Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

The proof is based on induction on double point number (with multiplicity). We count here an intersection point of multiplicity k as double points.

      Assume that the number of double points is minimal. Then the curves do not have self-intersections and any two curves intersect in at most one point.

      If there are closed curves, then take an innermost closed curve. This curve can be contracted to a point without intersecting other curves. Repeating this operation we homotop our set of curves to a set without closed curves. So we can suppose there are no closed curves in the set.

      Let {ai}, i = 1, . . . , n, be the set of curves and let di be the geodesics connecting the ends of the curve ai in the disc for each i = 1, . . . , n. We can isotop slightly (with respect to the boundary) the curves {ai} so that they lie in general position with {di}. So we can suppose that all intersections of {ai} and {di} are transversal.

      Assume that there are extra intersections of d₁ with the curves {ai}. Then there is a bigon formed by a subarc of d₁ and a subarc of some curve aj.

      Take an innermost bigon among all bigons adjacent to d₁. Its edges belong to d₁ and aj for some j. Since each two curves in {ai} intersect in at most one point, this bigon is an innermost bigon among all the bigons in the disc. If some curves {ai} intersect the bigon, then by Lemma 3.1 there is an innermost triangle in the bigon adjacent to aj. So, by moving aj, we can either decrease the number of intersection points in the bigon (if the triangle is adjacent to one edge of the bigon, see Fig. 3.23 left) or decrease the number of arcs intersecting the bigon (if the triangle is adjacent to two edges of the bigon, see Fig. 3.23 right).

      Fig. 3.23Removing intersecting curves from the bigon

      Repeating this operation, we obtain an empty bigon. Then we eliminate the bigon by moving the curve aj.

      In this manner we can homotop our curves to a position where the curves aj, j > 1, intersect d₁ in at most one point and the curves d₁ and a₁ form a bigon. This bigon is innermost so a₁ can be homotop to d₁ without increasing the number of intersection points.

      Applying these reasonings consequently to a₂ and d₂, a₃ and d₃ etc., we homotop the set of curves to the geodesics.

      Assume now that the number of double points is not minimal. Then there is either a self-intersection of some curve or two curves have two or more intersection points. Then there is an embedded loop or a bigon in the disc. Take an innermost loop or a bigon among all embedded loops and bigons.

      An innermost loop is empty and can be contracted, so the number of intersection points will reduce, see Fig. 3.24.

      Fig. 3.24Removing intersecting curves from the bigon

      An innermost bigon may contain only arcs without self-intersections, and any two of these arcs can have at most one intersection point. Then there is an innermost triangle adjacent to an edge of the bigon. Applying the triangle move, we can reduce either the number of intersection points inside the bigon or the number of arcs intersecting the bigon. After all, we shall have an empty bigon which can be removed with eliminating two intersection points in the disc.

      Thus, the configuration can be reduced to a case with the minimal number of intersection points and lemma is proved.

      Note that the constructed homotopy will be regular everywhere except the moment when a loop is contracted to a point.

      We can define the disc flow as follows. Consider a curve s = s₀. It can have several components and self-intersections. We define a family of curves st, t ≥ 0 inductively. Given a curve s_i−1, i ∈ , we reduce the radius of the disc Di by a factor so that s_i−1 is in general position with the boundary ∂Di. It means that s_i−1 and ∂Di intersect transversely and ∂Di contains no self-intersection points of s_i−1. Then we define st, t ∈ [i − 1, i] as the image of s_i−1 under the straightening homotopy of Lemma 3.2 in the (shrinked) disc Di. Note that the length of curves st is not monotonous but the length sequence of the curves si, i ∈ , is non increasing. The family of curves st, t ≥ 0, is called the disk flow of the curve s = s₀.

      The disc flow is not canonical since the choice of homotopy in Lemma 3.2 is not unique. Nonetheless, it possesses several useful properties whose formulation requires some additional notation. Let Δ be the map defined on the set of curves F which transforms a curve s = s₀ to the curve sn obtained from s by consecutive straightening in the discs D₁, . . . , Dn. When we talk about convergence of curves, we use the topology in the curve space such that the ε-neighbourhood of a curve γ consists of curves γ′ that admits a parametrisation for which the curves γ and γ′ are ε-close in the Frechet topology. This topology is induced by the Frechet metric in the space of curves.

      Theorem 3.5. Let γ, γ′ be transversally intersecting curve in a surface F. Let D₁, . . . , Dn be a disc covering of the surface F which is good with respect to γ ∪ γ′. Let γt be the images of the curves by the disc flow.

      (1)The number of self-intersection points of the curve γt does not increase with the grow of t ∈ [0, +∞).

      (2)The number of intersection points between the curves γt and does not increase with the grow of t, t ∈ [0, +∞).

      (3)Either γt disappears in a finite time or a subsequence of the curves {γt} converges to a geodesic with t → ∞. In the second case, if U is an open neighbourhood of the set of geodesics which are homotopic to γt, then there exists T > 0 such that γt belongs to U with t > T.

      (4)If the sequence {γi} converges to a geodesic γ_∞ with i → ∞, then length (Δ(γ)) converges to length (γ_∞).

      (5)Length(Δ(γ)) ≤ length(γ), and the equality takes place only if γ is a geodesic or a point.

      Proof. The first two properties follows from Lemma 3.2.

      Assume γi do not disappear, then