′(C′_{2}) ∩ F′(C′_{4}) ≠ ∅.

Let x′_{1} ∈ ℓ′_{1}, x′′_{1} ∈ ℓ′′_{1}, x′_{2} ∈ ℓ′_{2}, x′′_{2} ∈ ℓ′′_{2} be such that ξ(x′_{i}) = ξ(x′′_{i}) = x_{i}. It is not hard to find four path (see Figure 17) γ′_{i} : [0, 1] → S′, γ′′_{i} : [1, 2] → S′, such that:

\begin{matrix}(1) & γ′_{i}(1) = x′_{i} and γ′′_{i}(1) = x′′_{i} i = 1, 2; \\ (2) & ∀ t ≠ 1 \{ \begin{matrix}γ′_{1}(t) ∈ C′_{1}, & γ′′_{1}(t) ∈ C′_{2}, \\ γ′_{2}(t) ∈ C′_{3}, & γ′′_{2}(t) ∈ C′_{4}; \end{matrix} \\ (3) & ∀ t \{ \begin{matrix}F′(γ′_{1}(t)) = F′(γ′_{2}(t)), \\ F′(γ′′_{1}(t)) = F′(γ′′_{2}(t)). \end{matrix} \end{matrix}  Figure 17^{\begin{matrix}^{ℓ′_{1}  ℓ′′_{1}} & ℓ′_{2} \\ ℓ′′_{2} \end{matrix} ; x′_{1} ; x′′_{1} ; x′_{2} ; x′′_{2} ; γ′_{1} ; γ′′_{1} ; γ′_{2} ; γ′′_{2} ; C′_{1} ; C′_{2} ; C′_{3} ; C′_{4}}

Define:

g_{i}(t) = \{ \begin{matrix}h(γ′_{i}(t)) & if t ∈ [0, 1] \\ ρ(ω)h(γ′′_{i}(t)) & if t ∈ [1, 2] \end{matrix} i = 1, 2.

By definition of a strange height and (1), we see that g_{1} and g_{2} are continuous functions. Consider the two cases ω ∈ Ω^{+} = Ω − Ω^{−} and ω ∈ Ω^{−}. In the first case, suppose by contradiction that C′_{1} → C′_{3} and C′_{4} → C′_{3}. By definition of → and (2), (3), g_{1}(t) < g_{2}(t) for all t < 1 and g_{1}(t) > g_{2}(t) for all t > 1. This implies g_{1}(1) = g_{2}(1), hence F′(x′_{1}) = F′(x′_{2}) and h(x′_{1}) = h(x′_{2}) contradicting the fact (F′, h) is injective. A similar argument ends the proof in the second case too. □

As in the case of the trivial bundle, Proposition 2.30 and Proposition 2.31 gives necessary and sufficient conditions for the existence of a strange height for an excellent map F.

Theorem 2.32. Let F : S → N be an excellent map, R a system of segments for F _{Σ} and E = Ê_{ρ} a line bundle over N. F factors through π if and only if C can be given a structure of oriented graph with no loops verifying (A1) and (B1) of Proposition 2.31.

Proof. The proof is completely similar to that of Theorem 2.16. □

Remark 2.33. Repeating almost word by word the statements and proofs in this section, a factorization theorem for generic immersion can be proved. More precisely, let S be a compact surface with boundary, F : S → N a generic immersion and R a system of segments for F _{∂S}. Denote G(F) ⊂ S the graph made up by ∂S and those arcs in F^{−1}(R) having both vertices in ∂S ∪ F^{−1}(∞). G(F) is the 1-skeleton of a cell decomposition of S such that F restricted to the interior of each cell is one to one. Finally denote by C the set of 2-cells of such a decomposition.

Theorem 2.34. A generic immersion F factors through E = Ê_{ρ} if and only if C can be given a structure of oriented graph with no loops verifying (A1) and (B1) of Proposition 2.31.

Acknowledgment. This work was partially supported by M.U.R.S.T.

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Dipartimento di Matematica, Università di Trento, via Sommarive n. 14, 38050 Povo, Trento, Italy.

E-mail address: luminati@volterra.science.unitn.it, luminati@itnvax.science.unitn.it,