| Sbornik: Mathematics |
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Topological properties of caustics in five-dimensional spaces V. D. Sedykh National University of Oil and Gas "Gubkin University", Moscow, Russia
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    § 1. IntroductionCaustics are the sets of critical values of so-called Lagrangian maps (for all necessary definitions and facts, see [1] and [6]). Examples of caustics are light caustics, evolutes of plane curves, focal sets of hypersurfaces and other envelopes of systems of rays of different nature. A generic caustic is a singular hypersurface. The singular points of this hypersurface are described by Arnold’s theorem on Lagrangian singularities. Namely, the germs of a generic Lagrangian map $f\colon L\to V$ of a smooth manifold $L$ to a smooth manifold $V$ of the same dimension $n\leqslant5$ (both without boundary) are Lagrangian equivalent to the germs at the origin of the map
$$
\begin{equation*}
\mathbb{R}^n\to \mathbb{R}^n, \qquad (t,q)\mapsto \biggl(-\frac{\partial S(t,q)}{\partial t},q\biggr), \qquad t=(t_1,\dots,t_k), \qquad\! q=(q_{k+1},\dots,q_n),
\end{equation*}
\notag
$$
defined by the function $S=S(t,q)$ of one of the following types, depending on positive integers $\mu\leqslant n+1$:
$$
\begin{equation*}
\begin{aligned} \, &A_{\mu}^\pm\colon \qquad S=\pm t_1^{\mu+1}+q_{\mu-1}t_1^{\mu-1}+\dots+q_2t_1^2, \quad\mu\geqslant1, \\ &D_{\mu}^{\pm}\colon \qquad S=t_1^2t_2\pm t_2^{\mu-1}+q_{\mu-1}t_2^{\mu-2}+\dots+q_3t_2^2, \quad\mu\geqslant4, \\ &E_6^\pm\colon \qquad S=t_1^3\pm t_2^4+q_5t_1t_2^2+q_4t_1t_2+q_3t_2^2, \quad\mu=6. \end{aligned}
\end{equation*}
\notag
$$
The equivalence class of a Lagrangian map germ at a critical point with respect to Lagrangian equivalence is called a (Lagrangian) singularity. The type of the function $S$ determines the type of a Lagrangian map germ and its singularity. The number $\mu-1$ is called the codimension of the singularity. If $\mu$ is even or $\mu=1$, then germs of types $A_{\mu}^{+}$ and $A_{\mu}^{-}$ are Lagrangian equivalent (their types are denoted by $A_{\mu}$). In other cases, the germs of the types listed above are pairwise Lagrangian nonequivalent. Remark 1. We consider Lagrangian maps in the Whitney $C^{\infty}$-topology. Lagrangian singularities of types $A_{\mu}^\pm,D_{\mu}^\pm$ and $E_6^\pm$ are simple (that is, have zero modality) and stable. For $n>5$ there are singularities that have functional moduli depending on $n$ variables (coordinates in the target space; see the classification of Lagrangian singularities in [2] or Example 17.12 in [6]). These singularities cannot be removed by a small (Lagrangian) deformation of the map. Let $y$ be an arbitrary point in the target space $V$ of a generic proper Lagrangian map $f$. Consider the unordered set of symbols from Arnold’s theorem that are the types of the germs of $f$ at preimages of $y$. The formal commutative product $\mathcal{A}$ of these symbols is called the type of the multisingularity of $f$ at the point $y$ (or the type of the monosingularity if $y$ has only one preimage). If $f^{-1}(y)=\varnothing$, then $\mathcal{A}=\mathbf{1}$. The types of multisingularities belong to the free Abelian multiplicative semigroup $\mathbb{S}^+$ with identity element $\mathbf{1}$ and generators
$$
\begin{equation*}
A_1,\ A_{2},\ A_{4},\ A_{6},\ A_3^\pm,\ A_5^\pm,\ D_4^\pm,\ D_5^\pm,\ D_6^\pm\text{ and } E_6^\pm.
\end{equation*}
\notag
$$
The set $\mathcal{A}_f$ of points $y\in V$ at which $f$ has a multisingularity of type $\mathcal{A}\in\mathbb{S}^+$ is a smooth submanifold of the space $V$. It is called the manifold of multisingularities of type $\mathcal{A}$. The codimension of $\mathcal{A}_f$ in $V$ is equal to the sum of the codimensions of singularities of $f$ at all preimages of an arbitrary point $y\in \mathcal{A}_f$. This sum is called the codimension of a multisingularity of type $\mathcal{A}$ and is denoted by $\operatorname{codim}\mathcal{A}$. If $\mathcal{A}_f\neq\varnothing$, then $\operatorname{codim}\mathcal{A}\leqslant n$. The type of the multisingularity of a Lagrangian map $f$ at a critical value $y$ determines the type of the germ of its caustic at this point. Two germs of a caustic are diffeomorphic if and only if their types either coincide or differ in the number of factors $A_1$ and the signs in the superscripts of factors of the form $A_{2k+1}^\pm$, $D_{2k+1}^\pm$ and $E_6^\pm$. Therefore, the factors $A_1$ and the superscripts of the symbols of the forms indicated are often not written when we speak about the types of germs of a caustic and their singular points (but not about the types of multisingularities of Lagrangian maps). The set of points $y$ at which the caustic of $f$ has a singularity of a given type is a smooth submanifold of $V$. It is a disjoint union of manifolds $\mathcal{A}_f$ such that $\mathcal{A}\in\mathbb{S}^+$ determines the type of the singularity of a caustic that is under consideration. It is well known (Thom and Arnold) that the number of singularities of type $A_4$ (a swallowtail) and the total number of singularities of types $D_4^+$ (a purse) and $D_4^-$ (a pyramid) on a generic compact caustic in a three-dimensional space are even. Vassiliev proved the evenness of the following integers (see [3]): the number of type $D_5$ singularities on a generic compact caustic in a four-dimensional space; the total number of type $D_6^\pm$ singularities and the total number of type $A_{6}$ and type $E_6$ singularities on a generic compact caustic in a five-dimensional space. We proved in [5], Corollary 16.3, that the number of type $A_4A_2$ singularities and the total number of singularities of types $D_4^\pm A_2$ on a generic compact caustic in a four-dimensional space are even too. In this paper we find new coexistence conditions for isolated singularities of caustics in five-dimensional spaces. Let the numbers of isolated singularities of types $E_6$, $D_5A_2$, $D_4^+A_3$, $D_4^-A_3$, $D_4^+A_2^2$, $D_4^-A_2^2$, $A_4A_3$ and $A_4A_2^2$ of a caustic be denoted by the same respective symbols. Then the integers
$$
\begin{equation}
D_5A_2, \ A_4A_3, \ A_4A_2^2, \ D_4^+A_3+D_4^-A_3+E_6 \text{ and } D_4^+A_2^2+D_4^-A_2^2+\frac12A_4A_3
\end{equation}
\tag{1.1}
$$
are even for any generic compact caustic in a five-dimensional space. This is a corollary of Theorem 3, where two more new congruences modulo $2$ between the Euler characteristics of even-dimensional manifolds of caustic singularities are presented. Theorem 3 follows from the universal linear relations between the Euler characteristics of manifolds of singularities of a caustic listed in Theorem 2. The universality of these relations means that their coefficients (rational numbers) do not depend on the map. Finally, Theorem 2 follows from Theorem 1, where the list of universal linear relations between the Euler characteristics of manifolds of Lagrangian multisingularities is presented. This list is the main result of our paper. We obtain it by computer-assisted calculations using the results from [5] and [7] on the adjacency indices of Lagrangian multisingularities. The algorithm of calculations is described in § 3. § 2. The main resultSuppose that a generic proper Lagrangian map $f\colon L\to V$ has a multisingularity of type $X$ at a point $y\in V$, where $\operatorname{codim}X=c$. Fix a neighbourhood $U$ of the origin $0$ in $\mathbb{R}^{c}$, and consider a smooth embedding $h\colon U\to V$ such that ${h(0)=y}$ and the submanifold $h(U)\subset V$ is transversal to the manifold $X_f$ at $y$. Let ${B_{\varepsilon}\subset \mathbb{R}^{c}}$ be the open $c$-dimensional ball of radius $\varepsilon>0$ with centre $0$. Then there is a positive number $\varepsilon_0=\varepsilon_0(f,y,h)$ such that for all $\mathcal{A}\in\mathbb{S}^+$ and $\varepsilon<\varepsilon_0$ the set $h(B_{\varepsilon})\cap \mathcal{A}_f$ is a smooth manifold, and the equivalence class of this manifold under diffeomorphisms depends only on $\mathcal{A}$ and $X$. We denote this manifold by $\Xi_{\mathcal{A}}(X)$. Its Euler characteristic (the alternating sum of ranks of the usual homology groups, that is, homology groups with compact supports) is denoted by $J_{\mathcal{A}}(X)$. We say that a multisingularity of type $X$ is adjacent to a multisingularity of type $\mathcal{A}$ if $\mathcal{A}\neq X$ and $\Xi_\mathcal{A}(X)\neq\varnothing$. The number $J_{\mathcal{A}}(X)$ is called in this case the adjacency index of the multisingularity of type $X$ to the multisingularity of type $\mathcal{A}$. The adjacency indices of monosingularities are presented in [5] and [7]. The adjacency indices of multisingularities are calculated using Theorem 5.1 in [5]. Assume that the closure of the manifold $\mathcal{A}_f$ is compact. Let $\chi_f(\mathcal{A})$ denote the Euler characteristic of $\mathcal{A}_f$. Then
$$
\begin{equation}
\chi_f(\mathcal{A})=\frac12\sum_{X\in\mathbb{S}^+\setminus\{\mathcal{A}\}} (-1)^{n-\operatorname{codim} X}J_\mathcal{A}(X)\chi_f(X)
\end{equation}
\tag{2.1}
$$
if $\operatorname{codim}\mathcal{A}\equiv n-1\pmod{2}$. This is the formula (14.1) in [5]. It follows from the fact that the Euler characteristic of an odd-dimensional compact manifold with boundary is one half of the Euler characteristic of the boundary. Formula (2.1) defines a homogeneous system of linear equations between the Euler characteristics of manifolds of multisingularities of the map $f$. This system can easily be solved with respect to the Euler characteristics of odd-dimensional manifolds of multisingularities. The corresponding formulae for Lagrangian maps to three- and four-dimensional spaces are presented in [5], Theorems 14.1 and 14.3. The answer for five-dimensional spaces is given below. Remark 2. We adopt the following conventions: a generator $X_\mu^\delta$ of the semigroup $\mathbb{S}^+$ is $X_\mu^+$ if $\delta=+1$ and $X_\mu^-$ if $\delta=-1$; the number $\chi_f(\mathcal{A}A_1^k)$ is equal to zero if $\mathcal{A}$ does not contain generators $A_1$ and $k<0$. Let $L$ and $V$ be smooth manifolds of dimension $n=5$. To simplify formulae below we denote the Euler characteristic of the manifold $\mathcal{A}_f$ by $\mathcal{A}$. Theorem 1. The following relations between the Euler characteristics of the manifolds $\mathcal{A}_f$: The list of formulae in (2.2)–(2.12) was obtained by computer-assisted calculations. In § 3 we describe a quick way to a system of equations equivalent to the system of relations (2.1). Remark 3. Apparently, Theorem 1 describes a complete system (in the sense of [4]) of universal linear relations with real coefficients between the Euler characteristics of manifolds of multisingularities for generic Lagrangian maps of compact manifolds to five-dimensional spaces. To prove this, for each type of multisingularities of odd codimension one must construct a stable Lagrangian map that has multisingularities of only this type in the highest codimension and such that the modulus of the Euler characteristic of the manifold of all such multisingularities is greater than any prescribed positive number. This has not been done yet. Now assume that $\mathcal{A}\in\mathbb{S}^+$ does not contain the factors $A_1$, and let $\mathcal{A}_f^{\mathrm{ca}}$ be the disjoint union of manifolds $(\mathcal{A}A_1^k)_f$ over all nonnegative integers $k$. To simplify formulae below we also denote the Euler characteristic $\chi_f(\mathcal{A}^{\mathrm{ca}})$ of the manifold $\mathcal{A}_f^{\mathrm{ca}}$ by $\mathcal{A}$. Theorem 2. The following relations between the Euler characteristics of manifolds $\mathcal{A}_f^{\mathrm{ca}}$: This statement can be deduced from Theorem 1 using simple calculations. In particular, Theorem 2 implies the following result. Theorem 3. The following congruences modulo $2$ between the Euler characteristics of manifolds $\mathcal{A}_f^{\mathrm{ca}}$: and are valid for $\delta=\pm1$ and each generic proper Lagrangian map $f\colon L\to V$ such that the set of singular points of its caustic is compact. Remark 4. Congruences (2.24) and (2.25) imply the results of Vassiliev [3], mentioned in § 1, on singularities of types $A_6$, $E_6$ and $D_6^\pm$ on caustics in five-dimensional spaces. Remark 5. The evenness of the last number in the list (1.1) leads to the combination $2(D_4^+A_2^2+D_4^-A_2^2)+A_4A_3$, which is divisible by $4$. We do not know other integers $p>2$ for which there exist universal linear combinations with relatively prime integer coefficients not congruent to zero modulo $p$ that are composed of the numbers of isolated singularities of different types on generic compact caustics in five-dimensional spaces and are divisible by $p$. Remark 6. The completeness of the list of congruences from Theorem 3 remains at the level of a conjecture so far. To prove it, it suffices to construct stable Lagrangian maps to a five-dimensional space whose caustics are compact such that the parities of the numbers of multisingularities of the $17$ types form an arbitrary prescribed set. We transform formulae (2.13)–(2.23) using the following notation:
$$
\begin{equation*}
\begin{gathered} \, A_3^+X+A_3^-X=A_3X, \qquad A_5^+X+A_5^-X=A_5X, \qquad D_5^+X+D_5^-X=D_5X, \\ E_6^+X+E_6^-X=E_6X\quad\text{and} \quad (A_3^+)^2X+(A_3^-)^2X+A_3^+A_3^-X=A_3^2X \end{gathered}
\end{equation*}
\notag
$$
for any $X\in\mathbb{S}^+$ not containing factors equal to $A_1$. Then we obtain the following result. Theorem 4. The following relations between the Euler characteristics of manifolds of singularities of caustics: Theorem 4 is less informative than Theorem 2. For example, the evenness of the last two numbers in the list (1.1) does not follow from (2.33)–(2.42). Remark 7. The formulae for Lagrangian maps to three-dimensional spaces in [5], Theorem 14.1, follow from (2.2)–(2.12) if we put $\chi_f(\mathcal{A})=0$ for all $\mathcal{A}\in\mathbb{S}^+$ such that $\operatorname{codim} \mathcal{A}\!>\!3$. Similarly, the formulae in [5], Corollary 16.1, follow from (2.13)–(2.23). § 3. CalculationsLet $X_1,\dots,X_N$ be the generators of the semigroup $\mathbb{S}^+$. Let $\mathbb{Z}[\mathbb{S}^+]$ denote the semigroup algebra of $\mathbb{S}^+$. This is the algebra of polynomials in $X_1,\dots,X_N$ with integer coefficients. The types of multisingularities of a generic proper Lagrangian map $f\colon L\to V$ are monomials $X=X_1^{k_1},\dots,X_N^{k_N}$ in this polynomial algebra. The substitution $X_i\to Y_i$, $i=1,\dots,N$, determines an isomorphism ${\mathbb{Z}[\mathbb{S}^+]\to\mathbb{Z}[\mathbb{S}^+]}$. Consider an additive homomorphism
$$
\begin{equation*}
J\colon \mathbb{Z}[\mathbb{S}^+]\to\mathbb{Z}[\mathbb{S}^+]
\end{equation*}
\notag
$$
defined on monomials by
$$
\begin{equation*}
J(X)=\sum_{\mathcal{A}\in\mathbb{S}^+}(-1)^{\operatorname{codim} \mathcal{A}}J_{\mathcal{A}}(X)\mathcal{A},
\end{equation*}
\notag
$$
where $\mathcal{A}=Y_1^{l_1}\cdots Y_N^{l_N}$. By Corollary 5.3 in [5], $J$ is a ring homomorphism, and it is inverse to itself, that is,
$$
\begin{equation*}
J(\mathcal{B}\mathcal{C})=J(\mathcal{B})J(\mathcal{C})\quad \text{and}\quad J(J(\mathcal{B}))=\mathcal{B}
\end{equation*}
\notag
$$
for all $\mathcal{B},\mathcal{C}\in\mathbb{Z}[\mathbb{S}^+]$. The action of $J$ on generators of the semigroup $\mathbb{S}^+$ is described by the following formulae:
$$
\begin{equation*}
\begin{aligned} \, J(1)&=1, \\ J(A_1)&=A_1, \\ J(A_2)&=1+A_1^2-A_2, \\ J(A_3^\delta)&=A_1+A_1^3-2A_2A_1+A_3^\delta, \\ J(A_4)&=1+A_1^2+A_1^4-A_2-3A_2A_1^2+A_2^2+A_3^+A_1+A_3^-A_1-A_4, \\ J(D_4^\delta)&=\frac{1+\delta}{2}(1+2A_1^2-2A_2+2A_2^2)+\frac{3-\delta}{2}A_1^4 -(5-\delta)A_2A_1^2 \\ &\qquad+(2-\delta)(A_3^+A_1+A_3^-A_1)-D_4^{\delta}, \\ J(A_5^\delta)&=A_1+A_1^3+A_1^5-2A_2A_1-4A_2A_1^3+3A_2^2A_1+A_3^\delta+2A_3^\delta A_1^2+A_3^{-\delta}A_1^2 \\ &\qquad-2A_3^\delta A_2-2A_4A_1+A_5^\delta, \\ J(D_5^\delta)&=A_1+A_1^3+2A_1^5-3A_2A_1-9A_2A_1^3+6A_2^2A_1+A_3^++A_3^-+4A_3^+A_1^2 \\ &\qquad+4A_3^-A_1^2-2A_3^+A_2-2A_3^-A_2-2A_4A_1-D_4^+A_1-D_4^-A_1+D_5^\delta, \\ J(A_6)&=1+A_1^2+A_1^4+A_1^6-A_2-3A_2A_1^2-5A_2A_1^4+A_2^2+6A_2^2A_1^2+A_3^+A_1 \\ &\qquad+A_3^-A_1+2A_3^+A_1^3+2A_3^-A_1^3-A_2^3-3A_3^+A_2A_1-3A_3^-A_2A_1-A_4 \\ &\qquad-3A_4A_1^2+2A_4A_2+A_3^+A_3^-+A_5^+A_1+A_5^-A_1-A_6, \\ J(D_6^\delta)&=\frac{1+\delta}{2}(1+2A_1^2+2A_1^4-2A_2+3A_2^2-4A_2^3-2A_4+4A_4A_2)+\frac{5-\delta}{2}A_1^6 \\ &\qquad+\frac{33-\delta}{2}A_2^2A_1^2+\frac{5+\delta}{2}(A_3^+A_1+A_3^-A_1) +\frac{13-3\delta}{2}(A_3^+A_1^3+A_3^-A_1^3) \\ &\qquad-(5+3\delta)A_2A_1^2-2(7-\delta)A_2A_1^4-(9-2\delta)(A_3^+A_2A_1+A_3^-A_2A_1) \\ &\qquad+(A_3^+)^2+(A_3^-)^2+(1-\delta)(A_3^+A_3^-+A_5^+A_1+A_5^-A_1)-(5-\delta)A_4A_1^2 \\ &\qquad-D_4^\delta-2D_4^{\delta}A_1^2-D_4^{-\delta}A_1^2+2D_4^\delta A_2+D_5^+A_1+D_5^-A_1-D_6^\delta \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, J(E_6^\delta)&=1+2A_1^2+A_1^4+2A_1^6-2A_2-6A_2A_1^2-12A_2A_1^4+3A_2^2+16A_2^2A_1^2-4A_2^3 \\ &\qquad+2A_3^\delta A_1+3A_3^{-\delta}A_1+6A_3^\delta A_1^3+5A_3^{-\delta}A_1^3-8A_3^\delta A_2A_1 -6A_3^{-\delta}A_2A_1 \\ &\qquad+(A_3^\delta)^2-2A_4-6A_4A_1^2+4A_4A_2+A_5^+A_1+A_5^-A_1-D_4^+-2D_4^+A_1^2 \\ &\qquad-D_4^-A_1^2+2D_4^+A_2+2D_5^{\delta}A_1-E_6^{\delta}, \end{aligned}
\end{equation*}
\notag
$$
where $\delta=\pm1$. These formulae follow from Examples 4.2–4.4 and Corollaries 7.4, 7.5 and 7.11–7.13 in [5] and the corollary in [7]. Let
$$
\begin{equation*}
\Lambda=\sum_{X\in\mathbb{S}_{n,k}^+}(-1)^{\operatorname{codim} X}J(X)X,
\end{equation*}
\notag
$$
where $\mathbb{S}_{n,k}^+$ is the set of types of multisingularities $X\in\mathbb{S}^+$ such that $\operatorname{codim} X\leqslant n$ and the total number of the factors $X_1=A_1$ in $X$ does not exceed $k$. On the one hand $\Lambda$ is a polynomial in $Y_1,\dots,Y_N$,
$$
\begin{equation}
\Lambda=\sum_{\mathcal{A}\in\mathbb{S}^+}(-1)^{\operatorname{codim} \mathcal{A}}K(\mathcal{A})\mathcal{A},
\end{equation}
\tag{3.1}
$$
with coefficients
$$
\begin{equation*}
K(\mathcal{A})=\sum_{X\in\mathbb{S}_{n,k}^+}(-1)^{\operatorname{codim} X}J_{\mathcal{A}}(X)X
\end{equation*}
\notag
$$
which are polynomials in $X_1,\dots,X_N$. On the other hand $\Lambda$ is a polynomial in $X_1,\dots,X_N$ with coefficients which are polynomials in $Y_1,\dots,Y_N$. Namely, where and the sum is taken over all nonnegative integers $k_1,\dots,k_N$ such that
$$
\begin{equation*}
\sum_{i=1}^N k_i \operatorname{codim} X_i\leqslant n, \qquad k_1\leqslant k.
\end{equation*}
\notag
$$
This sum can easily be obtained from a partial sum of the formal power series in $x_1,\dots,x_N$
$$
\begin{equation*}
\sum_{k_1=0}^{+\infty}x_1^{k_1}\cdots \sum_{k_N=0}^{+\infty}x_N^{k_N}
\end{equation*}
\notag
$$
by means of the substitution (3.3). Now using software packages it is easy to transform the expression (3.2) for $\Lambda$ into an expression of the form (3.1). The coefficient $K(\mathcal{A})$ for each $\mathcal{A}\in \mathbb{S}_{n,k}^+$ such that $\operatorname{codim} \mathcal{A}\equiv n-1\pmod{2}$ determines the equation
$$
\begin{equation}
\sum_{X\in\mathbb{S}_{n,k}^+}(-1)^{\operatorname{codim} X}J_{\mathcal{A}}(X)\chi_f(X)=(-1)^n\chi_f(\mathcal{A}).
\end{equation}
\tag{3.4}
$$
The system of these equations is equivalent to the system of relations (2.1). The Euler characteristics of odd-dimensional manifolds of multisingularities is easily found from the resulting system using computer-assisted calculations. A program written for the Wolfram Mathematica package is available on the Internet.1
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