In the next three chapters we will be considering framed and/or oriented
fronts. The orientation of a front will be coming from the orientation of the
compact surfaceM in the Legendrian mapM →N. We show the orientation
of the front at its regular points by a co-orientation, so that this co-orientation followed by the orientation of the front is the orientation of N.
The framing of the fronts will be coming from the consideration of Leg-
endrian surfaces in the ST∗N: their projections to N provide our framed
fronts.
In more details, a contact element of the 3-manifold N is a hyperplane in
the tangent space to N at a point. A contact element with one of its sides
in the tangent space distinguished is usually called co-oriented, but we have
already mentioned that we will call it framed. The set of all framed contact
elements of N is the spherisation ST∗N of the cotangent bundle of N.
A point ofST∗N is a pair (ν,Π) consisting of a pointν ∈N and a framed hyperplane Π ⊂ TνN. In the tangent space to ST∗N at (ν,Π), consider the
hyperplane mapped to Π under the projection ρ:ST∗N →N.The field of all
such hyperplanes is the standard contact structure on ST∗N,and the projec-
tion ρ is a Legendrian fibration. The fronts in N of (immersed) Legendrian
at a point p ∈ L⊂ ST∗N, there is a framed contact 4-plane P tangent to L which is projected by ρ to a contact element of N tangent to the front ρ(L) at ρ(p) and inheriting the framing from that of P. Our fronts will be framed this way.
Remark 3. We would like once again to emphasise the use of the terminology in this thesis which may be somehow non-traditional:
• a front’s framing at a point is a choice of the positive side of the 2- plane tangent to the front at this point (this is, what is usually called a co-orientation of the front);
• our co-orientation of a front is a tool to inform aboutthe orientation of the projected Legendrian surface (see the first paragraph of this section);
• we will be also using another co-orientation, that of the discriminantal strata of codimension 1 in the space of all our Legendrian maps: the direction of this co-orientation is a fixed choice of one of the two possible directions in the corresponding generic 1-parameter local bifurcation of a front. All such bifurcations will be depicted and their positive directions will be explicitly specified.
Chapter 2
Local invariants of framed
oriented fronts
In this chapter we take our fronts to be framed and oriented as explained in Section 1.3. In all our figures, we assume that the ambient 3-space is taken with the right orientation (that is, the standard one). When writing the normal forms of generating families we are considering the contact structure onJ1(R2v,w,Ru) which is mentioned in the previous chapter. Very frequently in our figures we will be allowing various choices of the framing and orientation
like it is shown in Figure 2.1, where we are using α = ± for the framing
direction and β=± for showing the orientation.
In all our notations, the upper signs will be representing framings and the lower orientations.
α = −, β = + α = −, β = − α = +, β = − α = +, β = +
α β
Figure 2.1: Framing and showing the orientation of a smooth sheet.
2.1
Stratification of generic framed oriented
fronts in oriented 3-manifolds
A regular point of a generic front F corresponds to an A1 singularity of a
function in a generating family. Hence the set of all such points will be
denoted A1. Irregular points of F are:
• A21, transversal intersections of two smooth sheets. Taking the local region of the ambient space into which the framings of both sheets are directed, we distinguish 3 types of such strata A21,β1β2 with β1β2 =
++,+− or − − shown Figure 2.2;
• A31, same for three sheets. Considering the local quadrant of the ambient space into which the framings of all three sheets are directed, we get 4 types here: A31,β1β2β3, withβ1β2β3 = + + +,+ +−,+− −,− − −. Very
frequently it will be convenient to allow permutations of the subscripts with the understanding that, for example A31,−++ =A13,+−+ =A31,++−;
• A2±, cuspidal edges. The sign in the edge notation indicates the co-
Figure 2.2: Types of stable singularities of framed oriented fronts: A2 1,++, A21,+− and A21,−−.
natural orientation of the edge as shown in Figure 2.3;
1 3
2 Orientation of N
edge orientation
front framing
Figure 2.3: Edge orientation.
• A2A1, transversal intersections of cuspidal edges with regular sheets.
There are 8 types which differ by the sign of the edge, and the fram- ing and co-orientation of the smooth sheet. If the framing or the co- orientation of the sheet coincide with the edge orientation we decorate
Figure 2.4: Types of A2 singularities.
the A1 with a plus, and if it is opposite we decorate with a minus as
shown in Figure 2.5. In the left of the figure we show the general case A2σA1,βα; σ β α σ = + σ = − α = +, β = + α = +, β = − α = −, β = + α = −, β = − α = +, β = − α = −, β = + α = +, β = + α = −, β = −
Figure 2.5: Various A2A1 points of a front.
• A3,±±, swallowtail points. There are four types of swallowtails in this setting shown in Figure 2.6.
A
+ 3, −A
3, − +A
3, − −A
+ 3, +A
α 3, β α βFigure 2.6: Types of A3 singularities.