2. Stable Maps and Morse Descriptions of an Apparent Contour

See [23, 25, 26], the books [9, 10, 18, 24], the references quoted in [27], and also [1, 17].

The map \(\mathcal{F}\) keeps the orientation: for instance, when n = 2, a positively oriented Jordan curve in \(\mathbb{R}^{2}\) is mapped through \(\mathcal{F}\) into a positively oriented Jordan curve.

In the terminology of Thom, Morse functions are called correct, and Morse functions with distinct critical values are called excellent [6].

Not quite, since \(\mathbb{R}\) is not closed. However we require functions to behave “nicely” outside some interval [a, b], e.g. by forcing them to have constant nonzero derivative.

In this book a knot is a \(\mathcal{C}^{\infty }\) embedding of \(\mathbb{S}^{1}\) in \(\mathbb{R}^{3}\), hence in particular a tame knot in the usual terminology; see, for instance, [8, p. 5].

Therefore, each of these curves is the embedded image of \(\mathbb{S}^{1}\) into \(\mathcal{X}\).

We warn the reader that these cusps, belonging to the source manifold \(\mathcal{X}\), should not be confused with the cusps of an apparent contour, which lie in the target manifold \(\mathcal{Y}\).

Sometimes called the cone on a figure-eight curve.

See, for instance, [10, Chapter II, Proposition 5.8].

See [23] and [10, p. 162].

M is an abstract manifold, not necessarily oriented or connected. We shall be mostly interested (for instance, in Sect. 3.​2) in the case when M can be embedded in \(\mathbb{R}^{3}\), which gives, in particular, an orientation to \(M\).

Locally, each cusp of the apparent contour is diffeomorphic to the simple (or ordinary, see, for instance, [3, p. 115]) cusp, which has the form \(\{(x_{1},x_{2}): x_{2}^{2} = x_{1}^{3}\}\) or equivalently, in a parametric form, \((t^{2},t^{3})\) for a real parameter t in a neighbourhood of the origin.

The component is, sometimes, called “irreducible” (with cusps and double points); see, e.g., [22].

Compare, for instance, with Chaps. 1 (Fig. 1.​3) and 4. We specify below and in Definition 4.​1.​1 what is a closed arc, a nonstandard feature in graph theory.

In [12, Theorem 1] the author gives necessary and sufficient conditions for the factorization of an excellent map (see Example 2.1.8) through an immersion and a projection.

The notion of positive and negative cusps of Definition 2.2.12 (see, e.g., [21]) is different from the notion considered in Chap. 8 (compare Definition 8.​1.​2 and Remark 8.​1.​3).

The first property means that H admits an extension of class \(\mathcal{C}^{\infty }\) on an open set of \(\mathbb{R}^{n+1}\) containing \(\mathbb{R}^{n} \times [0, 1]\).

Consistently, we set h _t (⋅ ) = h(⋅ , t).

We recall that, if V is a \(\mathcal{C}^{\infty }\) orientable n-dimensional manifold without boundary, and if Diff⁺(V ) denotes the group of positive diffeomorphisms of V endowed with the \(\mathcal{C}^{\infty }(V,V )\) topology, then the orbits coincide with the connected components; see [7, p. 1]. If \(V = \mathbb{S}^{2}\), then the connected component of the identity coincides with the arcwise connected component of the identity (see [6, p. 1]).

Actually, even {0} could not be left fixed by \(\tilde{h}_{t}\).

Making use of (2.4), we define R(⋅ , t) as follows. Let us identify \(\mathbb{S}^{2}\) with the unit sphere in \(\mathbb{R}^{3}\) and endow it with parallels and meridians with the north pole identified with ∞ and \(0 \in \mathbb{S}^{2}\) identified with the south pole. The stereographic projection from the north pole to the tangent plane at the south pole provides an identification of points of \(\mathbb{R}^{2}\) with points of \(\mathbb{S}^{2}\setminus \{\infty \}\) (we employ a scale reduction of a factor \(2\) on the stereographic projection so that the equator is mapped onto the unit circle of \(\mathbb{R}^{2}\)). Suppose first that \(\tilde{h}_{t}(0)\) is not the south pole. Let P(t) be the intersection between the equator of \(\mathbb{S}^{2}\) and the meridian passing through (the poles and) \(\tilde{h}_{t}(0)\). Let r(t) be the line (in \(\mathbb{R}^{3}\)) joining p(t) to q(t), where p(t) [respectively q(t)] is the point on the equator having the longitude of P(t) plus (respectively minus) π∕2. Then R(⋅ , t) is the (smallest) rotation around r(t) sending \(\tilde{h}_{t}(0)\) into the origin. This is a rotation of angle given by the latitude of \(\tilde{h}_{t}(0)\) plus π∕2, clearly this rotation takes the above-mentioned meridian into itself. If \(\tilde{h}_{t}(0)\) is the south pole, we define \(R(\cdot,t):=\mathrm{ id}_{\mathbb{S}^{2}}(\cdot )\). Then, \(R(\cdot,t) \in \mathrm{ Diff}^{+}(\mathbb{S}^{2})\), and R is of class \(\mathcal{C}^{\infty }\) and satisfies the required properties.

This task is similar to describing a prime knot in knot theory by means of notations like the one introduced by Dowker-Thistletwaite [16, p. 7], see also the combinatorial description of knotted surfaces in [4, pp. 21,22].

See [11] for the definition of stratifications and stratified maps. Compare also with Chap. 6, Definition 6.​2.​3.

See, e.g., [4].

Notice that \(\mathfrak{m}\) can be thought of as an element of \(\mathrm{Diff}_{\mathrm{c}}(\mathbb{R}^{2})\).