Local Invariants of Framed Fronts in 3-Manifolds

We start with recalling a series of standard definitions which may be found, for example, in [Arnold et al.1985] or [Arnold1990].

A contact structure on a $(2n-1)$-dimensional manifold is a non-denerate field of tangent hyperplanes (called contact hyperplanes). Locally, this is a field of zeros of a differential 1-form $\alpha$ (called a contact form), and the non-degeneracy condition is that $\alpha{\wedge}(d\alpha)^{n-1}$ must be a local volume form on the manifold. A manifold equipped with a contact structure is called a contact manifold.

A Legendrian submanifold of a $(2n-1)$-dimensional contact manifold is its $(n-1)$-dimensional submanifold integral for the contact structure.

A fibration $p\colon E^{2n-1}\to B^{n}$ of a contact manifold $E$ over a base $B$ is called Legendrian if all its fibres are Legendrian submanifolds. A composition $M\stackrel{i}{\to}E\stackrel{p}{\to}B$ where $i$ is an embedding of a manifold $M^{n-1}$ into $E$ as a Legendrian submanifold is what is usually called a Legendrian map. However, in this paper we will allow $i$ to be a Legendrian immersion.

The image in $B$ of a Legendrian map $p\circ i$ in called a wave front or just a front.

A contact element of a manifold $N^{n}$ 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 of $ST^{*}\!N$ is a pair $(\nu,\pi)$ consisting of a point $\nu\in N$ and a framed hyperplane $\pi\subset T_{\nu}N.$ In the tangent space to $ST^{*}\!N$ at $(\nu,\pi)$, consider the hyperplane mapped to $\pi$ under the projection $\rho:ST^{*}\!N\to N.$ The field of all such hyperplanes is the standard contact structure on $ST^{*}\!N,$ and the projection $\rho$ is a Legendrain fibration. The fronts in $N$ of (immersed) Legendrian submanifolds of $ST^{*}\!N$ are naturally framed, at least at their regular points. These are the fronts considered in this paper.

All Legendrian fibrations of the same dimension are locally isomorphic. All through this paper we will be considering the $n=3$ case, and in all our local normal forms, we will be using the traditional local model for this dimension which is the space $J^{1}(\mathbb{R}^{2},\mathbb{R})$ of 1-jets of functions on the plane fibred over $J^{0}(\mathbb{R}^{2},\mathbb{R})\simeq\mathbb{R}_{u}\times\mathbb{R}^{2}_{v,w}$. The contact form here is $\alpha=du-Vdv-Wdw,$ where $V,W$ are coordinates along the fibres of the fibration. The fronts will be framed by the positive $u$-direction. For the orientation of the 3-space we take $du\wedge dv\wedge dw.$

A germ of a Legendrian surface $L\subset J^{1}(\mathbb{R}^{2}_{v,w},\mathbb{R}_{u})$ is defined by its generating family of functions $F(x,v,w)$:

Sometimes it is convenient to refer to this surface as defined by a generating family $u=F(x,v,w)$ of hypersurfaces. The minimal dimension of the variable $x$ here is the corank of the derivative of the projection $L\to\mathbb{R}^{2}_{v,w}$ at the base point. The smoothness of $L$ requires that the rank of the matrix $(F_{x})_{x,v,w}$ of the second derivatives at the base point must be equal to the dimension of $x$.

An equivalence of two Legendrian maps $L_{j}\to E_{j}\to B_{j},$ $j=1,2,$ is a commutative diagram

in which $\varphi$ is a contactomorphism, and the other two vertical arrows are diffeomorphisms. In terms of local generating families $\Phi(x,u,v,w)=0$ of hypersurfaces this corresponds to the contact equivalence of functions $\Phi$ preserving the fibration $(x,u,v,w)\mapsto(u,v,w)$ (see [Arnold et al.1985]).

The invariants we are going to construct will be expressed in terms of the geometry of the natural stratification of fronts. We shall now remind this stratification. The strata notation follows that of isolated function singularities whose versal deformations serve as generating families for the fronts (see, for example, [Arnold et al.1985] or [Arnold1990]).

A regular point of a generic front ${\mathcal{F}}$ corresponds to an $A_{1}$ singularity of a function in a generating family. Hence the set of all such points will be denoted $A_{1}$.

Irregular points of ${\mathcal{F}}$ are:

The signs in the notations of the last two pairs of singularities distinguish between different framing choices as shown in Fig. 1.

Local generating families of hypersurfaces for the one-component singularities are:

In what follows $M$ will always be a non-oriented compact surface, and $N$ an oriented 3-manifold, both without boundaries. Denote by ${\mathcal{L}}={\mathcal{L}}(M,N)$ the space of all Legendrian maps $M\looparrowright ST^{*}\!N\to N$, where the first arrow is a Legendrian immersion and the second the canonical projection. Maps whose fronts have more complicated singularities than those described above form the discriminantal hypersuface $\Xi$ in ${\mathcal{L}}.$ For example, if $M\looparrowright ST^{*}\!N$ is not an embedding, then the corresponding Legendrian map is in $\Xi.$

Consider connected components of ${\mathcal{L}}{\setminus}\Xi.$ A numerical invariant is a way to assign numbers to each of them. Along a generic path in ${\mathcal{L}},$ the values of an invariant change at the moments of discriminant crossings.

Considering invariants whose value set is different from $\mathbb{Z}_{2}$, we must have the discriminant locally co-oriented.

A local invariant $I$ defines its derivative $I^{\prime}=\sum_{i}x_{i}X_{i},$ where the $X_{i}\subset\Xi$ are the strata of codimension 1 in ${\mathcal{L}}$ we are able to distinguish for the needs of Definition 2.1, and the $x_{i}$ are the local increments of $I$ along generic paths in ${\mathcal{L}}$ crossing the $X_{i}$ in the co-orienting direction. On the other hand, $I$ is defined by $I^{\prime}$ on each connected component ${\mathcal{L}}_{j}$ of ${\mathcal{L}}$ up to a choice of ‘a constant of integration’, that is, up to an arbitrary choice of the value of $I$ at a non-discriminantal base point in ${\mathcal{L}}_{j}$.

Since the total increment of $I$ along any loop in $L_{j}$ vanishes, the derivative $\sum_{i}x_{i}X_{i}$ must be a trivial codimension 1 cycle in ${\mathcal{L}}_{j}$. The vanishing of the total increment on contractible loops (that is, the derivative being a cycle, maybe non-trivial) is equivalent to its vanishing on small loops in ${\mathcal{L}}$ around codimension 2 strata of the discriminant. Finding the relevant cyclicity constraints on the increments $x_{j}$ is the problem on which we concentrate in this paper. As it was said in Sect. 1, cycles of the form $\sum_{i}x_{i}X_{i}$ will be called discriminantal.

One of the ways to establish the triviality of a cycle $I^{\prime}=\sum_{i}x_{i}X_{i}$ is to find an integral (that is, path-independent) interpretation of its antiderivative $I$ in terms of the geometry of individual fronts.

Another obvious local invariant for fronts in $\mathbb{R}^{3}$ or $S^{3}$ is $I_{\ell}$ the self-linking number of the oriented framed link defined by the cuspidal edge of ${\mathcal{F}}$ (as shown in Fig. 2).

In Sect. 4.1 we will introduce yet another candidate for local invariants, $I_{\Sigma^{2}},$ counting an algebraic number of corank 2 points of Legendrian maps in homotopies. We do not have its integral interpretation: we only know its derivative which may happen to be a non-trivial cycle in ${\mathcal{L}}.$

All statements in this section refer to any compact surface $M$ and oriented 3-manifold $N.$

The first main result of this paper is

This implies

The exact value of the rank here depends on $I^{\prime}_{\Sigma^{2}}$ being or not being a trivial cycle.

Let ${\mathcal{L}}_{1}\subset{\mathcal{L}}(M,N)$ be the set of all Legendrian maps without corank 2 points. Discriminantal cycles in ${\mathcal{L}}_{1}$ do not contain any summands of $I^{\prime}_{\Sigma^{2}}.$ Our proof of Theorem 2.3 yields

One of the simplest events in generic one-parameter families of fronts is tangency of two smooth sheets. For framed fronts, we have two types of such self-tangencies: direct when the framings of the sheets at the touching point coincide, and opposite when they do not. We denote by $I^{\prime}_{dir}$ and $I^{\prime}_{opp}$ the derivatives of the $\mathbb{Z}_{2}$-valued functions counting the parities of the events in homotopies.

In the $\mathbb{Z}_{2}$ case we have

Prohibition of corank 2 points in homotopies cuts here the rank down to 7 and excludes $I^{\prime}_{\Sigma^{2}}$.

Depending on how many of the linear combinations of the mod2 discriminantal cycles $I^{\prime}_{dir},I^{\prime}_{opp}$ and $I^{\prime}_{\Sigma^{2}}$ turn out to be trivial, we have

Both theorems are proved in Sect. 5.4, with the preparations occupying Sect. 3 and the rest of Sect. 5.

We start with the bifurcations involving more than one local component of a front.

Following the notations introduced for singularities of a generic front, the notations of the bifurcations below are self-explanatory, with $T$ used for the tangency of the participating strata of a front. Letters $e$ and $h$ distinguish between elliptic and hyperbolic versions of similar bifurcations. Letter $r$ stays for the number of faces of the bounded region appearing after the bifurcation, which are framed outward the region. If there is no such bounded region, we make a special comment on the meaning of $r$. The figures illustrate only one particular value of $r$ from the range, or one particular choice of the sign. We use the orientation of the cuspidal edges as defined in Figure 2. We co-orient a discriminantal stratum in ${\mathcal{L}}$ by the direction of the shown sequence of the events in the bifurcation. In some cases we show only the ‘positive resolutions’ of the fronts and indicate by arrows how one of the local components has been moved during the bifurcations.

So, we have (see Figure 3):
$A_{1}^{4,r}$ $r=2,3,4$, intersection of four smooth sheets. The pre-bifurcation tetrahedral region has $4-r$ faces framed outwards. Therefore, the $r=2$ stratum $A_{1}^{4,2}$ is not co-orientable in ${\mathcal{L}}$. $TA_{1}^{3,r}$ $r=0,1,2,3$, three smooth sheets are pairwise transversal to each other, but the line of intersection of any two of them is tangent to the third sheet at the moment of bifurcation. $TA_{1}^{2,e,r}$ $r=0,1,2$, elliptic tangency of two smooth sheets. $TA_{1}^{2,h,r}$ $r=0,1$, same, but hyperbolic. We write $r=1$ if the sheets are framed to the same side, and $r=0$ if the framings are opposite. For $r=1$, the stratum is not co-orientable in ${\mathcal{L}}$.¡br¿ The strata $TA_{1}^{2,e,r}$ and $TA_{1}^{2,h,r}$ are exactly those responsible for the direct ($r=1$) and opposite ($r=0,2$) self-tangencies mentioned in Sect. 2.4. $A_{2}A_{1}^{2,\pm,\pm}$ cuspidal edge meets the intersection of two smooth sheets. The two signs are the signs of the two $A_{2}A_{1}$ points after the bifurcation, in the order in which the points lie locally on the cuspidal edge if we follow the edge orientation. $A_{2}^{2,e,\pm}$ two edges meet face-to-face. During the bifurcation the crossing sign of the two local oriented edges changes. The sign in the notation of the stratum is the crossing sign after the bifurcation. $A_{2}^{2,h}$ one of the edges is overtaking the other. The crossing sign of the two edges after the bifurcation is positive. $A_{3}^{\pm}A_{1}^{\pm}$ a smooth sheet passes through a swallowtail. The meaning of the $A_{1}$ sign is clear from the figure. $TA_{2}A_{1}^{e,\pm}$ cuspidal edge becomes tangent to a smooth sheet so that the two local components of ${\mathcal{F}}$ do not intersect before the bifurcation. The two cases differ by the framing of the smooth sheet. $TA_{2}A_{1}^{h,\pm}$ the hyperbolic version of the previous.

Our normal forms for generic 1-parameter uni-germ transformations follow [Arnold1976]; [Zakalyukin1983], and their illustrations in Fig. 4 correspond to [Arnold1990].

The sign of the real parameter $\lambda$ in the local normal forms for the generating families of hypersurfaces co-orients the strata in ${\mathcal{L}}$. Following what was said about the local normal forms in Sect. 2.2, the tangent plane to the $\lambda=0$ front at the origin is $u=0$ framed by the positive $u$-direction.

Altogether, we have identified 35 codimension 1 strata in ${\mathcal{L}}$, two of which, $A_{1}^{4,2}$ and $TA_{1}^{2,h,1}$, are not co-orientable.

It will be convenient for us to use sums of the strata differing only in certain indices in their notations. In such cases we will omit the corresponding signs or letter and assume that the summation is done across the whole range of the omitted symbols, for example:

To avoid confusion to which such abbreviations may lead in some cases, we will also use the notations similar to

Within these settings, inspection of Figs. 3 and 4 immediately implies

In particular, we see that in $\mathbb{R}^{3}$ or $S^{3}$, up to an additive constant, $I_{\ell}=(I_{c_{+}}-I_{c_{-}})/2.$

Within the lists of Sects. 3.1 and 3.2, the $\lambda=0$ members of the $D_{4}^{\pm}$ families are the only Legendrian maps having rank 0 at the origin. We will now express an algebraic number of such events in generic homotopies of fronts as an intersection number. The construction will imply that

for the possible local invariant $I_{\Sigma^{2}}$ mentioned in Sects. 2.3 and 2.4.

Every contact hyperplane of a contact structure is a symplectic linear space, with the symplectic form $d\alpha$ if $\alpha$ is the contact form. Moreover, the tangent space to a Legendrian submanifold at a point is a Legendrian subspace in the contact hyperplane. This provides a fibration

with a fibre $\Lambda_{2}$, the Lagrangian grassmannian of non-oriented Lagrangian planes of symplectic $\mathbb{R}^{4}$.

A family of Legendrian immersions $i_{t}\colon M\looparrowright ST^{*}\!N,$ $t\in[0,1],$ lifts to a family of embeddings $\widetilde{i}_{t}\colon M\to{\mathcal{E}},$ $m\mapsto T_{i_{t}(m)}i_{t}(M),$ $i_{t}=\phi\circ\widetilde{i}_{t}.$ Consider:

• •

  a 3-film ${\mathcal{Z}}\subset{\mathcal{E}},$ the union of the images of the embeddings $\widetilde{i}_{t},$ $t\in[0.1],$

• •

  the section ${\mathcal{V}}\subset{\mathcal{E}}$ of vertical Lagrangian planes, that is, the field of the kernels of the derivative of the projection $\rho\colon ST^{*}\!N\to N.$

In general position, ${\mathcal{Z}}$ and ${\mathcal{V}}$ meet at isolated non-boundary points of ${\mathcal{Z}}$, and $\phi$ projects these points exactly to the points of the Legendrian surfaces $i_{t}(M)$ at which their individual projections $\rho$ to the base $N$ have rank 0.

Our aim is to define an intersection number $\langle{\mathcal{V}},{\mathcal{Z}}\rangle$ in ${\mathcal{E}}$, that is, to assign signs to points of ${\mathcal{V}}\cap{\mathcal{Z}}.$ Consider one of these points, $z_{0}=\widetilde{i}_{t_{0}}(m_{0}),$ and the local trivialization $J^{1}(\mathbb{R}^{2}_{v,w},\mathbb{R}_{u})\times\Lambda_{2}$ of ${\mathcal{E}}$ near it, with $\phi(z_{0})=i_{t_{0}}(m_{0})$ being the origin in the first factor for which we will be using the coordinate description given in Sect. 2.1. Projection onto the second factor gives us a map germ $\pi\colon({\mathcal{Z}},z_{0})\to\Lambda_{2},$ and reduces the task to assigning a sign to $\pi.$ We will call $\pi(z_{0})$ the vertical point of $\Lambda_{2}.$

The 3-manifold $\Lambda_{2}$ is non-orientable [Vassiliev1988], and ${\mathcal{Z}}$ is not oriented since $M$ is not. However, $M\times\{t_{0}\}$ is co-oriented in $M\times[0,1]$ by the positive $t$-direction, and, according to the lemma, the family of the maps $\pi\circ\widetilde{i}_{t}$ will allow us to compare the two co-orientations, and thus obtain the sign we are looking for.

The intersection number $\langle{\mathcal{V}},{\mathcal{Z}}\rangle,$ calculated as the sum of the individual $\pm 1$ contributions across all points of ${\mathcal{V}}\cap{\mathcal{Z}},$ is an invariant of oriented paths in the space ${\mathcal{L}}(M,N)$ connecting maps $\rho\circ i_{0}$ and $\rho\circ i_{1}.$ In Sect. 5.4 we will show that the sum $D_{4}^{+}+D_{4}^{-}$ is a cycle in ${\mathcal{L}}.$ Therefore, the number $\langle{\mathcal{V}},{\mathcal{Z}}\rangle$ is preserved in homotopies of paths with fixed ends. Existence of a well-defined local invariant $I_{\Sigma^{2}}$ on ${\mathcal{L}}$ for which

is equivalent to vanishing of $\langle{\mathcal{V}},{\mathcal{Z}}\rangle$ on all non-contractible loops in ${\mathcal{L}}.$

We turn to the possible mod2 local invariant $I_{opp}$ of Sect. 2.4.

Opposite self-tangencies of fronts occur exactly at three types of the $TA_{1}^{2}$ strata: $TA_{1}^{e,0},$ $TA_{1}^{e,2}$ and $TA_{1}^{h,0}.$ In our proof of Theorem 2.6 in Sect. 5.4, we will show that the sum $TA_{1}^{2,opp}$ of these strata is a mod2 discriminantal cycle. We now give an example of a situation when this cycle is trivial.

Let $j$ be the involution of the space $ST^{*}\!N$ of framed contact elements of a manifold $N$ which changes the framing of each element to the opposite. An embedded Legendrian surface $i(M)\subset ST^{*}\!N$ has its antipodal surface $(j\circ i)(M).$

Assume $M$ is orientable and $N=\mathbb{R}^{3}.$ Take a direction in $\mathbb{R}^{3}$ and shift $(j\circ i)(M)$ along all vectors of this direction producing a semi-infinite cylinder ${\mathcal{C}}\subset ST^{*}\mathbb{R}^{3}$ parametrised by $M\times[0,\infty)$ and having the boundary $(j\circ i)(M).$ Choosing orientations on $ST^{*}\mathbb{R}^{3}$ and $M$, and taking the orientation on ${\mathcal{C}}$ so that $\partial C=-(j\circ i)(M),$ we obtain the integer intersection number $\ell_{opp}(i(M))=\langle{\mathcal{C}},(j\circ i)(M)\rangle.$ It does not depend on the choice of a direction in $\mathbb{R}^{3}$ and on orientation of $M.$

In generic homotopies of Legendrian surfaces, the value $\ell_{opp}(i(M))$ changes only at opposite self-tangencies of fronts—such moments correspond to intersections of a Legendrian surface with its antipodal surface. The increments are easily shown to be $\pm 2$ (see for example [Goryunov1997]). Therefore, the derivative of the mod2 local invariant $I_{opp}(\rho\circ i):=\ell_{opp}(i(M))/2$ is $TA_{1}^{2,opp}.$

We start with all 35 strata. Our initial goal is to reduce the number of unknown increments.

So far we have reduced the number of unknown increments down to 17, out of which 2 may be non-trivial only mod2.

The three 2-parameter deformations represented in Fig. 8 are of generating families induced from versal deformations of $A_{5},A_{4}$ and $D_{5}$ function singularities: the top left, top right and two bottom bifurcation diagrams are respectively of the families

In the $A_{5}$ case, the generating family is the most generic. In the $A_{4}$ case, it has codimension 1 within all families induced from $A_{4}$. Similar $D_{4}$ degenerations do not occur since they would correspond to a change of topology of the source surface $M.$

The local normal form of the $A_{4}$ family was obtained in [Goryunov1983], and the one for the uni-modal $A_{5}$ family may be found by very similar methods.

In the bottom row of Figure 8, the $\lambda_{1}=\lambda_{2}=0$ fronts are front singularities of infinite codimension. Moreover, the 2-parameter deformations chosen are of infinite codimension in the space of all 2-parameter deformations of these fronts: indeed, the $A_{4}^{+}$ and $A_{4}^{-}$ strata of the bifurcation diagrams coincide. However, the only difference of our bifurcation diagrams with those of generic 2-parameter families of finite-codimensional singularities defined by generating families with the same principal parts is a split of the two $A_{4}$ strata.

Figure 8 gives the following equations:

One can show that any further generic 2-parameter families of fronts (for example, passage of an extra $A_{1}$ sheet through a quadruple or $D_{4}$ bifurcation, or taking other framing options in Fig. 6, left) are not able to provide us with any equations on the increments independent from those we have already obtained.

We collect Eqs. 9–17 into two tables below, for arbitrary and mod2 coefficients. Each of the two tables consists of two parts. The left one contains the equations themselves in the order they have appeared (we are using dots for zero coefficients), while the right lists the discriminantal cycles as linear combinations of the strata (with the notations as in Lemma 3.1). The bars in the first table indicate the equations which are integer linear combinations of the others.

Here $I^{\prime}_{0}=\left(I^{\prime}_{s}/2+\widehat{I}^{\prime}_{s}/2+\widehat{I}^{% \prime}_{c}/2-3I^{\prime}_{\Sigma^{2}}\right)/2=\left(2I^{\prime}_{s_{+}}+I^{% \prime}_{c_{+}}-I^{\prime}_{c_{-}}-6I^{\prime}_{\Sigma^{2}}\right)/4.$

The rank count in the left part of the table shows that the space of integer discriminantal cycles is six-dimensional, and the right part of the table tells us that a basis of this integer space is formed by the six cycles excluding $\widehat{I}^{\prime}_{c}/2.$ This proves Theorem 2.3.

Adjusting the above table to the $\mathbb{Z}_{2}$ case, we use Eq. 11 to make a big stratum $A_{2}^{2.e/h}$, then exclude the $\left[A_{1}^{4}\right],$ $TA_{1}^{3}$ and $A_{2}A_{1}^{2}$ strata (along with the Eqs. 13, 12 and 14 in which exactly one of them appears now on its own), and drop the duplicate and $0=0$ equations. This leaves us with 3 independent equations in 11 unknowns:

The space of mod2 invariants has rank 8. There is one relation on the 9 discriminantal cycles listed in the table: $I^{\prime}_{s}/2+\widehat{I}^{\prime}_{s}/2+\widehat{I}^{\prime}_{c}/2+I^{% \prime}_{\Sigma^{2}}\equiv 0.$ Our proof of Theorem 2.6 is finished.